nerdamer.core.js

/*
 * Author : Martin Donk
 * Website : http://www.nerdamer.com
 * Email : martin.r.donk@gmail.com
 * Source : https://github.com/jiggzson/nerdamer
 */

/* global trig, trigh, Infinity, define, arguments2Array, NaN */
//externals ====================================================================
/* BigInterger.js v1.6.40 https://github.com/peterolson/BigInteger.js/blob/master/LICENSE */
//var nerdamerBigInt = typeof nerdamerBigInt !== 'undefined' ? nerdamerBigInt : require("big-integer");
/* big.js v5.2.2 https://github.com/MikeMcl/big.js/LICENCE */
//var nerdamerBigDecimal = typeof nerdamerBigDecimal !== 'undefined' ? nerdamerBigDecimal : require('big.js');

var nerdamer = (function (imports) {
    "use strict"; 

//version ======================================================================
    var version = '1.1.13';

//inits ========================================================================
    var _ = new Parser(); //nerdamer's parser

    //import bigInt
    var bigInt = imports.bigInt;
    var bigDec = imports.bigDec;

    //set the precision to js precision
    bigDec.set({
        precision: 250
    });

    var Groups = {};

    //container of pregenerated primes
    var PRIMES = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113
                , 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251,
        257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,
        401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557,
        563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701,
        709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863,
        877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031,
        1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171,
        1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303,
        1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471,
        1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607,
        1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753,
        1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931,
        1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083];

//Settings =====================================================================
    var CUSTOM_OPERATORS = {};

    var Settings = {
        //Enables/Disables call peekers. False means callPeekers are disabled and true means callPeekers are enabled.
        callPeekers: false,

        //the max number up to which to cache primes. Making this too high causes performance issues
        init_primes: 1000,

        exclude: [],
        //If you don't care about division by zero for example then this can be set to true.
        //Has some nasty side effects so choose carefully.
        suppress_errors: false,
        //the global used to invoke the libary to parse to a number. Normally cos(9) for example returns
        //cos(9) for convenience but parse to number will always try to return a number if set to true.
        PARSE2NUMBER: false,
        //this flag forces the a clone to be returned when add, subtract, etc... is called
        SAFE: false,
        //the symbol to use for imaginary symbols
        IMAGINARY: 'i',
        //the modules used to link numeric function holders
        FUNCTION_MODULES: [Math],
        //Allow certain characters
        ALLOW_CHARS: ['π'],
        //Allow nerdamer to convert multi-character variables
        USE_MULTICHARACTER_VARS: true,
        //Allow changing of power operator
        POWER_OPERATOR: '^',
        //The variable validation regex
        //VALIDATION_REGEX: /^[a-z_][a-z\d\_]*$/i
        VALIDATION_REGEX: /^[a-z_αAβBγΓδΔϵEζZηHθΘιIκKλΛμMνNξΞoOπΠρPσΣτTυϒϕΦχXψΨωΩ∞][0-9a-z_αAβBγΓδΔϵEζZηHθΘιIκKλΛμMνNξΞoOπΠρPσΣτTυϒϕΦχXψΨωΩ]*$/i,
        // The regex used to determine which characters should be included in implied multiplication
        IMPLIED_MULTIPLICATION_REGEX: /([\+\-\/\*]*[0-9]+)([a-z_αAβBγΓδΔϵEζZηHθΘιIκKλΛμMνNξΞoOπΠρPσΣτTυϒϕΦχXψΨωΩ]+[\+\-\/\*]*)/gi,
        //Aliases
        ALIASES: {
            'π': 'pi',
            '∞': 'Infinity'
        },
        POSITIVE_MULTIPLIERS: false,
        //Cached items
        CACHE: {},
        //Print out warnings or not
        SILENCE_WARNINGS: false,
        // Precision
        PRECISION: 21,
        // The Expression defaults to this value for decimal places
        EXPRESSION_DECP: 19,
        // The text function defaults to this value for decimal places
        DEFAULT_DECP: 16,
        //function mappings
        VECTOR: 'vector',
        PARENTHESIS: 'parens',
        SQRT: 'sqrt',
        ABS: 'abs',
        FACTORIAL: 'factorial',
        DOUBLEFACTORIAL: 'dfactorial',
        //reference pi and e
        LONG_PI: '3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214' +
                '808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196',
        LONG_E: '2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466' +
                '39193200305992181741359662904357290033429526059563073813232862794349076323382988075319525101901',
        PI: Math.PI,
        E: Math.E,
        LOG: 'log',
        LOG10: 'log10',
        LOG10_LATEX: 'log_{10}',
        MAX_EXP: 200000,
        //The number of scientific place to round to
        SCIENTIFIC_MAX_DECIMAL_PLACES: 14,
        //True if ints should not be converted to
        SCIENTIFIC_IGNORE_ZERO_EXPONENTS: true
    };

    (function () {
        Settings.CACHE.roots = {};
        var x = 40,
                y = 40;
        for(var i = 2; i <= x; i++) {
            for(var j = 2; j <= y; j++) {
                var nthpow = bigInt(i).pow(j);
                Settings.CACHE.roots[nthpow + '-' + j] = i;
            }
        }
    })();

    //Add the groups. These have been reorganized as of v0.5.1 to make CP the highest group
    //The groups that help with organizing during parsing. Note that for FN is still a function even
    //when it's raised to a symbol, which typically results in an EX
    var N = Groups.N = 1, // A number
            P = Groups.P = 2, // A number with a rational power e.g. 2^(3/5).
            S = Groups.S = 3, // A single variable e.g. x.
            EX = Groups.EX = 4, // An exponential
            FN = Groups.FN = 5, // A function
            PL = Groups.PL = 6, // A symbol/expression having same name with different powers e.g. 1/x + x^2
            CB = Groups.CB = 7, // A symbol/expression composed of one or more variables through multiplication e.g. x*y
            CP = Groups.CP = 8; // A symbol/expression composed of one variable and any other symbol or number x+1 or x+y

    var CONST_HASH = Settings.CONST_HASH = '#';

    var PARENTHESIS = Settings.PARENTHESIS;

    var SQRT = Settings.SQRT;

    var ABS = Settings.ABS;

    var FACTORIAL = Settings.FACTORIAL;

    var DOUBLEFACTORIAL = Settings.DOUBLEFACTORIAL;

    //the storage container "memory" for parsed expressions
    var EXPRESSIONS = [];

    //variables
    var VARS = {};

    //the container used to store all the reserved functions
    var RESERVED = [];

    var WARNINGS = [];

    /**
     * Use this when errors are suppressible
     * @param {String} msg
     * @param {object} ErrorObj
     */
    var err = function (msg, ErrorObj) {
        if(!Settings.suppress_errors) {
            if(ErrorObj)
                throw new ErrorObj(msg);
            else
                throw new Error(msg);
        }
    };

//Utils ========================================================================
    var customError = function (name) {
        var E = function (message) {
            this.name = name;
            this.message = message !== undefined ? message : '';
            var error = new Error(this.message);
            error.name = this.name;
            this.stack = error.stack;
        }; //create an empty error
        E.prototype = Object.create(Error.prototype);
        return E;
    };

    /**
     * Checks to see if value is one of nerdamer's reserved names
     * @param {String} value
     * @return boolean
     */
    var isReserved = function (value) {
        return RESERVED.indexOf(value) !== -1;
    };

    /**
     * Checks to see that all symbols in array are the same
     * @param {Symbol[]} arr
     * @returns {bool}
     */
    var allSame = function (arr) {
        var last = arr[0];
        for(var i = 1, l = arr.length; i < l; i++)
            if(!arr[i].equals(last))
                return false;
        return true;
    };

    /**
     * Used to pass warnings or low severity errors about the library
     * @param msg
     */
    var warn = function (msg) {
        WARNINGS.push(msg);
        if(Settings.SHOW_WARNINGS && console && console.warn) {
            console.warn(msg);
        }
    };

    /**
     * Enforces rule: "must start with a letter or underscore and
     * can have any number of underscores, letters, and numbers thereafter."
     * @param name The name of the symbol being checked
     * @param {String} typ - The type of symbols that's being validated
     * @throws {Exception}  - Throws an exception on fail
     */
    var validateName = function (name, typ) {
        typ = typ || 'variable';
        if(Settings.ALLOW_CHARS.indexOf(name) !== -1)
            return;
        var regex = Settings.VALIDATION_REGEX;
        if(!(regex.test(name))) {
            throw new InvalidVariableNameError(name + ' is not a valid ' + typ + ' name');
        }
    };

    /**
     * Convert number from scientific format to decimal format
     * @param {Number} num
     */
    var scientificToDecimal = function (num) {
        var nsign = Math.sign(num);
        //remove the sign
        num = Math.abs(num);
        //if the number is in scientific notation remove it
        if(/\d+\.?\d*e[\+\-]*\d+/i.test(num)) {
            var zero = '0',
                    parts = String(num).toLowerCase().split('e'), //split into coeff and exponent
                    e = parts.pop(), //store the exponential part
                    l = Math.abs(e), //get the number of zeros
                    sign = e / l,
                    coeff_array = parts[0].split('.');
            if(sign === -1) {
                l = l - coeff_array[0].length;
                if(l < 0) {
                    num = coeff_array[0].slice(0, l) + '.' + coeff_array[0].slice(l) + (coeff_array.length === 2 ? coeff_array[1] : '');
                }
                else {
                    num = zero + '.' + new Array(l + 1).join(zero) + coeff_array.join('');
                }
            }
            else {
                var dec = coeff_array[1];
                if(dec)
                    l = l - dec.length;
                if(l < 0) {
                    num = coeff_array[0] + dec.slice(0, l) + '.' + dec.slice(l);
                }
                else {
                    num = coeff_array.join('') + new Array(l + 1).join(zero);
                }
            }
        }

        return nsign < 0 ? '-' + num : num;
    };
    /**
     * Checks if number is a prime number
     * @param {Number} n - the number to be checked
     */
    var isPrime = function (n) {
        var q = Math.floor(Math.sqrt(n));
        for(var i = 2; i <= q; i++) {
            if(n % i === 0)
                return false;
        }
        return true;
    };

    /**
     * Generates an object with known variable value for evaluation
     * @param {String} variable
     * @param {any} value Any stringifyable object
     * @returns {Object} 
     */
    var knownVariable = function (variable, value) {
        var o = {};
        o[variable] = value;
        return o;
    };

    /**
     * Checks if n is a number
     * @param {any} n
     */
    var isNumber = function (n) {
        return /^\d+\.?\d*$/.test(n);
    };

    /**
     * Checks to see if an array contains only numeric values
     * @param {Array} arr
     */
    var allNumeric = function (arr) {
        for(var i = 0; i < arr.length; i++)
            if(!isNumber(arr[i]))
                return false;
        return true;
    };
    /**
     * Checks to see if a number or Symbol is a fraction
     * @param {Number|Symbol} num
     * @returns {boolean}
     */
    var isFraction = function (num) {
        if(isSymbol(num))
            return isFraction(num.multiplier.toDecimal());
        return (num % 1 !== 0);
    };

    /**
     * Checks to see if the object provided is a Symbol
     * @param {Object} obj
     */
    var isSymbol = function (obj) {
        return (obj instanceof Symbol);
    };

    /**
     * Checks to see if the object provided is an Expression
     * @param {Object} obj
     */
    var isExpression = function (obj) {
        return (obj instanceof Expression);
    };

    /**
     * This method traverses the symbol structure and grabs all the variables in a symbol. The variable
     * names are then returned in alphabetical order.
     * @param {Symbol} obj
     * @param {Boolean} poly
     * @param {Object} vars - An object containing the variables. Do not pass this in as it generated
     * automatically. In the future this will be a Collector object.
     * @returns {String[]} - An array containing variable names
     */
    var variables = function (obj, poly, vars) {
        vars = vars || {
            c: [],
            add: function (value) {
                if(this.c.indexOf(value) === -1 && isNaN(value))
                    this.c.push(value);
            }
        };

        if(isSymbol(obj)) {
            var group = obj.group,
                    prevgroup = obj.previousGroup;
            if(group === EX)
                variables(obj.power, poly, vars);

            if(group === CP || group === CB || prevgroup === CP || prevgroup === CB) {
                for(var x in obj.symbols) {
                    variables(obj.symbols[x], poly, vars);
                }
            }
            else if(group === S || prevgroup === S) {
                //very crude needs fixing. TODO
                if(!(obj.value === 'e' || obj.value === 'pi' || obj.value === Settings.IMAGINARY))
                    vars.add(obj.value);
            }
            else if(group === PL || prevgroup === PL) {
                variables(firstObject(obj.symbols), poly, vars);
            }
            else if(group === EX) {
                if(!isNaN(obj.value))
                    vars.add(obj.value);
                variables(obj.power, poly, vars);
            }
            else if(group === FN && !poly) {
                for(var i = 0; i < obj.args.length; i++) {
                    variables(obj.args[i], poly, vars);
                }
            }
        }

        return vars.c.sort();
    };

    /**
     * Returns the sum of an array
     * @param {Array} arr
     * @param {boolean} toNumber
     * @returns {Symbol}
     */
    var arraySum = function (arr, toNumber) {
        var sum = new Symbol(0);
        for(var i = 0; i < arr.length; i++) {
            var x = arr[i];
            // Convert to symbol if not
            sum = _.add(sum, !isSymbol(x) ? _.parse(x) : x);
        }

        return toNumber ? Number(sum) : sum;
    };

    /**
     * Separates out the variables into terms of variabls.
     * e.g. x+y+x*y+sqrt(2)+pi returns
     * {x: x, y: y, x y: x*y, constants: sqrt(2)+pi
     * @param {type} symbol
     * @param {type} o
     * @returns {undefined}
     * @throws {Error} for expontentials
     */
    var separate = function (symbol, o) {
        symbol = _.expand(symbol);
        o = o || {};
        var insert = function (key, sym) {
            if(!o[key])
                o[key] = new Symbol(0);
            o[key] = _.add(o[key], sym.clone());
        };
        symbol.each(function (x) {
            if(x.isConstant('all')) {
                insert('constants', x);
            }
            else if(x.group === S) {
                insert(x.value, x);
            }
            else if(x.group === FN && (x.fname === ABS || x.fname === '')) {
                separate(x.args[0]);
            }
            else if(x.group === EX || x.group === FN) {
                throw new Error('Unable to separate. Term cannot be a function!');
            }
            else {
                insert(variables(x).join(' '), x);
            }
        });

        return o;
    };

    /**
     * Fills holes in an array with zero symbol or generates one with n zeroes
     * @param {Array} arr
     * @param {Number} n
     */
    var fillHoles = function (arr, n) {
        n = n || arr.length;
        for(var i = 0; i < n; i++) {
            var sym = arr[i];
            if(!sym)
                arr[i] = new Symbol(0);
        }
        return arr;
    };

    /**
     *
     * Checks to see if the object provided is a Vector
     * @param {Object} obj
     */
    var isVector = function (obj) {
        return (obj instanceof Vector);
    };

    /**
     * Checks to see if the object provided is a Matrix
     * @param {Object} obj
     */
    var isMatrix = function (obj) {
        return (obj instanceof Matrix);
    };

    var isSet = function (obj) {
        return (obj instanceof Set);
    };

    /**
     * Checks to see if a symbol is in group N
     * @param {Symbol} symbol
     */
    var isNumericSymbol = function (symbol) {
        return symbol.group === N || symbol.group === P;
    };

    /**
     * Checks to see if a symbol is a variable with no multiplier nor power
     * @param {Symbol} symbol
     */
    var isVariableSymbol = function (symbol) {
        return symbol.group === S && symbol.multiplier.equals(1) && symbol.power.equals(1);
    };

    /**
     * Checks to see if the object provided is an Array
     * @param {Object} arr
     */
    var isArray = function (arr) {
        return Array.isArray(arr);
    };

    /**
     * Checks to see if a number is an integer
     * @param {Number} num
     */
    var isInt = function (num) {
        return /^[-+]?\d+e?\+?\d*$/gim.test(num.toString());
    };

    /**
     * @param {Number|Symbol} obj
     * @returns {boolean}
     */
    var isNegative = function (obj) {
        if(isSymbol(obj)) {
            obj = obj.multiplier;
        }
        return obj.lessThan(0);
    };
    /**
     * Safely stringify object
     * @param o
     */
    var stringify = function (o) {
        if(!o)
            return o;
        return String(o);
    };

    /**
     * @param {String} str
     * @returns {String} - returns a formatted string surrounded by brackets
     */
    var inBrackets = function (str) {
        return '(' + str + ')';
    };

    /**
     * A helper function to replace parts of string
     * @param {String} str - The original string
     * @param {Integer} from - The starting index
     * @param {Integer} to - The ending index
     * @param {String} with_str - The replacement string
     * @returns {String} - A formatted string
     */
    var stringReplace = function (str, from, to, with_str) {
        return str.substr(0, from) + with_str + str.substr(to, str.length);
    };

    /**
     * the Parser uses this to check if it's allowed to convert the obj to type Symbol
     * @param {Object} obj
     * @returns {boolean}
     */
    var customType = function (obj) {
        return obj !== undefined && obj.custom;
    };

    /**
     * Checks to see if numbers are both negative or are both positive
     * @param {Number} a
     * @param {Number} b
     * @returns {boolean}
     */
    var sameSign = function (a, b) {
        return (a < 0) === (b < 0);
    };

    /**
     * A helper function to replace multiple occurences in a string. Takes multiple arguments
     * @example format('{0} nice, {0} sweet', 'something')
     * //returns 'something nice, something sweet'
     */
    var format = function () {
        var args = [].slice.call(arguments),
                str = args.shift();
        var new_str = str.replace(/{(\d+)}/g, function (match, index) {
            var arg = args[index];
            return typeof arg === 'function' ? arg() : arg;
        });

        return new_str;
    };

    /**
     * Generates an array with values within a range. Multiplies by a step if provided
     * @param {Number} start
     * @param {Number} end
     * @param {Number} step
     */
    var range = function (start, end, step) {
        var arr = [];
        step = step || 1;
        for(var i = start; i <= end; i++)
            arr.push(i * step);
        return arr;
    };

    /**
     * Returns an array of all the keys in an array
     * @param {Object} obj
     * @returns {Array}
     */
    var keys = Object.keys;

    /**
     * Returns the first encountered item in an object. Items do not have a fixed order in objects
     * so only use if you need any first random or if there's only one item in the object
     * @param {Object} obj
     * @param {String} key Return this key as first object
     * @param {Boolean} both
     * @returns {*}
     */
    var firstObject = function (obj, key, both) {
        for(var x in obj)
            break;
        if(key)
            return x;
        if(both)
            return {
                key: x,
                obj: obj[x]
            };
        return obj[x];
    };

    /**
     * Substitutes out variables for two symbols, parses them to a number and them compares them numerically
     * @param {Symbol} sym1
     * @param {Symbol} sym2
     * @param {String[]} vars - an optional array of variables to use
     * @returns {bool}
     */
    var compare = function (sym1, sym2, vars) {
        var n = 5; //a random number between 1 and 5 is good enough
        var scope = {}; // scope object with random numbers generated using vars
        var comparison;
        for(var i = 0; i < vars.length; i++)
            scope[vars[i]] = new Symbol(Math.floor(Math.random() * n) + 1);
        block('PARSE2NUMBER', function () {
            comparison = _.parse(sym1, scope).equals(_.parse(sym2, scope));
        });
        return comparison;
    };

    /**
     * Is used to set a user defined function using the function assign operator
     * @param {String} name
     * @param {String[]} params_array
     * @param {String} body
     * @returns {Boolean}
     */
    var setFunction = function (name, params_array, body) {
        validateName(name);
        if(!isReserved(name)) {
            params_array = params_array || variables(_.parse(body));
            // The function gets set to PARSER.mapped function which is just
            // a generic function call.
            _.functions[name] = [_.mapped_function, params_array.length, {
                    name: name,
                    params: params_array,
                    body: body
                }];

            return body;
        }
        return null;
    };

    /**
     * Returns the minimum number in an array
     * @param {Array} arr
     * @returns {Number}
     */
    var arrayMax = function (arr) {
        return Math.max.apply(undefined, arr);
    };

    /**
     * Returns the maximum number in an array
     * @param {Array} arr
     * @returns {Number}
     */
    var arrayMin = function (arr) {
        return Math.min.apply(undefined, arr);
    };

    /**
     * Checks to see if two arrays are equal
     * @param {Array} arr1 
     * @param {Array} arr2 
     */
    var arrayEqual = function (arr1, arr2) {
        arr1.sort();
        arr2.sort();

        // The must be of the same length
        if(arr1.length === arr2.length) {
            for(var i = 0; i < arr1.length; i++) {
                // If any two items don't match we're done
                if(arr1[i] !== arr2[i]) {
                    return false;
                }
            }
            // Otherwise they're equal
            return true;
        }

        return false;
    };

    /**
     * Clones array with clonable items
     * @param {Array} arr
     * @returns {Array}
     */
    var arrayClone = function (arr) {
        var new_array = [], l = arr.length;
        for(var i = 0; i < l; i++)
            new_array[i] = arr[i].clone();
        return new_array;
    };

    /**
     * Fills numbers between array values
     * @param {Numbers[]} arr
     * @param {Integer} slices
     */
    var arrayAddSlices = function (arr, slices) {
        slices = slices || 20;
        var retval = [];
        var c, delta, e;
        retval.push(arr[0]); //push the beginning
        for(var i = 0; i < arr.length - 1; i++) {
            c = arr[i];
            delta = arr[i + 1] - c; //get the difference
            e = delta / slices; //chop it up in the desired number of slices
            for(var j = 0; j < slices; j++) {
                c += e; //add the mesh to the last slice
                retval.push(c);
            }
        }

        return retval;
    };

    /**
     * Gets nth roots of a number
     * @param {Symbol} symbol
     * @returns {Vector}
     */
    var nroots = function (symbol) {
        var a, b;

        if(symbol.group === FN && symbol.fname === '') {
            a = Symbol.unwrapPARENS(_.parse(symbol).toLinear());
            b = _.parse(symbol.power);
        }
        else if(symbol.group === P) {
            a = _.parse(symbol.value);
            b = _.parse(symbol.power);
        }

        if(a && b && (a.group === N) && b.group === N && a.multiplier.isNegative()) {
            var _roots = [];

            var parts = Symbol.toPolarFormArray(evaluate(symbol));
            var r = parts[0];

            //var r = _.parse(a).abs().toString();

            //https://en.wikipedia.org/wiki/De_Moivre%27s_formula
            var x = _.arg(a);
            var n = b.multiplier.den.toString();
            var p = b.multiplier.num.toString();

            var formula = '(({0})^({1})*(cos({3})+({2})*sin({3})))^({4})';

            for(var i = 0; i < n; i++) {
                var t = evaluate(_.parse(format("(({0})+2*pi*({1}))/({2})", x, i, n))).multiplier.toDecimal();
                _roots.push(evaluate(_.parse(format(formula, r, n, Settings.IMAGINARY, t, p))));
            }
            return Vector.fromArray(_roots);
        }
        else if(symbol.isConstant(true, true)) {
            var sign = symbol.sign();
            var x = evaluate(symbol.abs());
            var root = _.sqrt(x);

            var _roots = [root.clone(), root.negate()];

            if(sign < 0)
                _roots = _roots.map(function (x) {
                    return _.multiply(x, Symbol.imaginary());
                });

        }
        else {
            _roots = [_.parse(symbol)];
        }

        return Vector.fromArray(_roots);
    };

    /**
     * Sorts and array given 2 parameters
     * @param {String} a
     * @param {String} b
     */
    var comboSort = function (a, b) {
        var l = a.length,
                combined = []; //the linker
        for(var i = 0; i < a.length; i++) {
            combined.push([a[i], b[i]]); //create the map
        }

        combined.sort(function (x, y) {
            return x[0] - y[0];
        });

        var na = [], nb = [];

        for(i = 0; i < l; i++) {
            na.push(combined[i][0]);
            nb.push(combined[i][1]);
        }

        return [na, nb];
    };
    /**
     * TODO: Pick a more descriptive name and better description
     * Breaks a function down into it's parts wrt to a variable, mainly coefficients
     * Example a*x^2+b wrt x
     * @param {Symbol} fn
     * @param {String} wrt
     * @param {bool} as_obj
     */
    var decompose_fn = function (fn, wrt, as_obj) {
        wrt = String(wrt); //convert to string
        var ax, a, x, b;
        if(fn.group === CP) {
            var t = _.expand(fn.clone()).stripVar(wrt);
            ax = _.subtract(fn.clone(), t.clone());
            b = t;
        }
        else
            ax = fn.clone();
        a = ax.stripVar(wrt);
        x = _.divide(ax.clone(), a.clone());
        b = b || new Symbol(0);
        if(as_obj)
            return {
                a: a,
                x: x,
                ax: ax,
                b: b
            };
        return [a, x, ax, b];
    };
    /**
     * Rounds a number up to x decimal places
     * @param {Number} x
     * @param {Number} s
     */
    var nround = function (x, s) {
        if(isInt(x)) {
            if(x >= Number.MAX_VALUE)
                return x.toString();
            return Number(x);
        }

        s = typeof s === 'undefined' ? 14 : s;
        return Math.round(x * Math.pow(10, s)) / Math.pow(10, s);
    };

    /**
     * Is used for u-substitution. Gets a suitable u for substitution. If for
     * instance a is used in the symbol then it keeps going down the line until
     * one is found that's not in use. If all letters are taken then it
     * starts appending numbers.
     * IMPORTANT! It assumes that the substitution will be undone
     * beore the user gets to interact with the object again.
     * @param {Symbol} symbol
     */
    var getU = function (symbol) {
        //start with u
        var u = 'u', //start with u
                v = u, //init with u
                c = 0, //postfix number
                vars = variables(symbol);
        //make sure this variable isn't reserved and isn't in the variable list
        while(!(RESERVED.indexOf(v) === - 1 && vars.indexOf(v) === - 1))
            v = u + c++;
        //get an empty slot. It seems easier to just push but the
        //problem is that we may have some which are created by clearU
        for(var i = 0, l = RESERVED.length; i <= l; i++)
            //reserved cannot equals false or 0 so we can safely check for a falsy type
            if(!RESERVED[i]) {
                RESERVED[i] = v; //reserve the variable
                break;
            }
        return v;
    };

    /**
     * Clears the u variable so it's no longer reserved
     * @param {String} u
     */
    var clearU = function (u) {
        var indx = RESERVED.indexOf(u);
        if(indx !== -1)
            RESERVED[indx] = undefined;
    };

    /**
     * Loops through each item in object and calls function with item as param
     * @param {Object|Array} obj
     * @param {Function} fn
     */
    var each = function (obj, fn) {
        if(isArray(obj)) {
            var l = obj.length;
            for(var i = 0; i < l; i++)
                fn.call(obj, i);
        }
        else {
            for(var x in obj)
                if(obj.hasOwnProperty(x))
                    fn.call(obj, x);
        }
    };

    /**
     * Checks to see if a number is an even number
     * @param {Number} num
     * @returns {boolean}
     */
    var even = function (num) {
        return num % 2 === 0;
    };

    /**
     * Checks to see if a fraction is divisible by 2
     * @param {Number} num
     * @returns {boolean}
     */
    var evenFraction = function (num) {
        return 1 / (num % 1) % 2 === 0;
    };

    /**
     * Strips duplicates out of an array
     * @param {Array} arr
     */
    var arrayUnique = function (arr) {
        var l = arr.length, a = [];
        for(var i = 0; i < l; i++) {
            var item = arr[i];
            if(a.indexOf(item) === -1)
                a.push(item);
        }
        return a;
    };

    /**
     * Gets all the variables in an array of Symbols
     * @param {Symbol[]} arr 
     */
    var arrayGetVariables = function (arr) {
        var vars = variables(arr[0], null, null, true);

        //get all variables
        for(var i = 1, l = arr.length; i < l; i++)
            vars = vars.concat(variables(arr[i]));
        //remove duplicates
        vars = arrayUnique(vars).sort();

        //done
        return vars;
    };

    /**
     * Removes duplicates from an array. Returns a new array
     * @param {Array} arr
     * @param {Function} condition
     */
    var removeDuplicates = function (arr, condition) {
        var conditionType = typeof condition;

        if(conditionType !== 'function' || conditionType === 'undefined') {
            condition = function (a, b) {
                return a === b;
            };
        }

        var seen = [];

        while(arr.length) {
            var a = arr[0];
            //only one element left so we're done
            if(arr.length === 1) {
                seen.push(a);
                break;
            }
            var temp = [];
            seen.push(a); //we already scanned these
            for(var i = 1; i < arr.length; i++) {
                var b = arr[i];
                //if the number is outside the specified tolerance
                if(!condition(a, b))
                    temp.push(b);
            }
            //start over with the remainder
            arr = temp;
        }

        return seen;
    };

    /**
     * Reserves the names in an object so they cannot be used as function names
     * @param {Object} obj
     */
    var reserveNames = function (obj) {
        var add = function (item) {
            if(RESERVED.indexOf(item) === -1)
                RESERVED.push(item);
        };

        if(typeof obj === 'string')
            add(obj);
        else {
            each(obj, function (x) {
                add(x);
            });
        }
    };

    /**
     * Removes an item from either an array or an object. If the object is an array, the index must be
     * specified after the array. If it's an object then the key must be specified
     * @param {Object|Array} obj
     * @param {Integer} indexOrKey
     */
    var remove = function (obj, indexOrKey) {
        var result;
        if(isArray(obj)) {
            result = obj.splice(indexOrKey, 1)[0];
        }
        else {
            result = obj[indexOrKey];
            delete obj[indexOrKey];
        }
        return result;
    };

    /**
     * Creates a temporary block in which one of the global settings is temporarily modified while
     * the function is called. For instance if you want to parse directly to a number rather than have a symbolic
     * answer for a period you would set PARSE2NUMBER to true in the block.
     * @example block('PARSE2NUMBER', function(){//symbol being parsed to number}, true);
     * @param {String} setting - The setting being accessed
     * @param {Function} f
     * @param {boolean} opt - The value of the setting in the block
     * @param {String} obj - The obj of interest. Usually a Symbol but could be any object
     */
    var block = function (setting, f, opt, obj) {
        var current_setting = Settings[setting];
        Settings[setting] = opt === undefined ? true : !!opt;
        var retval = f.call(obj);
        Settings[setting] = current_setting;
        return retval;
    };

    /**
     * provide a mechanism for accessing functions directly. Not yet complete!!!
     * Some functions will return undefined. This can maybe just remove the
     * function object at some point when all functions are eventually
     * housed in the global function object. Returns ALL parser available
     * functions. Parser.functions may not contain all functions
     */
    var importFunctions = function () {
        var o = {};
        for(var x in _.functions)
            o[x] = _.functions[x][0];
        return o;
    };

    /**
     * Converts function arguments to an array. Now used by gcd and lcm in Algebra.js :)
     * @param {Array|object} obj
     */
    var arguments2Array = function (obj) {
        return [].slice.call(obj);
    };

    /**
     * Returns the coefficients of a symbol given a variable. Given ax^2+b^x+c, it divides
     * each nth term by x^n.
     * @param {Symbol} symbol
     * @param {Symbol} wrt
     */
    var getCoeffs = function (symbol, wrt, info) {
        var coeffs = [];
        //we loop through the symbols and stick them in their respective
        //containers e.g. y*x^2 goes to index 2
        symbol.each(function (term) {
            if(term.contains(wrt)) {
                //we want only the coefficient which in this case will be everything but the variable
                //e.g. a*b*x -> a*b if the variable to solve for is x
                var coeff = term.stripVar(wrt),
                        x = _.divide(term.clone(), coeff.clone()),
                        p = x.power.toDecimal();
            }
            else {
                coeff = term;
                p = 0;
            }
            var e = coeffs[p];
            //if it exists just add it to it
            coeffs[p] = e ? _.add(e, coeff) : coeff;

        }, true);

        for(var i = 0; i < coeffs.length; i++)
            if(!coeffs[i])
                coeffs[i] = new Symbol(0);
        //fill the holes
        return coeffs;
    };

    /**
     * As the name states. It forces evaluation of the expression
     * @param {Symbol} symbol
     * @param {Symbol} o
     */
    var evaluate = function (symbol, o) {
        return block('PARSE2NUMBER', function () {
            return _.parse(symbol, o);
        }, true);
    };

    /**
     * Converts an array to a vector. Consider moving this to Vector.fromArray
     * @param {String[]|String|Symbol|Number|Number[]} x
     */
    var convertToVector = function (x) {
        if(isArray(x)) {
            var vector = new Vector([]);
            for(var i = 0; i < x.length; i++)
                vector.elements.push(convertToVector(x[i]));
            return vector;
        }
        //Ensure that a nerdamer ready object is returned
        if(!isSymbol(x))
            return _.parse(x);
        return x;
    };

    /**
     * Generates prime numbers up to a specified number
     * @param {Number} upto
     */
    var generatePrimes = function (upto) {
        //get the last prime in the array
        var last_prime = PRIMES[PRIMES.length - 1] || 2;
        //no need to check if we've already encountered the number. Just check the cache.
        for(var i = last_prime; i < upto; i++) {
            if(isPrime(i))
                PRIMES.push(i);
        }
    };
    /**
     * Checks to see if all arguments are numbers
     * @param {object} args
     */
    var allNumbers = function (args) {
        for(var i = 0; i < args.length; i++)
            if(args[i].group !== N)
                return false;
        return true;
    };
    /*
     * Checks if all arguments aren't just all number but if they
     * are constants as well e.g. pi, e.
     * @param {object} args
     */
    var allConstants = function (args) {
        for(var i = 0; i < args.length; i++) {
            if(args[i].isPi() || args[i].isE())
                continue;
            if(!args[i].isConstant(true))
                return false;
        }
        return true;
    };

    /**
     * Used to multiply two expression in expanded form
     * @param {Symbol} a
     * @param {Symbol} b
     */
    var mix = function (a, b, opt) {
        // Flip them if b is a CP or PL and a is not
        if(b.isComposite() && !a.isComposite() || b.isLinear() && !a.isLinear()) {
            [a, b] = [b, a];
        }
        // A temporary variable to hold the expanded terms
        var t = new Symbol(0);
        if(a.isLinear()) {
            a.each(function (x) {
                // If b is not a PL or a CP then simply multiply it
                if(!b.isComposite()) {
                    var term = _.multiply(_.parse(x), _.parse(b));
                    t = _.add(t, _.expand(term, opt));
                }
                // Otherwise multiply out each term.
                else if(b.isLinear()) {
                    b.each(function (y) {
                        var term = _.multiply(_.parse(x), _.parse(y));
                        var expanded = _.expand(_.parse(term), opt);
                        t = _.add(t, expanded);
                    }, true);
                }
                else {
                    t = _.add(t, _.multiply(x, _.parse(b)));
                }
            }, true);
        }
        else {
            // Just multiply them together
            t = _.multiply(a, b);
        }

        // The expanded function is now t
        return t;
    };

//Exceptions ===================================================================
    //Is thrown for division by zero
    var DivisionByZero = customError('DivisionByZero');
    // Is throw if an error occured during parsing
    var ParseError = customError('ParseError');
    // Is thrown if the expression results in undefined
    var UndefinedError = customError('UndefinedError');
    // Is throw input is out of the function domain
    var OutOfFunctionDomainError = customError('OutOfFunctionDomainError');
    // Is throw if a function exceeds x amount of iterations
    var MaximumIterationsReached = customError('MaximumIterationsReached');
    // Is thrown if the parser receives an incorrect type
    var NerdamerTypeError = customError('NerdamerTypeError');
    // Is thrown if bracket parity is not correct
    var ParityError = customError('ParityError');
    // Is thrown if an unexpectd or incorrect operator is encountered
    var OperatorError = customError('OperatorError');
    // Is thrown if an index is out of range.
    var OutOfRangeError = customError('OutOfRangeError');
    // Is thrown if dimensions are incorrect. Mostly for matrices
    var DimensionError = customError('DimensionError');
    // Is thrown if variable name violates naming rule
    var InvalidVariableNameError = customError('InvalidVariableNameError');
    // Is thrown if the limits of the library are exceeded for a function
    // This can be that the function become unstable passed a value
    var ValueLimitExceededError = customError('ValueLimitExceededError');
    // Is throw if the value is an incorrect LH or RH value
    var NerdamerValueError = customError('NerdamerValueError');
    // Is thrown if the value is an incorrect LH or RH value
    var SolveError = customError('SolveError');
    // Is thrown for an infinite loop
    var InfiniteLoopError = customError('InfiniteLoopError');
    // Is thrown if an operator is found when there shouldn't be one
    var UnexpectedTokenError = customError('UnexpectedTokenError');

    var exceptions = {
        DivisionByZero: DivisionByZero,
        ParseError: ParseError,
        OutOfFunctionDomainError: OutOfFunctionDomainError,
        UndefinedError: UndefinedError,
        MaximumIterationsReached: MaximumIterationsReached,
        NerdamerTypeError: NerdamerTypeError,
        ParityError: ParityError,
        OperatorError: OperatorError,
        OutOfRangeError: OutOfRangeError,
        DimensionError: DimensionError,
        InvalidVariableNameError: InvalidVariableNameError,
        ValueLimitExceededError: ValueLimitExceededError,
        NerdamerValueError: NerdamerValueError,
        SolveError: SolveError,
        InfiniteLoopError: InfiniteLoopError,
        UnexpectedTokenError: UnexpectedTokenError
    };
//Math2 ========================================================================
    //This object holds additional functions for nerdamer. Think of it as an extension of the Math object.
    //I really don't like touching objects which aren't mine hence the reason for Math2. The names of the
    //functions within are pretty self-explanatory.
    //NOTE: DO NOT USE INLINE COMMENTS WITH THE MATH2 OBJECT! THIS BREAK DURING COMPILATION OF BUILDFUNCTION.
    var Math2 = {
        csc: function (x) {
            return 1 / Math.sin(x);
        },
        sec: function (x) {
            return 1 / Math.cos(x);
        },
        cot: function (x) {
            return 1 / Math.tan(x);
        },
        acsc: function (x) {
            return Math.asin(1 / x);
        },
        asec: function (x) {
            return Math.acos(1 / x);
        },
        acot: function (x) {
            return (Math.PI / 2) - Math.atan(x);
        },
        // https://gist.github.com/jiggzson/df0e9ae8b3b06ff3d8dc2aa062853bd8
        erf: function (x) {
            var t = 1 / (1 + 0.5 * Math.abs(x));
            var result = 1 - t * Math.exp(-x * x - 1.26551223 +
                    t * (1.00002368 +
                            t * (0.37409196 +
                                    t * (0.09678418 +
                                            t * (-0.18628806 +
                                                    t * (0.27886807 +
                                                            t * (-1.13520398 +
                                                                    t * (1.48851587 +
                                                                            t * (-0.82215223 +
                                                                                    t * (0.17087277)))))))))
                    );
            return x >= 0 ? result : -result;
        },
        diff: function (f) {
            var h = 0.001;

            var derivative = function (x) {
                return (f(x + h) - f(x - h)) / (2 * h);
            };

            return derivative;
        },
        median: function (...values) {
            values.sort(function (a, b) {
                return a - b;
            });

            var half = Math.floor(values.length / 2);

            if(values.length % 2)
                return values[half];

            return (values[half - 1] + values[half]) / 2.0;
        },
        /*
         * Reverses continued fraction calculation
         * @param {obj} contd
         * @returns {Number}
         */
        fromContinued: function (contd) {
            var arr = contd.fractions.slice();
            var e = 1 / arr.pop();
            for(var i = 0, l = arr.length; i < l; i++) {
                e = 1 / (arr.pop() + e);
            }
            return contd.sign * (contd.whole + e);
        },
        /*
         * Calculates continued fractions
         * @param {Number} n
         * @param {Number} x The number of places
         * @returns {Number}
         */
        continuedFraction: function (n, x) {
            x = x || 20;
            var sign = Math.sign(n); /*store the sign*/
            var absn = Math.abs(n); /*get the absolute value of the number*/
            var whole = Math.floor(absn); /*get the whole*/
            var ni = absn - whole; /*subtract the whole*/
            var c = 0; /*the counter to keep track of iterations*/
            var done = false;
            var epsilon = 1e-14;
            var max = 1e7;
            var e, w;
            var retval = {
                whole: whole,
                sign: sign,
                fractions: []
            };
            /*start calculating*/
            while(!done && ni !== 0) {
                /*invert and get the whole*/
                e = 1 / ni;
                w = Math.floor(e);
                if(w > max) {
                    /*this signals that we may have already gone too far*/
                    var d = Math2.fromContinued(retval) - n;
                    if(d <= Number.EPSILON)
                        break;
                }
                /*add to result*/
                retval.fractions.push(w);
                /*move the ni to the decimal*/
                ni = e - w;
                /*ni should always be a decimal. If we have a whole number then we're in the rounding errors*/
                if(ni <= epsilon || c >= x - 1)
                    done = true;
                c++;
            }
            /*cleanup 1/(n+1/1) = 1/(n+1) so just move the last digit one over if it's one*/
            var idx = retval.fractions.length - 1;
            if(retval.fractions[idx] === 1) {
                retval.fractions.pop();
                /*increase the last one by one*/
                retval.fractions[--idx]++;
            }
            return retval;
        },
        bigpow: function (n, p) {
            if(!(n instanceof Frac))
                n = Frac.create(n);
            if(!(p instanceof Frac))
                p = Frac.create(p);
            var retval = new Frac(0);
            if(p.isInteger()) {
                retval.num = n.num.pow(p.toString());
                retval.den = n.den.pow(p.toString());
            }
            else {
                var num = Frac.create(Math.pow(n.num, p.num));
                var den = Frac.create(Math.pow(n.den, p.num));

                retval.num = Math2.nthroot(num, p.den.toString());
                retval.den = Math2.nthroot(den, p.den);
            }
            return retval;
        },
        //http://stackoverflow.com/questions/15454183/how-to-make-a-function-that-computes-the-factorial-for-numbers-with-decimals
        gamma: function (z) {
            var g = 7;
            var C = [
                0.99999999999980993,
                676.5203681218851,
                -1259.1392167224028,
                771.32342877765313,
                -176.61502916214059,
                12.507343278686905,
                -0.13857109526572012,
                9.9843695780195716e-6,
                1.5056327351493116e-7]
                    ;

            if(z < 0.5)
                return Math.PI / (Math.sin(Math.PI * z) * Math2.gamma(1 - z));
            else {
                z -= 1;

                var x = C[0];
                for(var i = 1; i < g + 2; i++)
                    x += C[i] / (z + i);

                var t = z + g + 0.5;
                return Math.sqrt(2 * Math.PI) * Math.pow(t, (z + 0.5)) * Math.exp(-t) * x;
            }
        },
        //factorial
        bigfactorial: function (x) {
            var retval = new bigInt(1);
            for(var i = 2; i <= x; i++)
                retval = retval.times(i);
            return new Frac(retval);
        },
        //https://en.wikipedia.org/wiki/Logarithm#Calculation
        bigLog: function (x) {
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"13240077436443988749179508462267267187169441948722358165090554769250505713747934643200804819418670147225695324432684266924694524337920816452346599774452681831320005286326986675907899608537972384924882996757503264622991355949039882526389342174307168805166215838138277557052303430492669193939212362638263582899713198716541723383138016564027766560215944409353427176135895982596327685665844815618402881202645610620284792793420780517248/2544223084468158291883698813309541801455311468982232546872485444308211415529998472787377800559884210837213042932180479090277285630234238711851480232520137856848809986631784843528381778520727465146661792797924458540957133423665746229799675650290296217658444899605236550972043549278128087645211909479009099766619355677984218929672461506691980442071860591767266913041147587815452007726513853820116629482732060593116624596368806566625", "1953999166296955830935495158735359200362904181792947794529339487489730042568305997099959302322956898299616194932283060554261566410988618045107398092345476532371402134206635235570281738377188438407703089325315446371127042537576093536896282955524842632708645655481028161471313608974238110718242273935956977555610147714316158486553633871312187084618154014921190595222799283957140353/375191165084882521037046014569185165885459082629136124177286500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"];
            if(CACHE[x]) {
                return Frac.quick.apply(null, CACHE[x].split('/'));
            }
            x = new Frac(x);
            var n = 80;
            var retval = new Frac(0);
            var a = x.subtract(new Frac(1));
            var b = x.add(new Frac(1));
            for(var i = 0; i < n; i++) {
                var t = new Frac(2 * i + 1);
                var k = Math2.bigpow(a.divide(b), t);
                var r = t.clone().invert().multiply(k);
                retval = retval.add(r);

            }
            return retval.multiply(new Frac(2));
        },
        //the factorial function but using the big library instead
        factorial: function (x) {
            var is_int = x % 1 === 0;

            /*factorial for negative integers is complex infinity according to Wolfram Alpha*/
            if(is_int && x < 0)
                return NaN;

            if(!is_int)
                return Math2.gamma(x + 1);

            var retval = 1;
            for(var i = 2; i <= x; i++)
                retval = retval * i;
            return retval;
        },
        //double factorial
        //http://mathworld.wolfram.com/DoubleFactorial.html
        dfactorial: function (x) {
            if(isInt(x)) {
                var even = x % 2 === 0;
                /* If x = even then n = x/2 else n = (x-1)/2*/
                var n = even ? x / 2 : (x + 1) / 2;
                /*the return value*/
                var r = new Frac(1);
                /*start the loop*/
                if(even)
                    for(var i = 1; i <= n; i++)
                        r = r.multiply(new Frac(2).multiply(new Frac(i)));
                else
                    for(var i = 1; i <= n; i++)
                        r = r.multiply(new Frac(2).multiply(new Frac(i)).subtract(new Frac(1)));
            }
            else {
                /*Not yet extended to bigNum*/
                r = Math.pow(2, (1 + 2 * x - Math.cos(Math.PI * x)) / 4) * Math.pow(Math.PI, (Math.cos(Math.PI * x) - 1) / 4) * Math2.gamma(1 + x / 2);
            }

            /*done*/
            return r;
        },
        GCD: function () {
            var args = arrayUnique([].slice.call(arguments)
                    .map(function (x) {
                        return Math.abs(x);
                    })).sort(),
                    a = Math.abs(args.shift()),
                    n = args.length;

            while(n-- > 0) {
                var b = Math.abs(args.shift());
                while(true) {
                    a %= b;
                    if(a === 0) {
                        a = b;
                        break;
                    }
                    b %= a;
                    if(b === 0)
                        break;
                    ;
                }
            }
            return a;
        },
        QGCD: function () {
            var args = [].slice.call(arguments);
            var a = args[0];
            for(var i = 1; i < args.length; i++) {
                var b = args[i];
                var sign = a.isNegative() && b.isNegative() ? -1 : 1;
                a = b.gcd(a);
                if(sign < 0)
                    a.negate();
            }
            return a;
        },
        LCM: function (a, b) {
            return (a * b) / Math2.GCD(a, b);
        },
        //pow but with the handling of negative numbers
        //http://stackoverflow.com/questions/12810765/calculating-cubic-root-for-negative-number
        pow: function (b, e) {
            if(b < 0) {
                if(Math.abs(e) < 1) {
                    /*nth root of a negative number is imaginary when n is even*/
                    if(1 / e % 2 === 0)
                        return NaN;
                    return -Math.pow(Math.abs(b), e);
                }
            }
            return Math.pow(b, e);
        },
        factor: function (n) {
            n = Number(n);
            var sign = Math.sign(n); /*store the sign*/
            /*move the number to absolute value*/
            n = Math.abs(n);
            var ifactors = Math2.ifactor(n);
            var factors = new Symbol();
            factors.symbols = {};
            factors.group = CB;
            for(var x in ifactors) {
                var factor = new Symbol(1);
                factor.group = P; /*cheat a little*/
                factor.value = x;
                factor.power = new Symbol(ifactors[x]);
                factors.symbols[x] = factor;
            }
            factors.updateHash();

            if(n === 1) {
                factors = new Symbol(n);
            }

            /*put back the sign*/
            if(sign < 0)
                factors.negate();

            return factors;
        },
        /**
         * Uses trial division
         * @param {Integer} n - the number being factored
         * @param {object} factors -  the factors object
         * @returns {object}
         */
        sfactor: function (n, factors) {
            factors = factors || {};
            var r = Math.floor(Math.sqrt(n));
            var lcprime = PRIMES[PRIMES.length - 1];
            /*a one-time cost... Hopefully ... And don't bother for more than a million*/
            /*takes too long*/
            if(r > lcprime && n < 1e6)
                generatePrimes(r);
            var l = PRIMES.length;
            for(var i = 0; i < l; i++) {
                var prime = PRIMES[i];
                /*trial division*/
                while(n % prime === 0) {
                    n = n / prime;
                    factors[prime] = (factors[prime] || 0) + 1;
                }
            }
            if(n > 1)
                factors[n] = 1;
            return factors;
        },
        /**
         * Pollard's rho
         * @param {Integer} n
         * @returns {object}
         */
        ifactor: function (n) {
            var input = new bigInt(n);

            n = String(n);

            if(n === '0')
                return {'0': 1};
            n = new bigInt(n); /*convert to bigInt for safety*/
            var sign = n.sign ? -1 : 1;
            n = n.abs();
            var factors = {}; /*factor object being returned.*/
            if(n.lt('65536')) { /*less than 2^16 just use trial division*/
                factors = Math2.sfactor(n, factors);
            }
            else {
                var add = function (e) {
                    if(!e.isPrime()) {
                        factors = Math2.sfactor(e, factors);
                    }
                    else
                        factors[e] = (factors[e] || 0) + 1;
                };

                try {
                    //set a safety
                    var max = 1e3;
                    var safety = 0;

                    while(!n.abs().equals(1)) {
                        if(n.isPrime()) {
                            add(n);
                            break;
                        }
                        else {
                            function rho(c) {
                                var xf = new bigInt(c),
                                        cz = 2,
                                        x = new bigInt(c),
                                        factor = new bigInt(1);

                                while(factor.equals(1)) {
                                    for(var i = 0; i <= cz && factor.equals(1); i++) {
                                        //trigger the safety
                                        if(safety++ > max)
                                            throw new Error('stopping');

                                        x = x.pow(2).add(1).mod(n);
                                        factor = bigInt.gcd(x.minus(xf).abs(), n);
                                    }

                                    cz = cz * 2;
                                    xf = x;
                                }
                                if(factor.equals(n)) {
                                    return rho(c + 1);
                                }
                                return factor;
                            }
                            var factor = rho(2);
                            add(factor);
                            /*divide out the factor*/
                            n = n.divide(factor);
                        }
                    }
                }
                catch(e) {
                    //reset factors
                    factors = {};
                    add(input);
                }

            }

            /*put the sign back*/
            if(sign === -1) {
                var sm = arrayMin(keys(factors)); /*/get the smallest number*/
                factors['-' + sm] = factors[sm];
                delete factors[sm];
            }

            return factors;
        },
        //factors a number into rectangular box. If sides are primes that this will be
        //their prime factors. e.g. 21 -> (7)(3), 133 -> (7)(19)
        boxfactor: function (n, max) {
            max = max || 200; //stop after this number of iterations
            var c, r,
                    d = Math.floor((5 / 12) * n), //the divisor
                    i = 0, //number of iterations
                    safety = false;
            while(true) {
                c = Math.floor(n / d);
                r = n % d;
                if(r === 0)
                    break; //we're done
                if(safety)
                    return [n, 1];
                d = Math.max(r, d - r);
                i++;
                safety = i > max;
            }
            return [c, d, i];
        },
        fib: function (n) {
            var sign = Math.sign(n);
            n = Math.abs(n);
            sign = even(n) ? sign : Math.abs(sign);
            var a = 0, b = 1, f = 1;
            for(var i = 2; i <= n; i++) {
                f = a + b;
                a = b;
                b = f;
            }
            return f * sign;
        },
        mod: function (x, y) {
            return x % y;
        },
        //http://mathworld.wolfram.com/IntegerPart.html
        integer_part: function (x) {
            var sign = Math.sign(x);
            return sign * Math.floor(Math.abs(x));
        },
        simpson: function (f, a, b, step) {
            var get_value = function (f, x, side) {
                var v = f(x);
                var d = 0.000000000001;
                if(isNaN(v)) {
                    v = f(side === 1 ? x + d : x - d);
                }
                return v;
            };

            step = step || 0.0001;
            //calculate the number of intervals
            var n = Math.abs(Math.floor((b - a) / step));
            //simpson's rule requires an even number of intervals. If it's not then add 1
            if(n % 2 !== 0)
                n++;
            //get the interval size
            var dx = (b - a) / n;
            //get x0
            var retval = get_value(f, a, 1);

            //get the middle part 4x1+2x2+4x3 ...
            //but first set a flag to see if it's even or odd.
            //The first one is odd so we start there
            var even = false;
            //get x1
            var xi = a + dx;
            //the coefficient
            var c, k;
            //https://en.wikipedia.org/wiki/Simpson%27s_rule
            for(var i = 1; i < n; i++) {
                c = even ? 2 : 4;
                k = c * get_value(f, xi, 1);
                retval += k;
                //flip the even flag
                even = !even;
                //increment xi
                xi += dx;
            }

            //add xn
            return (retval + get_value(f, xi, 2)) * (dx / 3);

        },
        /**
         * https://github.com/scijs/integrate-adaptive-simpson
         * @param {Function} f - the function being integrated
         * @param {Number} a - lower bound
         * @param {Number} b - upper bound
         * @param {Number} tol - step width
         * @param {Number} maxdepth
         * @returns {Number}
         */
        num_integrate: function (f, a, b, tol, maxdepth) {
            if(maxdepth < 0)
                throw new Error('max depth cannot be negative');

            /* This algorithm adapted from pseudocode in:*/
            /* http://www.math.utk.edu/~ccollins/refs/Handouts/rich.pdf*/
            function adsimp(f, a, b, fa, fm, fb, V0, tol, maxdepth, depth, state) {
                if(state.nanEncountered) {
                    return NaN;
                }
                var h, f1, f2, sl, sr, s2, m, V1, V2, err;
                h = b - a;
                f1 = f(a + h * 0.25);
                f2 = f(b - h * 0.25);
                /* Simple check for NaN:*/
                if(isNaN(f1)) {
                    state.nanEncountered = true;
                    return;
                }
                /* Simple check for NaN:*/
                if(isNaN(f2)) {
                    state.nanEncountered = true;
                    return;
                }

                sl = h * (fa + 4 * f1 + fm) / 12;
                sr = h * (fm + 4 * f2 + fb) / 12;
                s2 = sl + sr;
                err = (s2 - V0) / 15;

                if(state.maxDepthCount > 1000 * maxdepth) {
                    return;
                }


                if(depth > maxdepth) {
                    state.maxDepthCount++;
                    return s2 + err;
                }
                else if(Math.abs(err) < tol) {
                    return s2 + err;
                }
                else {
                    m = a + h * 0.5;
                    V1 = adsimp(f, a, m, fa, f1, fm, sl, tol * 0.5, maxdepth, depth + 1, state);
                    if(isNaN(V1)) {
                        state.nanEncountered = true;
                        return NaN;
                    }
                    V2 = adsimp(f, m, b, fm, f2, fb, sr, tol * 0.5, maxdepth, depth + 1, state);

                    if(isNaN(V2)) {
                        state.nanEncountered = true;
                        return NaN;
                    }

                    return V1 + V2;
                }
            }

            function integrate(f, a, b, tol, maxdepth) {
                var state = {
                    maxDepthCount: 0,
                    nanEncountered: false
                };

                if(tol === undefined) {
                    tol = 1e-9;
                }
                if(maxdepth === undefined) {
                    /*Issue #458 - This was lowered because of performance issues. */
                    /*This was suspected from before but is now confirmed with this issue*/
                    maxdepth = 45;
                }

                var fa = f(a);
                var fm = f(0.5 * (a + b));
                var fb = f(b);

                var V0 = (fa + 4 * fm + fb) * (b - a) / 6;

                var result = adsimp(f, a, b, fa, fm, fb, V0, tol, maxdepth, 1, state);

                if(state.maxDepthCount > 0) {
                    warn('integrate-adaptive-simpson: Warning: maximum recursion depth (' + maxdepth + ') reached ' + state.maxDepthCount + ' times');
                }

                if(state.nanEncountered) {
                    throw new Error('Function does not converge over interval!');
                }

                return result;
            }
            var retval;

            try {
                retval = integrate(f, a, b, tol, maxdepth);
            }
            catch(e) {
                /*fallback to non-adaptive*/
                return Math2.simpson(f, a, b);
            }
            return nround(retval, 12);
        },
        //https://en.wikipedia.org/wiki/Trigonometric_integral
        //CosineIntegral
        Ci: function (x) {
            var n = 20,
                    /*roughly Euler–Mascheroni*/
                    g = 0.5772156649015329,
                    sum = 0;
            for(var i = 1; i < n; i++) {
                /*cache 2n*/
                var n2 = 2 * i;
                sum += (Math.pow(-1, i) * Math.pow(x, n2)) / (n2 * Math2.factorial(n2));
            }
            return Math.log(x) + g + sum;
        },
        /*SineIntegral*/
        Si: function (x) {
            var n = 20,
                    sum = 0;
            for(var i = 0; i < n; i++) {
                var n2 = 2 * i;
                sum += (Math.pow(-1, i) * Math.pow(x, n2 + 1)) / ((n2 + 1) * Math2.factorial(n2 + 1));
            }
            return sum;
        },
        /*ExponentialIntegral*/
        Ei: function (x) {
            if(Number(x) === 0)
                return -Infinity;
            var n = 30,
                    g = 0.5772156649015328606, /*roughly Euler–Mascheroni*/
                    sum = 0;
            for(var i = 1; i < n; i++) {
                sum += Math.pow(x, i) / (i * Math2.factorial(i));
            }
            return g + Math.abs(Math.log(x)) + sum;
        },
        /*Hyperbolic Sine Integral*/
        /*http://mathworld.wolfram.com/Shi.html*/
        Shi: function (x) {
            var n = 30,
                    sum = 0,
                    k, t;
            for(var i = 0; i < n; i++) {
                k = 2 * i;
                t = k + 1;
                sum += Math.pow(x, t) / (t * t * Math2.factorial(k));
            }
            return sum;
        },
        /*the cosine integral function*/
        Chi: function (x) {
            var dx, g, f;
            dx = 0.001;
            g = 0.5772156649015328606;
            f = function (t) {
                return (Math.cosh(t) - 1) / t;
            };
            return Math.log(x) + g + Math2.num_integrate(f, 0.002, x, dx);
        },
        /*the log integral*/
        Li: function (x) {
            return Math2.Ei(Math2.bigLog(x));
        },
        /*the gamma incomplete function*/
        gamma_incomplete: function (n, x) {
            var t = n - 1,
                    sum = 0,
                    x = x || 0;
            for(var i = 0; i < t; i++) {
                sum += Math.pow(x, i) / Math2.factorial(i);
            }
            return Math2.factorial(t) * Math.exp(-x) * sum;
        },
        /*
         * Heaviside step function - Moved from Special.js (originally contributed by Brosnan Yuen)
         * Specification : http://mathworld.wolfram.com/HeavisideStepFunction.html
         * if x > 0 then 1
         * if x == 0 then 1/2
         * if x < 0 then 0
         */
        step: function (x) {
            if(x > 0)
                return 1;
            if(x < 0)
                return 0;
            return 0.5;
        },
        /*
         * Rectangle function - Moved from Special.js (originally contributed by Brosnan Yuen)
         * Specification : http://mathworld.wolfram.com/RectangleFunction.html
         * if |x| > 1/2 then 0
         * if |x| == 1/2 then 1/2
         * if |x| < 1/2 then 1
         */
        rect: function (x) {
            var x = Math.abs(x);
            if(x === 0.5)
                return x;
            if(x > 0.5)
                return 0;
            return 1;
        },
        /*
         * Sinc function - Moved from Special.js (originally contributed by Brosnan Yuen)
         * Specification : http://mathworld.wolfram.com/SincFunction.html
         * if x == 0 then 1
         * otherwise sin(x)/x
         */
        sinc: function (x) {
            if(x.equals(0))
                return 1;
            return Math.sin(x) / x;
        },
        /*
         * Triangle function - Moved from Special.js (originally contributed by Brosnan Yuen)
         * Specification : http://mathworld.wolfram.com/TriangleFunction.html
         * if |x| >= 1 then 0
         * if |x| < then 1-|x|
         */
        tri: function (x) {
            x = Math.abs(x);
            if(x >= 1)
                return 0;
            return 1 - x;
        },
        //https://en.wikipedia.org/wiki/Nth_root_algorithm
        nthroot: function (A, n) {
            /*make sure the input is of type Frac*/
            if(!(A instanceof Frac))
                A = new Frac(A.toString());
            if(!(n instanceof Frac))
                n = new Frac(n.toString());
            if(n.equals(1))
                return A;
            /*begin algorithm*/
            var xk = A.divide(new Frac(2)); /*x0*/
            var e = new Frac(1e-15);
            var dk, dk0, d0;
            var a = n.clone().invert(),
                    b = n.subtract(new Frac(1));
            do {
                var powb = Math2.bigpow(xk, b);
                var dk_dec = a.multiply(A.divide(powb).subtract(xk)).toDecimal(25);
                dk = Frac.create(dk_dec);
                if(d0)
                    break;

                xk = xk.add(dk);
                /*check to see if there's no change from the last xk*/
                var dk_dec = dk.toDecimal();
                d0 = dk0 ? dk0 === dk_dec : false;
                dk0 = dk_dec;
            }
            while(dk.abs().gte(e))

            return xk;
        },
        /*https://gist.github.com/jiggzson/0c5b33cbcd7b52b36132b1e96573285f*/
        /*Just the square root function but big :)*/
        sqrt: function (n) {
            if(!(n instanceof Frac))
                n = new Frac(n);
            var xn, d, ld, same_delta;
            var c = 0; /*counter*/
            var done = false;
            var delta = new Frac(1e-20);
            xn = n.divide(new Frac(2));
            var safety = 1000;
            do {
                /*break if we're not converging*/
                if(c > safety)
                    throw new Error('Unable to calculate square root for ' + n);
                xn = xn.add(n.divide(xn)).divide(new Frac(2));
                xn = new Frac(xn.decimal(30));
                /*get the difference from the true square*/
                d = n.subtract(xn.multiply(xn));
                /*if the square of the calculated number is close enough to the number*/
                /*we're getting the square root or the last delta was the same as the new delta*/
                /*then we're done*/
                same_delta = ld ? ld.equals(d) : false;
                if(d.clone().abs().lessThan(delta) || same_delta)
                    done = true;
                /*store the calculated delta*/
                ld = d;
                c++; /*increase the counter*/
            }
            while(!done)

            return xn;
        }
    };
    //link the Math2 object to Settings.FUNCTION_MODULES
    Settings.FUNCTION_MODULES.push(Math2);
    reserveNames(Math2); //reserve the names in Math2


//Polyfills ====================================================================
    //https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/
    Math.sign = Math.sign || function (x) {
        x = +x; // convert to a number
        if(x === 0 || isNaN(x)) {
            return x;
        }
        return x > 0 ? 1 : -1;
    };

    Math.cosh = Math.cosh || function (x) {
        var y = Math.exp(x);
        return (y + 1 / y) / 2;
    };

    Math.sech = Math.sech || function (x) {
        return 1 / Math.cosh(x);
    };

    Math.csch = Math.csch || function (x) {
        return 1 / Math.sinh(x);
    };

    Math.coth = Math.coth || function (x) {
        return 1 / Math.tanh(x);
    };

    Math.sinh = Math.sinh || function (x) {
        var y = Math.exp(x);
        return (y - 1 / y) / 2;
    };

    Math.tanh = Math.tanh || function (x) {
        if(x === Infinity) {
            return 1;
        }
        else if(x === -Infinity) {
            return -1;
        }
        else {
            var y = Math.exp(2 * x);
            return (y - 1) / (y + 1);
        }
    };

    Math.asinh = Math.asinh || function (x) {
        if(x === -Infinity) {
            return x;
        }
        else {
            return Math.log(x + Math.sqrt(x * x + 1));
        }
    };

    Math.acosh = Math.acosh || function (x) {
        return Math.log(x + Math.sqrt(x * x - 1));
    };

    Math.atanh = Math.atanh || function (x) {
        return Math.log((1 + x) / (1 - x)) / 2;
    };

    Math.log10 = Math.log10 || function (x) {
        return Math.log(x) * Math.LOG10E;
    };

    Math.trunc = Math.trunc || function (x) {
        if(isNaN(x)) {
            return NaN;
        }
        if(x > 0) {
            return Math.floor(x);
        }
        return Math.ceil(x);
    };

//Global functions =============================================================
    /**
     * This method will return a hash or a text representation of a Symbol, Matrix, or Vector.
     * If all else fails it *assumes* the object has a toString method and will call that.
     *
     * @param {Object} obj
     * @param {String} option get is as a hash
     * @param {int} useGroup
     * @returns {String}
     */
    function text(obj, option, useGroup, decp) { 
        var asHash = option === 'hash',
                //whether to wrap numbers in brackets
                wrapCondition = undefined,
                opt = asHash ? undefined : option,
                asDecimal = opt === 'decimal' || opt === 'decimals';

        if(asDecimal && typeof decp === 'undefined')
            decp = Settings.DEFAULT_DECP;

        function toString(obj) {
            switch(option)
            {
                case 'decimals':
                case 'decimal':
                    wrapCondition = wrapCondition || function (str) {
                        return false;
                    };
                    return obj.valueOf();
                case 'recurring':
                    wrapCondition = wrapCondition || function (str) {
                        return str.indexOf("'") !== -1;
                    };

                    var str = obj.toString();
                    //verify that the string is actually a fraction
                    var frac = /^-?\d+(?:\/\d+)?$/.exec(str);
                    if(frac.length === 0)
                        return str;

                    //split the fraction into the numerator and denominator
                    var parts = frac[0].split('/');
                    var negative = false;
                    var m = Number(parts[0]);
                    if(m < 0) {
                        m = -m;
                        negative = true;
                    }
                    var n = Number(parts[1]);
                    if(!n)
                        n = 1;

                    //https://softwareengineering.stackexchange.com/questions/192070/what-is-a-efficient-way-to-find-repeating-decimal#comment743574_192081
                    var quotient = Math.floor(m / n), c = 10 * (m - quotient * n);
                    quotient = quotient.toString() + ".";
                    while(c && c < n) {
                        c *= 10;
                        quotient += "0";
                    }
                    var digits = "", passed = [], i = 0;
                    while(true) {
                        if(typeof passed[c] !== 'undefined') {
                            var prefix = digits.slice(0, passed[c]),
                                    cycle = digits.slice(passed[c]),
                                    result = quotient + prefix + "'" + cycle + "'";
                            return (negative ? "-" : "") + result.replace("'0'", "").replace(/\.$/, "");
                        }
                        var q = Math.floor(c / n), r = c - q * n;
                        passed[c] = i;
                        digits += q.toString();
                        i += 1;
                        c = 10 * r;
                    }
                case 'mixed':
                    wrapCondition = wrapCondition || function (str) {
                        return str.indexOf('/') !== -1;
                    };

                    var str = obj.toString();
                    //verify that the string is actually a fraction
                    var frac = /^-?\d+(?:\/\d+)?$/.exec(str);
                    if(frac.length === 0)
                        return str;

                    //split the fraction into the numerator and denominator
                    var parts = frac[0].split('/');
                    var numer = new bigInt(parts[0]);
                    var denom = new bigInt(parts[1]);
                    if(denom.equals(0))
                        denom = new bigInt(1);

                    //return the quotient plus the remainder
                    var divmod = numer.divmod(denom);
                    var quotient = divmod.quotient;
                    var remainder = divmod.remainder;
                    var operator = parts[0][0] === '-' || quotient.equals(0) || remainder.equals(0) ? '' : '+';
                    return (quotient.equals(0) ? '' : quotient.toString()) + operator + (remainder.equals(0) ? '' : (remainder.toString() + '/' + parts[1]));
                case 'scientific':
                    wrapCondition = wrapCondition || function (str) {
                        return false;
                    };
                    return new Scientific(obj.valueOf()).toString(Settings.SCIENTIFIC_MAX_DECIMAL_PLACES);
                default:
                    wrapCondition = wrapCondition || function (str) {
                        return str.indexOf('/') !== -1;
                    };

                    return obj.toString();
            }
        }

        //if the object is a symbol
        if(isSymbol(obj)) {
            var multiplier = '',
                    power = '',
                    sign = '',
                    group = obj.group || useGroup,
                    value = obj.value;

            //if the value is to be used as a hash then the power and multiplier need to be suppressed
            if(!asHash) {
                //use asDecimal to get the object back as a decimal
                var om = toString(obj.multiplier);
                if(om == '-1' && String(obj.multiplier) === '-1') {
                    sign = '-';
                    om = '1';
                }
                //only add the multiplier if it's not 1
                if(om != '1')
                    multiplier = om;
                //use asDecimal to get the object back as a decimal
                var p = obj.power ? toString(obj.power) : '';
                //only add the multiplier
                if(p != '1') {
                    //is it a symbol
                    if(isSymbol(p)) {
                        power = text(p, opt);
                    }
                    else {
                        power = p;
                    }
                }
            }

            switch(group) {
                case N:
                    multiplier = '';
                    //round if requested
                    var m = decp && asDecimal ? obj.multiplier.toDecimal(decp) : toString(obj.multiplier);
                    //if it's numerical then all we need is the multiplier
                    value = String(obj.multiplier) == '-1' ? '1' : m;
                    power = '';
                    break;
                case PL:
                    value = obj.collectSymbols().map(function (x) {
                        var txt = text(x, opt, useGroup, decp);
                        if(txt == '0')
                            txt = '';
                        return txt;
                    }).sort().join('+').replace(/\+\-/g, '-');
                    break;
                case CP:
                    value = obj.collectSymbols().map(function (x) {
                        var txt = text(x, opt, useGroup, decp);
                        if(txt == '0')
                            txt = '';
                        return txt;
                    }).sort().join('+').replace(/\+\-/g, '-');
                    break;
                case CB:
                    value = obj.collectSymbols(function (symbol) {
                        var g = symbol.group;
                        //both groups will already be in brackets if their power is greater than 1
                        //so skip it.
                        if((g === PL || g === CP) && (symbol.power.equals(1) && symbol.multiplier.equals(1))) {
                            return inBrackets(text(symbol, opt));
                        }
                        return text(symbol, opt);
                    }).join('*');
                    break;
                case EX:
                    var pg = obj.previousGroup,
                            pwg = obj.power.group;

                    //PL are the exception. It's simpler to just collect and set the value
                    if(pg === PL)
                        value = obj.collectSymbols(text, opt).join('+').replace('+-', '-');
                    if(!(pg === N || pg === S || pg === FN) && !asHash) {
                        value = inBrackets(value);
                    }

                    if((pwg === CP || pwg === CB || pwg === PL || obj.power.multiplier.toString() != '1') && power) {
                        power = inBrackets(power);
                    }
                    break;
            }

            if(group === FN) {
                value = obj.fname + inBrackets(obj.args.map(function (symbol) {
                    return text(symbol, opt);
                }).join(','));
            }
            //TODO: Needs to be more efficient. Maybe.
            if(group === FN && obj.fname in CUSTOM_OPERATORS) {
                var a = text(obj.args[0]);
                var b = text(obj.args[1]);
                if(obj.args[0].isComposite()) //preserve the brackets
                    a = inBrackets(a);
                if(obj.args[1].isComposite()) //preserve the brackets
                    b = inBrackets(b);
                value = a + CUSTOM_OPERATORS[obj.fname] + b;
            }
            //wrap the power since / is less than ^
            //TODO: introduce method call isSimple
            if(power && group !== EX && wrapCondition(power)) {
                power = inBrackets(power);
            }

            //the following groups are held together by plus or minus. They can be raised to a power or multiplied
            //by a multiplier and have to be in brackets to preserve the order of precedence
            if(((group === CP || group === PL) && (multiplier && multiplier != '1' || sign === '-'))
                    || ((group === CB || group === CP || group === PL) && (power && power != '1'))
                    || !asHash && group === P && value == -1
                    || obj.fname === PARENTHESIS) {

                value = inBrackets(value);
            }

            if(decp && (option === 'decimal' || option === 'decimals' && multiplier)) {
                multiplier = nround(multiplier, decp);
            }


            //add the sign back
            var c = sign + multiplier;

            if(multiplier && wrapCondition(multiplier))
                c = inBrackets(c);

            if(power < 0)
                power = inBrackets(power);

            //add the multiplication back
            if(multiplier)
                c = c + '*';

            if(power) {
                if(value === 'e' && Settings.E_TO_EXP) {
                    return c + 'exp' + inBrackets(power);
                }
                power = Settings.POWER_OPERATOR + power;
            }

            //this needs serious rethinking. Must fix
            if(group === EX && value.charAt(0) === '-') {
                value = inBrackets(value);
            }

            var cv = c + value;

            if(obj.parens) {
                cv = inBrackets(cv);
            }

            return cv + power;
        }
        else if(isVector(obj)) {
            var l = obj.elements.length,
                    c = [];
            for(var i = 0; i < l; i++)
                c.push(obj.elements[i].text(option));
            return '[' + c.join(',') + ']';
        }
        else {
            try {
                return obj.toString();
            }
            catch(e) {
                return '';
            }
        }
    }
    /**
     * Calculates prime factors for a number. It first checks if the number
     * is a prime number. If it's not then it will calculate all the primes 
     * for that number.
     * @param {int} num
     * @returns {Array}
     */

    function primeFactors(num) {
        if(isPrime(num)) {
            return [num];
        }

        var l = num, i = 1, factors = [],
                epsilon = 2.2204460492503130808472633361816E-16;
        while(i < l) {
            var quotient = num / i;
            var whole = Math.floor(quotient);
            var remainder = quotient - whole;

            if(remainder <= epsilon && i > 1) {
                // If the prime wasn't found but calculated then save it and
                // add it as a factor.
                if(isPrime(i)) {
                    if(PRIMES.indexOf(i) === -1) {
                        PRIMES.push(i);
                    }
                    factors.push(i);
                }

                // Check if the remainder is a prime
                if(isPrime(whole)) {
                    factors.push(whole);
                    break;
                }

                l = whole;
            }
            i++;
        }

        return factors.sort(function (a, b) {
            return a - b;
        });
    }
    ;
    primeFactors(314146179365)
//Expression ===================================================================
    /**
     * This is what nerdamer returns. It's sort of a wrapper around the symbol class and
     * provides the user with some useful functions. If you want to provide the user with extra
     * library functions then add them to this class's prototype.
     * @param {Symbol} symbol
     * @returns {Expression} wraps around the Symbol class
     */
    function Expression(symbol) {
        //we don't want arrays wrapped
        this.symbol = symbol;
    }
    /**
     * Returns stored expression at index. For first index use 1 not 0.
     * @param {bool} asType
     * @param {Integer} expression_number
     */
    Expression.getExpression = function (expression_number, asType) {
        if(expression_number === 'last' || !expression_number)
            expression_number = EXPRESSIONS.length;
        if(expression_number === 'first')
            expression_number = 1;
        var index = expression_number - 1,
                expression = EXPRESSIONS[index],
                retval = expression ? new Expression(expression) : expression;
        return retval;
    };
    Expression.prototype = {
        /**
         * Returns the text representation of the expression
         * @param {String} opt - option of formatting numbers
         * @param {Number} n The number of significant figures
         * @returns {String}
         */
        text: function (opt, n) {
            n = n || Settings.EXPRESSION_DECP;
            opt = opt || 'decimals';
            if(this.symbol.text_)
                return this.symbol.text_(opt);

            return text(this.symbol, opt, undefined, n);
        },
        /**
         * Returns the latex representation of the expression
         * @param {String} option - option for formatting numbers
         * @returns {String}
         */
        latex: function (option) {
            if(this.symbol.latex)
                return this.symbol.latex(option);
            return LaTeX.latex(this.symbol, option);
        },
        valueOf: function () {
            return this.symbol.valueOf();
        },

        /**
         * Evaluates the expression and tries to reduce it to a number if possible.
         * If an argument is given in the form of %{integer} it will evaluate that expression.
         * Other than that it will just use it's own text and reparse
         * @returns {Expression}
         */
        evaluate: function () {

            // Don't evaluate an empty vector
            if(isVector(this.symbol) && this.symbol.dimensions() === 0) {
                return this;
            }

            var first_arg = arguments[0], expression, idx = 1;

            //Enable getting of expressions using the % so for example %1 should get the first expression
            if(typeof first_arg === 'string') {
                expression = (first_arg.charAt(0) === '%') ? Expression.getExpression(first_arg.substr(1)).text() : first_arg;
            }
            else if(first_arg instanceof Expression || isSymbol(first_arg)) {
                expression = first_arg.text();
            }
            else {
                expression = this.symbol.text();
                idx--;
            }

            var subs = arguments[idx] || {};

            var retval = new Expression(block('PARSE2NUMBER', function () {
                return _.parse(expression, subs);
            }, true));

            return retval;
        },
        /**
         * Converts a symbol to a JS function. Pass in an array of variables to use that order instead of
         * the default alphabetical order
         * @param vars {Array}
         */
        buildFunction: function (vars) {
            return Build.build(this.symbol, vars);
        },
        /**
         * Checks to see if the expression is just a plain old number
         * @returns {boolean}
         */
        isNumber: function () {
            return isNumericSymbol(this.symbol);
        },
        /**
         * Checks to see if the expression is infinity
         * @returns {boolean}
         */
        isInfinity: function () {
            return Math.abs(this.symbol.multiplier) === Infinity;
        },
        /**
         * Checks to see if the expression contains imaginary numbers
         * @returns {boolean}
         */
        isImaginary: function () {
            return evaluate(_.parse(this.symbol)).isImaginary();
        },
        /**
         * Returns all the variables in the expression
         * @returns {Array}
         */
        variables: function () {
            return variables(this.symbol);
        },

        toString: function () {
            try {
                if(isArray(this.symbol))
                    return '[' + this.symbol.toString() + ']';
                return this.symbol.toString();
            }
            catch(e) {
                return '';
            }
        },
        //forces the symbol to be returned as a decimal
        toDecimal: function (prec) {
            Settings.precision = prec;
            var dec = text(this.symbol, 'decimals');
            Settings.precision = undefined;
            return dec;
        },
        //checks to see if the expression is a fraction
        isFraction: function () {
            return isFraction(this.symbol);
        },
        //checks to see if the symbol is a multivariate polynomial
        isPolynomial: function () {
            return this.symbol.isPoly();
        },
        //performs a substitution
        sub: function (symbol, for_symbol) {
            return new Expression(this.symbol.sub(_.parse(symbol), _.parse(for_symbol)));
        },
        operation: function (otype, symbol) {
            if(isExpression(symbol))
                symbol = symbol.symbol;
            else if(!isSymbol(symbol))
                symbol = _.parse(symbol);
            return new Expression(_[otype](this.symbol.clone(), symbol.clone()));
        },
        add: function (symbol) {
            return this.operation('add', symbol);
        },
        subtract: function (symbol) {
            return this.operation('subtract', symbol);
        },
        multiply: function (symbol) {
            return this.operation('multiply', symbol);
        },
        divide: function (symbol) {
            return this.operation('divide', symbol);
        },
        pow: function (symbol) {
            return this.operation('pow', symbol);
        },
        expand: function () {
            return new Expression(_.expand(this.symbol));
        },
        each: function (callback, i) {
            if(this.symbol.each)
                this.symbol.each(callback, i);
            else if(isArray(this.symbol)) {
                for(var i = 0; i < this.symbol.length; i++)
                    callback.call(this.symbol, this.symbol[i], i);
            }
            else
                callback.call(this.symbol);
        },
        eq: function (value) {
            if(!isSymbol(value))
                value = _.parse(value);
            try {
                var d = _.subtract(this.symbol.clone(), value);
                return d.equals(0);
            }
            catch(e) {
                return false;
            }
            ;
        },
        lt: function (value) {
            if(!isSymbol(value))
                value = _.parse(value);
            try {
                var d = evaluate(_.subtract(this.symbol.clone(), value));
                return d.lessThan(0);
            }
            catch(e) {
                return false;
            }
            ;
        },
        gt: function (value) {
            if(!isSymbol(value))
                value = _.parse(value);
            try {
                var d = evaluate(_.subtract(this.symbol.clone(), value));
                return d.greaterThan(0);
            }
            catch(e) {
                return false;
            }
        },
        gte: function (value) {
            return this.gt(value) || this.eq(value);
        },
        lte: function (value) {
            return this.lt(value) || this.eq(value);
        },

        numerator: function () {
            return new Expression(this.symbol.getNum());
        },
        denominator: function () {
            return new Expression(this.symbol.getDenom());
        },
        hasFunction: function (f) {
            return this.symbol.containsFunction(f);
        },
        contains: function (variable) {
            return this.symbol.contains(variable);
        }
    };
    //Aliases
    Expression.prototype.toTeX = Expression.prototype.latex;

    
//Scientific ===================================================================
    /*
     * Javascript has the toExponential method but this allows you to work with string and therefore any number of digits of your choosing
     * For example Scientific('464589498449496467924197545625247695464569568959124568489548454');
     */

    function Scientific(num) {
        if(!(this instanceof Scientific))
            return new Scientific(num);

        num = String(typeof num === 'undefined' ? 0 : num); //convert to a string

        //remove the sign
        if(num.startsWith('-')) {
            this.sign = -1;
            //remove the sign
            num = num.substr(1, num.length);
        }
        else {
            this.sign = 1;
        }

        if(Scientific.isScientific(num)) {
            this.fromScientific(num);
        }
        else {
            this.convert(num);
        }
        return this;
    }

    Scientific.prototype = {
        fromScientific: function (num) {
            var parts = String(num).toLowerCase().split('e');
            this.coeff = parts[0];
            this.exponent = parts[1];

            return this;
        },
        convert: function (num) {
            //get wholes and decimals
            var parts = num.split('.');
            //make zero go away
            var w = parts[0] || '';
            var d = parts[1] || '';
            //convert zero to blank strings
            w = Scientific.removeLeadingZeroes(w);
            d = Scientific.removeTrailingZeroes(d);
            //find the location of the decimal place which is right after the wholes
            var dot_location = w.length;
            //add them together so we can move the dot
            var n = w + d;
            //find the next number
            var zeroes = Scientific.leadingZeroes(n).length;
            //set the exponent
            this.exponent = dot_location - (zeroes + 1);
            //set the coeff but first remove leading zeroes
            var coeff = Scientific.removeLeadingZeroes(n);
            this.coeff = coeff.charAt(0) + '.' + (coeff.substr(1, coeff.length) || '0');

            //the coeff decimal places
            var dec = this.coeff.split('.')[1] || ''; //if it's undefined or zero it's going to blank

            this.decp = dec === '0' ? 0 : dec.length;
            //decimals
            this.dec = d;
            //wholes
            this.wholes = w;

            return this;
        },
        round: function (num) {
            var n = this.copy();

            num = Number(num); //cast to number for safety
            //since we know it guaranteed to be in the format {digit}{optional dot}{optional digits}
            //we can round based on this
            if(num === 0)
                n.coeff = n.coeff.charAt(0);
            else {
                //get up to n-1 digits
                var rounded = this.coeff.substring(0, num + 1);
                //get the next two
                var next_two = this.coeff.substring(num + 1, num + 3);
                //the extra digit
                var ed = next_two.charAt(0);

                if(next_two.charAt(1) > 4)
                    ed++;

                n.coeff = rounded + ed;
            }

            return n;
        },
        copy: function () {
            var n = new Scientific(0);
            n.coeff = this.coeff;
            n.exponent = this.exponent;
            n.sign = this.sign;
            return n;
        },
        toString: function (n) {
            var retval;

            if(Settings.SCIENTIFIC_IGNORE_ZERO_EXPONENTS && this.exponent === 0 && this.decp < n) {
                if(this.decp === 0)
                    retval = this.wholes;
                else
                    retval = this.coeff;
            }
            else {
                var coeff = typeof n === 'undefined' ? this.coeff : Scientific.round(this.coeff, Math.min(n, this.decp || 1));
                retval = this.exponent === 0 ? coeff : coeff + 'e' + this.exponent;
            }

            return (this.sign === -1 ? '-' : '') + retval;
        }
    };

    Scientific.isScientific = function (num) {
        return /\d+\.?\d*e[\+\-]*\d+/i.test(num);
    };
    Scientific.leadingZeroes = function (num) {
        var match = num.match(/^(0*).*$/);
        return match ? match[1] : '';
    };
    Scientific.removeLeadingZeroes = function (num) {
        var match = num.match(/^0*(.*)$/);
        return match ? match[1] : '';
    };

    Scientific.removeTrailingZeroes = function (num) {
        var match = num.match(/0*$/);
        return match ? num.substring(0, num.length - match[0].length) : '';
    };

    Scientific.round = function (c, n) {
        var coeff = nround(c, n);
        var m = String(coeff).split('.').pop();
        var d = n - m.length;
        //if we're asking for more significant figures
        if(d > 0) {
            coeff = coeff + (new Array(d + 1).join(0));
        }
        return coeff;
    };

//Frac =========================================================================
    function Frac(n) {
        if(n instanceof Frac)
            return n;
        if(n === undefined)
            return this;
        try {
            if(isInt(n)) {
                try {
                    this.num = bigInt(n);
                    this.den = bigInt(1);
                }
                catch(e) {
                    return Frac.simple(n);
                }
            }
            else {
                var frac = n instanceof bigDec ? Fraction.quickConversion(n) : Fraction.convert(n);
                this.num = new bigInt(frac[0]);
                this.den = new bigInt(frac[1]);
            }
        }
        catch(e) {
            return Frac.simple(n);
        }

    }
    //safe to use with negative numbers or other types
    Frac.create = function (n) {
        if(n instanceof Frac)
            return n;
        n = n.toString();
        var is_neg = n.charAt(0) === '-'; //check if it's negative
        if(is_neg)
            n = n.substr(1, n.length - 1); //remove the sign
        var frac = new Frac(n);
        //put the sign back
        if(is_neg)
            frac.negate();
        return frac;
    };
    Frac.isFrac = function (o) {
        return (o instanceof Frac);
    };
    Frac.quick = function (n, d) {
        var frac = new Frac();
        frac.num = new bigInt(n);
        frac.den = new bigInt(d);
        return frac;
    };
    Frac.simple = function (n) {
        var nstr = String(scientificToDecimal(n)),
                m_dc = nstr.split('.'),
                num = m_dc.join(''),
                den = 1,
                l = (m_dc[1] || '').length;
        for(var i = 0; i < l; i++)
            den += '0';
        var frac = Frac.quick(num, den);
        return frac.simplify();
    };
    Frac.prototype = {
        multiply: function (m) {
            if(this.isOne()) {
                return m.clone();
            }
            if(m.isOne()) {
                return this.clone();
            }

            var c = this.clone();
            c.num = c.num.multiply(m.num);
            c.den = c.den.multiply(m.den);

            return c.simplify();
        },
        divide: function (m) {
            if(m.equals(0))
                throw new DivisionByZero('Division by zero not allowed!');
            return this.clone().multiply(m.clone().invert()).simplify();
        },
        subtract: function (m) {
            return this.clone().add(m.clone().neg());
        },
        neg: function () {
            this.num = this.num.multiply(-1);
            return this;
        },
        add: function (m) {
            var n1 = this.den, n2 = m.den, c = this.clone();
            var a = c.num, b = m.num;
            if(n1.equals(n2)) {
                c.num = a.add(b);
            }
            else {
                c.num = a.multiply(n2).add(b.multiply(n1));
                c.den = n1.multiply(n2);
            }

            return c.simplify();
        },
        mod: function (m) {
            var a = this.clone(),
                    b = m.clone();
            //make their denominators even and return the mod of their numerators
            a.num = a.num.multiply(b.den);
            a.den = a.den.multiply(b.den);
            b.num = b.num.multiply(this.den);
            b.den = b.den.multiply(this.den);
            a.num = a.num.mod(b.num);
            return a.simplify();
        },
        simplify: function () {
            var gcd = bigInt.gcd(this.num, this.den);

            this.num = this.num.divide(gcd);
            this.den = this.den.divide(gcd);
            return this;
        },
        clone: function () {
            var m = new Frac();
            m.num = new bigInt(this.num);
            m.den = new bigInt(this.den);
            return m;
        },
        decimal: function (prec) {
            var sign = this.num.isNegative() ? '-' : '';
            if(this.num.equals(this.den)) {
                return '1';
            }
            //go plus one for rounding
            prec = prec || Settings.PRECISION;
            prec++;
            var narr = [],
                    n = this.num.abs(),
                    d = this.den;
            for(var i = 0; i < prec; i++) {
                var w = n.divide(d), //divide out whole
                        r = n.subtract(w.multiply(d)); //get remainder

                narr.push(w);
                if(r.equals(0))
                    break;
                n = r.times(10); //shift one dec place
            }
            var whole = narr.shift();
            if(narr.length === 0) {
                return sign + whole.toString();
            }

            if(i === prec) {
                var lt = [];
                //get the last two so we can round it
                for(var i = 0; i < 2; i++)
                    lt.unshift(narr.pop());
                //put the last digit back by rounding the last two
                narr.push(Math.round(lt.join('.')));
            }

            var dec = whole.toString() + '.' + narr.join('');
            return sign + dec;
        },
        toDecimal: function (prec) { 
            prec = prec || Settings.PRECISION;
            if(prec) {
                return this.decimal(prec);
            }
            else {
                return this.num / this.den;
            }
        },
        qcompare: function (n) {
            return [this.num.multiply(n.den), n.num.multiply(this.den)];
        },
        equals: function (n) {
            if(!isNaN(n))
                n = new Frac(n);
            var q = this.qcompare(n);

            return q[0].equals(q[1]);
        },
        absEquals: function (n) {
            if(!isNaN(n))
                n = new Frac(n);
            var q = this.qcompare(n);

            return q[0].abs().equals(q[1]);
        },
        //lazy check to be fixed. Sufficient for now but will cause future problems
        greaterThan: function (n) {
            if(!isNaN(n))
                n = new Frac(n);
            var q = this.qcompare(n);

            return q[0].gt(q[1]);
        },
        gte: function (n) {
            return this.greaterThan(n) || this.equals(n);
        },
        lte: function (n) {
            return this.lessThan(n) || this.equals(n);
        },
        lessThan: function (n) {
            if(!isNaN(n))
                n = new Frac(n);
            var q = this.qcompare(n);

            return q[0].lt(q[1]);
        },
        isInteger: function () {
            return this.den.equals(1);
        },
        negate: function () {
            this.num = this.num.multiply(-1);
            return this;
        },
        invert: function () {
            var t = this.den;
            //why invert 0/1? It'll become 1/0 and that's a lie.
            if(!this.num.equals(0)) {
                var isnegative = this.num.isNegative();
                this.den = this.num.abs();
                this.num = t;
                if(isnegative)
                    this.num = this.num.multiply(-1);
            }

            return this;
        },
        isOne: function () {
            return this.num.equals(1) && this.den.equals(1);
        },
        sign: function () {
            return this.num.isNegative() ? -1 : 1;
        },
        abs: function () {
            this.num = this.num.abs();
            return this;
        },
        gcd: function (f) {
            return Frac.quick(bigInt.gcd(f.num, this.num), bigInt.lcm(f.den, this.den));
        },
        toString: function () {
            return !this.den.equals(1) ? this.num.toString() + '/' + this.den.toString() : this.num.toString();
        },
        valueOf: function () {
//            if(this.num == 24) throw new Error(999)
            if(Settings.USE_BIG)
                return new bigDec(this.num.toString()).div(new bigDec(this.den.toString()));
            var retval = this.num / this.den;
            return retval;
        },
        isNegative: function () {
            return this.toDecimal() < 0;
        }
    };

//Symbol =======================================================================
    /**
     * All symbols e.g. x, y, z, etc or functions are wrapped in this class. All symbols have a multiplier and a group.
     * All symbols except for "numbers (group N)" have a power.
     * @class Primary data type for the Parser.
     * @param {String} obj
     * @returns {Symbol}
     */
    function Symbol(obj) {
        var isInfinity = obj === 'Infinity';
        // This enables the class to be instantiated without the new operator
        if(!(this instanceof Symbol)) {
            return new Symbol(obj);
        }
        // Convert big numbers to a string
        if(obj instanceof bigDec) {
            obj = obj.toString();
        }
        //define numeric symbols
        if(/^(\-?\+?\d+)\.?\d*e?\-?\+?\d*/i.test(obj) || obj instanceof bigDec) {
            this.group = N;
            this.value = CONST_HASH;
            this.multiplier = new Frac(obj);
        }
        //define symbolic symbols
        else {
            this.group = S;
            validateName(obj);
            this.value = obj;
            this.multiplier = new Frac(1);
            this.imaginary = obj === Settings.IMAGINARY;
            this.isInfinity = isInfinity;
        }

        //As of 6.0.0 we switched to infinite precision so all objects have a power
        //Although this is still redundant in constants, it simplifies the logic in
        //other parts so we'll keep it
        this.power = new Frac(1);

        // Added to silence the strict warning.
        return this;
    }
    /**
     * Returns vanilla imaginary symbol
     * @returns {Symbol}
     */
    Symbol.imaginary = function () {
        var s = new Symbol(Settings.IMAGINARY);
        s.imaginary = true;
        return s;
    };
    /**
     * Return nerdamer's representation of Infinity
     * @param {int} negative -1 to return negative infinity
     * @returns {Symbol}
     */
    Symbol.infinity = function (negative) {
        var v = new Symbol('Infinity');
        if(negative === -1)
            v.negate();
        return v;
    };
    Symbol.shell = function (group, value) {
        var symbol = new Symbol(value);
        symbol.group = group;
        symbol.symbols = {};
        symbol.length = 0;
        return symbol;
    };
    //sqrt(x) -> x^(1/2)
    Symbol.unwrapSQRT = function (symbol, all) {
        var p = symbol.power;
        if(symbol.fname === SQRT && (symbol.isLinear() || all)) {
            var t = symbol.args[0].clone();
            t.power = t.power.multiply(new Frac(1 / 2));
            t.multiplier = t.multiplier.multiply(symbol.multiplier);
            symbol = t;
            if(all)
                symbol.power = p.multiply(new Frac(1 / 2));
        }

        return symbol;
    };
    Symbol.hyp = function (a, b) {
        a = a || new Symbol(0);
        b = b || new Symbol(0);
        return _.sqrt(_.add(_.pow(a.clone(), new Symbol(2)), _.pow(b.clone(), new Symbol(2))));
    };
    //converts to polar form array
    Symbol.toPolarFormArray = function (symbol) {
        var re, im, r, theta;
        re = symbol.realpart();
        im = symbol.imagpart();
        r = Symbol.hyp(re, im);
        theta = re.equals(0) ? _.parse('pi/2') : _.trig.atan(_.divide(im, re));
        return [r, theta];
    };
    //removes parentheses
    Symbol.unwrapPARENS = function (symbol) {
        if(symbol.fname === '') {
            var r = symbol.args[0];
            r.power = r.power.multiply(symbol.power);
            r.multiplier = r.multiplier.multiply(symbol.multiplier);
            if(symbol.fname === '')
                return Symbol.unwrapPARENS(r);
            return r;
        }
        return symbol;
    };
    //quickly creates a Symbol
    Symbol.create = function (value, power) {
        power = power === undefined ? 1 : power;
        return _.parse('(' + value + ')^(' + power + ')');
    };
    Symbol.prototype = {
        /**
         * Gets nth root accounting for rounding errors
         * @param {Number} n
         * @return {Number}
         */
        getNth: function (n) {
            // First calculate the root
            var root = evaluate(_.pow(_.parse(this.multiplier), _.parse(n).invert()));
            // Round of any errors
            var rounded = _.parse(nround(root));
            // Reverse the root
            var e = evaluate(_.pow(rounded, _.parse(n)));
            // If the rounded root equals the original number then we're good 
            if(e.equals(_.parse(this.multiplier))) {
                return rounded;
            }
            // Otherwise return the unrounded version
            return root;
        },
        /**
         * Checks if symbol is to the nth power
         * @returns {Boolean}
         */
        isToNth: function (n) {
            // Start by check in the multiplier for squareness
            // First get the root but round it because currently we still depend 
            var root = this.getNth(n);
            var nthMultiplier = isInt(root);
            var nthPower;

            if(this.group === CB) {
                // Start by assuming that all will be square.
                nthPower = true;
                // All it takes is for one of the symbols to not have an even power
                // e.g. x^n1*y^n2 requires that both n1 and n2 are even
                this.each(function (x) {
                    var isNth = x.isToNth(n);

                    if(!isNth) {
                        nthPower = false;
                    }
                });
            }
            else {
                // Check if the power is divisible by n if it's not a number.
                nthPower = this.group === N ? true : isInt(_.divide(_.parse(this.power), _.parse(n)));
            }

            return nthMultiplier && nthPower;
        },
        /**
         * Checks if a symbol is square
         * @return {Boolean}
         */
        isSquare: function () {
            return this.isToNth(2);
        },
        /**
         * Checks if a symbol is cube
         * @return {Boolean}
         */
        isCube: function () {
            return this.isToNth(3);
        },
        /**
         * Checks if a symbol is a bare variable
         * @return {Boolean}
         */
        isSimple: function () {
            return this.power.equals(1) && this.multiplier.equals(1);
        },
        /**
         * Simplifies the power of the symbol
         * @returns {Symbol} a clone of the symbol
         */
        powSimp: function () {
            if(this.group === CB) {
                var powers = [],
                        sign = this.multiplier.sign();
                this.each(function (x) {
                    var p = x.power;
                    //why waste time if I can't do anything anyway
                    if(isSymbol(p) || p.equals(1))
                        return this.clone();
                    powers.push(p);
                });
                var min = new Frac(arrayMin(powers));

                //handle the coefficient
                //handle the multiplier
                var sign = this.multiplier.sign(),
                        m = this.multiplier.clone().abs(),
                        mfactors = Math2.ifactor(m);
                //if we have a multiplier of 6750 and a min of 2 then the factors are 5^3*5^3*2
                //we can then reduce it to 2*3*5*(15)^2
                var out_ = new Frac(1);
                var in_ = new Frac(1);

                for(var x in mfactors) {
                    var n = new Frac(mfactors[x]);
                    if(!n.lessThan(min)) {
                        n = n.divide(min).subtract(new Frac(1));
                        in_ = in_.multiply(new Frac(x)); //move the factor inside the bracket
                    }

                    out_ = out_.multiply(_.parse(inBrackets(x) + '^' + inBrackets(n)).multiplier);
                }
                var t = new Symbol(in_);
                this.each(function (x) {
                    x = x.clone();
                    x.power = x.power.divide(min);
                    t = _.multiply(t, x);
                });

                var xt = _.symfunction(PARENTHESIS, [t]);
                xt.power = min;
                xt.multiplier = sign < 0 ? out_.negate() : out_;

                return xt;
            }
            return this.clone();
        },
        /**
         * Checks to see if two functions are of equal value
         * @param {Symbol} symbol
         */
        equals: function (symbol) {
            if(!isSymbol(symbol))
                symbol = new Symbol(symbol);
            return this.value === symbol.value && this.power.equals(symbol.power)
                    && this.multiplier.equals(symbol.multiplier)
                    && this.group === symbol.group;
        },
        abs: function () {
            var e = this.clone();
            e.multiplier.abs();
            return e;
        },
        // Greater than
        gt: function (symbol) {
            if(!isSymbol(symbol))
                symbol = new Symbol(symbol);
            return this.isConstant() && symbol.isConstant() && this.multiplier.greaterThan(symbol.multiplier);
        },
        // Greater than
        gte: function (symbol) {
            if(!isSymbol(symbol))
                symbol = new Symbol(symbol);
            return this.equals(symbol) ||
                    this.isConstant() && symbol.isConstant() && this.multiplier.greaterThan(symbol.multiplier);
        },
        // Less than
        lt: function (symbol) {
            if(!isSymbol(symbol))
                symbol = new Symbol(symbol);
            return this.isConstant() && symbol.isConstant() && this.multiplier.lessThan(symbol.multiplier);
        },
        // Less than
        lte: function (symbol) {
            if(!isSymbol(symbol))
                symbol = new Symbol(symbol);
            return this.equals(symbol) ||
                    this.isConstant() && symbol.isConstant() && this.multiplier.lessThan(symbol.multiplier);
        },
        /**
         * Because nerdamer doesn't group symbols by polynomials but
         * rather a custom grouping method, this has to be
         * reinserted in order to make use of most algorithms. This function
         * checks if the symbol meets the criteria of a polynomial.
         * @param {bool} multivariate
         * @returns {boolean}
         */
        isPoly: function (multivariate) {
            var g = this.group,
                    p = this.power;
            //the power must be a integer so fail if it's not
            if(!isInt(p) || p < 0)
                return false;
            //constants and first orders
            if(g === N || g === S || this.isConstant(true))
                return true;
            var vars = variables(this);
            if(g === CB && vars.length === 1) {
                //the variable is assumed the only one that was found
                var v = vars[0];
                //if no variable then guess what!?!? We're done!!! We have a polynomial.
                if(!v)
                    return true;
                for(var x in this.symbols) {
                    var sym = this.symbols[x];
                    //sqrt(x)
                    if(sym.group === FN && !sym.args[0].isConstant())
                        return false;
                    if(!sym.contains(v) && !sym.isConstant(true))
                        return false;
                }
                return true;
            }
            //PL groups. These only fail if a power is not an int
            //this should handle cases such as x^2*t
            if(this.isComposite() || g === CB && multivariate) {
                //fail if we're not checking for multivariate polynomials
                if(!multivariate && vars.length > 1)
                    return false;
                //loop though the symbols and check if they qualify
                for(var x in this.symbols) {
                    //we've already the symbols if we're not checking for multivariates at this point
                    //so we check the sub-symbols
                    if(!this.symbols[x].isPoly(multivariate))
                        return false;
                }
                return true;
            }
            else
                return false;

            /*
             //all tests must have passed so we must be dealing with a polynomial
             return true;
             */
        },
        //removes the requested variable from the symbol and returns the remainder
        stripVar: function (x, exclude_x) {
            var retval;
            if((this.group === PL || this.group === S) && this.value === x)
                retval = new Symbol(exclude_x ? 0 : this.multiplier);
            else if(this.group === CB && this.isLinear()) {
                retval = new Symbol(1);
                this.each(function (s) {
                    if(!s.contains(x, true))
                        retval = _.multiply(retval, s.clone());
                });
                retval.multiplier = retval.multiplier.multiply(this.multiplier);
            }
            else if(this.group === CP && !this.isLinear()) {
                retval = new Symbol(this.multiplier);
            }
            else if(this.group === CP && this.isLinear()) {
                retval = new Symbol(0);
                this.each(function (s) {
                    if(!s.contains(x)) {
                        var t = s.clone();
                        t.multiplier = t.multiplier.multiply(this.multiplier);
                        retval = _.add(retval, t);
                    }
                });
                //BIG TODO!!! It doesn't make much sense
                if(retval.equals(0))
                    retval = new Symbol(this.multiplier);
            }
            else if(this.group === EX && this.power.contains(x, true)) {
                retval = new Symbol(this.multiplier);
            }
            else if(this.group === FN && this.contains(x)) {
                retval = new Symbol(this.multiplier);
            }
            else
                //wth? This should technically be the multiplier.
                //Unfortunately this method wasn't very well thought out :`(.
                //should be: retval = new Symbol(this.multiplier);
                //use: ((1+x^2)*sqrt(-1+x^2))^(-1) for correction.
                //this will break a bunch of unit tests so be ready to for the long haul
                retval = this.clone();


            return retval;
        },
        //returns symbol in array form with x as base e.g. a*x^2+b*x+c = [c, b, a].
        toArray: function (v, arr) {
            arr = arr || {
                arr: [],
                add: function (x, idx) {
                    var e = this.arr[idx];
                    this.arr[idx] = e ? _.add(e, x) : x;
                }
            };
            var g = this.group;

            if(g === S && this.contains(v)) {
                arr.add(new Symbol(this.multiplier), this.power);
            }
            else if(g === CB) {
                var a = this.stripVar(v),
                        x = _.divide(this.clone(), a.clone());
                var p = x.isConstant() ? 0 : x.power;
                arr.add(a, p);
            }
            else if(g === PL && this.value === v) {
                this.each(function (x, p) {
                    arr.add(x.stripVar(v), p);
                });
            }
            else if(g === CP) {
                //the logic: they'll be broken into symbols so e.g. (x^2+x)+1 or (a*x^2+b*x+c)
                //each case is handled above
                this.each(function (x) {
                    x.toArray(v, arr);
                });
            }
            else if(this.contains(v)) {
                throw new NerdamerTypeError('Cannot convert to array! Exiting');
            }
            else {
                arr.add(this.clone(), 0); //it's just a constant wrt to v
            }
            //fill the holes
            arr = arr.arr; //keep only the array since we don't need the object anymore
            for(var i = 0; i < arr.length; i++)
                if(!arr[i])
                    arr[i] = new Symbol(0);
            return arr;
        },
        //checks to see if a symbol contans a function
        hasFunc: function (v) {
            var fn_group = this.group === FN || this.group === EX;
            if(fn_group && !v || fn_group && this.contains(v))
                return true;
            if(this.symbols) {
                for(var x in this.symbols) {
                    if(this.symbols[x].hasFunc(v))
                        return true;
                }
            }
            return false;
        },
        sub: function (a, b) {
            a = !isSymbol(a) ? _.parse(a) : a.clone();
            b = !isSymbol(b) ? _.parse(b) : b.clone();
            if(a.group === N || a.group === P)
                err('Cannot substitute a number. Must be a variable');
            var same_pow = false,
                    a_is_unit_multiplier = a.multiplier.equals(1),
                    m = this.multiplier.clone(),
                    retval;
            /*
             * In order to make the substitution the bases have to first match take
             * (x+1)^x -> (x+1)=y || x^2 -> x=y^6
             * In both cases the first condition is that the bases match so we begin there
             * Either both are PL or both are not PL but we cannot have PL and a non-PL group match
             */
            if(this.value === a.value && (this.group !== PL && a.group !== PL || this.group === PL && a.group === PL)) {
                //we cleared the first hurdle but a subsitution may not be possible just yet
                if(a_is_unit_multiplier || a.multiplier.equals(this.multiplier)) {
                    if(a.isLinear()) {
                        retval = b;
                    }
                    else if(a.power.equals(this.power)) {
                        retval = b;
                        same_pow = true;
                    }
                    if(a.multiplier.equals(this.multiplier))
                        m = new Frac(1);
                }
            }
            //the next thing is to handle CB
            else if(this.group === CB || this.previousGroup === CB) {
                retval = new Symbol(1);
                this.each(function (x) {
                    var subbed = _.parse(x.sub(a, b)); //parse it again for safety
                    retval = _.multiply(retval, subbed);

                });
            }
            else if(this.isComposite()) {
                var symbol = this.clone();

                if(a.isComposite() && symbol.isComposite() && symbol.isLinear() && a.isLinear()) {
                    var find = function (stack, needle) {
                        for(var x in stack.symbols) {
                            var sym = stack.symbols[x];
                            //if the symbol equals the needle or it's within the sub-symbols we're done
                            if(sym.isComposite() && find(sym, needle) || sym.equals(needle))
                                return true;
                        }
                        return false;
                    };
                    //go fish
                    for(var x in a.symbols) {
                        if(!find(symbol, a.symbols[x]))
                            return symbol.clone();
                    }
                    retval = _.add(_.subtract(symbol.clone(), a), b);
                }
                else {
                    retval = new Symbol(0);
                    symbol.each(function (x) {
                        retval = _.add(retval, x.sub(a, b));
                    });
                }
            }
            else if(this.group === EX) {
                // the parsed value could be a function so parse and sub
                retval = _.parse(this.value).sub(a, b);
            }
            else if(this.group === FN) {
                var nargs = [];
                for(var i = 0; i < this.args.length; i++) {
                    var arg = this.args[i];
                    if(!isSymbol(arg))
                        arg = _.parse(arg);
                    nargs.push(arg.sub(a, b));
                }
                retval = _.symfunction(this.fname, nargs);
            }
            //if we did manage a substitution
            if(retval) {
                if(!same_pow) {
                    //substitute the power
                    var p = this.group === EX ? this.power.sub(a, b) : _.parse(this.power);
                    //now raise the symbol to that power
                    retval = _.pow(retval, p);
                }

                //transfer the multiplier
                retval.multiplier = retval.multiplier.multiply(m);

                //done
                return retval;
            }
            //if all else fails
            return this.clone();
        },
        isMonomial: function () {
            if(this.group === S)
                return true;
            if(this.group === CB) {
                for(var x in this.symbols)
                    if(this.symbols[x].group !== S)
                        return false;
            }
            else
                return false;
            return true;
        },
        isPi: function () {
            return this.group === S && this.value === 'pi';
        },
        sign: function () {
            return this.multiplier.sign();
        },
        isE: function () {
            return this.value === 'e';
        },
        isSQRT: function () {
            return this.fname === SQRT;
        },
        isConstant: function (check_all, check_symbols) {
            if(check_symbols && this.group === CB) {
                for(var x in this.symbols) {
                    if(this.symbols[x].isConstant(true))
                        return true;
                }
            }

            if(check_all === 'functions' && this.isComposite()) {
                var isConstant = true;

                this.each(function (x) {
                    if(!x.isConstant(check_all, check_symbols)) {
                        isConstant = false;
                    }
                }, true);

                return isConstant;
            }

            if(check_all === 'all' && (this.isPi() || this.isE())) {
                return true;
            }

            if(check_all && this.group === FN) {
                for(var i = 0; i < this.args.length; i++) {
                    if(!this.args[i].isConstant(check_all))
                        return false;
                }
                return true;
            }

            if(check_all)
                return isNumericSymbol(this);
            return this.value === CONST_HASH;
        },
        //the symbols is imaginary if
        //1. n*i
        //2. a+b*i
        //3. a*i
        isImaginary: function () {
            if(this.imaginary)
                return true;
            else if(this.symbols) {
                for(var x in this.symbols)
                    if(this.symbols[x].isImaginary())
                        return true;
            }
            return false;
        },
        /**
         * Returns the real part of a symbol
         * @returns {Symbol}
         */
        realpart: function () {
            if(this.isConstant()) {
                return this.clone();
            }
            else if(this.imaginary)
                return new Symbol(0);
            else if(this.isComposite()) {
                var retval = new Symbol(0);
                this.each(function (x) {
                    retval = _.add(retval, x.realpart());
                });
                return retval;
            }
            else if(this.isImaginary())
                return new Symbol(0);
            return this.clone();
        },
        /*
         * Return imaginary part of a symbol
         * @returns {Symbol}
         */
        imagpart: function () {
            if(this.group === S && this.isImaginary())
                return new Symbol(this.multiplier);
            if(this.isComposite()) {
                var retval = new Symbol(0);
                this.each(function (x) {
                    retval = _.add(retval, x.imagpart());
                });
                return retval;
            }
            if(this.group === CB)
                return this.stripVar(Settings.IMAGINARY);
            return new Symbol(0);
        },
        isInteger: function () {
            return this.isConstant() && this.multiplier.isInteger();
        },
        isLinear: function (wrt) {
            if(wrt) {
                if(this.isConstant())
                    return true;
                if(this.group === S) {
                    if(this.value === wrt)
                        return this.power.equals(1);
                    else
                        return true;
                }

                if(this.isComposite() && this.power.equals(1)) {
                    for(var x in this.symbols) {
                        if(!this.symbols[x].isLinear(wrt))
                            return false;
                    }
                    return true;
                }

                if(this.group === CB && this.symbols[wrt])
                    return this.symbols[wrt].isLinear(wrt);
                return false;
            }
            else
                return this.power.equals(1);
        },
        /**
         * Checks to see if a symbol has a function by a specified name or within a specified list
         * @param {String|String[]} names
         * @returns {Boolean}
         */
        containsFunction: function (names) {
            if(typeof names === 'string')
                names = [names];
            if(this.group === FN && names.indexOf(this.fname) !== -1)
                return true;
            if(this.symbols) {
                for(var x in this.symbols) {
                    if(this.symbols[x].containsFunction(names))
                        return true;
                }
            }
            return false;
        },
        multiplyPower: function (p2) {
            //leave out 1
            if(this.group === N && this.multiplier.equals(1))
                return this;

            var p1 = this.power;

            if(this.group !== EX && p2.group === N) {
                var p = p2.multiplier;
                if(this.group === N && !p.isInteger()) {
                    this.convert(P);
                }

                this.power = p1.equals(1) ? p.clone() : p1.multiply(p);

                if(this.group === P && isInt(this.power)) {
                    //bring it back to an N
                    this.value = Math.pow(this.value, this.power);
                    this.toLinear();
                    this.convert(N);
                }
            }
            else {
                if(this.group !== EX) {
                    p1 = new Symbol(p1);
                    this.convert(EX);
                }
                this.power = _.multiply(p1, p2);
            }

            return this;
        },
        setPower: function (p, retainSign) {
            //leave out 1
            if(this.group === N && this.multiplier.equals(1)) {
                return this;
            }
            if(this.group === EX && !isSymbol(p)) {
                this.group = this.previousGroup;
                delete this.previousGroup;
                if(this.group === N) {
                    this.multiplier = new Frac(this.value);
                    this.value = CONST_HASH;
                }
                else
                    this.power = p;
            }
            else {
                var isSymbolic = false;
                if(isSymbol(p)) {
                    if(p.group === N) {
                        //p should be the multiplier instead
                        p = p.multiplier;

                    }
                    else {
                        isSymbolic = true;
                    }
                }
                var group = isSymbolic ? EX : P;
                this.power = p;
                if(this.group === N && group)
                    this.convert(group, retainSign);
            }

            return this;
        },
        /**
         * Checks to see if symbol is located in the denominator
         * @returns {boolean}
         */
        isInverse: function () {
            if(this.group === EX)
                return (this.power.multiplier.lessThan(0));
            return this.power < 0;
        },
        /**
         * Make a duplicate of a symbol by copying a predefined list of items.
         * The name 'copy' would probably be a more appropriate name.
         * to a new symbol
         * @param {Symbol} c
         * @returns {Symbol}
         */
        clone: function (c) {
            var clone = c || new Symbol(0),
                    //list of properties excluding power as this may be a symbol and would also need to be a clone.
                    properties = [
                        'value', 'group', 'length', 'previousGroup', 'imaginary', 'fname', 'args', 'isInfinity', 'scientific'],
                    l = properties.length, i;
            if(this.symbols) {
                clone.symbols = {};
                for(var x in this.symbols) {
                    clone.symbols[x] = this.symbols[x].clone();
                }
            }

            for(i = 0; i < l; i++) {
                if(this[properties[i]] !== undefined) {
                    clone[properties[i]] = this[properties[i]];
                }
            }

            clone.power = this.power.clone();
            clone.multiplier = this.multiplier.clone();
            //add back the flag to track if this symbol is a conversion symbol
            if(this.isConversion)
                clone.isConversion = this.isConversion;

            if(this.isUnit)
                clone.isUnit = this.isUnit;

            return clone;
        },
        /**
         * Converts a symbol multiplier to one.
         * @param {Boolean} keepSign Keep the multiplier as negative if the multiplier is negative and keepSign is true
         * @returns {Symbol}
         */
        toUnitMultiplier: function (keepSign) {
            this.multiplier.num = new bigInt(this.multiplier.num.isNegative() && keepSign ? -1 : 1);
            this.multiplier.den = new bigInt(1);
            return this;
        },
        /**
         * Converts a Symbol's power to one.
         * @returns {Symbol}
         */
        toLinear: function () {
            // Do nothing if it's already linear
            if(this.power.equals(1)) {
                return this;
            }
            this.setPower(new Frac(1));
            return this;
        },
        /**
         * Iterates over all the sub-symbols. If no sub-symbols exist then it's called on itself
         * @param {Function} fn
         * @@param {Boolean} deep If true it will itterate over the sub-symbols their symbols as well
         */
        each: function (fn, deep) {
            if(!this.symbols) {
                fn.call(this, this, this.value);
            }
            else {
                for(var x in this.symbols) {
                    var sym = this.symbols[x];
                    if(sym.group === PL && deep) {
                        for(var y in sym.symbols) {
                            fn.call(x, sym.symbols[y], y);
                        }
                    }
                    else
                        fn.call(this, sym, x);
                }
            }
        },
        /**
         * A numeric value to be returned for Javascript. It will try to
         * return a number as far a possible but in case of a pure symbolic
         * symbol it will just return its text representation
         * @returns {String|Number}
         */
        valueOf: function () {
            if(this.group === N)
                return this.multiplier.valueOf();
            else if(this.power === 0) {
                return 1;
            }
            else if(this.multiplier === 0) {
                return 0;
            }
            else {
                return text(this, 'decimals');
            }
        },
        /**
         * Checks to see if a symbols has a particular variable within it.
         * Pass in true as second argument to include the power of exponentials
         * which aren't check by default.
         * @example var s = _.parse('x+y+z'); s.contains('y');
         * //returns true
         * @param {any} variable
         * @param {boolean} all
         * @returns {boolean}
         */
        contains: function (variable, all) {
            //contains expects a string
            variable = String(variable);
            var g = this.group;
            if(this.value === variable)
                return true;
            if(this.symbols) {
                for(var x in this.symbols) {
                    if(this.symbols[x].contains(variable, all))
                        return true;
                }
            }
            if(g === FN || this.previousGroup === FN) {
                for(var i = 0; i < this.args.length; i++) {
                    if(this.args[i].contains(variable, all))
                        return true;
                }
            }

            if(g === EX) {
                //exit only if it does
                if(all && this.power.contains(variable, all)) {
                    return true;
                }
                if(this.value === variable)
                    return true;

            }

            return this.value === variable;
        },
        /**
         * Negates a symbols
         * @returns {boolean}
         */
        negate: function () {
            this.multiplier.negate();
            if(this.group === CP || this.group === PL)
                this.distributeMultiplier();
            return this;
        },
        /**
         * Inverts a symbol
         * @param {boolean} power_only
         * @param {boolean} all
         * @returns {boolean}
         */
        invert: function (power_only, all) {
            //invert the multiplier
            if(!power_only)
                this.multiplier = this.multiplier.invert();
            //invert the rest
            if(isSymbol(this.power)) {
                this.power.negate();
            }
            else if(this.group === CB && all) {
                this.each(function (x) {
                    return x.invert();
                });
            }
            else {
                if(this.power && this.group !== N)
                    this.power.negate();
            }
            return this;
        },
        /**
         * Symbols of group CP or PL may have the multiplier being carried by
         * the top level symbol at any given time e.g. 2*(x+y+z). This is
         * convenient in many cases, however in some cases the multiplier needs
         * to be carried individually e.g. 2*x+2*y+2*z.
         * This method distributes the multiplier over the entire symbol
         * @param {boolean} all
         * @returns {Symbol}
         */
        distributeMultiplier: function (all) {
            var is_one = all ? this.power.absEquals(1) : this.power.equals(1);
            if(this.symbols && is_one && this.group !== CB && !this.multiplier.equals(1)) {
                for(var x in this.symbols) {
                    var s = this.symbols[x];
                    s.multiplier = s.multiplier.multiply(this.multiplier);
                    s.distributeMultiplier();
                }
                this.toUnitMultiplier();
            }

            return this;
        },
        /**
         * This method expands the exponent over the entire symbol just like
         * distributeMultiplier
         * @returns {Symbol}
         */
        distributeExponent: function () {
            if(!this.power.equals(1)) {
                var p = this.power;
                for(var x in this.symbols) {
                    var s = this.symbols[x];
                    if(s.group === EX) {
                        s.power = _.multiply(s.power, new Symbol(p));
                    }
                    else {
                        this.symbols[x].power = this.symbols[x].power.multiply(p);
                    }
                }
                this.toLinear();
            }
            return this;
        },
        /**
         * This method will attempt to up-convert or down-convert one symbol
         * from one group to another. Not all symbols are convertible from one
         * group to another however. In that case the symbol will remain
         * unchanged.
         * @param {int} group
         * @param {string} imaginary
         */
        convert: function (group, imaginary) {
            if(group > FN) {
                //make a clone of this symbol;
                var cp = this.clone();

                //attach a symbols object and upgrade the group
                this.symbols = {};

                if(group === CB) {
                    //symbol of group CB hold symbols bound together through multiplication
                    //because of commutativity this multiplier can technically be anywhere within the group
                    //to keep track of it however it's easier to always have the top level carry it
                    cp.toUnitMultiplier();
                }
                else {
                    //reset the symbol
                    this.toUnitMultiplier();
                }

                if(this.group === FN) {
                    cp.args = this.args;
                    delete this.args;
                    delete this.fname;
                }

                //the symbol may originate from the symbol i but this property no longer holds true
                //after copying
                if(this.isImgSymbol)
                    delete this.isImgSymbol;

                this.toLinear();
                //attach a clone of this symbol to the symbols object using its proper key
                this.symbols[cp.keyForGroup(group)] = cp;
                this.group = group;
                //objects by default don't have a length property. However, in order to keep track of the number
                //of sub-symbols we have to impliment our own.
                this.length = 1;
            }
            else if(group === EX) {
                //1^x is just one so check and make sure
                if(!(this.group === N && this.multiplier.equals(1))) {
                    if(this.group !== EX)
                        this.previousGroup = this.group;
                    if(this.group === N) {
                        this.value = this.multiplier.num.toString();
                        this.toUnitMultiplier();
                    }
                    //update the hash to reflect the accurate hash
                    else
                        this.value = text(this, 'hash');

                    this.group = EX;
                }
            }
            else if(group === N) {
                var m = this.multiplier.toDecimal();
                if(this.symbols)
                    this.symbols = undefined;
                new Symbol(this.group === P ? m * Math.pow(this.value, this.power) : m).clone(this);
            }
            else if(group === P && this.group === N) {
                this.value = imaginary ? this.multiplier.num.toString() : Math.abs(this.multiplier.num.toString());
                this.toUnitMultiplier(!imaginary);
                this.group = P;
            }
            return this;
        },
        /**
         * This method is one of the principal methods to make it all possible.
         * It performs cleanup and prep operations whenever a symbols is
         * inserted. If the symbols results in a 1 in a CB (multiplication)
         * group for instance it will remove the redundant symbol. Similarly
         * in a symbol of group PL or CP (symbols glued by multiplication) it
         * will remove any dangling zeroes from the symbol. It will also
         * up-convert or down-convert a symbol if it detects that it's
         * incorrectly grouped. It should be noted that this method is not
         * called directly but rather by the 'attach' method for addition groups
         * and the 'combine' method for multiplication groups.
         * @param {Symbol} symbol
         * @param {String} action
         */
        insert: function (symbol, action) {
            //this check can be removed but saves a lot of aggravation when trying to hunt down
            //a bug. If left, you will instantly know that the error can only be between 2 symbols.
            if(!isSymbol(symbol))
                err('Object ' + symbol + ' is not of type Symbol!');
            if(this.symbols) {
                var group = this.group;
                if(group > FN) {
                    var key = symbol.keyForGroup(group);
                    var existing = key in this.symbols ? this.symbols[key] : false; //check if there's already a symbol there
                    if(action === 'add') {
                        var hash = key;
                        if(existing) {
                            //add them together using the parser
                            this.symbols[hash] = _.add(existing, symbol);
                            //if the addition resulted in a zero multiplier remove it
                            if(this.symbols[hash].multiplier.equals(0)) {
                                delete this.symbols[hash];
                                this.length--;

                                if(this.length === 0) {
                                    this.convert(N);
                                    this.multiplier = new Frac(0);
                                }
                            }
                        }
                        else {
                            this.symbols[key] = symbol;
                            this.length++;
                        }
                    }
                    else {
                        //check if this is of group P and unwrap before inserting
                        if(symbol.group === P && isInt(symbol.power)) {
                            symbol.convert(N);
                        }

                        //transfer the multiplier to the upper symbol but only if the symbol numeric
                        if(symbol.group !== EX) {
                            this.multiplier = this.multiplier.multiply(symbol.multiplier);
                            symbol.toUnitMultiplier();
                        }
                        else {
                            symbol.parens = symbol.multiplier.lessThan(0);
                            this.multiplier = this.multiplier.multiply(symbol.multiplier.clone().abs());
                            symbol.toUnitMultiplier(true);
                        }

                        if(existing) {
                            //remove because the symbol may have changed
                            symbol = _.multiply(remove(this.symbols, key), symbol);
                            if(symbol.isConstant()) {
                                this.multiplier = this.multiplier.multiply(symbol.multiplier);
                                symbol = new Symbol(1); //the dirty work gets done down the line when it detects 1
                            }

                            this.length--;
                            //clean up
                        }

                        //don't insert the symbol if it's 1
                        if(!symbol.isOne(true)) {
                            this.symbols[key] = symbol;
                            this.length++;
                        }
                        else if(symbol.multiplier.lessThan(0)) {
                            this.negate(); //put back the sign
                        }
                    }

                    //clean up
                    if(this.length === 0)
                        this.convert(N);
                    //update the hash
                    if(this.group === CP || this.group === CB) {
                        this.updateHash();
                    }
                }
            }

            return this;
        },
        //the insert method for addition
        attach: function (symbol) {
            if(isArray(symbol)) {
                for(var i = 0; i < symbol.length; i++)
                    this.insert(symbol[i], 'add');
                return this;
            }
            return this.insert(symbol, 'add');
        },
        //the insert method for multiplication
        combine: function (symbol) {
            if(isArray(symbol)) {
                for(var i = 0; i < symbol.length; i++)
                    this.insert(symbol[i], 'multiply');
                return this;
            }
            return this.insert(symbol, 'multiply');
        },
        /**
         * This method should be called after any major "surgery" on a symbol.
         * It updates the hash of the symbol for example if the fname of a
         * function has changed it will update the hash of the symbol.
         */
        updateHash: function () {
            if(this.group === N)
                return;

            if(this.group === FN) {
                var contents = '',
                        args = this.args,
                        is_parens = this.fname === PARENTHESIS;
                for(var i = 0; i < args.length; i++)
                    contents += (i === 0 ? '' : ',') + text(args[i]);
                var fn_name = is_parens ? '' : this.fname;
                this.value = fn_name + (is_parens ? contents : inBrackets(contents));
            }
            else if(!(this.group === S || this.group === PL)) {
                this.value = text(this, 'hash');
            }
        },
        /**
         * this function defines how every group in stored within a group of
         * higher order think of it as the switchboard for the library. It
         * defines the hashes for symbols.
         * @param {int} group
         */
        keyForGroup: function (group) {
            var g = this.group;
            var key;

            if(g === N) {
                key = this.value;
            }
            else if(g === S || g === P) {
                if(group === PL)
                    key = this.power.toDecimal();
                else
                    key = this.value;
            }
            else if(g === FN) {
                if(group === PL)
                    key = this.power.toDecimal();
                else
                    key = text(this, 'hash');
            }
            else if(g === PL) {
                //if the order is reversed then we'll assume multiplication
                //TODO: possible future dilemma
                if(group === CB)
                    key = text(this, 'hash');
                else if(group === CP) {
                    if(this.power.equals(1))
                        key = this.value;
                    else
                        key = inBrackets(text(this, 'hash')) + Settings.POWER_OPERATOR + this.power.toDecimal();
                }
                else if(group === PL)
                    key = this.power.toString();
                else
                    key = this.value;
                return key;
            }
            else if(g === CP) {
                if(group === CP) {
                    key = text(this, 'hash');
                }
                if(group === PL)
                    key = this.power.toDecimal();
                else
                    key = this.value;
            }
            else if(g === CB) {
                if(group === PL)
                    key = this.power.toDecimal();
                else
                    key = text(this, 'hash');
            }
            else if(g === EX) {
                if(group === PL)
                    key = text(this.power);
                else
                    key = text(this, 'hash');
            }

            return key;
        },
        /**
         * Symbols are typically stored in an object which works fine for most
         * cases but presents a problem when the order of the symbols makes
         * a difference. This function simply collects all the symbols and
         * returns them as an array. If a function is supplied then that
         * function is called on every symbol contained within the object.
         * @param {Function} fn
         * @param {Object} opt
         * @param {Function} sort_fn
         * @@param {Boolean} expand_symbol
         * @returns {Array}
         */
        collectSymbols: function (fn, opt, sort_fn, expand_symbol) {
            var collected = [];
            if(!this.symbols)
                collected.push(this);
            else {
                for(var x in this.symbols) {
                    var symbol = this.symbols[x];
                    if(expand_symbol && (symbol.group === PL || symbol.group === CP)) {
                        collected = collected.concat(symbol.collectSymbols());
                    }
                    else
                        collected.push(fn ? fn(symbol, opt) : symbol);
                }
            }
            if(sort_fn === null)
                sort_fn = undefined; //WTF Firefox? Seriously?

            return collected.sort(sort_fn);//sort hopefully gives us some sort of consistency
        },
        /**
         * Returns the latex representation of the symbol
         * @param {String} option
         * @returns {String}
         */
        latex: function (option) {
            return LaTeX.latex(this, option);
        },
        /**
         * Returns the text representation of a symbol
         * @param {String} option
         * @returns {String}
         */
        text: function (option) {
            return text(this, option);
        },
        /**
         * Checks if the function evaluates to 1. e.g. x^0 or 1 :)
         * @@param {bool} abs Compares the absolute value
         */
        isOne: function (abs) {
            var f = abs ? 'absEquals' : 'equals';
            if(this.group === N)
                return this.multiplier[f](1);
            else
                return this.power.equals(0);
        },
        isComposite: function () {
            var g = this.group,
                    pg = this.previousGroup;
            return g === CP || g === PL || pg === PL || pg === CP;
        },
        isCombination: function () {
            var g = this.group,
                    pg = this.previousGroup;
            return g === CB || pg === CB;
        },
        lessThan: function (n) {
            return this.multiplier.lessThan(n);
        },
        greaterThan: function (n) {
            if(!isSymbol(n)) {
                n = new Symbol(n);
            }

            // We can't tell for sure if a is greater than be if they're not both numbers
            if(!this.isConstant(true) || !n.isConstant(true)) {
                return false;
            }

            return this.multiplier.greaterThan(n.multiplier);
        },
        /**
         * Get's the denominator of the symbol if the symbol is of class CB (multiplication)
         * with other classes the symbol is either the denominator or not.
         * Take x^-1+x^-2. If the symbol was to be mixed such as x+x^-2 then the symbol doesn't have have an exclusive
         * denominator and has to be found by looking at the actual symbols themselves.
         */
        getDenom: function () {
            var retval, symbol;
            symbol = this.clone();
            //e.g. 1/(x*(x+1))
            if(this.group === CB && this.power.lessThan(0))
                symbol = _.expand(symbol);

            //if the symbol already is the denominator... DONE!!!
            if(symbol.power.lessThan(0) || symbol.group === EX && symbol.power.multiplier.lessThan(0)) {
                var d = _.parse(symbol.multiplier.den);
                retval = symbol.toUnitMultiplier();
                retval.power.negate();
                retval = _.multiply(d, retval); //put back the coeff
            }
            else if(symbol.group === CB) {
                retval = _.parse(symbol.multiplier.den);
                for(var x in symbol.symbols) {
                    var s = symbol.symbols[x];
                    if(s.power < 0 || s.group === EX && s.power.multiplier.lessThan(0))
                        retval = _.multiply(retval, symbol.symbols[x].clone().invert());
                }
            }
            else {
                retval = _.parse(symbol.multiplier.den);
            }
            return retval;
        },
        getNum: function () {
            var retval, symbol;
            symbol = this.clone();
            //e.g. 1/(x*(x+1))
            if(symbol.group === CB && symbol.power.lessThan(0))
                symbol = _.expand(symbol);
            //if the symbol already is the denominator... DONE!!!
            if(symbol.power.greaterThan(0) && symbol.group !== CB || symbol.group === EX && symbol.power.multiplier.greaterThan(0)) {
                retval = _.multiply(_.parse(symbol.multiplier.num), symbol.toUnitMultiplier());
            }
            else if(symbol.group === CB) {
                retval = _.parse(symbol.multiplier.num);
                symbol.each(function (x) {
                    if(x.power > 0 || x.group === EX && x.power.multiplier > 0) {
                        retval = _.multiply(retval, x.clone());
                    }
                });
            }
//            else if(symbol.group === EX && this.previousGroup === S) {
//                retval = _.multiply(_.parse(symbol.multiplier.num), symbol.toUnitMultiplier());
//            }
            else {
                retval = _.parse(symbol.multiplier.num);
            }
            return retval;
        },
        toString: function () {
            return this.text();
        }
    };

//Parser =======================================================================
    //Uses modified Shunting-yard algorithm. http://en.wikipedia.org/wiki/Shunting-yard_algorithm
    function Parser() {
        //Point to the local parser instead of the global one
        var _ = this;
        var bin = {};
        var preprocessors = {names: [], actions: []};

//Parser.classes ===============================================================
        function Slice(upper, lower) {
            this.start = upper;
            this.end = lower;
        }
        ;
        Slice.prototype.isConstant = function () {
            return this.start.isConstant() && this.end.isConstant();
        };
        Slice.prototype.text = function () {
            return text(this.start) + ':' + text(this.end);
        };

        /**
         * Class used to collect arguments for functions
         * @returns {Parser.Collection}
         */
        function Collection() {
            this.elements = [];
        }
        Collection.prototype.append = function (e) {
            this.elements.push(e);
        };
        Collection.prototype.getItems = function () {
            return this.elements;
        };
        Collection.prototype.toString = function () {
            return _.pretty_print(this.elements);
        };
        Collection.create = function (e) {
            var collection = new Collection();
            if(e)
                collection.append(e);
            return collection;
        };

        function Token(node, node_type, column) {
            this.type = node_type;
            this.value = node;
            if(column !== undefined)
                this.column = column + 1;
            if(node_type === Token.OPERATOR) {
                //copy everything over from the operator
                var operator = operators[node];
                for(var x in operator)
                    this[x] = operator[x];

            }
            else if(node_type === Token.FUNCTION) {
                this.precedence = Token.MAX_PRECEDENCE; //leave enough roon
                this.leftAssoc = false;
            }
        }
        Token.prototype.toString = function () {
            return this.value;
        };
        Token.prototype.toString = function () {
            if(this.is_prefix)
                return '`' + this.value;
            return this.value;
        };
        //some constants
        Token.OPERATOR = 'OPERATOR';
        Token.VARIABLE_OR_LITERAL = 'VARIABLE_OR_LITERAL';
        Token.FUNCTION = 'FUNCTION';
        Token.UNIT = 'UNIT';
        Token.KEYWORD = 'KEYWORD';
        Token.MAX_PRECEDENCE = 999;
        //create link to classes
        this.classes = {
            Collection: Collection,
            Slice: Slice,
            Token: Token
        };
//Parser.modules ===============================================================
        //object for functions which handle complex number
        var complex = {
            prec: undefined,
            cos: function (r, i) {
                var re, im;
                re = _.parse(Math.cos(r) * Math.cosh(i));
                im = _.parse(Math.sin(r) * Math.sinh(i));
                return _.subtract(re, _.multiply(im, Symbol.imaginary()));
            },
            sin: function (r, i) {
                var re, im;
                re = _.parse(Math.sin(r) * Math.cosh(i));
                im = _.parse(Math.cos(r) * Math.sinh(i));
                return _.subtract(re, _.multiply(im, Symbol.imaginary()));
            },
            tan: function (r, i) {
                var re, im;
                re = _.parse(Math.sin(2 * r) / (Math.cos(2 * r) + Math.cosh(2 * i)));
                im = _.parse(Math.sinh(2 * i) / (Math.cos(2 * r) + Math.cosh(2 * i)));
                return _.add(re, _.multiply(im, Symbol.imaginary()));
            },
            sec: function (r, i) {
                var t = this.removeDen(this.cos(r, i));
                return _.subtract(t[0], _.multiply(t[1], Symbol.imaginary()));
            },
            csc: function (r, i) {
                var t = this.removeDen(this.sin(r, i));
                return _.add(t[0], _.multiply(t[1], Symbol.imaginary()));
            },
            cot: function (r, i) {
                var t = this.removeDen(this.tan(r, i));
                return _.subtract(t[0], _.multiply(t[1], Symbol.imaginary()));
            },
            acos: function (r, i) {
                var symbol, sq, a, b, c, squared;
                symbol = this.fromArray([r, i]);
                squared = _.pow(symbol.clone(), new Symbol(2));
                sq = _.expand(squared); //z*z
                a = _.multiply(sqrt(_.subtract(new Symbol(1), sq)), Symbol.imaginary());
                b = _.expand(_.add(symbol.clone(), a));
                c = log(b);
                return _.expand(_.multiply(Symbol.imaginary().negate(), c));
            },
            asin: function (r, i) {
                return _.subtract(_.parse('pi/2'), this.acos(r, i));
            },
            atan: function (r, i) {
                // Handle i and -i
                if(r.equals(0) && (i.equals(1) || i.equals(-1))) {
                    // Just copy Wolfram Alpha for now. The parenthesis 
                    return _.parse(`${Symbol.infinity()}*${Settings.IMAGINARY}*${i}`);
                }
                var a, b, c, symbol;
                symbol = complex.fromArray([r, i]);
                a = _.expand(_.multiply(Symbol.imaginary(), symbol.clone()));
                b = log(_.expand(_.subtract(new Symbol(1), a.clone())));
                c = log(_.expand(_.add(new Symbol(1), a.clone())));
                return _.expand(_.multiply(_.divide(Symbol.imaginary(), new Symbol(2)), _.subtract(b, c)));
            },
            asec: function (r, i) {
                var d = this.removeDen([r, i]);
                d[1].negate();
                return this.acos.apply(this, d);
            },
            acsc: function (r, i) {
                var d = this.removeDen([r, i]);
                d[1].negate();
                return this.asin.apply(this, d);
            },
            acot: function (r, i) {
                var d = this.removeDen([r, i]);
                d[1].negate();
                return this.atan.apply(this, d);
            },
            //Hyperbolic trig
            cosh: function (r, i) {
                var re, im;
                re = _.parse(Math.cosh(r) * Math.cos(i));
                im = _.parse(Math.sinh(r) * Math.sin(i));
                return _.add(re, _.multiply(im, Symbol.imaginary()));
            },
            sinh: function (r, i) {
                var re, im;
                re = _.parse(Math.sinh(r) * Math.cos(i));
                im = _.parse(Math.cosh(r) * Math.sin(i));
                return _.add(re, _.multiply(im, Symbol.imaginary()));
            },
            tanh: function (r, i) {
                var re, im;
                re = _.parse(Math.sinh(2 * r) / (Math.cos(2 * i) + Math.cosh(2 * r)));
                im = _.parse(Math.sin(2 * i) / (Math.cos(2 * i) + Math.cosh(2 * r)));
                return _.subtract(re, _.multiply(im, Symbol.imaginary()));
            },
            sech: function (r, i) {
                var t = this.removeDen(this.cosh(r, i));
                return _.subtract(t[0], _.multiply(t[1], Symbol.imaginary()));
            },
            csch: function (r, i) {
                var t = this.removeDen(this.sinh(r, i));
                return _.subtract(t[0], _.multiply(t[1], Symbol.imaginary()));
            },
            coth: function (r, i) {
                var t = this.removeDen(this.tanh(r, i));
                return _.add(t[0], _.multiply(t[1], Symbol.imaginary()));
            },
            acosh: function (r, i) {
                var a, b, z;
                z = this.fromArray([r, i]);
                a = sqrt(_.add(z.clone(), new Symbol(1)));
                b = sqrt(_.subtract(z.clone(), new Symbol(1)));
                return _.expand(log(_.add(z, _.expand(_.multiply(a, b)))));
            },
            asinh: function (r, i) {
                var a, z;
                z = this.fromArray([r, i]);
                a = sqrt(_.add(new Symbol(1), _.expand(_.pow(z.clone(), new Symbol(2)))));
                return _.expand(log(_.add(z, a)));
            },
            atanh: function (r, i) {
                var a, b, z;
                z = this.fromArray([r, i]);
                a = log(_.add(z.clone(), new Symbol(1)));
                b = log(_.subtract(new Symbol(1), z));
                return _.expand(_.divide(_.subtract(a, b), new Symbol(2)));
            },
            asech: function (r, i) {
                var t = this.removeDen([r, i]);
                t[1].negate();
                return this.acosh.apply(this, t);
            },
            acsch: function (r, i) {
                var t = this.removeDen([r, i]);
                t[1].negate();
                return this.asinh.apply(this, t);
            },
            acoth: function (r, i) {
                var t = this.removeDen([r, i]);
                t[1].negate();
                return this.atanh.apply(this, t);
            },
            sqrt: function (symbol) {
                var re, im, h, a, d;
                re = symbol.realpart();
                im = symbol.imagpart();
                h = Symbol.hyp(re, im);
                a = _.add(re.clone(), h);
                d = sqrt(_.multiply(new Symbol(2), a.clone()));
                return _.add(_.divide(a.clone(), d.clone()), _.multiply(_.divide(im, d), Symbol.imaginary()));
            },
            log: function (r, i) {
                var re, im, phi;
                re = log(Symbol.hyp(r, i));
                phi = Settings.USE_BIG ? Symbol(bigDec.atan2(i.multiplier.toDecimal(), r.multiplier.toDecimal())) : Math.atan2(i, r);
                im = _.parse(phi);
                return _.add(re, _.multiply(Symbol.imaginary(), im));
            },
            erf(symbol, n) {
                //Do nothing for now. Revisit this in the future.
                return _.symfunction('erf', [symbol]);

                n = n || 30;

                var f = function (R, I) {
                    return block('PARSE2NUMBER', function () {
                        var retval = new Symbol(0);
                        for(var i = 0; i < n; i++) {
                            var a, b;
                            a = _.parse(bigDec.exp(bigDec(i).toPower(2).neg().dividedBy(bigDec(n).pow(2).plus(bigDec(R).toPower(2).times(4)))));
                            b = _.parse(format('2*({1})-e^(-(2*{0}*{1}*{2}))*(2*{1}*cosh({2}*{3})-{0}*{3}*sinh({3}*{2}))', Settings.IMAGINARY, R, I, i));
                            retval = _.add(retval, _.multiply(a, b));
                        }
                        return _.multiply(retval, new Symbol(2));
                    }, true);
                };
                var re, im, a, b, c, k;
                re = symbol.realpart();
                im = symbol.imagpart();

                k = _.parse(format('(e^(-{0}^2))/pi', re));
                a = _.parse(format('(1-e^(-(2*{0}*{1}*{2})))/(2*{1})', Settings.IMAGINARY, re, im));
                b = f(re.toString(), im.toString());

                return _.add(_.parse(Math2.erf(re.toString())), _.multiply(k, _.add(a, b)));
            },
            removeDen: function (symbol) {
                var den, r, i, re, im;
                if(isArray(symbol)) {
                    r = symbol[0];
                    i = symbol[1];
                }
                else {
                    r = symbol.realpart();
                    i = symbol.imagpart();
                }

                den = Math.pow(r, 2) + Math.pow(i, 2);
                re = _.parse(r / den);
                im = _.parse(i / den);
                return [re, im];
            },
            fromArray: function (arr) {
                return _.add(arr[0], _.multiply(Symbol.imaginary(), arr[1]));
            },
            evaluate: function (symbol, f) {
                var re, im, sign;

                sign = symbol.power.sign();
                //remove it from under the denominator
                symbol.power = symbol.power.abs();
                //expand
                if(symbol.power.greaterThan(1))
                    symbol = _.expand(symbol);
                //remove the denominator
                if(sign < 0) {
                    var d = this.removeDen(symbol);
                    re = d[0];
                    im = d[1];
                }
                else {
                    re = symbol.realpart();
                    im = symbol.imagpart();
                }

                if(re.isConstant('all') && im.isConstant('all'))
                    return this[f].call(this, re, im);

                return _.symfunction(f, [symbol]);
            }
        };
        //object for functions which handle trig
        var trig = this.trig = {
            //container for trigonometric function
            cos: function (symbol) {
                if(symbol.equals('pi') && symbol.multiplier.den.equals(2))
                    return new Symbol(0);

                if(Settings.PARSE2NUMBER) {
                    if(symbol.equals(new Symbol(Settings.PI / 2)))
                        return new Symbol(0);
                    if(symbol.isConstant()) {
                        if(Settings.USE_BIG) {
                            return new Symbol(bigDec.cos(symbol.multiplier.toDecimal()));
                        }

                        return new Symbol(Math.cos(symbol.valueOf()));
                    }
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'cos');
                    }
                }
                if(symbol.equals(0))
                    return new Symbol(1);

                var retval,
                        c = false,
                        q = getQuadrant(symbol.multiplier.toDecimal()),
                        m = symbol.multiplier.abs();
                symbol.multiplier = m;

                if(symbol.isPi() && symbol.isLinear()) {
                    //return for 1 or -1 for multiples of pi
                    if(isInt(m)) {
                        retval = new Symbol(even(m) ? 1 : -1);
                    }
                    else {
                        var n = Number(m.num), d = Number(m.den);
                        if(d === 2)
                            retval = new Symbol(0);
                        else if(d === 3) {
                            retval = _.parse('1/2');
                            c = true;
                        }
                        else if(d === 4) {
                            retval = _.parse('1/sqrt(2)');
                            c = true;
                        }
                        else if(d === 6) {
                            retval = _.parse('sqrt(3)/2');
                            c = true;
                        }
                        else
                            retval = _.symfunction('cos', [symbol]);
                    }
                }

                if(c && (q === 2 || q === 3))
                    retval.negate();

                if(!retval)
                    retval = _.symfunction('cos', [symbol]);

                return retval;
            },
            sin: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        if(symbol % Math.PI === 0) {
                            return new Symbol(0);
                        }

                        if(Settings.USE_BIG) {
                            return new Symbol(bigDec.sin(symbol.multiplier.toDecimal()));
                        }

                        return new Symbol(Math.sin(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'sin');
                }

                if(symbol.equals(0))
                    return new Symbol(0);

                var retval,
                        c = false,
                        q = getQuadrant(symbol.multiplier.toDecimal()),
                        sign = symbol.multiplier.sign(),
                        m = symbol.multiplier.abs();
                symbol.multiplier = m;
                if(symbol.equals('pi'))
                    retval = new Symbol(0);
                else if(symbol.isPi() && symbol.isLinear()) {
                    //return for 0 for multiples of pi
                    if(isInt(m)) {
                        retval = new Symbol(0);
                    }
                    else {
                        var n = m.num, d = m.den;
                        if(d == 2) {
                            retval = new Symbol(1);
                            c = true;
                        }
                        else if(d == 3) {
                            retval = _.parse('sqrt(3)/2');
                            c = true
                        }
                        else if(d == 4) {
                            retval = _.parse('1/sqrt(2)');
                            c = true;
                        }
                        else if(d == 6) {
                            retval = _.parse('1/2');
                            c = true;
                        }
                        else
                            retval = _.multiply(new Symbol(sign), _.symfunction('sin', [symbol]));
                    }
                }

                if(!retval)
                    retval = _.multiply(new Symbol(sign), _.symfunction('sin', [symbol]));

                if(c && (q === 3 || q === 4))
                    retval.negate();

                return retval;
            },
            tan: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol % Math.PI === 0 && symbol.isLinear()) {
                        return new Symbol(0);
                    }
                    if(symbol.isConstant()) {
                        if(Settings.USE_BIG) {
                            return new Symbol(bigDec.tan(symbol.multiplier.toDecimal()));
                        }

                        return new Symbol(Math.tan(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'tan');
                }
                var retval,
                        c = false,
                        q = getQuadrant(symbol.multiplier.toDecimal()),
                        m = symbol.multiplier;

                symbol.multiplier = m;

                if(symbol.isPi() && symbol.isLinear()) {
                    //return 0 for all multiples of pi
                    if(isInt(m)) {
                        retval = new Symbol(0);
                    }
                    else {
                        var n = m.num, d = m.den;
                        if(d == 2)
                            throw new UndefinedError('tan is undefined for ' + symbol.toString());
                        else if(d == 3) {
                            retval = _.parse('sqrt(3)');
                            c = true;
                        }
                        else if(d == 4) {
                            retval = new Symbol(1);
                            c = true;
                        }
                        else if(d == 6) {
                            retval = _.parse('1/sqrt(3)');
                            c = true;
                        }
                        else
                            retval = _.symfunction('tan', [symbol]);
                    }
                }

                if(!retval)
                    retval = _.symfunction('tan', [symbol]);

                if(c && (q === 2 || q === 4))
                    retval.negate();

                return retval;
            },
            sec: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        if(Settings.USE_BIG) {
                            return new Symbol(new bigDec(1).dividedBy(bigDec.cos(symbol.multiplier.toDecimal())));
                        }

                        return new Symbol(Math2.sec(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'sec');
                    return _.parse(format('1/cos({0})', symbol));
                }

                var retval,
                        c = false,
                        q = getQuadrant(symbol.multiplier.toDecimal()),
                        m = symbol.multiplier.abs();
                symbol.multiplier = m;

                if(symbol.isPi() && symbol.isLinear()) {
                    //return for 1 or -1 for multiples of pi
                    if(isInt(m)) {
                        retval = new Symbol(even(m) ? 1 : -1);
                    }
                    else {
                        var n = m.num, d = m.den;
                        if(d == 2)
                            throw new UndefinedError('sec is undefined for ' + symbol.toString());
                        else if(d == 3) {
                            retval = new Symbol(2);
                            c = true;
                        }
                        else if(d == 4) {
                            retval = _.parse('sqrt(2)');
                            c = true;
                        }
                        else if(d == 6) {
                            retval = _.parse('2/sqrt(3)');
                            c = true;
                        }
                        else
                            retval = _.symfunction('sec', [symbol]);
                    }
                }

                if(c && (q === 2 || q === 3))
                    retval.negate();

                if(!retval)
                    retval = _.symfunction('sec', [symbol]);

                return retval;
            },
            csc: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        if(Settings.USE_BIG) {
                            return new Symbol(new bigDec(1).dividedBy(bigDec.sin(symbol.multiplier.toDecimal())));
                        }

                        return new Symbol(Math2.csc(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'csc');
                    return _.parse(format('1/sin({0})', symbol));
                }

                var retval,
                        c = false,
                        q = getQuadrant(symbol.multiplier.toDecimal()),
                        sign = symbol.multiplier.sign(),
                        m = symbol.multiplier.abs();

                symbol.multiplier = m;

                if(symbol.isPi() && symbol.isLinear()) {
                    //return for 0 for multiples of pi
                    if(isInt(m)) {
                        throw new UndefinedError('csc is undefined for ' + symbol.toString());
                    }
                    else {
                        var n = m.num, d = m.den;
                        if(d == 2) {
                            retval = new Symbol(1);
                            c = true;
                        }
                        else if(d == 3) {
                            retval = _.parse('2/sqrt(3)');
                            c = true
                        }
                        else if(d == 4) {
                            retval = _.parse('sqrt(2)');
                            c = true;
                        }
                        else if(d == 6) {
                            retval = new Symbol(2);
                            c = true;
                        }
                        else
                            retval = _.multiply(new Symbol(sign), _.symfunction('csc', [symbol]));
                    }
                }

                if(!retval)
                    retval = _.multiply(new Symbol(sign), _.symfunction('csc', [symbol]));

                if(c && (q === 3 || q === 4))
                    retval.negate();

                return retval;
            },
            cot: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol % (Math.PI/2) === 0) {
                        return new Symbol(0);
                    }
                    if(symbol.isConstant()) {
                        if(Settings.USE_BIG) {
                            return new Symbol(new bigDec(1).dividedBy(bigDec.tan(symbol.multiplier.toDecimal())));
                        }

                        return new Symbol(Math2.cot(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'cot');
                    return _.parse(format('1/tan({0})', symbol));
                }
                var retval,
                        c = false,
                        q = getQuadrant(symbol.multiplier.toDecimal()),
                        m = symbol.multiplier;

                symbol.multiplier = m;

                if(symbol.isPi() && symbol.isLinear()) {
                    //return 0 for all multiples of pi
                    if(isInt(m)) {
                        throw new UndefinedError('cot is undefined for ' + symbol.toString());
                    }
                    else {
                        var n = m.num, d = m.den;
                        if(d == 2)
                            retval = new Symbol(0);
                        else if(d == 3) {
                            retval = _.parse('1/sqrt(3)');
                            c = true;
                        }
                        else if(d == 4) {
                            retval = new Symbol(1);
                            c = true;
                        }
                        else if(d == 6) {
                            retval = _.parse('sqrt(3)');
                            c = true;
                        }
                        else
                            retval = _.symfunction('cot', [symbol]);
                    }
                }

                if(!retval)
                    retval = _.symfunction('cot', [symbol]);

                if(c && (q === 2 || q === 4))
                    retval.negate();

                return retval;
            },
            acos: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        // Handle values in the complex domain
                        if(symbol.gt(1) || symbol.lt(-1)) {
                            var x = symbol.toString();
                            return expand(evaluate(`pi/2-asin(${x})`));
                        }
                        // Handle big numbers
                        if(Settings.USE_BIG) {
                            return new Symbol(bigDec.acos(symbol.multiplier.toDecimal()));
                        }

                        return new Symbol(Math.acos(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'acos');
                }
                return _.symfunction('acos', arguments);
            },
            asin: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        // Handle values in the complex domain
                        if(symbol.gt(1) || symbol.lt(-1)) {
                            var i = Settings.IMAGINARY;
                            var x = symbol.multiplier.toDecimal();
                            return expand(evaluate(`${i}*log(sqrt(1-${x}^2)-${i}*${x})`));
                        }
                        // Handle big numbers
                        if(Settings.USE_BIG) {
                            return new Symbol(bigDec.asin(symbol.multiplier.toDecimal()));
                        }

                        return new Symbol(Math.asin(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'asin');
                }
                return _.symfunction('asin', arguments);
            },
            atan: function (symbol) {
                var retval;
                if(symbol.equals(0))
                    retval = new Symbol(0);
                else if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        // Handle big numbers
                        if(Settings.USE_BIG) {
                            return new Symbol(bigDec.atan(symbol.multiplier.toDecimal()));
                        }

                        return new Symbol(Math.atan(symbol.valueOf()));
                    }
                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'atan');
                    return _.symfunction('atan', arguments);
                }
                else if(symbol.equals(-1))
                    retval = _.parse('-pi/4');
                else
                    retval = _.symfunction('atan', arguments);
                return retval;
            },
            asec: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.equals(0)) {
                        throw new OutOfFunctionDomainError('Input is out of the domain of sec!');
                    }
                    if(symbol.isConstant()) {
                        return trig.acos(symbol.invert());
                    }
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'asec');
                    }
                }
                return _.symfunction('asec', arguments);
            },
            acsc: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        return trig.asin(symbol.invert());
                    }

                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'acsc');
                }
                return _.symfunction('acsc', arguments);
            },
            acot: function (symbol) {
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        return new _.add(_.parse('pi/2'), trig.atan(symbol).negate());
                    }

                    if(symbol.isImaginary())
                        return complex.evaluate(symbol, 'acot');
                }
                return _.symfunction('acot', arguments);
            },
            atan2: function (a, b) {
                if(a.equals(0) && b.equals(0))
                    throw new UndefinedError('atan2 is undefined for 0, 0');

                if(Settings.PARSE2NUMBER && a.isConstant() && b.isConstant()) {
                    return new Symbol(Math.atan2(a, b));
                }
                return _.symfunction('atan2', arguments);
            }
        };
        //object for functions which handle hyperbolic trig
        var trigh = this.trigh = {
            //container for hyperbolic trig function
            cosh: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant())
                        return new Symbol(Math.cosh(symbol.valueOf()));
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'cosh');
                    }
                }

                return retval = _.symfunction('cosh', arguments);
            },
            sinh: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant())
                        return new Symbol(Math.sinh(symbol.valueOf()));
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'sinh');
                    }
                }

                return retval = _.symfunction('sinh', arguments);
            },
            tanh: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant())
                        return new Symbol(Math.tanh(symbol.valueOf()));
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'tanh');
                    }
                }

                return retval = _.symfunction('tanh', arguments);
            },
            sech: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant()) {
                        return new Symbol(Math.sech(symbol.valueOf()));
                    }
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'sech');
                    }
                    return _.parse(format('1/cosh({0})', symbol));
                }

                return retval = _.symfunction('sech', arguments);
            },
            csch: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant())
                        return new Symbol(Math.csch(symbol.valueOf()));
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'csch');
                    }
                    return _.parse(format('1/sinh({0})', symbol));
                }

                return retval = _.symfunction('csch', arguments);
            },
            coth: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER) {
                    if(symbol.isConstant())
                        return new Symbol(Math.coth(symbol.valueOf()));
                    if(symbol.isImaginary()) {
                        return complex.evaluate(symbol, 'coth');
                    }
                    return _.parse(format('1/tanh({0})', symbol));
                }

                return retval = _.symfunction('coth', arguments);
            },
            acosh: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER && symbol.isImaginary())
                    retval = complex.evaluate(symbol, 'acosh');
                else if(Settings.PARSE2NUMBER)
                    retval = evaluate(_.parse(format(Settings.LOG + '(({0})+sqrt(({0})^2-1))', symbol.toString())));
                else
                    retval = _.symfunction('acosh', arguments);
                return retval;
            },
            asinh: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER && symbol.isImaginary())
                    retval = complex.evaluate(symbol, 'asinh');
                else if(Settings.PARSE2NUMBER)
                    retval = evaluate(_.parse(format(Settings.LOG + '(({0})+sqrt(({0})^2+1))', symbol.toString())));
                else
                    retval = _.symfunction('asinh', arguments);
                return retval;
            },
            atanh: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER && symbol.isImaginary())
                    retval = complex.evaluate(symbol, 'atanh');
                else if(Settings.PARSE2NUMBER) {
                    retval = evaluate(_.parse(format('(1/2)*' + Settings.LOG + '((1+({0}))/(1-({0})))', symbol.toString())));
                }
                else
                    retval = _.symfunction('atanh', arguments);
                return retval;
            },
            asech: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER && symbol.isImaginary())
                    retval = complex.evaluate(symbol, 'asech');
                else if(Settings.PARSE2NUMBER)
                    retval = evaluate(log(_.add(symbol.clone().invert(), sqrt(_.subtract(_.pow(symbol, new Symbol(-2)), new Symbol(1))))));
                else
                    retval = _.symfunction('asech', arguments);
                return retval;
            },
            acsch: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER && symbol.isImaginary())
                    retval = complex.evaluate(symbol, 'acsch');
                else if(Settings.PARSE2NUMBER)
                    retval = evaluate(_.parse(format(Settings.LOG + '((1+sqrt(1+({0})^2))/({0}))', symbol.toString())));
                else
                    retval = _.symfunction('acsch', arguments);
                return retval;
            },
            acoth: function (symbol) {
                var retval;
                if(Settings.PARSE2NUMBER && symbol.isImaginary())
                    retval = complex.evaluate(symbol, 'acoth');
                else if(Settings.PARSE2NUMBER) {
                    if(symbol.equals(1))
                        retval = Symbol.infinity();
                    else
                        retval = evaluate(
                                _.divide(
                                        log(_.divide(_.add(symbol.clone(), new Symbol(1)), _.subtract(symbol.clone(), new Symbol(1)))),
                                        new Symbol(2)));
                }
                else
                    retval = _.symfunction('acoth', arguments);
                return retval;
            }
        };
        //list of supported units
        this.units = {};
        //list all the supported operators
        var operators = {
            '\\': {
                precedence: 8,
                operator: '\\',
                action: 'slash',
                prefix: true,
                postfix: false,
                leftAssoc: true,
                operation: function (e) {
                    return e; //bypass the slash
                }
            },
            '!!': {
                precedence: 7,
                operator: '!!',
                action: 'dfactorial',
                prefix: false,
                postfix: true,
                leftAssoc: true,
                operation: function (e) {
                    return _.symfunction(Settings.DOUBLEFACTORIAL, [e]); //wrap it in a factorial function
                }
            },
            '!': {
                precedence: 7,
                operator: '!',
                action: 'factorial',
                prefix: false,
                postfix: true,
                leftAssoc: true,
                operation: function (e) {
                    return factorial(e); //wrap it in a factorial function
                }
            },
            '^': {
                precedence: 6,
                operator: '^',
                action: 'pow',
                prefix: false,
                postfix: false,
                leftAssoc: true
            },
            '**': {
                precedence: 6,
                operator: '**',
                action: 'pow',
                prefix: false,
                postfix: false,
                leftAssoc: true
            },
            '%': {
                precedence: 4,
                operator: '%',
                action: 'percent',
                prefix: false,
                postfix: true,
                leftAssoc: true,
                overloaded: true,
                overloadAction: 'mod',
                overloadLeftAssoc: false,
                operation: function (x) {
                    return _.divide(x, new Symbol(100));
                }
            },
            '*': {
                precedence: 4,
                operator: '*',
                action: 'multiply',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            '/': {
                precedence: 4,
                operator: '/',
                action: 'divide',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            '+': {
                precedence: 3,
                operator: '+',
                action: 'add',
                prefix: true,
                postfix: false,
                leftAssoc: false,
                operation: function (x) {
                    return x;
                }
            },
            'plus': {
                precedence: 3,
                operator: 'plus',
                action: 'add',
                prefix: true,
                postfix: false,
                leftAssoc: false,
                operation: function (x) {
                    return x;
                }
            },
            '-': {
                precedence: 3,
                operator: '-',
                action: 'subtract',
                prefix: true,
                postfix: false,
                leftAssoc: false,
                operation: function (x) {
                    return x.negate();
                }
            },
            '=': {
                precedence: 2,
                operator: '=',
                action: 'equals',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            '==': {
                precedence: 1,
                operator: '==',
                action: 'eq',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            '<': {
                precedence: 1,
                operator: '<',
                action: 'lt',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            '<=': {
                precedence: 1,
                operator: '<=',
                action: 'lte',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            '>': {
                precedence: 1,
                operator: '>',
                action: 'gt',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            '=>': {
                precedence: 1,
                operator: '=>',
                action: 'gte',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            ',': {
                precedence: 0,
                operator: ',',
                action: 'comma',
                prefix: false,
                postfix: false,
                leftAssoc: false
            },
            ':': {
                precedence: 0,
                operator: ',',
                action: 'assign',
                prefix: false,
                postfix: false,
                leftAssoc: false,
                vectorFn: 'slice'
            },
            ':=': {
                precedence: 0,
                operator: ',',
                action: 'function_assign',
                prefix: false,
                postfix: false,
                leftAssoc: true
            }
        };
        //brackets
        var brackets = {
            '(': {
                type: 'round',
                id: 1,
                is_open: true,
                is_close: false
            },
            ')': {
                type: 'round',
                id: 2,
                is_open: false,
                is_close: true
            },
            '[': {
                type: 'square',
                id: 3,
                is_open: true,
                is_close: false,
                maps_to: 'vector'
            },
            ']': {
                type: 'square',
                id: 4,
                is_open: false,
                is_close: true
            },
            '{': {
                type: 'curly',
                id: 5,
                is_open: true,
                is_close: false,
                maps_to: 'Set'
            },
            '}': {
                type: 'curly',
                id: 6,
                is_open: false,
                is_close: true
            }
        };
        // Supported functions.
        // Format: function_name: [mapped_function, number_of_parameters]
        var functions = this.functions = {
            'cos': [trig.cos, 1],
            'sin': [trig.sin, 1],
            'tan': [trig.tan, 1],
            'sec': [trig.sec, 1],
            'csc': [trig.csc, 1],
            'cot': [trig.cot, 1],
            'acos': [trig.acos, 1],
            'asin': [trig.asin, 1],
            'atan': [trig.atan, 1],
            'arccos': [trig.acos, 1],
            'arcsin': [trig.asin, 1],
            'arctan': [trig.atan, 1],
            'asec': [trig.asec, 1],
            'acsc': [trig.acsc, 1],
            'acot': [trig.acot, 1],
            'atan2': [trig.atan2, 2],
            'acoth': [trigh.acoth, 1],
            'asech': [trigh.asech, 1],
            'acsch': [trigh.acsch, 1],
            'sinh': [trigh.sinh, 1],
            'cosh': [trigh.cosh, 1],
            'tanh': [trigh.tanh, 1],
            'asinh': [trigh.asinh, 1],
            'sech': [trigh.sech, 1],
            'csch': [trigh.csch, 1],
            'coth': [trigh.coth, 1],
            'acosh': [trigh.acosh, 1],
            'atanh': [trigh.atanh, 1],
            'log10': [, 1],
            'exp': [exp, 1],
            'radians': [radians, 1],
            'degrees': [degrees, 1],
            'min': [min, -1],
            'max': [max, -1],
            'erf': [, 1],
            'floor': [, 1],
            'ceil': [, 1],
            'trunc': [, 1],
            'Si': [, 1],
            'step': [, 1],
            'rect': [, 1],
            'sinc': [sinc, 1],
            'tri': [, 1],
            'sign': [sign, 1],
            'Ci': [, 1],
            'Ei': [, 1],
            'Shi': [, 1],
            'Chi': [, 1],
            'Li': [, 1],
            'fib': [, 1],
            'fact': [factorial, 1],
            'factorial': [factorial, 1],
            'continued_fraction': [continued_fraction, [1, 2]],
            'dfactorial': [, 1],
            'gamma_incomplete': [, [1, 2]],
            'round': [round, [1, 2]],
            'scientific': [scientific, [1, 2]],
            'mod': [mod, 2],
            'pfactor': [pfactor, 1],
            'vector': [vector, -1],
            'matrix': [matrix, -1],
            'Set': [set, -1],
            'imatrix': [imatrix, -1],
            'parens': [parens, -1],
            'sqrt': [sqrt, 1],
            'cbrt': [cbrt, 1],
            'nthroot': [nthroot, 2],
            'log': [log, [1, 2]],
            'expand': [expandall, 1],
            'abs': [abs, 1],
            'invert': [invert, 1],
            'determinant': [determinant, 1],
            'size': [size, 1],
            'transpose': [transpose, 1],
            'dot': [dot, 2],
            'cross': [cross, 2],
            'vecget': [vecget, 2],
            'vecset': [vecset, 3],
            'vectrim': [vectrim, [1, 2]],
            'matget': [matget, 3],
            'matset': [matset, 4],
            'matgetrow': [matgetrow, 2],
            'matsetrow': [matsetrow, 3],
            'matgetcol': [matgetcol, 2],
            'matsetcol': [matsetcol, 3],
            'rationalize': [rationalize, 1],
            'IF': [IF, 3],
            'is_in': [is_in, 2],
            //imaginary support
            'realpart': [realpart, 1],
            'imagpart': [imagpart, 1],
            'conjugate': [conjugate, 1],
            'arg': [arg, 1],
            'polarform': [polarform, 1],
            'rectform': [rectform, 1],
            'sort': [sort, [1, 2]],
            'integer_part': [, 1],
            'union': [union, 2],
            'contains': [contains, 2],
            'intersection': [intersection, 2],
            'difference': [difference, 2],
            'intersects': [intersects, 2],
            'is_subset': [is_subset, 2],
            //system support
            'print': [print, -1]
        };

        //error handler
        this.error = err;
        //this function is used to comb through the function modules and find a function given its name
        var findFunction = function (fname) {
            var fmodules = Settings.FUNCTION_MODULES,
                    l = fmodules.length;
            for(var i = 0; i < l; i++) {
                var fmodule = fmodules[i];
                if(fname in fmodule)
                    return fmodule[fname];
            }
            err('The function ' + fname + ' is undefined!');
        };

        /**
         * This method gives the ability to override operators with new methods.
         * @param {String} which
         * @param {Function} with_what
         */
        this.override = function (which, with_what) {
            if(!bin[which])
                bin[which] = [];
            bin[which].push(this[which]);
            this[which] = with_what;
        };

        /**
         * Restores a previously overridden operator
         * @param {String} what
         */
        this.restore = function (what) {
            if(this[what])
                this[what] = bin[what].pop();
        };

        /**
         * This method is supposed to behave similarly to the override method but it does not override
         * the existing function rather it only extends it
         * @param {String} what
         * @param {Function} with_what
         * @param {boolean} force_call
         */
        this.extend = function (what, with_what, force_call) {
            var _ = this,
                    extended = this[what];
            if(typeof extended === 'function' && typeof with_what === 'function') {
                var f = this[what];
                this[what] = function (a, b) {
                    if(isSymbol(a) && isSymbol(b) && !force_call)
                        return f.call(_, a, b);
                    else
                        return with_what.call(_, a, b, f);
                };
            }
        };

        /**
         * Generates library's representation of a function. It's a fancy way of saying a symbol with
         * a few extras. The most important thing is that that it gives a fname and
         * an args property to the symbols in addition to changing its group to FN
         * @param {String} fn_name
         * @param {Array} params
         * @returns {Symbol}
         */
        this.symfunction = function (fn_name, params) {
            //call the proper function and return the result;
            var f = new Symbol(fn_name);
            f.group = FN;
            if(typeof params === 'object')
                params = [].slice.call(params);//ensure an array
            f.args = params;
            f.fname = fn_name === PARENTHESIS ? '' : fn_name;
            f.updateHash();
            return f;
        };

        /**
         * An internal function call for the Parser. This will either trigger a real
         * function call if it can do so or just return a symbolic representation of the
         * function using symfunction.
         * @param {String} fn_name
         * @param {Array} args
         * @param {int} allowed_args
         * @returns {Symbol}
         */
        this.callfunction = function (fn_name, args, allowed_args) {
            var fn_settings = functions[fn_name];

            if(!fn_settings)
                err('Nerdamer currently does not support the function ' + fn_name);

            var num_allowed_args = fn_settings[1] || allowed_args, //get the number of allowed arguments
                    fn = fn_settings[0], //get the mapped function
                    retval;
            //We want to be able to call apply on the arguments or create a symfunction. Both require
            //an array so make sure to wrap the argument in an array.
            if(!(args instanceof Array))
                args = args !== undefined ? [args] : [];

            if(num_allowed_args !== -1) {
                var is_array = isArray(num_allowed_args),
                        min_args = is_array ? num_allowed_args[0] : num_allowed_args,
                        max_args = is_array ? num_allowed_args[1] : num_allowed_args,
                        num_args = args.length;

                var error_msg = fn_name + ' requires a {0} of {1} arguments. {2} provided!';

                if(num_args < min_args)
                    err(format(error_msg, 'minimum', min_args, num_args));
                if(num_args > max_args)
                    err(format(error_msg, 'maximum', max_args, num_args));
            }

            /*
             * The following are very important to the how nerdamer constructs functions!
             * Assumption 1 - if fn is undefined then handling of the function is purely numeric. This
             *     enables us to reuse Math, Math2, ..., any function from Settings.FUNCTIONS_MODULES entry
             * Assumption 2 - if fn is defined then that function takes care of EVERYTHING including symbolics
             * Assumption 3 - if the user calls symbolics on a function that returns a numeric value then
             *     they are expecting a symbolic output.
             */
            //check if arguments are all numers
            var numericArgs = allNumbers(args);
            //Big number support. Check if Big number is requested and the arguments are all numeric and, not imaginary
//            if (Settings.USE_BIG && numericArgs) {
//                retval = Big[fn_name].apply(undefined, args);
//            }
//            else {
            if(!fn) {
                //Remember assumption 1. No function defined so it MUST be numeric in nature
                fn = findFunction(fn_name);
                if(Settings.PARSE2NUMBER && numericArgs)
                    retval = bigConvert(fn.apply(fn, args));
                else
                    retval = _.symfunction(fn_name, args);
            }
            else {
                //Remember assumption 2. The function is defined so it MUST handle all aspects including numeric values
                retval = fn.apply(fn_settings[2], args);
            }
//            }

            return retval;
        };
        /**
         * Build a regex based on the operators currently loaded. These operators are to be ignored when
         * substituting spaces for multiplication
         */
        this.operator_filter_regex = (function () {
            //we only want the operators which are singular since those are the ones
            //that nerdamer uses anyway
            var ostr = '^\\' + Object.keys(operators).filter(function (x) {
                if(x.length === 1)
                    return x;
            }).join('\\');
            //create a regex which captures all spaces between characters except those
            //have an operator on one end
            return new RegExp('([' + ostr + '])\\s+([' + ostr + '])');
        })();

        /**
         * Replaces nerdamer.setOperator
         * @param {object} operator
         * @param {boolean} shift
         */
        this.setOperator = function (operator, action, shift) {
            var name = operator.operator; //take the name to be the symbol
            operators[name] = operator;
            if(action)
                this[operator.action] = action;
            //make the parser aware of the operator
            _[name] = operator.operation;
            //make the action available to the parser if infix
            if(!operator.action && !(operator.prefix || operator.postif)) {
                operator.action = name;
            }
            //if this operator is exclusive then all successive operators should be shifted
            if(shift === 'over' || shift === 'under') {
                var precedence = operator.precedence;

                for(var x in operators) {
                    var o = operators[x];
                    var condition = shift === 'over' ? o.precedence >= precedence : o.precedence > precedence;
                    if(condition)
                        o.precedence++;
                }
                ;
            }
        };

        /**
         * Gets an opererator by its symbol
         * @param {String} operator
         * @returns {Object}
         */
        this.getOperator = function (operator) {
            return operators[operator];
        };

        this.aliasOperator = function (o, n) {
            var t = {};
            var operator = operators[o];
            //copy everything over to the new operator
            for(var x in operator) {
                t[x] = operator[x];
            }
            //update the symbol
            t.operator = n;

            this.setOperator(t);
        };

        /**
         * Returns the list of operators. Caution! Can break parser!
         * @returns {object}
         */
        this.getOperators = function () {
            //will replace this with some cloning action in the future
            return operators;
        };

        this.getBrackets = function () {
            return brackets;
        };
        /*
         * Preforms preprocessing on the string. Useful for making early modification before
         * sending to the parser
         * @param {String} e
         */
        var prepare_expression = function (e) {
            /*
             * Since variables cannot start with a number, the assumption is made that when this occurs the
             * user intents for this to be a coefficient. The multiplication symbol in then added. The same goes for
             * a side-by-side close and open parenthesis
             */
            e = String(e);
            //apply preprocessors
            for(var i = 0; i < preprocessors.actions.length; i++)
                e = preprocessors.actions[i].call(this, e);

            //e = e.split(' ').join('');//strip empty spaces
            //replace multiple spaces with one space
            e = e.replace(/\s+/g, ' ');

            //only even bother to check if the string contains e. This regex is painfully slow and might need a better solution. e.g. hangs on (0.06/3650))^(365)
            if(/e/gi.test(e)) {
                e = e.replace(/\-*\d+\.*\d*e\+?\-?\d+/gi, function (x) {
                    return scientificToDecimal(x);
                });
            }
            //replace scientific numbers

            //allow omission of multiplication after coefficients
            e = e.replace(Settings.IMPLIED_MULTIPLICATION_REGEX, function () {
                var str = arguments[4],
                        group1 = arguments[1],
                        group2 = arguments[2],
                        start = arguments[3],
                        first = str.charAt(start),
                        before = '',
                        d = '*';
                if(!first.match(/[\+\-\/\*]/))
                    before = str.charAt(start - 1);
                if(before.match(/[a-z]/i))
                    d = '';
                return group1 + d + group2;
            })
                    .replace(/([a-z0-9_]+)/gi, function (match, a) {
                        if(Settings.USE_MULTICHARACTER_VARS === false && !(a in functions)) {
                            if(!isNaN(a))
                                return a;
                            return a.split('').join('*');
                        }
                        return a;
                    })
                    //allow omission of multiplication sign between brackets
                    .replace(/\)\(/g, ')*(') || '0';
            //replace x(x+a) with x*(x+a)
            while(true) {
                var e_org = e; //store the original
                e = e.replace(/([a-z0-9_]+)(\()|(\))([a-z0-9]+)/gi, function (match, a, b, c, d) {
                    var g1 = a || c,
                            g2 = b || d;
                    if(g1 in functions) //create a passthrough for functions
                        return g1 + g2;
                    return g1 + '*' + g2;
                });
                //if the original equals the replace we're done
                if(e_org === e)
                    break;
            }
            return e;
        };
        //delay setting of constants until Settings is ready
        this.initConstants = function () {
            this.CONSTANTS = {
                E: new Symbol(Settings.E),
                PI: new Symbol(Settings.PI)
            };
        };
        /*
         * Debugging method used to better visualize vector and arrays
         * @param {object} o
         * @returns {String}
         */
        this.pretty_print = function (o) {
            if(Array.isArray(o)) {
                var s = o.map(x => _.pretty_print(x)).join(', ');
                if(o.type === 'vector')
                    return 'vector<' + s + '>';
                return '(' + s + ')';
            }
            return o.toString();
        };
        this.peekers = {
            pre_operator: [],
            post_operator: [],
            pre_function: [],
            post_function: []
        };

        this.callPeekers = function (name) {
            if(Settings.callPeekers) {
                var peekers = this.peekers[name];
                //remove the first items and stringify
                var args = arguments2Array(arguments).slice(1).map(stringify);
                //call each one of the peekers
                for(var i = 0; i < peekers.length; i++) {
                    peekers[i].apply(null, args);
                }
            }
        };
        /*
         * Tokenizes the string
         * @param {String} e
         * @returns {Token[]}
         */
        this.tokenize = function (e) {
            //cast to String
            e = String(e);
            //remove multiple white spaces and spaces at beginning and end of string
            e = e.trim().replace(/\s+/g, ' ');
            //remove spaces before and after brackets
            for(var x in brackets) {
                var regex = new RegExp(brackets[x].is_close ? '\\s+\\' + x : '\\' + x + '\\s+', 'g');
                e = e.replace(regex, x);
            }

            var col = 0; //the column position
            var L = e.length; //expression length
            var lpos = 0; //marks beginning of next token
            var tokens = []; //the tokens container
            var scopes = [tokens]; //initiate with the tokens as the highest scope
            var target = scopes[0]; //the target to which the tokens are added. This can swing up or down
            var depth = 0;
            var open_brackets = [];
            var has_space = false; //marks if an open space character was found
            var SPACE = ' ';
            var EMPTY_STRING = '';
            var COMMA = ',';
            var MINUS = '-';
            var MULT = '*';
            //Possible source of bug. Review
            /*
             //gets the next space
             var next_space = function(from) {
             for(var i=from; i<L; i++) {
             if(e.charAt(i) === ' ')
             return i;
             }
             
             return L; //assume the end of the string instead
             };
             */
            /**
             * Adds a scope to tokens
             * @param {String} scope_type
             * @param {int} column
             * @returns {undefined}
             */
            var addScope = function (scope_type, column) {
                var new_scope = []; //create a new scope
                if(scope_type !== undefined) {
                    new_scope.type = scope_type;
                }
                new_scope.column = column; //mark the column of the scope
                scopes.push(new_scope); //add it to the list of scopes
                target.push(new_scope); //add it to the tokens list since now it's a scope
                target = new_scope; //point to it
                depth++; //go down one in scope
            };
            /**
             * Goes up in scope by one
             * @returns {undefined}
             */
            var goUp = function () {
                scopes.pop(); //remove the scope from the scopes stack
                target = scopes[--depth]; //point the above scope
            };
            /**
             * Extracts all the operators from the expression string starting at postion start_at
             * @param {int} start_at
             * @returns {String}
             */
            var get_operator_str = function (start_at) {
                start_at = start_at !== undefined ? start_at : col;
                //mark the end of the operator as the start since we're just going
                //to be walking along the string
                var end = start_at + 1;
                //just keep moving along
                while(e.charAt(end++) in operators) {
                }
                //remember that we started at one position ahead. The beginning operator is what triggered
                //this function to be called in the first place. String.CharAt is zero based so we now
                //have to correct two places. The initial increment + the extra++ at the end of end during
                //the last iteration.
                return e.substring(start_at, end - 1);
            };
            /**
             * Breaks operator up in to several different operators as defined in operators
             * @param {String} operator_str
             * @returns {String[]}
             */
            var chunkify = function (operator_str) {
                var start = col - operator_str.length; //start of operator
                var _operators = [];
                var operator = operator_str.charAt(0);
                //grab the largest possible chunks but start at 2 since we already know
                //that the first character is an operator

                for(var i = 1, L = operator_str.length; i < L; i++) {
                    var ch = operator_str.charAt(i);
                    var o = operator + ch;
                    //since the operator now is undefined then the last operator
                    //was the largest possible combination.
                    if(!(o in operators)) {
                        _operators.push(new Token(operator, Token.OPERATOR, start + i));
                        operator = ch;
                    }
                    else
                        operator = o;//now the operator is the larger chunk
                }
                //add the last operator
                _operators.push(new Token(operator, Token.OPERATOR, start + i));
                return _operators;
            };

            /**
             * Is used to add a token to the tokens array. Makes sure that no empty token is added
             * @param {int} at
             * @param {String} token
             * @returns {undefined}
             */
            var add_token = function (at, token) {
                //grab the token if we're not supplied one
                if(token === undefined)
                    token = e.substring(lpos, at);
                //only add it if it's not an empty string
                if(token in _.units)
                    target.push(new Token(token, Token.UNIT, lpos));
                else if(token !== '')
                    target.push(new Token(token, Token.VARIABLE_OR_LITERAL, lpos));
            };
            /**
             * Adds a function to the output
             * @param {String} f
             * @returns {undefined}
             */
            var add_function = function (f) {
                target.push(new Token(f, Token.FUNCTION, lpos));
            };
            /**
             * Tokens are found between operators so this marks the location of where the last token was found
             * @param {int} position
             * @returns {undefined}
             */
            var set_last_position = function (position) {
                lpos = position + 1;
            };
            /**
             * When a operator is found and added, especially a combo operator, then the column location
             * has to be adjusted to the end of the operator
             * @returns {undefined}
             */
            var adjust_column_position = function () {
                lpos = lpos + operator_str.length - 2;
                col = lpos - 1;
            };
            for(; col < L; col++) {
                var ch = e.charAt(col);
                if(ch in operators) {
                    add_token(col);
                    //is the last token numeric?
                    var last_token_is_numeric = target[0] && isNumber(target[0]);
                    //is this character multiplication?
                    var is_multiplication = last_token_is_numeric && ch === MULT;
                    //if we're in a new scope then go up by one but if the space
                    //is right befor an operator then it makes no sense to go up in scope
                    //consider sin -x. The last position = current position at the minus sign
                    //this means that we're going for sin(x) -x which is wrong
                    //Ignore comma since comma is still part of the existing scope.
                    if(has_space && lpos < col && !(ch === COMMA || is_multiplication)) {
                        has_space = false;
                        goUp();
                    }
                    //mark the last position that a
                    set_last_position(col + 1);
                    var operator_str = get_operator_str(col);

                    adjust_column_position();
                    target.push.apply(target, chunkify(operator_str));
                }
                else if(ch in brackets) {
                    var bracket = brackets[ch];

                    if(bracket.is_open) {
                        //mark the bracket
                        open_brackets.push([bracket, lpos]);
                        var f = e.substring(lpos, col);
                        if(f in functions) {
                            add_function(f);
                        }
                        else if(f !== '') {
                            //assume multiplication
                            //TODO: Add the multiplication to stack
                            target.push(new Token(f, Token.VARIABLE_OR_LITERAL, lpos));
                        }
                        //go down one in scope
                        addScope(bracket.maps_to, col);
                    }
                    else if(bracket.is_close) {
                        //get the matching bracket
                        var pair = open_brackets.pop();
                        //throw errors accordingly
                        //missing open bracket
                        if(!pair)
                            throw new ParityError('Missing open bracket for bracket at: ' + (col + 1));
                        //incorrect pair
                        else if(pair[0].id !== bracket.id - 1)
                            throw new ParityError('Parity error');

                        add_token(col);
                        goUp();
                    }
                    set_last_position(col);
                }
                else if(ch === SPACE) {
                    var prev = e.substring(lpos, col); //look back
                    var nxt = e.charAt(col + 1); //look forward
                    if(has_space) {

                        if(prev in operators) {
                            target.push(new Token(prev, Token.OPERATOR, col));
                        }
                        else {
                            add_token(undefined, prev);
                            //we're at the closing space
                            goUp(); //go up in scope if we're at a space

                            //assume multiplication if it's not an operator except for minus
                            var is_operator = nxt in operators;

                            if((is_operator && operators[nxt].value === MINUS) || !is_operator) {
                                target.push(new Token(MULT, Token.OPERATOR, col));
                            }
                        }
                        has_space = false; //remove the space
                    }
                    else {
                        //we're at the closing space
                        //check if it's a function
                        var f = e.substring(lpos, col);

                        if(f in functions) {
                            //there's no need to go up in scope if the next character is an operator
                            has_space = true; //mark that a space was found
                            add_function(f);
                            addScope();
                        }
                        else if(f in operators) {
                            target.push(new Token(f, Token.OPERATOR, col));
                        }
                        else {
                            add_token(undefined, f);
                            //peek ahead to the next character
                            var nxt = e.charAt(col + 1);

                            //If it's a number then add the multiplication operator to the stack but make sure that the next character
                            //is not an operator

                            if(prev !== EMPTY_STRING && nxt !== EMPTY_STRING && !(prev in operators) && !(nxt in operators))
                                target.push(new Token(MULT, Token.OPERATOR, col));
                        }
                        //Possible source of bug. Review
                        /*
                         //space can mean multiplication so add the symbol if the is encountered
                         if(/\d+|\d+\.?\d*e[\+\-]*\d+/i.test(f)) {
                         var next = e.charAt(col+1);
                         var next_is_operator = next in operators;
                         var ns = next_space(col+1);
                         var next_word = e.substring(col+1, ns);
                         //the next can either be a prefix operator or no operator
                         if((next_is_operator && operators[next].prefix) || !(next_is_operator || next_word in operators))
                         target.push(new Token('*', Token.OPERATOR, col));
                         }
                         */
                    }
                    set_last_position(col); //mark this location
                }
            }
            //check that all brackets were closed
            if(open_brackets.length) {
                var b = open_brackets.pop();
                throw new ParityError('Missing closed bracket for bracket at ' + (b[1] + 1));
            }
            //add the last token
            add_token(col);

            return tokens;
        };
        /*
         * Puts token array in Reverse Polish Notation
         * @param {Token[]} tokens
         * @returns {Token[]}
         */
        this.toRPN = function (tokens) {
            var fn = tokens.type;
            var l = tokens.length, i;
            var output = [];
            var stack = [];
            var prefixes = [];
            var collapse = function (target, destination) {
                while(target.length)
                    destination.push(target.pop());
            };
            //mark all the prefixes and add them to the stack
            for(i = 0; i < l; i++) {
                var token = tokens[i];
                if(token.type !== Token.OPERATOR)
                    break;
                if(!token.prefix)
                    throw new OperatorError('Not a prefix operator');
                token.is_prefix = true;
                stack.push(token);
            }
            //begin with remaining tokens
            for(; i < l; i++) {
                var e = tokens[i];
                if(e.type === Token.OPERATOR) {
                    var operator = e;

                    //create the option for the operator being overloaded
                    if(operator.overloaded) {
                        var next = tokens[i + 1];
                        //if it's followed by a number or variable then we assume it's not a postfix operator
                        if(next && next.type === Token.VARIABLE_OR_LITERAL) {
                            operator.postfix = false;
                            //override the original function with the overload function
                            operator.action = operator.overloadAction;
                            operator.leftAssoc = operator.overloadLeftAssoc;
                        }
                    }

                    //if the stack is not empty
                    while(stack.length) {
                        var last = stack[stack.length - 1];
                        //if (there is an operator at the top of the operator stack with greater precedence)
                        //or (the operator at the top of the operator stack has equal precedence and is left associative)) ~ wikipedia
                        //the !prefixes.length makes sure that the operator on stack isn't prematurely taken fromt he stack.
                        if(!(last.precedence > operator.precedence || !operator.leftAssoc && last.precedence === operator.precedence))
                            break;
                        output.push(stack.pop());
                    }

                    //change the behavior of the operator if it's a vector and we've been asked to do so
                    if((fn === 'vector' || fn === 'set') && 'vectorFn' in operator)
                        operator.action = operator.vectorFn;


                    //if the operator is a postfix operator then we're ready to go since it belongs
                    //to the preceding token. However the output cannot be empty. It must have either
                    //an operator or a variable/literal
                    if(operator.postfix) {
                        var previous = tokens[i - 1];
                        if(!previous)
                            throw new OperatorError("Unexpected prefix operator '" + e.value + "'! at " + e.column);
                        else if(previous.type === Token.OPERATOR) {
                            //a postfix can only be followed by a postfix
                            if(!previous.postfix)
                                throw new OperatorError("Unexpected prefix operator '" + previous.value + "'! at " + previous.column);
                        }
                    }
                    else {
                        //we must be at an infix so point the operator this
                        do {
                            //the first one is an infix operator all others have to be prefix operators so jump to the end
                            var next = tokens[i + 1]; //take a look ahead
                            var next_is_operator = next ? next.type === Token.OPERATOR : false; //check if it's an operator
                            if(next_is_operator) {
                                //if it's not a prefix operator then it not in the right place
                                if(!next.prefix) {
                                    throw new OperatorError('A prefix operator was expected at ' + next.column);
                                }
                                //mark it as a confirmed prefix
                                next.is_prefix = true;
                                //add it to the prefixes
                                prefixes.push(next);
                                i++;
                            }
                        }
                        while(next_is_operator)
                    }

                    //if it's a prefix it should be on a special stack called prefixes
                    //we do this to hold on to prefixes because of left associative operators.
                    //they belong to the variable/literal but if placed on either the stack
                    //or output there's no way of knowing this. I might be wrong so I welcome
                    //any discussion about this.

                    if(operator.is_prefix) //ADD ALL EXCEPTIONS FOR ADDING TO PREFIX STACK HERE. !!!
                        prefixes.push(operator);
                    else
                        stack.push(operator);
                    //move the prefixes to the stack
                    while(prefixes.length) {
                        if(operator.leftAssoc || !operator.leftAssoc && prefixes[prefixes.length - 1].precedence >= operator.precedence) //revisit for commas
                            stack.push(prefixes.pop());
                        else
                            break;
                    }
                }
                else if(e.type === Token.VARIABLE_OR_LITERAL) {
                    //move prefixes to stack at beginning of scope
                    if(output.length === 0)
                        collapse(prefixes, stack);
                    //done with token
                    output.push(e);
                    var last_on_stack = stack[stack.length - 1];
                    //then move all the prefixes to the output
                    if(!last_on_stack || !last_on_stack.leftAssoc)
                        collapse(prefixes, output);
                }
                else if(e.type === Token.FUNCTION) {
                    stack.push(e);
                }
                else if(e.type === Token.UNIT) {
                    //if it's a unit it belongs on the stack since it's tied to the previous token
                    output.push(e);
                }
                //if it's an additonal scope then put that into RPN form
                if(Array.isArray(e)) {
                    output.push(this.toRPN(e));
                    if(e.type)
                        output.push(new Token(e.type, Token.FUNCTION, e.column)); //since it's hidden it needs no column

                }
            }
            //collapse the remainder of the stack and prefixes to output
            collapse(stack, output);
            collapse(prefixes, output);

            return output;
        };
        /*
         * Parses the tokens
         * @param {Tokens[]} rpn
         * @param {object} substitutions
         * @returns {Symbol}
         */
        this.parseRPN = function (rpn, substitutions) {
            try {
                //default substitutions
                substitutions = substitutions || {};
                //prepare the substitutions.
                //we first parse them out as-is
                for(var x in substitutions)
                    substitutions[x] = _.parse(substitutions[x], {});

                //Although technically constants,
                //pi and e are only available when evaluating the expression so add to the subs.
                //Doing this avoids rounding errors
                //link e and pi
                if(Settings.PARSE2NUMBER) {
                    //use the value provided if the individual for some strange reason prefers this.
                    //one reason could be to sub e but not pi or vice versa
                    if(!('e' in substitutions))
                        substitutions.e = new Symbol(Settings.E);
                    if((!('pi' in substitutions)))
                        substitutions.pi = new Symbol(Settings.PI);
                }

                var Q = [];
                for(var i = 0, l = rpn.length; i < l; i++) {
                    var e = rpn[i];

                    //Arrays indicate a new scope so parse that out
                    if(Array.isArray(e)) {
                        e = this.parseRPN(e, substitutions);
                    }

                    if(e) {
                        if(e.type === Token.OPERATOR) {
                            if(e.is_prefix || e.postfix)
                                //resolve the operation assocated with the prefix
                                Q.push(e.operation(Q.pop()));
                            else {
                                var b = Q.pop();
                                var a = Q.pop();
                                //Throw an error if the RH value is empty. This cannot be a postfix since we already checked
                                if(typeof a === 'undefined')
                                    throw new OperatorError(e + ' is not a valid postfix operator at ' + e.column);

                                var is_comma = e.action === 'comma';
                                //convert Sets to Vectors on all operations at this point. Sets are only recognized functions or individually
                                if(a instanceof Set && !is_comma)
                                    a = Vector.fromSet(a);

                                if(b instanceof Set && !is_comma)
                                    b = Vector.fromSet(b);

                                //call all the pre-operators
                                this.callPeekers('pre_operator', a, b, e);

                                var ans = _[e.action](a, b);

                                //call all the pre-operators
                                this.callPeekers('post_operator', ans, a, b, e);

                                Q.push(ans);
                            }
                        }
                        else if(e.type === Token.FUNCTION) {
                            var args = Q.pop();
                            var parent = args.parent; //make a note of the parent
                            if(!(args instanceof Collection))
                                args = Collection.create(args);
                            //the return value may be a vector. If it is then we check
                            //Q to see if there's another vector on the stack. If it is then
                            //we check if has elements. If it does then we know that we're dealing
                            //with an "getter" object and return the requested values

                            //call the function. This is the _.callfunction method in nerdamer
                            //call the function. This is the _.callfunction method in nerdamer
                            var fn_name = e.value;
                            var fn_args = args.getItems();

                            //call the pre-function peekers
                            this.callPeekers('pre_function', fn_name, fn_args);

                            var ret = _.callfunction(fn_name, fn_args);

                            //call the post-function peekers
                            this.callPeekers('post_function', ret, fn_name, fn_args);

                            var last = Q[Q.length - 1];
                            var next = rpn[i + 1];
                            var next_is_comma = next && next.type === Token.OPERATOR && next.value === ',';

                            if(!next_is_comma && ret instanceof Vector && last && last.elements && !(last instanceof Collection)) {
                                //remove the item from the queue
                                var item = Q.pop();

                                var getter = ret.elements[0];
                                //check if it's symbolic. If so put it back and add the item to the stack
                                if(!getter.isConstant()) {
                                    item.getter = getter;
                                    Q.push(item);
                                    Q.push(ret);
                                }
                                else if(getter instanceof Slice) {
                                    //if it's a Slice return the slice
                                    Q.push(Vector.fromArray(item.elements.slice(getter.start, getter.end)));
                                }
                                else {
                                    var index = Number(getter);
                                    var il = item.elements.length;
                                    //support for negative indices
                                    if(index < 0)
                                        index = il + index;
                                    //it it's still out of bounds
                                    if(index < 0 || index >= il) //index should no longer be negative since it's been reset above
                                        //range error
                                        throw new OutOfRangeError('Index out of range ' + (e.column + 1));

                                    var element = item.elements[index];
                                    //cyclic but we need to mark this for future reference
                                    item.getter = index;
                                    element.parent = item;

                                    Q.push(element);
                                }
                            }
                            else {
                                //extend the parent reference
                                if(parent)
                                    ret.parent = parent;
                                Q.push(ret);
                            }

                        }
                        else {
                            var subbed;
                            var v = e.value;

                            if(v in Settings.ALIASES)
                                e = _.parse(Settings.ALIASES[e]);
                            //wrap it in a symbol if need be
                            else if(e.type === Token.VARIABLE_OR_LITERAL)
                                e = new Symbol(v);
                            else if(e.type === Token.UNIT) {
                                e = new Symbol(v);
                                e.isUnit = true;
                            }

                            //make substitutions
                            //Always constants first. This avoids the being overridden
                            if(v in _.CONSTANTS) {
                                subbed = e;
                                e = new Symbol(_.CONSTANTS[v]);
                            }
                            //next substitutions. This allows declared variable to be overridden
                            //check if the values match to avoid erasing the multiplier.
                            //Example:/e = 3*a. substutiting a for a will wipe out the multiplier.
                            else if(v in substitutions && v !== substitutions[v].toString()) {
                                subbed = e;
                                e = substitutions[v].clone();
                            }
                            //next declare variables
                            else if(v in VARS) {
                                subbed = e;
                                e = VARS[v].clone();
                            }
                            //make notation of what it was before
                            if(subbed)
                                e.subbed = subbed;

                            Q.push(e);
                        }
                    }
                }

                var retval = Q[0];

                if(['undefined', 'string', 'number'].indexOf(typeof retval) !== -1) {
                    throw new UnexpectedTokenError('Unexpected token!');
                }

                return retval;
            }
            catch(error) {
                var rethrowErrors = [OutOfFunctionDomainError];
                // Rethrow certain errors in the same class to preserve them
                rethrowErrors.forEach(function (E) {
                    if(error instanceof E) {
                        throw new E(error.message + ': ' + e.column);
                    }
                });

                throw new ParseError(error.message + ': ' + e.column);
            }
        };
        /**
         * This is the method that triggers the parsing of the string. It generates a parse tree but processes
         * it right away. The operator functions are called when their respective operators are reached. For instance
         * + with cause this.add to be called with the left and right hand values. It works by walking along each
         * character of the string and placing the operators on the stack and values on the output. When an operator
         * having a lower order than the last is reached then the stack is processed from the last operator on the
         * stack.
         * @param {String} token
         */

        function Node(token) {
            this.type = token.type;
            this.value = token.value;
            //the incoming token may already be a Node type
            this.left = token.left;
            this.right = token.right;
        }

        Node.prototype.toString = function () {
            var left = this.left ? this.left.toString() + '---' : '';
            var right = this.right ? '---' + this.right.toString() : '';
            return left + '(' + this.value + ')' + right;
        };

        Node.prototype.toHTML = function (depth, indent) {
            depth = depth || 0;
            indent = typeof indent === 'undefined' ? 4 : indent;
            var tab = function (n) {
                return ' '.repeat(indent * n);
            };
            var html = '';
            var left = this.left ? tab(depth + 1) + '<li>\n' + this.left.toHTML(depth + 2, indent) + tab(depth + 1) + '</li> \n' : '';
            var right = this.right ? tab(depth + 1) + '<li>\n' + this.right.toHTML(depth + 2, indent) + tab(depth + 1) + '</li>\n' : '';
            var html = tab(depth) + '<div class="' + this.type.toLowerCase() + '"><span>' + this.value + '</span></div>' + tab(depth) + '\n';
            if(left || right) {
                html += tab(depth) + '<ul>\n' + left + right + tab(depth) + '</ul>\n';
            }
            html += '';
            return html;
        };

        this.tree = function (tokens) {
            var Q = [];
            for(var i = 0; i < tokens.length; i++) {
                var e = tokens[i];
                //Arrays indicate a new scope so parse that out
                if(Array.isArray(e)) {
                    e = this.tree(e);
                    //if it's a comma then it's just arguments
                    Q.push(e);
                    continue;
                }
                if(e.type === Token.OPERATOR) {
                    if(e.is_prefix || e.postfix) {
                        //prefixes go to the left, postfix to the right
                        var location = e.is_prefix ? 'left' : 'right';
                        var last = Q.pop();
                        e = new Node(e);
                        e[location] = last;
                        Q.push(e);
                    }
                    else {
                        e = new Node(e);
                        e.right = Q.pop();
                        e.left = Q.pop();
                        Q.push(e);
                    }
                }
                else if(e.type === Token.FUNCTION) {
                    e = new Node(e);
                    var args = Q.pop();
                    e.right = args;
                    if(e.value === 'object') {
                        //check if Q has a value
                        var last = Q[Q.length - 1];
                        if(last) {
                            while(last.right) {
                                last = last.right;
                            }
                            last.right = e;
                            continue;
                        }
                    }

                    Q.push(e);
                }
                else {
                    Q.push(new Node(e));
                }
            }

            return Q[0];
        };
        this.parse = function (e, substitutions) {
            e = prepare_expression(e);
            substitutions = substitutions || {};
            //three passes but easier to debug
            var tokens = this.tokenize(e);
            var rpn = this.toRPN(tokens);
            return this.parseRPN(rpn, substitutions);
        };
        /**
         * TODO: Switch to Parser.tokenize for this method
         * Reads a string into an array of Symbols and operators
         * @param {String} expression_string
         * @returns {Array}
         */
        this.toObject = function (expression_string) {
            var objectify = function (tokens) {
                var output = [];
                for(var i = 0, l = tokens.length; i < l; i++) {
                    var token = tokens[i];
                    var v = token.value;
                    if(token.type === Token.VARIABLE_OR_LITERAL) {
                        output.push(new Symbol(v));
                    }
                    else if(token.type === Token.FUNCTION) {
                        //jump ahead since the next object are the arguments
                        i++;
                        //create a symbolic function and stick it on output
                        var f = _.symfunction(v, objectify(tokens[i]));
                        f.isConversion = true;
                        output.push(f);
                    }
                    else if(token.type === Token.OPERATOR) {
                        output.push(v);
                    }
                    else {
                        output.push(objectify(token));
                    }
                }

                return output;
            };
            return objectify(_.tokenize(expression_string));
        };

        // A helper method for toTeX
        var chunkAtCommas = function (arr) {
            var j, k = 0, chunks = [[]];
            for(var j = 0, l = arr.length; j < l; j++) {
                if(arr[j] === ',') {
                    k++;
                    chunks[k] = [];
                }
                else {
                    chunks[k].push(arr[j]);
                }
            }
            return chunks;
        };

        // Helper method for toTeX
        var rem_brackets = function (str) {
            return str.replace(/^\\left\((.+)\\right\)$/g, function (str, a) {
                if(a)
                    return a;
                return str;
            });
        };

        var remove_redundant_powers = function (arr) {
            // The filtered array
            var narr = [];

            while(arr.length) {
                // Remove the element from the front
                var e = arr.shift();
                var next = arr[0];
                var next_is_array = isArray(next);
                var next_is_minus = next === '-';

                // Remove redundant plusses 
                if(e === '^') {
                    if(next === '+') {
                        arr.shift();
                    }
                    else if(next_is_array && next[0] === '+') {
                        next.shift();
                    }

                    // Remove redundant parentheses
                    if(next_is_array && next.length === 1) {
                        arr.unshift(arr.shift()[0]);
                    }
                }

                // Check if it's a negative power
                if(e === '^' && (next_is_array && next[0] === '-' || next_is_minus)) {
                    // If so:
                    // - Remove it from the new array, place a one and a division sign in that array and put it back
                    var last = narr.pop();
                    // Check if it's something multiplied by
                    var before = narr[narr.length - 1];
                    var before_last = '1';

                    if(before === '*') {
                        narr.pop();
                        // For simplicity we just pop it. 
                        before_last = narr.pop();
                    }
                    // Implied multiplication
                    else if(isArray(before)) {
                        before_last = narr.pop();
                    }

                    narr.push(before_last, '/', last, e);

                    // Remove the negative sign from the power 
                    if(next_is_array) {
                        next.shift();
                    }
                    else {
                        arr.shift();
                    }

                    // Remove it from the array so we don't end up with redundant parentheses if we can
                    if(next_is_array && next.length === 1) {
                        narr.push(arr.shift()[0]);
                    }
                }
                else {
                    narr.push(e);
                }
            }

            return narr;
        };
        /*
         * Convert expression or object to LaTeX
         * @param {String} expression_or_obj
         * @param {object} opt
         * @returns {String}
         */
        this.toTeX = function (expression_or_obj, opt) {
            opt = opt || {};
            // Add decimal option as per issue #579. Consider passing an object to Latex.latex as option instead of string
            var decimals = opt.decimals === true ? 'decimals' : undefined;

            var obj = typeof expression_or_obj === 'string' ? this.toObject(expression_or_obj) : expression_or_obj,
                    TeX = [],
                    cdot = typeof opt.cdot === 'undefined' ? '\\cdot' : opt.cdot; //set omit cdot to true by default

            // Remove negative powers as per issue #570
            obj = remove_redundant_powers(obj);

            if(isArray(obj)) {
                var nobj = [], a, b;
                //first handle ^
                for(var i = 0; i < obj.length; i++) {
                    a = obj[i];

                    if(obj[i + 1] === '^') {
                        b = obj[i + 2];
                        nobj.push(LaTeX.braces(this.toTeX([a])) + '^' + LaTeX.braces(this.toTeX([b])));
                        i += 2;
                    }
                    else {
                        nobj.push(a);
                    }
                }
                obj = nobj;
            }

            for(var i = 0, l = obj.length; i < l; i++) {
                var e = obj[i];

                // Convert * to cdot
                if(e === '*') {
                    e = cdot;
                }

                if(isSymbol(e)) {
                    if(e.group === FN) {
                        var fname = e.fname, f;

                        if(fname === SQRT) {
                            f = '\\sqrt' + LaTeX.braces(this.toTeX(e.args));
                        }
                        else if(fname === ABS) {
                            f = LaTeX.brackets(this.toTeX(e.args), 'abs');
                        }
                        else if(fname === PARENTHESIS) {
                            f = LaTeX.brackets(this.toTeX(e.args), 'parens');
                        }
                        else if(fname === Settings.LOG10) {
                            f = '\\' + Settings.LOG10_LATEX + '\\left( ' + this.toTeX(e.args) + '\\right)';
                        }
                        else if(fname === 'integrate') {
                            /* Retrive [Expression, x] */
                            var chunks = chunkAtCommas(e.args);
                            /* Build TeX */
                            var expr = LaTeX.braces(this.toTeX(chunks[0])),
                                    dx = this.toTeX(chunks[1]);
                            f = '\\int ' + expr + '\\, d' + dx;
                        }
                        else if(fname === 'defint') {
                            var chunks = chunkAtCommas(e.args),
                                    expr = LaTeX.braces(this.toTeX(chunks[0])),
                                    dx = this.toTeX(chunks[3]),
                                    lb = this.toTeX(chunks[1]),
                                    ub = this.toTeX(chunks[2]);
                            f = '\\int\\limits_{' + lb + '}^{' + ub + '} ' + expr + '\\, d' + dx;

                        }
                        else if(fname === 'diff') {
                            var chunks = chunkAtCommas(e.args);
                            var dx = '', expr = LaTeX.braces(this.toTeX(chunks[0]));
                            /* Handle cases: one argument provided, we need to guess the variable, and assume n = 1 */
                            if(chunks.length === 1) {
                                var vars = [];
                                for(j = 0; j < chunks[0].length; j++) {
                                    if(chunks[0][j].group === 3) {
                                        vars.push(chunks[0][j].value);
                                    }
                                }
                                vars.sort();
                                dx = vars.length > 0 ? ('\\frac{d}{d ' + vars[0] + '}') : '\\frac{d}{d x}';
                            }
                            /* If two arguments, we have expression and variable, we assume n = 1 */
                            else if(chunks.length === 2) {
                                dx = '\\frac{d}{d ' + chunks[1] + '}';
                            }
                            /* If we have more than 2 arguments, we assume we've got everything */
                            else {
                                dx = '\\frac{d^{' + chunks[2] + '}}{d ' + this.toTeX(chunks[1]) + '^{' + chunks[2] + '}}';
                            }

                            f = dx + '\\left(' + expr + '\\right)';

                        }
                        else if(fname === 'sum' || fname === 'product') {
                            // Split e.args into 4 parts based on locations of , symbols.
                            var argSplit = [[], [], [], []], j = 0, i;
                            for(i = 0; i < e.args.length; i++) {
                                if(e.args[i] === ',') {
                                    j++;
                                    continue;
                                }
                                argSplit[j].push(e.args[i]);
                            }
                            // Then build TeX string.
                            f = (fname === 'sum' ? '\\sum_' : '\\prod_') + LaTeX.braces(this.toTeX(argSplit[1]) + ' = ' + this.toTeX(argSplit[2]));
                            f += '^' + LaTeX.braces(this.toTeX(argSplit[3])) + LaTeX.braces(this.toTeX(argSplit[0]));
                        }
                        else if(fname === 'limit') {
                            var args = chunkAtCommas(e.args).map(function (x) {
                                if(Array.isArray(x))
                                    return _.toTeX(x.join(''));
                                return _.toTeX(String(x));
                            });
                            f = '\\lim_' + LaTeX.braces(args[1] + '\\to ' + args[2]) + ' ' + LaTeX.braces(args[0]);
                        }
                        else if(fname === FACTORIAL || fname === DOUBLEFACTORIAL) {
                            f = this.toTeX(e.args) + (fname === FACTORIAL ? '!' : '!!');
                        }
                        else {
                            f = LaTeX.latex(e, decimals);
                            //f = '\\mathrm'+LaTeX.braces(fname.replace(/_/g, '\\_')) + LaTeX.brackets(this.toTeX(e.args), 'parens');
                        }

                        TeX.push(f);
                    }
                    else {
                        TeX.push(LaTeX.latex(e, decimals));
                    }
                }
                else if(isArray(e)) {
                    TeX.push(LaTeX.brackets(this.toTeX(e)));
                }
                else {
                    if(e === '/')
                        TeX.push(LaTeX.frac(rem_brackets(TeX.pop()), rem_brackets(this.toTeX([obj[++i]]))));
                    else
                        TeX.push(e);
                }
            }

            return TeX.join(' ');
        };

//Parser.functions ==============================================================
        /* Although parens is not a "real" function it is important in some cases when the
         * symbol must carry parenthesis. Once set you don't have to worry about it anymore
         * as the parser will get rid of it at the first opportunity
         */
        function parens(symbol) {
            if(Settings.PARSE2NUMBER) {
                return symbol;
            }
            return _.symfunction('parens', [symbol]);
        }

        function abs(symbol) {

            //|-∞| = ∞
            if(symbol.isInfinity) {
                return Symbol.infinity();
            }
            if(symbol.multiplier.lessThan(0))
                symbol.multiplier.negate();

            if(symbol.isImaginary()) {
                var re = symbol.realpart();
                var im = symbol.imagpart();
                if(re.isConstant() && im.isConstant())
                    return sqrt(_.add(_.pow(re, new Symbol(2)), _.pow(im, new Symbol(2))));
            }
            else if(isNumericSymbol(symbol) || even(symbol.power)) {
                return symbol;
            }

            if(symbol.isComposite()) {
                var ms = [];
                symbol.each(function (x) {
                    ms.push(x.multiplier);
                });
                var gcd = Math2.QGCD.apply(null, ms);
                if(gcd.lessThan(0)) {
                    symbol.multiplier = symbol.multiplier.multiply(new Frac(-1));
                    symbol.distributeMultiplier();
                }
            }

            //convert |n*x| to n*|x|
            var m = _.parse(symbol.multiplier);
            symbol.toUnitMultiplier();

            return _.multiply(m, _.symfunction(ABS, [symbol]));
        }
        /**
         * The factorial function
         * @param {Symbol} symbol
         * @return {Symbol}
         */
        function factorial(symbol) {
            var retval;
            if(isVector(symbol)) {
                var V = new Vector();
                symbol.each(function (x, i) {
                    //i start at one.
                    V.set(i - 1, factorial(x));
                });
                return V;
            }
            if(isMatrix(symbol)) {
                var M = new Matrix();
                symbol.each(function (x, i, j) {
                    //i start at one.
                    M.set(i, j, factorial(x));
                });
                return M;
            }
            if(Settings.PARSE2NUMBER && symbol.isConstant()) {
                if(isInt(symbol)) {
                    retval = Math2.bigfactorial(symbol);
                }
                else {
                    retval = Math2.gamma(symbol.multiplier.add(new Frac(1)).toDecimal());
                }

                retval = bigConvert(retval);
                return retval;
            }
            else if(symbol.isConstant()) {
                var den = symbol.getDenom();
                if(den.equals(2)) {
                    var num = symbol.getNum();
                    var a, b, c, n;

                    if(!symbol.multiplier.isNegative()) {
                        n = _.add(num, new Symbol(1)).multiplier.divide(new Frac(2));
                        a = Math2.bigfactorial(new Frac(2).multiply(n));
                        b = _.pow(new Symbol(4), new Symbol(n)).multiplier.multiply(Math2.bigfactorial(n));
                    }
                    else {
                        n = _.subtract(num.negate(), new Symbol(1)).multiplier.divide(new Frac(2));
                        a = _.pow(new Symbol(-4), new Symbol(n)).multiplier.multiply(Math2.bigfactorial(n));
                        b = Math2.bigfactorial(new Frac(2).multiply(n));
                    }
                    c = a.divide(b);
                    return _.multiply(_.parse('sqrt(pi)'), new Symbol(c));
                }
            }
            return _.symfunction(FACTORIAL, [symbol]);
        }
        ;
        /**
         * Returns the continued fraction of a number
         * @param {Symbol} symbol
         * @param {Symbol} n
         * @returns {Symbol}
         */
        function continued_fraction(symbol, n) {
            var _symbol = evaluate(symbol);
            if(_symbol.isConstant()) {
                var cf = Math2.continuedFraction(_symbol, n);
                //convert the fractions array to a new Vector
                var fractions = Vector.fromArray(cf.fractions.map(function (x) {
                    return new Symbol(x);
                }));
                return Vector.fromArray([new Symbol(cf.sign), new Symbol(cf.whole), fractions]);
            }
            return _.symfunction('continued_fraction', arguments);
        }
        /**
         * Returns the error function
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function erf(symbol) {
            var _symbol = evaluate(symbol);

            if(_symbol.isConstant()) {
                return Math2.erf(_symbol);
            }
            else if(_symbol.isImaginary()) {
                return complex.erf(symbol);
            }
            return _.symfunction('erf', arguments);
        }
        ;
        /**
         * The mod function
         * @param {Symbol} symbol1
         * @param {Symbol} symbol2
         * @returns {Symbol}
         */
        function mod(symbol1, symbol2) {
            if(symbol1.isConstant() && symbol2.isConstant()) {
                var retval = new Symbol(1);
                retval.multiplier = retval.multiplier.multiply(symbol1.multiplier.mod(symbol2.multiplier));
                return retval;
            }
            //try to see if division has remainder of zero
            var r = _.divide(symbol1.clone(), symbol2.clone());
            if(isInt(r))
                return new Symbol(0);
            return _.symfunction('mod', [symbol1, symbol2]);
        }
        /**
         * A branghing function
         * @param {Boolean} condition
         * @param {Symbol} a
         * @param {Symbol} b
         * @returns {Symbol}
         */
        function IF(condition, a, b) {
            if(typeof condition !== 'boolean')
                if(isNumericSymbol(condition))
                    condition = !!Number(condition);
            if(condition)
                return a;
            return b;
        }
        /**
         *
         * @param {Matrix|Vector|Set|Collection} obj
         * @param {Symbol} item
         * @returns {Boolean}
         */
        function is_in(obj, item) {
            if(isMatrix(obj)) {
                for(var i = 0, l = obj.rows(); i < l; i++) {
                    for(var j = 0, l2 = obj.cols(); j < l2; j++) {
                        var element = obj.elements[i][j];
                        if(element.equals(item))
                            return new Symbol(1);
                    }
                }
            }
            else if(obj.elements) {
                for(var i = 0, l = obj.elements.length; i < l; i++) {
                    if(obj.elements[i].equals(item))
                        return new Symbol(1);
                }
            }

            return new Symbol(0);
        }

        /**
         * A symbolic extension for sinc
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function sinc(symbol) {
            if(Settings.PARSE2NUMBER) {
                if(symbol.isConstant()) {
                    return new Symbol(Math2.sinc(symbol));
                }
                return _.parse(format('sin({0})/({0})', symbol));
            }
            return _.symfunction('sinc', [symbol]);
        }

        /**
         * A symbolic extension for exp. This will auto-convert all instances of exp(x) to e^x.
         * Thanks @ Happypig375
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function exp(symbol) {
            if(symbol.fname === Settings.LOG && symbol.isLinear()) {
                return _.pow(symbol.args[0], Symbol.create(symbol.multiplier));
            }
            return _.parse(format('e^({0})', symbol));
        }

        /**
         * Converts value degrees to radians
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function radians(symbol) {
            return _.parse(format('({0})*pi/180', symbol));
        }

        /**
         * Converts value from radians to degrees
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function degrees(symbol) {
            return _.parse(format('({0})*180/pi', symbol));
        }

        function nroots(symbol) {
            var a, b;
            if(symbol.group === FN && symbol.fname === '') {
                a = Symbol.unwrapPARENS(_.parse(symbol).toLinear());
                b = _.parse(symbol.power);
            }
            else if(symbol.group === P) {
                a = _.parse(symbol.value);
                b = _.parse(symbol.power);
            }

            if(a && b && a.group === N && b.group === N) {
                var _roots = [];
                var parts = Symbol.toPolarFormArray(symbol);
                var r = _.parse(a).abs().toString();
                //https://en.wikipedia.org/wiki/De_Moivre%27s_formula
                var x = arg(a).toString();
                var n = b.multiplier.den.toString();
                var p = b.multiplier.num.toString();

                var formula = "(({0})^({1})*(cos({3})+({2})*sin({3})))^({4})";
                for(var i = 0; i < n; i++) {
                    var t = evaluate(_.parse(format("(({0})+2*pi*({1}))/({2})", x, i, n))).multiplier.toDecimal();
                    _roots.push(evaluate(_.parse(format(formula, r, n, Settings.IMAGINARY, t, p))));
                }
                return Vector.fromArray(_roots);
            }
            else if(symbol.isConstant(true)) {
                var sign = symbol.sign();
                var x = evaluate(symbol.abs());
                var root = _.sqrt(x);

                var _roots = [root.clone(), root.negate()];

                if(sign < 0)
                    _roots = _roots.map(function (x) {
                        return _.multiply(x, Symbol.imaginary());
                    });
            }
            else {
                _roots = [_.parse(symbol)];
            }

            return Vector.fromArray(_roots);
        }

        /**
         * Rationalizes a symbol
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function rationalize(symbol) {
            if(symbol.isComposite()) {
                var retval = new Symbol(0);
                var num, den, retnum, retden, a, b, n, d;
                symbol.each(function (x) {
                    num = x.getNum();
                    den = x.getDenom();
                    retnum = retval.getNum();
                    retden = retval.getDenom();
                    a = _.multiply(den, retnum);
                    b = _.multiply(num, retden);
                    n = _.expand(_.add(a, b));
                    d = _.multiply(retden, den);
                    retval = _.divide(n, d);
                }, true);

                return retval;
            }
            return symbol;
        }

        /**
         * The square root function
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function sqrt(symbol) {
            if(!isSymbol(symbol)) {
                symbol = _.parse(symbol);
            }
            
            // Exit early for EX
            if(symbol.group === EX) {
                return _.symfunction(SQRT, [symbol]);
            }

            if(symbol.fname === '' && symbol.power.equals(1))
                symbol = symbol.args[0];

            var is_negative = symbol.multiplier.sign() < 0;

            if(Settings.PARSE2NUMBER) {
                if(symbol.isConstant() && !is_negative) {
                    return new Symbol(bigDec.sqrt(symbol.multiplier.toDecimal()));
                }
                else if(symbol.isImaginary()) {
                    return complex.sqrt(symbol);
                }
                else if(symbol.group === S) {
                    return _.symfunction('sqrt', [symbol]);
                }
            }

            var img, retval,
                    isConstant = symbol.isConstant();

            if(symbol.group === CB && symbol.isLinear()) {
                var m = sqrt(Symbol(symbol.multiplier));
                for(var s in symbol.symbols) {
                    var x = symbol.symbols[s];
                    m = _.multiply(m, sqrt(x));
                }

                retval = m;
            }
            //if the symbol is already sqrt then it's that symbol^(1/4) and we can unwrap it
            else if(symbol.fname === SQRT) {
                var s = symbol.args[0];
                var ms = symbol.multiplier;
                s.setPower(symbol.power.multiply(new Frac(0.25)));
                retval = s;
                //grab the multiplier
                if(!ms.equals(1))
                    retval = _.multiply(sqrt(_.parse(ms)), retval);
            }
            //if the symbol is a fraction then we don't keep can unwrap it. For instance
            //no need to keep sqrt(x^(1/3))
            else if(!symbol.power.isInteger()) {
                symbol.setPower(symbol.power.multiply(new Frac(0.5)));
                retval = symbol;
            }
            else if(symbol.multiplier < 0 && symbol.group === S) {
                var a = _.parse(symbol.multiplier).negate();
                var b = _.parse(symbol).toUnitMultiplier().negate();
                retval = _.multiply(_.symfunction(Settings.SQRT, [b]), sqrt(a));
            }
            else {

                //Related to issue #401. Since sqrt(a)*sqrt(b^-1) relates in issues, we'll change the form
                //to sqrt(a)*sqrt(b)^1 for better simplification
                //the sign of the power
                var sign = symbol.power.sign();
                //remove the sign
                symbol.power = symbol.power.abs();

                //if the symbols is imagary then we place in the imaginary part. We'll return it
                //as a product
                if(isConstant && symbol.multiplier.lessThan(0)) {
                    img = Symbol.imaginary();
                    symbol.multiplier = symbol.multiplier.abs();
                }

                var q = symbol.multiplier.toDecimal(),
                        qa = Math.abs(q),
                        t = Math.sqrt(qa);

                var m;
                //it's a perfect square so take the square
                if(isInt(t)) {
                    m = new Symbol(t);
                }
                else if(isInt(q)) {
                    var factors = Math2.ifactor(q);
                    var tw = 1;
                    for(var x in factors) {
                        var n = factors[x],
                                nn = (n - (n % 2)); //get out the whole numbers
                        if(nn) { //if there is a whole number ...
                            var w = Math.pow(x, nn);
                            tw *= Math.pow(x, nn / 2); //add to total wholes
                            q /= w; //reduce the number by the wholes
                        }
                    }
                    m = _.multiply(_.symfunction(SQRT, [new Symbol(q)]), new Symbol(tw));
                }
                else {
                    //reduce the numerator and denominator using prime factorization
                    var c = [new Symbol(symbol.multiplier.num), new Symbol(symbol.multiplier.den)];
                    var r = [new Symbol(1), new Symbol(1)];
                    var sq = [new Symbol(1), new Symbol(1)];
                    for(var i = 0; i < 2; i++) {
                        var n = c[i];
                        //get the prime factors and loop through each.
                        pfactor(n).each(function (x) {
                            x = Symbol.unwrapPARENS(x);
                            var b = x.clone().toLinear();
                            var p = Number(x.power);
                            //We'll consider it safe to use the native Number since 2^1000 is already a pretty huge number
                            var rem = p % 2; //get the remainder. This will be 1 if 3 since sqrt(n^2) = n where n is positive
                            var w = (p - rem) / 2; //get the whole numbers of n/2
                            r[i] = _.multiply(r[i], _.pow(b, new Symbol(w)));
                            sq[i] = _.multiply(sq[i], sqrt(_.pow(b, new Symbol(rem))));
                        });
                    }
                    m = _.divide(_.multiply(r[0], sq[0]), _.multiply(r[1], sq[1]));
                }


                //strip the multiplier since we already took the sqrt
                symbol = symbol.toUnitMultiplier(true);
                //if the symbol is one just return one and not the sqrt function
                if(symbol.isOne()) {
                    retval = symbol;
                }
                else if(even(symbol.power.toString())) {
                    //just raise it to the 1/2
                    retval = _.pow(symbol.clone(), new Symbol(0.5));
                }
                else {
                    retval = _.symfunction(SQRT, [symbol]);
                }

                //put back the sign that was removed earlier
                if(sign < 0)
                    retval.power.negate();

                if(m)
                    retval = _.multiply(m, retval);

                if(img)
                    retval = _.multiply(img, retval);
            }

            if(is_negative && Settings.PARSE2NUMBER)
                return _.parse(retval);

            return retval;
        }

        /**
         * The cube root function
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function cbrt(symbol) {
            if(!symbol.isConstant(true)) {
                var retval;

                var n = symbol.power / 3;
                //take the cube root of the multplier
                var m = _.pow(_.parse(symbol.multiplier), new Symbol(1 / 3));
                //strip the multiplier
                var sym = symbol.toUnitMultiplier();

                //simplify the power
                if(isInt(n)) {
                    retval = _.pow(sym.toLinear(), _.parse(n));
                }
                else {
                    if(sym.group === CB) {
                        retval = new Symbol(1);
                        sym.each(function (x) {
                            retval = _.multiply(retval, cbrt(x));
                        });
                    }
                    else {
                        retval = _.symfunction('cbrt', [sym]);
                    }
                }

                return _.multiply(m, retval);
            }
            return nthroot(symbol, new Symbol(3));
        }

        function scientific(symbol, sigfigs) {
            //Just set the flag and keep it moving. Symbol.toString will deal with how to
            //display this
            symbol.scientific = sigfigs || 10;
            return symbol;
        }

        /**
         *
         * @param {Symbol} num - the number being raised
         * @param {Symbol} p - the exponent
         * @param {type} prec - the precision wanted
         * @param {bool} asbig - true if a bigDecimal is wanted
         * @returns {Symbol}
         */
        function nthroot(num, p, prec, asbig) {
            //clone p and convert to a number if possible
            p = evaluate(_.parse(p));

            //cannot calculate if p = 0. nthroot(0, 0) => 0^(1/0) => undefined
            if(p.equals(0)) {
                throw new UndefinedError('Unable to calculate nthroots of zero');
            }

            //Stop computation if it negative and even since we have an imaginary result
            if(num < 0 && even(p))
                throw new Error('Cannot calculate nthroot of negative number for even powers');

            //return non numeric values unevaluated
            if(!num.isConstant(true)) {
                return _.symfunction('nthroot', arguments);
            }

            //evaluate numeric values
            if(num.group !== N) {
                num = evaluate(num);
            }

            //default is to return a big value
            if(typeof asbig === 'undefined')
                asbig = true;

            prec = prec || 25;

            var sign = num.sign();
            var retval;
            var ans;

            if(sign < 0) {
                num = abs(num); //remove the sign
            }

            if(isInt(num) && p.isConstant()) {

                if(num < 18446744073709551616) {
                    //2^64
                    ans = Frac.create(Math.pow(num, 1 / p));
                }
                else {
                    ans = Math2.nthroot(num, p);
                }

                var retval;
                if(asbig) {
                    retval = new Symbol(ans);
                }
                retval = new Symbol(ans.toDecimal(prec));

                return _.multiply(new Symbol(sign), retval);
            }
        }

        function pfactor(symbol) {
            //Fix issue #458 | nerdamer("sqrt(1-(3.3333333550520926e-7)^2)").evaluate().text()
            //More Big Number issues >:(
            if(symbol.greaterThan(9.999999999998891e+41) || symbol.equals(-1))
                return symbol;
            //Fix issue #298
            if(symbol.equals(Math.PI))
                return new Symbol(Math.PI);
            //evaluate the symbol to merge constants
            symbol = evaluate(symbol.clone());

            if(symbol.isConstant()) {
                var retval = new Symbol(1);
                var m = symbol.toString();
                if(isInt(m)) {
                    var factors = Math2.ifactor(m);
                    for(var factor in factors) {
                        var p = factors[factor];
                        retval = _.multiply(retval, _.symfunction('parens', [new Symbol(factor).setPower(new Frac(p))]));
                    }
                }
                else {
                    var n = pfactor(new Symbol(symbol.multiplier.num));
                    var d = pfactor(new Symbol(symbol.multiplier.den));
                    retval = _.multiply(_.symfunction('parens', [n]), _.symfunction('parens', [d]).invert());
                }
            }
            else
                retval = _.symfunction('pfactor', arguments);
            return retval;
        }

        /**
         * Get's the real part of a complex number. Return number if real
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function realpart(symbol) {
            return symbol.realpart();
        }

        /**
         * Get's the imaginary part of a complex number
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function imagpart(symbol) {
            return symbol.imagpart();
        }

        /**
         * Computes the conjugate of a complex number
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function conjugate(symbol) {
            var re = symbol.realpart();
            var im = symbol.imagpart();
            return _.add(re, _.multiply(im.negate(), Symbol.imaginary()));
        }

        /**
         * Returns the arugment of a complex number
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function arg(symbol) {
            var re = symbol.realpart();
            var im = symbol.imagpart();
            if(re.isConstant() && im.isConstant())
                return new Symbol(Math.atan2(im, re));
            return _.symfunction('atan2', [im, re]);
        }

        /**
         * Returns the arugment of a complex number
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function arg(symbol) {
            var re = symbol.realpart();
            var im = symbol.imagpart();
            if(re.isConstant() && im.isConstant()) {
                if(im.equals(0) && re.equals(-1)) {
                    return _.parse('pi');
                }
                else if(im.equals(1) && re.equals(0)) {
                    return _.parse('pi/2');
                }
                else if(im.equals(1) && re.equals(1)) {
                    return _.parse('pi/4');
                }
                return new Symbol(Math.atan2(im, re));
            }
            return _.symfunction('atan2', [im, re]);
        }

        /**
         * Returns the polarform of a complex number
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function polarform(symbol) {
            var p, r, e, theta;
            p = Symbol.toPolarFormArray(symbol);
            theta = p[1];
            r = p[0];
            e = _.parse(format('e^({0}*({1}))', Settings.IMAGINARY, theta));
            return _.multiply(r, e);
        }

        /**
         * Returns the rectangular form of a complex number. Does not work for symbolic coefficients
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function rectform(symbol) {
            //TODO: e^((i*pi)/4)
            var original = symbol.clone();
            try {
                var f, p, q, s, h, d, n;
                f = decompose_fn(symbol, 'e', true);
                p = _.divide(f.x.power, Symbol.imaginary());
                q = evaluate(trig.tan(p));
                s = _.pow(f.a, new Symbol(2));
                d = q.getDenom(true);
                n = q.getNum();
                h = Symbol.hyp(n, d);
                //check
                if(h.equals(f.a)) {
                    return _.add(d, _.multiply(Symbol.imaginary(), n));
                }
                else {
                    return original;
                }
            }
            catch(e) {
                return original;
            }
        }

        function symMinMax(f, args) {
            args.map(function (x) {
                x.numVal = evaluate(x).multiplier;
            });
            var l, a, b, a_val, b_val;
            while(true) {
                l = args.length;
                if(l < 2)
                    return args[0];
                a = args.pop();
                b = args[l - 2];
                if(f === 'min' ? a.numVal < b.numVal : a.numVal > b.numVal) {
                    args.pop();
                    args.push(a);
                }
            }
        }

        /**
         * Returns maximum of a set of numbers
         * @returns {Symbol}
         */
        function max() {
            var args = [].slice.call(arguments);
            if(allSame(args))
                return args[0];
            if(allNumbers(args))
                return new Symbol(Math.max.apply(null, args));
            if(Settings.SYMBOLIC_MIN_MAX && allConstants(args))
                return symMinMax('max', args);
            return _.symfunction('max', args);
        }

        /**
         * Returns minimum of a set of numbers
         * @returns {Symbol}
         */
        function min() {
            var args = [].slice.call(arguments);
            if(allSame(args))
                return args[0];
            if(allNumbers(args))
                return new Symbol(Math.min.apply(null, args));
            if(Settings.SYMBOLIC_MIN_MAX && allConstants(args))
                return symMinMax('min', args);
            return _.symfunction('min', args);
        }

        /**
         * Returns the sign of a number
         * @param {Symbol} x
         * @returns {Symbol}
         */
        function sign(x) {
            if(x.isConstant(true))
                return new Symbol(Math.sign(evaluate(x)));
            return _.symfunction('sign', arguments);
        }

        function sort(symbol, opt) {
            opt = opt ? opt.toString() : 'asc';
            var getval = function (e) {
                if(e.group === N)
                    return e.multiplier;
                if(e.group === FN) {
                    if(e.fname === '')
                        return getval(e.args[0]);
                    return e.fname;
                }
                if(e.group === S)
                    return e.power;

                return e.value;
            };
            var symbols = isVector(symbol) ? symbol.elements : symbol.collectSymbols();
            return new Vector(symbols.sort(function (a, b) {
                var aval = getval(a),
                        bval = getval(b);
                if(opt === 'desc')
                    return bval - aval;
                return aval - bval;
            }));
        }

        /**
         * The log function
         * @param {Symbol} symbol
         * @param {Symbol} base
         * @returns {Symbol}
         */
        function log(symbol, base) {

            if(symbol.equals(1)) {
                return new Symbol(0);
            }

            var retval;

            if(symbol.fname === SQRT && symbol.multiplier.equals(1)) {
                retval = _.divide(log(symbol.args[0]), new Symbol(2));

                if(symbol.power.sign() < 0) {
                    retval.negate();
                }

                // Exit early
                return retval;
            }

            //log(0) is undefined so complain
            if(symbol.equals(0)) {
                throw new UndefinedError(Settings.LOG + '(0) is undefined!');
            }

            //deal with imaginary values
            if(symbol.isImaginary()) {
                return complex.evaluate(symbol, Settings.LOG);
            }

            if(symbol.isConstant() && typeof base !== 'undefined' && base.isConstant()) {
                var log_sym = Math.log(symbol);
                var log_base = Math.log(base);
                retval = new Symbol(log_sym / log_base);
            }
            else if(symbol.group === EX && symbol.power.multiplier.lessThan(0) || symbol.power.toString() === '-1') {
                symbol.power.negate();
                //move the negative outside but keep the positive inside :)
                retval = log(symbol).negate();
            }
            else if(symbol.value === 'e' && symbol.multiplier.equals(1)) {
                var p = symbol.power;
                retval = isSymbol(p) ? p : new Symbol(p);
            }
            else if(symbol.group === FN && symbol.fname === 'exp') {
                var s = symbol.args[0];
                if(symbol.multiplier.equals(1))
                    retval = _.multiply(s, new Symbol(symbol.power));
                else
                    retval = _.symfunction(Settings.LOG, [symbol]);
            }
            else if(Settings.PARSE2NUMBER && isNumericSymbol(symbol)) {
                // Parse for safety.
                symbol = _.parse(symbol);

                var img_part;
                if(symbol.multiplier.lessThan(0)) {
                    symbol.negate();
                    img_part = _.multiply(new Symbol(Math.PI), new Symbol('i'));
                }

                retval = new Symbol(Math.log(symbol.multiplier.toDecimal()));

                if(img_part) {
                    retval = _.add(retval, img_part);
                }

            }
            else {
                var s;
                if(!symbol.power.equals(1) && !symbol.contains('e')) {
                    s = symbol.group === EX ? symbol.power : new Symbol(symbol.power);
                    symbol.toLinear();
                }
                //log(a,a) = 1 since the base is allowed to be changed.
                //This was pointed out by Happypig375 in issue #280
                if(arguments.length > 1 && allSame(arguments)) {
                    retval = new Symbol(1);
                }
                else {
                    retval = _.symfunction(Settings.LOG, arguments);
                }

                if(s)
                    retval = _.multiply(s, retval);
            }

            return retval;
        }

        /**
         * Round a number up to s decimal places
         * @param {Number} x
         * @param {int} s - the number of decimal places
         * @returns {undefined}
         */
        function round(x, s) {
            var sIsConstant = s && s.isConstant() || typeof s === 'undefined';
            if(x.isConstant() && sIsConstant) {
                var v, e, exp, retval;
                v = x;
                //round the coefficient of then number but not the actual decimal value
                //we know this because a negative number was passed
                if(s && s.lessThan(0)) {
                    s = abs(s);
                    //convert the number to exponential form
                    e = Number(x).toExponential().toString().split('e');
                    //point v to the coefficient of then number
                    v = e[0];
                    //set the expontent
                    exp = e[1];
                }
                //round the number to the requested precision
                retval = new Symbol(nround(v, Number(s || 0)));
                //if there's a exponent then put it back
                return _.multiply(retval, _.pow(new Symbol(10), new Symbol(exp || 0)))
            }


            return _.symfunction('round', arguments);
        }

        /**
         * Gets the quadrant of the trig function
         * @param {Frac} m
         * @returns {Int}
         */
        function getQuadrant(m) {
            var v = m % 2, quadrant;

            if(v < 0)
                v = 2 + v; //put it in terms of pi

            if(v >= 0 && v <= 0.5)
                quadrant = 1;
            else if(v > 0.5 && v <= 1)
                quadrant = 2;
            else if(v > 1 && v <= 1.5)
                quadrant = 3;
            else
                quadrant = 4;
            return quadrant;
        }

        /*
         * Serves as a bridge between numbers and bigNumbers
         * @param {Frac|Number} n
         * @returns {Symbol}
         */
        function bigConvert(n) {
            if(!isFinite(n)) {
                var sign = Math.sign(n);
                var r = new Symbol(String(Math.abs(n)));
                r.multiplier = r.multiplier.multiply(new Frac(sign));
                return r;
            }
            if(isSymbol(n))
                return n;
            if(typeof n === 'number') {
                try {
                    n = Frac.simple(n);
                }
                catch(e) {
                    n = new Frac(n);
                }
            }

            var symbol = new Symbol(0);
            symbol.multiplier = n;
            return symbol;
        }
        ;

        function clean(symbol) {
            // handle functions with numeric values
            // handle denominator within denominator
            // handle trig simplifications
            var g = symbol.group, retval;
            //Now let's get to work
            if(g === CP) {
                var num = symbol.getNum(),
                        den = symbol.getDenom() || new Symbol(1),
                        p = Number(symbol.power),
                        factor = new Symbol(1);
                if(Math.abs(p) === 1) {
                    den.each(function (x) {
                        if(x.group === CB) {
                            factor = _.multiply(factor, clean(x.getDenom()));
                        }
                        else if(x.power.lessThan(0)) {
                            factor = _.multiply(factor, clean(x.clone().toUnitMultiplier()));
                        }
                    });

                    var new_den = new Symbol(0);
                    //now divide out the factor and add to new den
                    den.each(function (x) {
                        new_den = _.add(_.divide(x, factor.clone()), new_den);
                    });

                    factor.invert(); //invert so it can be added to the top
                    var new_num;
                    if(num.isComposite()) {
                        new_num = new Symbol(0);
                        num.each(function (x) {
                            new_num = _.add(_.multiply(clean(x), factor.clone()), new_num);
                        });
                    }
                    else
                        new_num = _.multiply(factor, num);

                    retval = _.divide(new_num, new_den);
                }
            }
            else if(g === CB) {
                retval = new Symbol(1);
                symbol.each(function (x) {
                    retval = _.multiply(retval, _.clean(x));
                });
            }
            else if(g === FN) {
                if(symbol.args.length === 1 && symbol.args[0].isConstant())
                    retval = block('PARSE2NUMBER', function () {
                        return _.parse(symbol);
                    }, true);
            }

            if(!retval)
                retval = symbol;

            return retval;
        }

        /**
         * A wrapper for the expand function
         * @param {Symbol} symbol
         * @returns {Symbol}
         */
        function expandall(symbol, opt) {
            opt = opt || {
                expand_denominator: true,
                expand_functions: true
            };
            return expand(symbol, opt);
        }
        /**
         * Expands a symbol
         * @param symbol
         */
        // Old expand
        function expand(symbol, opt) {
            if(Array.isArray(symbol)) {
                return symbol.map(function (x) {
                    return expand(x, opt);
                });
            }
            opt = opt || {};
            //deal with parenthesis
            if(symbol.group === FN && symbol.fname === '') {
                var f = expand(symbol.args[0], opt);
                var x = expand(_.pow(f, _.parse(symbol.power)), opt);
                return _.multiply(_.parse(symbol.multiplier), x).distributeMultiplier();
            }
            // We can expand these groups so no need to waste time. Just return and be done.
            if([N, P, S].indexOf(symbol.group) !== -1) {
                return symbol; //nothing to do
            }

            var original = symbol.clone();

            // Set up a try-catch block. If anything goes wrong then we simply return the original symbol
            try {
                // Store the power and multiplier
                var m = symbol.multiplier.toString();
                var p = Number(symbol.power);
                var retval = symbol;

                // Handle (a+b)^2 | (x+x^2)^2
                if(symbol.isComposite() && isInt(symbol.power) && symbol.power > 0) {
                    var n = p - 1;
                    // Strip the expression of it's multiplier and power. We'll call it f. The power will be p and the multiplier m.
                    var f = new Symbol(0);

                    symbol.each(function (x) {
                        f = _.add(f, expand(_.parse(x), opt));
                    });

                    var expanded = _.parse(f);

                    for(var i = 0; i < n; i++) {
                        expanded = mix(expanded, f, opt);
                    }

                    retval = _.multiply(_.parse(m), expanded).distributeMultiplier();
                }
                else if(symbol.group === FN && opt.expand_functions === true) {
                    var args = [];
                    // Expand function the arguments
                    symbol.args.forEach(function (x) {
                        args.push(expand(x, opt));
                    });
                    // Put back the power and multiplier
                    retval = _.pow(_.symfunction(symbol.fname, args), _.parse(symbol.power));
                    retval = _.multiply(retval, _.parse(symbol.multiplier));
                }
                else if(symbol.isComposite() && isInt(symbol.power) && symbol.power < 0 && opt.expand_denominator === true) {
                    // Invert it. Expand it and then re-invert it.
                    symbol = symbol.invert();
                    retval = expand(symbol, opt);
                    retval.invert();
                }
                else if(symbol.group === CB) {
                    var rank = function (s) {
                        switch(s.group) {
                            case CP:
                                return 0;
                            case PL:
                                return 1;
                            case CB:
                                return 2;
                            case FN:
                                return 3;
                            default:
                                return 4;
                        }
                    };
                    // Consider (a+b)(c+d). The result will be (a*c+a*d)+(b*c+b*d).
                    // We start by moving collecting the symbols. We want others>FN>CB>PL>CP
                    var symbols = symbol.collectSymbols().sort(function (a, b) {
                        return rank(b) - rank(a);
                    })
                            // Distribute the power to each symbol and expand
                            .map(function (s) {
                                var x = _.pow(s, _.parse(p));
                                var e = expand(x, opt);
                                return e;
                            });

                    var f = symbols.pop();

                    // If the first symbols isn't a composite then we're done
                    if(f.isComposite() && f.isLinear()) {
                        symbols.forEach(function (s) {
                            f = mix(f, s, opt);
                        });

                        // If f is of group PL or CP then we can expand some more
                        if(f.isComposite()) {
                            if(f.power > 1) {
                                f = expand(_.pow(f, _.parse(f.power)), opt);
                            }
                            // Put back the multiplier
                            retval = _.multiply(_.parse(m), f).distributeMultiplier();
                            ;
                        }
                        else {
                            // Everything is expanded at this point so if it's still a CB
                            // then just return the symbol
                            retval = f;
                        }
                    }
                    else {
                        // Just multiply back in the expanded form of each
                        retval = f;
                        symbols.forEach(function (s) {
                            retval = _.multiply(retval, s);
                        });
                        // Put back the multiplier
                        retval = _.multiply(retval, _.parse(m)).distributeMultiplier();
                    }

                    // TODO: This exists solely as a quick fix for sqrt(11)*sqrt(33) not simplifying.
                    if(retval.group === CB) {
                        retval = _.parse(retval);
                    }
                }
                else {
                    // Otherwise just return the expression
                    retval = symbol;
                }
                // Final cleanup and return
                return retval;
            }
            catch(e) {
                return original;
            }

            return original;
        }

        /**
         * Returns an identity matrix of nxn
         * @param {Number} n
         * @returns {Matrix}
         */
        function imatrix(n) {
            return Matrix.identity(n);
        }

        /**
         * Retrieves and item from a vector
         * @param {Vector} vector
         * @param {Number} index
         * @returns {Vector|Symbol}
         */
        function vecget(vector, index) {
            if(index.isConstant() && isInt(index))
                return vector.elements[index];
            return _.symfunction('vecget', arguments);
        }

        /**
         * Removes duplicates from a vector
         * @param {Vector} vector
         * @param {Number} tolerance
         * @returns {Vector}
         */
        function vectrim(vector, tolerance) {
            tolerance = typeof tolerance === 'undefined' ? 1e-14 : tolerance;

            vector = vector.clone();

            tolerance = Number(tolerance);
            //place algebraic solutions first
            vector.elements.sort(function (a, b) {
                return b.group - a.group;
            });
            //depending on the start point we may have duplicates so we need to clean those up a bit.
            //start by creating an object with the solution and the numeric value. This way we don't destroy algebraic values
            vector.elements = removeDuplicates(vector.elements, function (a, b) {
                var diff = Number(_.subtract(evaluate(a), evaluate(b)).abs());
                return diff <= tolerance;
            });

            return vector;
        }

        /**
         * Set a value for a vector at a given index
         * @param {Vector} vector
         * @param {Number} index
         * @param {Symbol} value
         * @returns {Vector}
         */
        function vecset(vector, index, value) {
            if(!index.isConstant)
                return _.symfunction('vecset', arguments);
            vector.elements[index] = value;
            return vector;
        }

        function matget(matrix, i, j) {
            if(i.isConstant() && j.isConstant())
                return matrix.elements[i][j];
            return _.symfunction('matget', arguments);
        }

        function matgetrow(matrix, i) {
            if(i.isConstant())
                return new Matrix(matrix.elements[i]);
            return _.symfunction('matgetrow', arguments);
        }

        function matsetrow(matrix, i, x) {
            //handle symbolics
            if(!i.isConstant())
                return _.symfunction('matsetrow', arguments);
            if(matrix.elements[i].length !== x.elements.length)
                throw new DimensionError('Matrix row must match row dimensions!');
            var M = matrix.clone();
            M.elements[i] = x.clone().elements;
            return M;
        }

        function matgetcol(matrix, col_index) {
            //handle symbolics
            if(!col_index.isConstant())
                return _.symfunction('matgetcol', arguments);
            col_index = Number(col_index);
            var M = Matrix.fromArray([]);
            matrix.each(function (x, i, j) {
                if(j === col_index) {
                    M.elements.push([x.clone()]);
                }
            });
            return M;
        }

        function matsetcol(matrix, j, col) {
            //handle symbolics
            if(!j.isConstant())
                return _.symfunction('matsetcol', arguments);
            j = Number(j);
            if(matrix.rows() !== col.elements.length)
                throw new DimensionError('Matrix columns must match number of columns!');
            col.each(function (x, i) {
                matrix.set(i - 1, j, x.elements[0].clone());
            });
            return matrix;
        }


        function matset(matrix, i, j, value) {
            matrix.elements[i][j] = value;
            return matrix;
        }

        //the constructor for vectors
        function vector() {
            return new Vector([].slice.call(arguments));
        }

        //the constructor for matrices
        function matrix() {
            return Matrix.fromArray(arguments);
        }

        //the constructor for sets
        function set() {
            return Set.fromArray(arguments);
        }

        function determinant(symbol) {
            if(isMatrix(symbol)) {
                return symbol.determinant();
            }
            return symbol;
        }

        function size(symbol) {
            var retval;
            if(isMatrix(symbol))
                retval = [new Symbol(symbol.cols()), new Symbol(symbol.rows())];
            else if(isVector(symbol) || isSet(symbol))
                retval = new Symbol(symbol.elements.length);
            else
                err('size expects a matrix or a vector');
            return retval;
        }

        function dot(vec1, vec2) {
            if(isVector(vec1) && isVector(vec2))
                return vec1.dot(vec2);
            err('function dot expects 2 vectors');
        }

        function cross(vec1, vec2) {
            if(isVector(vec1) && isVector(vec2))
                return vec1.cross(vec2);
            err('function cross expects 2 vectors');
        }

        function transpose(mat) {
            if(isMatrix(mat))
                return mat.transpose();
            err('function transpose expects a matrix');
        }

        function invert(mat) {
            if(isMatrix(mat))
                return mat.invert();
            err('invert expects a matrix');
        }

        //basic set functions
        function union(set1, set2) {
            return set1.union(set2);
        }

        function intersection(set1, set2) {
            return set1.intersection(set2);
        }

        function contains(set1, e) {
            return set1.contains(e);
        }

        function difference(set1, set2) {
            return set1.difference(set2);
        }

        function intersects(set1, set2) {
            return new Symbol(Number(set1.intersects(set2)));
        }

        function is_subset(set1, set2) {
            return new Symbol(Number(set1.is_subset(set2)));
        }

        function print() {
            arguments2Array(arguments).map(function (x) {
                console.log(x.toString());
            });
        }

        function testSQRT(symbol) {
            //wrap the symbol in sqrt. This eliminates one more check down the line.
            if(!isSymbol(symbol.power) && symbol.power.absEquals(0.5)) {
                var sign = symbol.power.sign();
                //don't devide the power directly. Notice the use of toString. This makes it possible
                //to use a bigNumber library in the future
                var retval = sqrt(symbol.group === P ? new Symbol(symbol.value) : symbol.toLinear());
                //place back the sign of the power
                if(sign < 0)
                    retval.invert();
                return retval;
            }
            return symbol;
        }

        //try to reduce a symbol by pulling its power
        function testPow(symbol) {
            if(symbol.group === P) {
                var v = symbol.value;

                var fct = primeFactors(v)[0];

                //safety
                if(!fct) {
                    warn('Unable to compute prime factors. This should not happen. Please review and report.');
                    return symbol;
                }

                var n = new Frac(Math.log(v) / Math.log(fct)),
                        p = n.multiply(symbol.power);

                //we don't want a more complex number than before
                if(p.den > symbol.power.den)
                    return symbol;

                if(isInt(p))
                    symbol = Symbol(Math.pow(fct, p));
                else
                    symbol = new Symbol(fct).setPower(p);
            }

            return symbol;
        }

        //Link the functions to the parse so they're available outside of the library.
        //This is strictly for convenience and may be deprecated.
        this.expand = expand;
        this.round = round;
        this.clean = clean;
        this.sqrt = sqrt;
        this.cbrt = cbrt;
        this.abs = abs;
        this.log = log;
        this.rationalize = rationalize;
        this.nthroot = nthroot;
        this.arg = arg;
        this.conjugate = conjugate;
        this.imagpart = imagpart;
        this.realpart = realpart;

        //TODO:
        //Utilize the function below instead of the linked function
        this.getFunction = function (name) {
            return functions[name][0];
        };

//Parser.methods ===============================================================
        this.addPreprocessor = function (name, action, order, shift_cells) {
            var names = preprocessors.names;
            var actions = preprocessors.actions;
            if((typeof action !== 'function')) //the person probably forgot to specify a name
                throw new PreprocessorError('Incorrect parameters. Function expected!');
            if(!order) {
                names.push(name);
                actions.push(action);
            }
            else {
                if(shift_cells) {
                    names.splice(order, 0, name);
                    actions.splice(order, 0, action);
                }
                else {
                    names[order] = name;
                    actions[order] = action;
                }
            }
        };

        this.getPreprocessors = function () {
            var preprocessors = {};
            for(var i = 0, l = preprocessors.names.length; i < l; i++) {
                var name = preprocessors.names[i];
                preprocessors[name] = {
                    order: i,
                    action: preprocessors.actions[i]
                };
            }
            return preprocessors;
        };

        this.removePreprocessor = function (name, shift_cells) {
            var i = preprocessors.names.indexOf(name);
            if(shift_cells) {
                remove(preprocessors.names, i);
                remove(preprocessors.actions, i);
            }
            else {
                preprocessors.names[i] = undefined;
                preprocessors.actions[i] = undefined;
            }
        };

        //The loader for functions which are not part of Math2
        this.mapped_function = function () {
            var subs = {},
                    params = this.params;

            for(var i = 0; i < params.length; i++) {
                subs[params[i]] = String(arguments[i]);
            }

            return _.parse(this.body, subs);
        };
        /**
         * Adds two symbols
         * @param {Symbol} a
         * @param {Symbol} b
         * @returns {Symbol}
         */
        this.add = function (a, b) {
            var aIsSymbol = isSymbol(a),
                    bIsSymbol = isSymbol(b);
            //we're dealing with two symbols
            if(aIsSymbol && bIsSymbol) {
                //forward the adding of symbols with units to the Unit module
                if(a.unit || b.unit) {
                    return _.Unit.add(a, b);
                }
                //handle Infinity
                //https://www.encyclopediaofmath.org/index.php/Infinity
                if(a.isInfinity || b.isInfinity) {
                    var aneg = a.multiplier.lessThan(0),
                            bneg = b.multiplier.lessThan(0);

                    if(a.isInfinity && b.isInfinity && aneg !== bneg) {
                        throw new UndefinedError('(' + a + ')+(' + b + ') is not defined!');
                    }

                    var inf = Symbol.infinity();
                    if(bneg)
                        inf.negate();
                    return inf;
                }

                if(a.isComposite() && a.isLinear() && b.isComposite() && b.isLinear()) {
                    a.distributeMultiplier();
                    b.distributeMultiplier();
                    // Fix for issue #606
                    if(b.length > a.length && a.group === b.group) {
                        [a, b] = [b, a];
                    }
                }
                
                //no need to waste time on zeroes
                if(a.multiplier.equals(0))
                    return b;
                if(b.multiplier.equals(0))
                    return a;

                if(a.isConstant() && b.isConstant() && Settings.PARSE2NUMBER) {
                    var result = new Symbol(a.multiplier.add(b.multiplier).toDecimal(Settings.PRECISION));
                    return result;
                }

                var g1 = a.group,
                        g2 = b.group,
                        ap = a.power.toString(),
                        bp = b.power.toString();

                //always keep the greater group on the left.
                if(g1 < g2 || (g1 === g2 && ap > bp && bp > 0)) {
                    return this.add(b, a);
                }

                /*note to self: Please don't forget about this dilemma ever again. In this model PL and CB goes crazy
                 * because it doesn't know which one to prioritize. */
                //correction to PL dilemma
                if(g1 === CB && g2 === PL && a.value === b.value) {
                    //swap
                    var t = a;
                    a = b;
                    b = t;
                    g1 = a.group;
                    g2 = b.group;
                    ap = a.power.toString();
                    bp = b.power.toString();
                }

                var powEQ = ap === bp,
                        v1 = a.value,
                        v2 = b.value,
                        aIsComposite = a.isComposite(),
                        bIsComposite = b.isComposite(),
                        h1, h2, result;

                if(aIsComposite)
                    h1 = text(a, 'hash');
                if(bIsComposite)
                    h2 = text(b, 'hash');

                if(g1 === CP && g2 === CP && b.isLinear() && !a.isLinear() && h1 !== h2) {
                    return this.add(b, a);
                }

                //PL & PL should compare hashes and not values e.g. compare x+x^2 with x+x^3 and not x with x
                if(g1 === PL && g2 === PL) {
                    v1 = h1;
                    v2 = h2;
                }

                var PN = g1 === P && g2 === N,
                        PNEQ = a.value === b.multiplier.toString(),
                        valEQ = (v1 === v2 || h1 === h2 && h1 !== undefined || (PN && PNEQ));

                //equal values, equal powers
                if(valEQ && powEQ && g1 === g2) {
                    //make sure to convert N to something P can work with
                    if(PN)
                        b = b.convert(P);//CL

                    //handle PL
                    if(g1 === PL && (g2 === S || g2 === P)) {
                        a.distributeMultiplier();
                        result = a.attach(b);
                    }
                    else {
                        result = a;//CL
                        if(a.multiplier.isOne() && b.multiplier.isOne() && g1 === CP && a.isLinear() && b.isLinear()) {
                            for(var s in b.symbols) {
                                var x = b.symbols[s];
                                result.attach(x);
                            }
                        }
                        else
                            result.multiplier = result.multiplier.add(b.multiplier);
                    }
                }
                //equal values uneven powers
                else if(valEQ && g1 !== PL) {
                    //break the tie for e.g. (x+1)+((x+1)^2+(x+1)^3)
                    if(g1 === CP && g2 === PL) {
                        b.insert(a, 'add');
                        result = b;
                    }
                    else {
                        result = Symbol.shell(PL).attach([a, b]);
                        //update the hash
                        result.value = g1 === PL ? h1 : v1;
                    }
                }
                else if(aIsComposite && a.isLinear()) {
                    var canIterate = g1 === g2,
                            bothPL = g1 === PL && g2 === PL;

                    //we can only iterate group PL if they values match
                    if(bothPL)
                        canIterate = a.value === b.value;
                    //distribute the multiplier over the entire symbol
                    a.distributeMultiplier();

                    if(b.isComposite() && b.isLinear() && canIterate) {
                        b.distributeMultiplier();
                        //CL
                        for(var s in b.symbols) {
                            var x = b.symbols[s];
                            a.attach(x);
                        }
                        result = a;
                    }
                    //handle cases like 2*(x+x^2)^2+2*(x+x^2)^3+4*(x+x^2)^2
                    else if(bothPL && a.value !== h2 || g1 === PL && !valEQ) {
                        result = Symbol.shell(CP).attach([a, b]);
                        result.updateHash();

                    }
                    else {
                        result = a.attach(b);
                    }
                }
                else {
                    if(g1 === FN && a.fname === SQRT && g2 !== EX && b.power.equals(0.5)) {
                        var m = b.multiplier.clone();
                        b = sqrt(b.toUnitMultiplier().toLinear());
                        b.multiplier = m;
                    }
                    //fix for issue #3 and #159
                    if(a.length === 2 && b.length === 2 && even(a.power) && even(b.power)) {
                        result = _.add(expand(a), expand(b));
                    }
                    else {
                        result = Symbol.shell(CP).attach([a, b]);
                        result.updateHash();
                    }
                }

                if(result.multiplier.equals(0))
                    result = new Symbol(0);

                //make sure to remove unnecessary wraps
                if(result.length === 1) {
                    var m = result.multiplier;
                    result = firstObject(result.symbols);
                    result.multiplier = result.multiplier.multiply(m);
                }

                return result;
            }
            else {
                //keep symbols to the right
                if(bIsSymbol && !aIsSymbol) {
                    var t = a;
                    a = b;
                    b = t; //swap
                    t = bIsSymbol;
                    bIsSymbol = aIsSymbol;
                    aIsSymbol = t;
                }

                var bIsMatrix = isMatrix(b);

                if(aIsSymbol && bIsMatrix) {
                    var M = new Matrix();
                    b.eachElement(function (e, i, j) {
                        M.set(i, j, _.add(a.clone(), e));
                    });

                    b = M
                }
                else {
                    if(isMatrix(a) && bIsMatrix) {
                        b = a.add(b);
                    }
                    else if(aIsSymbol && isVector(b)) {
                        b.each(function (x, i) {
                            i--;
                            b.elements[i] = _.add(a.clone(), b.elements[i]);
                        });
                    }
                    else {
                        if(isVector(a) && isVector(b)) {
                            b.each(function (x, i) {
                                i--;
                                b.elements[i] = _.add(a.elements[i], b.elements[i]);
                            });
                        }
                        else if(isVector(a) && isMatrix(b)) {
                            //try to convert a to a matrix
                            return _.add(b, a);
                        }
                        else if(isMatrix(a) && isVector(b)) {
                            if(b.elements.length === a.rows()) {
                                var M = new Matrix(), l = a.cols();
                                b.each(function (e, i) {
                                    var row = [];
                                    for(var j = 0; j < l; j++) {
                                        row.push(_.add(a.elements[i - 1][j].clone(), e.clone()));
                                    }
                                    M.elements.push(row);
                                });
                                return M;
                            }
                            else
                                err('Dimensions must match!');
                        }
                    }
                }
                return b;
            }

        };
        /**
         * Gets called when the parser finds the - operator. Not the prefix operator. See this.add
         * @param {Symbol} a
         * @param {Symbol} b
         * @returns {Symbol}
         */
        this.subtract = function (a, b) {
            var aIsSymbol = aIsSymbol = isSymbol(a),
                    bIsSymbol = isSymbol(b), t;

            if(aIsSymbol && bIsSymbol) {
                if(a.unit || b.unit) {
                    return _.Unit.subtract(a, b);
                }
                return this.add(a, b.negate());
            }
            else {
                if(bIsSymbol && isVector(a)) {
                    b = a.map(function (x) {
                        return _.subtract(x, b.clone());
                    });
                }
                else if(aIsSymbol && isVector(b)) {
                    b = b.map(function (x) {
                        return _.subtract(a.clone(), x);
                    });
                }
                else if(isVector(a) && isVector(b)) {
                    if(a.dimensions() === b.dimensions())
                        b = a.subtract(b);
                    else
                        _.error('Unable to subtract vectors. Dimensions do not match.');
                }
                else if(isMatrix(a) && isVector(b)) {
                    if(b.elements.length === a.rows()) {
                        var M = new Matrix(), l = a.cols();
                        b.each(function (e, i) {
                            var row = [];
                            for(var j = 0; j < l; j++) {
                                row.push(_.subtract(a.elements[i - 1][j].clone(), e.clone()));
                            }
                            M.elements.push(row);
                        });
                        return M;
                    }
                    else
                        err('Dimensions must match!');
                }
                else if(isVector(a) && isMatrix(b)) {
                    var M = b.clone().negate();
                    return _.add(M, a);
                }
                else if(isMatrix(a) && isMatrix(b)) {
                    b = a.subtract(b);
                }
                else if(isMatrix(a) && bIsSymbol) {
                    var M = new Matrix();
                    a.each(function (x, i, j) {
                        M.set(i, j, _.subtract(x, b.clone()));
                    });
                    b = M;
                }
                else if(aIsSymbol && isMatrix(b)) {
                    var M = new Matrix();
                    b.each(function (x, i, j) {
                        M.set(i, j, _.subtract(a.clone(), x));
                    });
                    b = M;
                }
                return b;
            }
        };
        /**
         * Gets called when the parser finds the * operator. See this.add
         * @param {Symbol} a
         * @param {Symbol} b
         * @returns {Symbol}
         */
        this.multiply = function (a, b) {
            var aIsSymbol = isSymbol(a),
                    bIsSymbol = isSymbol(b);
            //we're dealing with function assignment here
            if(aIsSymbol && b instanceof Collection) {
                b.elements.push(a);
                return b;
            }
            if(aIsSymbol && bIsSymbol) {
                //if it has a unit then add it and return it right away.
                if(b.isUnit) {
                    var result = a.clone();
                    a.unit = b;
                    return result;
                }
                
                //if it has units then just forward that problem to the unit module
                if(a.unit || b.unit) {
                    return _.Unit.multiply(a, b);
                }

                //handle Infinty
                if(a.isInfinity || b.isInfinity) {
                    if(a.equals(0) || b.equals(0))
                        throw new UndefinedError(a + '*' + b + ' is undefined!');
                    //x/infinity
                    if(b.power.lessThan(0)) {
                        if(!a.isInfinity) {
                            return new Symbol(0);
                        }
                        else {
                            throw new UndefinedError('Infinity/Infinity is not defined!');
                        }
                    }

                    var sign = a.multiplier.multiply(b.multiplier).sign(),
                            inf = Symbol.infinity();
                    if(a.isConstant() || b.isConstant() || (a.isInfinity && b.isInfinity)) {
                        if(sign < 0)
                            inf.negate();

                        return inf;
                    }
                }
                //the quickies
                if(a.isConstant() && b.isConstant() && Settings.PARSE2NUMBER) {
                    var t = new bigDec(a.multiplier.toDecimal()).times(new bigDec(b.multiplier.toDecimal())).toFixed();
                    var retval = new Symbol(t);
                    return retval;
                }

                //don't waste time
                if(a.isOne()) {
                    return b.clone();
                }
                if(b.isOne()) {
                    return a.clone();
                }

                if(a.multiplier.equals(0) || b.multiplier.equals(0))
                    return new Symbol(0);

                if(b.group > a.group && !(b.group === CP))
                    return this.multiply(b, a);
                //correction for PL/CB dilemma
                if(a.group === CB && b.group === PL && a.value === b.value) {
                    var t = a;
                    a = b;
                    b = t;//swap
                }

                var g1 = a.group,
                        g2 = b.group,
                        bnum = b.multiplier.num,
                        bden = b.multiplier.den;

                if(g1 === FN && a.fname === SQRT && !b.isConstant() && a.args[0].value === b.value && !a.args[0].multiplier.lessThan(0)) {
                    //unwrap sqrt
                    var a_pow = a.power;
                    var a_multiplier = _.parse(a.multiplier);
                    a = _.multiply(a_multiplier, a.args[0].clone());
                    a.setPower(new Frac(0.5).multiply(a_pow));
                    g1 = a.group;
                }
                //simplify n/sqrt(n). Being very specific
                else if(g1 === FN && a.fname === SQRT && a.multiplier.equals(1) && a.power.equals(-1) && b.isConstant() && a.args[0].equals(b)) {
                    a = _.symfunction(SQRT, [b.clone()]);
                    b = new Symbol(1);
                }
                ;

                var v1 = a.value,
                        v2 = b.value,
                        sign = new Frac(a.sign()),
                        //since P is just a morphed version of N we need to see if they relate
                        ONN = (g1 === P && g2 === N && b.multiplier.equals(a.value)),
                        //don't multiply the multiplier of b since that's equal to the value of a
                        m = ONN ? new Frac(1).multiply(a.multiplier).abs() : a.multiplier.multiply(b.multiplier).abs(),
                        result = a.clone().toUnitMultiplier();
                b = b.clone().toUnitMultiplier(true);

                //further simplification of sqrt
                if(g1 === FN && g2 === FN) {
                    var u = a.args[0].clone();
                    var v = b.args[0].clone();
                    if(a.fname === SQRT && b.fname === SQRT && a.isLinear() && b.isLinear()) {

                        var q = _.divide(u, v).invert();
                        if(q.gt(1) && isInt(q)) {
                            //b contains a factor a which can be moved to a
                            result = _.multiply(a.args[0].clone(), sqrt(q.clone()));
                            b = new Symbol(1);
                        }
                    }
                    //simplify factorial but only if
                    //1 - It's division so b will have a negative power
                    //2 - We're not dealing with factorials of numbers
                    else if(a.fname === FACTORIAL && b.fname === FACTORIAL && !u.isConstant() && !v.isConstant() && b.power < 0) {
                        //assume that n = positive
                        var d = _.subtract(u.clone(), v.clone());

                        //if it's not numeric then we don't know if we can simplify so just return
                        if(d.isConstant()) {

                            //there will never be a case where d == 0 since this will already have
                            //been handled at the beginning of this function
                            t = new Symbol(1);
                            if(d < 0) {
                                //If d is negative then the numerator is larger so expand that
                                for(var i = 0, n = Math.abs(d); i <= n; i++) {
                                    var s = _.add(u.clone(), new Symbol(i));
                                    t = _.multiply(t, s);
                                }

                                result = _.multiply(_.pow(u, new Symbol(a.power)), _.pow(t, new Symbol(b.power)));

                                b = new Symbol(1);
                            }
                            else {
                                //Otherwise the denominator is larger so expand that
                                for(var i = 0, n = Math.abs(d); i <= n; i++) {
                                    var s = _.add(v.clone(), new Symbol(i));
                                    t = _.multiply(t, s);
                                }

                                result = _.multiply(_.pow(t, new Symbol(a.power)), _.pow(v, new Symbol(b.power)));

                                b = new Symbol(1);
                            }
                        }
                    }
                }


                //if both are PL then their hashes have to match
                if(v1 === v2 && g1 === PL && g1 === g2) {
                    v1 = a.text('hash');
                    v2 = b.text('hash');
                }

                //same issue with (x^2+1)^x*(x^2+1)
                //EX needs an exception when multiplying because it needs to recognize
                //that (x+x^2)^x has the same hash as (x+x^2). The latter is kept as x
                if(g2 === EX && b.previousGroup === PL && g1 === PL) {
                    v1 = text(a, 'hash', EX);
                }

                if((v1 === v2 || ONN) && !(g1 === PL && (g2 === S || g2 === P || g2 === FN)) && !(g1 === PL && g2 === CB)) {
                    var p1 = a.power,
                            p2 = b.power,
                            isSymbolP1 = isSymbol(p1),
                            isSymbolP2 = isSymbol(p2),
                            toEX = (isSymbolP1 || isSymbolP2);
                    //TODO: this needs cleaning up
                    if(g1 === PL && g2 !== PL && b.previousGroup !== PL && p1.equals(1)) {
                        result = new Symbol(0);
                        a.each(function (x) {
                            result = _.add(result, _.multiply(x, b.clone()));
                        }, true);
                    }
                    else {
                        //add the powers
                        result.power = toEX ? _.add(
                                !(isSymbol(p1)) ? new Symbol(p1) : p1,
                                !(isSymbol(p2)) ? new Symbol(p2) : p2
                                ) : (g1 === N /*don't add powers for N*/ ? p1 : p1.add(p2));

                        //eliminate zero power values and convert them to numbers
                        if(result.power.equals(0))
                            result = result.convert(N);

                        //properly convert to EX
                        if(toEX)
                            result.convert(EX);

                        //take care of imaginaries
                        if(a.imaginary && b.imaginary) {
                            var isEven = even(result.power % 2);
                            if(isEven) {
                                result = new Symbol(1);
                                m.negate();
                            }
                        }

                        //cleanup: this causes the LaTeX generator to get confused as to how to render the symbol
                        if(result.group !== EX && result.previousGroup)
                            result.previousGroup = undefined;
                        //the sign for b is floating around. Remember we are assuming that the odd variable will carry
                        //the sign but this isn't true if they're equals symbols
                        result.multiplier = result.multiplier.multiply(b.multiplier);
                    }
                }
                else if(g1 === CB && a.isLinear()) {
                    if(g2 === CB)
                        b.distributeExponent();
                    if(g2 === CB && b.isLinear()) {
                        for(var s in b.symbols) {
                            var x = b.symbols[s];
                            result = result.combine(x);
                        }
                        result.multiplier = result.multiplier.multiply(b.multiplier);
                    }
                    else {
                        result.combine(b);
                    }
                }
                else {
                    //the multiplier was already handled so nothing left to do
                    if(g1 !== N) {
                        if(g1 === CB) {
                            result.distributeExponent();
                            result.combine(b);
                        }
                        else if(!b.isOne()) {
                            var bm = b.multiplier.clone();
                            b.toUnitMultiplier();
                            result = Symbol.shell(CB).combine([result, b]);
                            //transfer the multiplier to the outside
                            result.multiplier = result.multiplier.multiply(bm);
                        }
                    }
                    else {
                        result = b.clone().toUnitMultiplier();
                    }
                }

                if(result.group === P) {
                    var logV = Math.log(result.value),
                            n1 = Math.log(bnum) / logV,
                            n2 = Math.log(bden) / logV,
                            ndiv = m.num / bnum,
                            ddiv = m.den / bden;
                    //we don't want to divide by zero no do we? Strange things happen.
                    if(n1 !== 0 && isInt(n1) && isInt(ndiv)) {
                        result.power = result.power.add(new Frac(n1));
                        m.num /= bnum; //BigInt? Keep that in mind for the future.
                    }
                    if(n2 !== 0 && isInt(n2) && isInt(ddiv)) {
                        result.power = result.power.subtract(new Frac(n2));
                        m.den /= bden; //BigInt? Keep that in mind for the future.
                    }
                }

                //unpack CB if length is only one
                if(result.length === 1) {
                    var t = result.multiplier;
                    //transfer the multiplier
                    result = firstObject(result.symbols);
                    result.multiplier = result.multiplier.multiply(t);
                }

                //reduce square root
                var ps = result.power.toString();
                if(even(ps) && result.fname === SQRT) {
                    //grab the sign of the symbol
                    sign = sign * result.sign();
                    var p = result.power;
                    result = result.args[0];
                    result = _.multiply(new Symbol(m), _.pow(result, new Symbol(p.divide(new Frac(2)))));
                    //flip it back to the correct sign
                    if(sign < 0)
                        result.negate()
                }
                else {
                    result.multiplier = result.multiplier.multiply(m).multiply(sign);
                    if(result.group === CP && result.isImaginary())
                        result.distributeMultiplier();
                }

                //back convert group P to a simpler group N if possible
                if(result.group === P && isInt(result.power.toDecimal()))
                    result = result.convert(N);

                return result;
            }
            else {
                //****** Matrices & Vector *****//
                if(bIsSymbol && !aIsSymbol) { //keep symbols to the right
                    t = a;
                    a = b;
                    b = t; //swap
                    t = bIsSymbol;
                    bIsSymbol = aIsSymbol;
                    aIsSymbol = t;
                }

                var isMatrixB = isMatrix(b), isMatrixA = isMatrix(a);
                if(aIsSymbol && isMatrixB) {
                    var M = new Matrix();
                    b.eachElement(function (e, i, j) {
                        M.set(i, j, _.multiply(a.clone(), e));
                    });

                    b = M;
                }
                else {
                    if(isMatrixA && isMatrixB) {
                        b = a.multiply(b);
                    }
                    else if(aIsSymbol && isVector(b)) {
                        b.each(function (x, i) {
                            i--;
                            b.elements[i] = _.multiply(a.clone(), b.elements[i]);
                        });
                    }
                    else {
                        if(isVector(a) && isVector(b)) {
                            b.each(function (x, i) {
                                i--;
                                b.elements[i] = _.multiply(a.elements[i], b.elements[i]);
                            });
                        }
                        else if(isVector(a) && isMatrix(b)) {
                            //try to convert a to a matrix
                            return this.multiply(b, a);
                        }
                        else if(isMatrix(a) && isVector(b)) {
                            if(b.elements.length === a.rows()) {
                                var M = new Matrix(), l = a.cols();
                                b.each(function (e, i) {
                                    var row = [];
                                    for(var j = 0; j < l; j++) {
                                        row.push(_.multiply(a.elements[i - 1][j].clone(), e.clone()));
                                    }
                                    M.elements.push(row);
                                });
                                return M;
                            }
                            else
                                err('Dimensions must match!');
                        }
                    }
                }

                return b;
            }
        };
        /**
         * Gets called when the parser finds the / operator. See this.add
         * @param {Symbol} a
         * @param {Symbol} b
         * @returns {Symbol}
         */
        this.divide = function (a, b) {
            var aIsSymbol = isSymbol(a),
                    bIsSymbol = isSymbol(b);

            if(aIsSymbol && bIsSymbol) {
                //forward to Unit division
                if(a.unit || b.unit) {
                    return _.Unit.divide(a, b);
                }
                var result;
                if(b.equals(0))
                    throw new DivisionByZero('Division by zero not allowed!');

                if(a.isConstant() && b.isConstant()) {
                    result = a.clone();
                    result.multiplier = result.multiplier.divide(b.multiplier);
                }
                else {
                    b.invert();
                    result = _.multiply(a, b);
                }
                return result;
            }
            else {
                //******* Vectors & Matrices *********//
                var isVectorA = isVector(a), isVectorB = isVector(b);
                if(aIsSymbol && isVectorB) {
                    b = b.map(function (x) {
                        return _.divide(a.clone(), x);
                    });
                }
                else if(isVectorA && bIsSymbol) {
                    b = a.map(function (x) {
                        return _.divide(x, b.clone());
                    });
                }
                else if(isVectorA && isVectorB) {
                    if(a.dimensions() === b.dimensions()) {
                        b = b.map(function (x, i) {
                            return _.divide(a.elements[--i], x);
                        });
                    }
                    else
                        _.error('Cannot divide vectors. Dimensions do not match!');
                }
                else {
                    var isMatrixA = isMatrix(a), isMatrixB = isMatrix(b);
                    if(isMatrixA && bIsSymbol) {
                        var M = new Matrix();
                        a.eachElement(function (x, i, j) {
                            M.set(i, j, _.divide(x, b.clone()));
                        });
                        b = M;
                    }
                    else if(aIsSymbol && isMatrixB) {
                        var M = new Matrix();
                        b.eachElement(function (x, i, j) {
                            M.set(i, j, _.divide(a.clone(), x));
                        });
                        b = M;
                    }
                    else if(isMatrixA && isMatrixB) {
                        var M = new Matrix();
                        if(a.rows() === b.rows() && a.cols() === b.cols()) {
                            a.eachElement(function (x, i, j) {
                                M.set(i, j, _.divide(x, b.elements[i][j]));
                            });
                            b = M;
                        }
                        else {
                            _.error('Dimensions do not match!');
                        }
                    }
                    else if(isMatrixA && isVectorB) {
                        if(a.cols() === b.dimensions()) {
                            var M = new Matrix();
                            a.eachElement(function (x, i, j) {
                                M.set(i, j, _.divide(x, b.elements[i].clone()));
                            });
                            b = M;
                        }
                        else {
                            _.error('Unable to divide matrix by vector.');
                        }
                    }
                }
                return b;
            }
        };
        /**
         * Gets called when the parser finds the ^ operator. See this.add
         * @param {Symbol} a
         * @param {Symbol} b
         * @returns {Symbol}
         */
        this.pow = function (a, b) {
            var aIsSymbol = isSymbol(a),
                    bIsSymbol = isSymbol(b);
            if(aIsSymbol && bIsSymbol) {
                //it has units then it's the Unit module's problem
                if(a.unit || b.unit) {
                    return _.Unit.pow(a, b);
                }
                
                // Handle abs
                if(a.group === FN && a.fname === ABS && even(b)) {
                    var m = a.multiplier.clone();
                    var raised = _.pow(a.args[0], b);
                    raised.multiplier = m;
                    return raised;
                }
                
                // Handle infinity
                if(a.isInfinity || b.isInfinity) {
                    if(a.isInfinity && b.isInfinity)
                        throw new UndefinedError('(' + a + ')^(' + b + ') is undefined!');

                    if(a.isConstant() && b.isInfinity) {
                        if(a.equals(0)) {
                            if(b.lessThan(0))
                                throw new UndefinedError('0^Infinity is undefined!');
                            return new Symbol(0);
                        }
                        if(a.equals(1))
                            throw new UndefinedError('1^' + b.toString() + ' is undefined!');
                        //a^-oo
                        if(b.lessThan(0))
                            return new Symbol(0);
                        //a^oo
                        if(!a.lessThan(0))
                            return Symbol.infinity();
                    }

                    if(a.isInfinity && b.isConstant()) {
                        if(b.equals(0))
                            throw new UndefinedError(a + '^0 is undefined!');
                        if(b.lessThan(0))
                            return new Symbol(0);
                        return _.multiply(Symbol.infinity(), _.pow(new Symbol(a.sign()), b.clone()));
                    }
                }

                var aIsZero = a.equals(0);
                var bIsZero = b.equals(0);
                if(aIsZero && bIsZero)
                    throw new UndefinedError('0^0 is undefined!');

                // Return 0 right away if possible
                if(aIsZero && b.isConstant() && b.multiplier.greaterThan(0))
                    return new Symbol(0);

                if(bIsZero)
                    return new Symbol(1);

                var bIsConstant = b.isConstant(),
                        aIsConstant = a.isConstant(),
                        bIsInt = b.isInteger(),
                        m = a.multiplier,
                        result = a.clone();

                // 0^0, 1/0, etc. Complain.
                if(aIsConstant && bIsConstant && a.equals(0) && b.lessThan(0))
                    throw new UndefinedError('Division by zero is not allowed!');

                // Compute imaginary numbers right away
                if(Settings.PARSE2NUMBER && aIsConstant && bIsConstant && a.sign() < 0 && evenFraction(b)) {
                    var k, re, im;
                    k = Math.PI * b;
                    re = new Symbol(Math.cos(k));
                    im = _.multiply(Symbol.imaginary(), new Symbol(Math.sin(k)));
                    return _.add(re, im);
                }

                // Imaginary number under negative nthroot or to the n
                if(Settings.PARSE2NUMBER && a.isImaginary() && bIsConstant && isInt(b) && !b.lessThan(0)) {
                    var re, im, r, theta, nre, nim, phi;
                    re = a.realpart();
                    im = a.imagpart();
                    if(re.isConstant('all') && im.isConstant('all')) {
                        phi = Settings.USE_BIG ? Symbol(bigDec.atan2(i.multiplier.toDecimal(), r.multiplier.toDecimal()).times(b.toString())) : Math.atan2(im, re) * b;
                        theta = new Symbol(phi);
                        r = _.pow(Symbol.hyp(re, im), b);
                        nre = _.multiply(r.clone(), _.trig.cos(theta.clone()));
                        nim = _.multiply(r, _.trig.sin(theta));
                        return _.add(nre, _.multiply(Symbol.imaginary(), nim));
                    }
                }

                // Take care of the symbolic part
                result.toUnitMultiplier();
                //simpifly sqrt
                if(result.group === FN && result.fname === SQRT && !bIsConstant) {
                    var s = result.args[0];
                    s.multiplyPower(new Symbol(0.5));
                    s.multiplier.multiply(result.multiplier);
                    s.multiplyPower(b);
                    result = s;
                }
                else {
                    var sign = m.sign();
                    //handle cases such as (-a^3)^(1/4)
                    if(evenFraction(b) && sign < 0) {
                        // Swaperoo
                        // First put the sign back on the symbol
                        result.negate();
                        // Wrap it in brackets
                        result = _.symfunction(PARENTHESIS, [result]);
                        // Move the sign back the exterior and let nerdamer handle the rest
                        result.negate();
                    }

                    result.multiplyPower(b);
                }

                if(aIsConstant && bIsConstant && Settings.PARSE2NUMBER) {
                    var c;
                    //remove the sign
                    if(sign < 0) {
                        a.negate();
                        if(b.multiplier.den.equals(2))
                            //we know that the numerator has to be odd and therefore it's i
                            c = new Symbol(Settings.IMAGINARY);
                        else if(isInt(b.multiplier)) {
                            if(even(b.multiplier))
                                c = new Symbol(1);
                            else
                                c = new Symbol(-1);
                        }
                        else if(!even(b.multiplier.den)) {
                            c = new Symbol(Math.pow(sign, b.multiplier.num));
                        }
                        else {
                            c = _.pow(_.symfunction(PARENTHESIS, [new Symbol(sign)]), b.clone());
                        }
                    }

                    result = new Symbol(Math.pow(a.multiplier.toDecimal(), b.multiplier.toDecimal()));

                    //result = new Symbol(Math2.bigpow(a.multiplier, b.multiplier));
                    //put the back sign
                    if(c)
                        result = _.multiply(result, c);
                }
                else if(bIsInt && !m.equals(1)) {
                    var abs_b = b.abs();
                    // Provide fall back to JS until big number implementation is improved
                    if(abs_b.gt(Settings.MAX_EXP)) {
                        if(b.sign() < 0)
                            return new Symbol(0);
                        return Symbol.infinity();
                    }
                    else {
                        var p = b.multiplier.toDecimal();
                        var sgn = Math.sign(p);
                        p = Math.abs(p);
                        var multiplier = new Frac(1);
                        multiplier.num = m.num.pow(p);
                        multiplier.den = m.den.pow(p);
                        if(sgn < 0)
                            multiplier.invert();
                        //multiplying is justified since after mulltiplyPower if it was of group P it will now be of group N
                        result.multiplier = result.multiplier.multiply(multiplier);
                    }
                }
                else {
                    var sign = a.sign();
                    if(b.isConstant() && a.isConstant() && !b.multiplier.den.equals(1) && sign < 0) {
                        //we know the sign is negative so if the denominator for b == 2 then it's i
                        if(b.multiplier.den.equals(2)) {
                            var i = new Symbol(Settings.IMAGINARY);
                            a.negate();//remove the sign
                            //if the power is negative then i is negative
                            if(b.lessThan(0)) {
                                i.negate();
                                b.negate();//remove the sign from the power
                            }
                            //pull the power normally and put back the imaginary
                            result = _.multiply(_.pow(a, b), i);
                        }
                        else {
                            var aa = a.clone();
                            aa.multiplier.negate();
                            result = _.pow(_.symfunction(PARENTHESIS, [new Symbol(sign)]), b.clone());
                            var _a = _.pow(new Symbol(aa.multiplier.num), b.clone());
                            var _b = _.pow(new Symbol(aa.multiplier.den), b.clone());
                            var r = _.divide(_a, _b);
                            result = _.multiply(result, r);
                        }
                    }
                    else if(Settings.PARSE2NUMBER && b.isImaginary()) {
                        //4^(i + 2) = e^(- (2 - 4 i) π n + (2 + i) log(4))

                        var re = b.realpart();
                        var im = b.imagpart();
                        /*
                         if(b.group === CP && false) {
                         var ex = _.pow(a.clone(), re);
                         var xi = _.multiply(_.multiply(ex.clone(), trig.sin(im.clone())), Symbol.imaginary());
                         var xa = _.multiply(trig.cos(im), ex);
                         result = _.add(xi, xa);
                         }
                         else {
                         */
                        var aa = a.clone().toLinear();
                        var a1 = _.pow(aa.clone(), re);
                        var log_a = log(aa.clone());
                        var b1 = trig.cos(_.multiply(im.clone(), log_a));
                        var c1 = _.multiply(trig.sin(_.multiply(im, log(aa))), Symbol.imaginary());
                        result = _.multiply(a1, _.add(b1, c1));
                        result = _.expand(_.parse(result));
                        /*
                         }   
                         */
                    }
                    else {
                        //b is a symbol
                        var neg_num = a.group === N && sign < 0,
                            num = testSQRT(new Symbol(neg_num ? m.num : Math.abs(m.num)).setPower(b.clone())),
                            den = testSQRT(new Symbol(m.den).setPower(b.clone()).invert());

                        //eliminate imaginary if possible
                        if(a.imaginary) {
                            if(bIsInt) {
                                var s, p, n;
                                s = Math.sign(b);
                                p = abs(b);
                                n = p % 4;
                                result = new Symbol(even(n) ? -1 : Settings.IMAGINARY);
                                if(n === 0 || s < 0 && (n === 1) || s > 0 && (n === 3)) {
                                    result.negate();
                                }
                            }
                            else {
                                //assume i = sqrt(-1) -> (-1)^(1/2)
                                var nr = b.multiplier.multiply(Frac.quick(1, 2)),
                                        //the denominator denotes the power so raise to it. It will turn positive it round
                                        tn = Math.pow(-1, nr.num);
                                result = even(nr.den) ? new Symbol(-1).setPower(nr, true) : new Symbol(tn);
                            }
                        }
                        //ensure that the sign is carried by the symbol and not the multiplier
                        //this enables us to check down the line if the multiplier can indeed be transferred
                        if(sign < 0 && !neg_num)
                            result.negate();

                        //retain the absolute value
                        if(bIsConstant && a.group !== EX) {
                            var evenr = even(b.multiplier.den),
                                    evenp = even(a.power),
                                    n = result.power.toDecimal(),
                                    evennp = even(n);
                            if(evenr && evenp && !evennp) {
                                if(n === 1)
                                    result = _.symfunction(ABS, [result]);
                                else if(!isInt(n)) {
                                    var p = result.power;
                                    result = _.symfunction(ABS, [result.toLinear()]).setPower(p);
                                }
                                else {
                                    result = _.multiply(_.symfunction(ABS, [result.clone().toLinear()]),
                                            result.clone().setPower(new Frac(n - 1)));
                                }
                                //quick workaround. Revisit
                                if(Settings.POSITIVE_MULTIPLIERS && result.fname === ABS)
                                    result = result.args[0];
                            }
                        }
                        //multiply out sqrt
                        if(b.equals(2) && result.group === CB) {
                            var _result = new Symbol(1);
                            result.each(function (sym) {
                                _result = _.multiply(_result, _.pow(sym, b));
                            });
                            result = _result;
                        }
                    }
                }

                result = testSQRT(result);

                // Don't multiply until we've tested the remaining symbol
                if(num && den) {
                    result = _.multiply(result, testPow(_.multiply(num, den)));
                }

                // Reduce square root
                if(result.fname === SQRT) {
                    var isEX = result.group === EX;
                    var t = isEX ? result.power.multiplier.toString() : result.power.toString();
                    if(even(t)) {
                        var pt = isEX ? _.divide(result.power, new Symbol(2)) : new Symbol(result.power.divide(new Frac(2))),
                                m = result.multiplier;
                        result = _.pow(result.args[0], pt);
                        result.multiplier = result.multiplier.multiply(m);
                    }
                }
                // Detect Euler's identity
                else if(!Settings.IGNORE_E && result.isE() && result.group === EX && result.power.contains('pi')
                        && result.power.contains(Settings.IMAGINARY) && b.group === CB) {
                    var theta = b.stripVar(Settings.IMAGINARY);
                    result = _.add(trig.cos(theta), _.multiply(Symbol.imaginary(), trig.sin(theta)));
                }

                return result;
            }
            else {
                if(isVector(a) && bIsSymbol) {
                    a = a.map(function (x) {
                        return _.pow(x, b.clone());
                    });
                }
                else if(isMatrix(a) && bIsSymbol) {
                    var M = new Matrix();
                    a.eachElement(function (x, i, j) {
                        M.set(i, j, _.pow(x, b.clone()));
                    });
                    a = M;
                }
                else if(aIsSymbol && isMatrix(b)) {
                    var M = new Matrix();
                    b.eachElement(function (x, i, j) {
                        M.set(i, j, _.pow(a.clone(), x));
                    });
                    a = M;
                }
                return a;
            }
        };
        // Gets called when the parser finds the , operator.
        // Commas return a Collector object which is roughly an array
        this.comma = function (a, b) {
            if(!(a instanceof Collection))
                a = Collection.create(a);
            a.append(b);
            return a;
        };
        // Link to modulus
        this.mod = function (a, b) {
            return mod(a, b);
        };
        // Used to slice elements from arrays
        this.slice = function (a, b) {
            return new Slice(a, b);
        };
        // The equality setter
        this.equals = function (a, b) {
            // Equality can only be set for group S so complain it's not
            if(a.group !== S && !a.isLinear())
                err('Cannot set equality for ' + a.toString());
            VARS[a.value] = b.clone();
            return b;
        };
        // Percent
        this.percent = function (a) {
            return _.divide(a, new Symbol(100));
        };
        // Set variable
        this.assign = function (a, b) {
            if(a instanceof Collection && b instanceof Collection) {
                a.elements.map(function (x, i) {
                    return _.assign(x, b.elements[i]);
                });
                return Vector.fromArray(b.elements);
            }
            if(a.parent) {
                // It's referring to the parent instead. The current item can be discarded
                var e = a.parent;
                e.elements[e.getter] = b;
                delete e.getter;
                return e;
            }

            if(a.group !== S)
                throw new NerdamerValueError('Cannot complete operation. Incorrect LH value for ' + a);
            VARS[a.value] = b;
            return b;
        };
        this.function_assign = function (a, b) {
            var f = a.elements.pop();
            return setFunction(f, a.elements, b);
        };
        // Function to quickly convert bools to Symbols
        var bool2Symbol = function (x) {
            return new Symbol(x === true ? 1 : 0);
        };
        //check for equality
        this.eq = function (a, b) {
            return bool2Symbol(a.equals(b));
        };
        //checks for greater than
        this.gt = function (a, b) {
            return bool2Symbol(a.gt(b));
        };
        //checks for greater than equal
        this.gte = function (a, b) {
            return bool2Symbol(a.gte(b));
        };
        //checks for less than
        this.lt = function (a, b) {
            return bool2Symbol(a.lt(b));
        };
        //checks for less than equal
        this.lte = function (a, b) {
            return bool2Symbol(a.lte(b));
        };
        // wraps the factorial
        this.factorial = function (a) {
            return this.symfunction(FACTORIAL, [a]);
        };
        // wraps the double factorial
        this.dfactorial = function (a) {
            return this.symfunction(DOUBLEFACTORIAL, [a]);
        };
    }
    ;

    /* "STATIC" */
    // converts a number to a fraction.
    var Fraction = {
        /**
         * Converts a decimal to a fraction
         * @param {number} value
         * @param {object} opts
         * @returns {Array} - an array containing the denominator and the numerator
         */
        convert: function (value, opts) {
            var frac;
            if(value === 0) {
                frac = [0, 1];
            }
            else {
                if(value < 1e-6 || value > 1e20) {
                    var qc = this.quickConversion(Number(value));
                    if(qc[1] <= 1e20) {
                        var abs = Math.abs(value);
                        var sign = value / abs;
                        frac = this.fullConversion(abs.toFixed((qc[1] + '').length - 1));
                        frac[0] = frac[0] * sign;
                    }
                    else {
                        frac = qc;
                    }
                }
                else {
                    frac = this.fullConversion(value);
                }
            }
            return frac;
        },
        /**
         * If the fraction is too small or too large this gets called instead of fullConversion method
         * @param {number} dec
         * @returns {Array} - an array containing the denominator and the numerator
         */
        quickConversion: function (value) {
            var stripSign = function (s) {
                // Explicitely convert to a string
                if(typeof s !== 'string') {
                    s = s.toString();
                }

                var sign = '';

                // Remove and store the sign
                var start = s.charAt(0);
                if(start === '-') {
                    s = s.substr(1, s.length);
                    sign = '-';
                }
                else if(start === '+') {
                    // Just remove the plus sign
                    s = s.substr(1, s.length);
                }

                return {
                    sign: sign,
                    value: s
                };
            };


            function convert(value) {
                // Explicitely convert to a decimal
                if(Scientific.isScientific(value)) {
                    value = scientificToDecimal(value);
                }

                // Split the value into the sign and the value
                var nparts = stripSign(value);

                // Split it at the decimal. We'll refer to it as the coeffient parts
                var cparts = nparts.value.split('.');

                // Combine the entire number by removing leading zero and adding the decimal part
                // This would be teh same as moving the decimal point to the end
                var num;
                // We're dealing with integers
                if(cparts.length === 1) {
                    num = cparts[0];
                }
                else {
                    num = cparts[0] + cparts[1];
                }
                var n = cparts[1] ? cparts[1].length : 0;
                // Generate the padding for the zeros
                var den = `1${'0'.repeat(n)}`;

                if(num !== '0') {
                    num = num.replace(/^0+/, '');
                }
                return [nparts.sign + num, den];
            }

            return convert(value);
        },
        /**
         * Returns a good approximation of a fraction. This method gets called by convert
         * http://mathforum.org/library/drmath/view/61772.html
         * Decimal To Fraction Conversion - A Simpler Version
         * Dr Peterson
         * @param {number} dec
         * @returns {Array} - an array containing the denominator and the numerator
         */
        fullConversion: function (dec) {
            var done = false;
            // you can adjust the epsilon to a larger number if you don't need very high precision
            var n1 = 0, d1 = 1, n2 = 1, d2 = 0, n = 0, q = dec, epsilon = 1e-16;
            while(!done) {
                n++;
                if(n > 10000) {
                    done = true;
                }
                var a = Math.floor(q);
                var num = n1 + a * n2;
                var den = d1 + a * d2;
                var e = (q - a);
                if(e < epsilon) {
                    done = true;
                }
                q = 1 / e;
                n1 = n2;
                d1 = d2;
                n2 = num;
                d2 = den;
                if(Math.abs(num / den - dec) < epsilon || n > 30) {
                    done = true;
                }
            }
            return [num, den];
        }
    };
    //Depends on Fraction

    //The latex generator
    var LaTeX = {
        parser: (function () {
            // create a parser and strip it from everything except the items that you need
            var keep = ['classes', 'setOperator', 'getOperators', 'getBrackets', 'tokenize', 'toRPN', 'tree', 'units'];
            var parser = new Parser();
            for(var x in parser) {
                if(keep.indexOf(x) === -1)
                    delete parser[x];
            }
            // declare the operators
            parser.setOperator({
                precedence: 8,
                operator: '\\',
                action: 'slash',
                prefix: true,
                postfix: false,
                leftAssoc: true,
                operation: function (e) {
                    return e; // bypass the slash
                }
            });
            parser.setOperator({
                precedence: 8,
                operator: '\\,',
                action: 'slash_comma',
                prefix: true,
                postfix: false,
                leftAssoc: true,
                operation: function (e) {
                    return e; // bypass the slash
                }
            });
            // have braces not map to anything. We want them to be return as-is
            var brackets = parser.getBrackets();
            brackets['{'].maps_to = undefined;
            return parser;
        })(),
        space: '~',
        dot: ' \\cdot ',
        // grab a list of supported functions but remove the excluded ones found in exclFN

        latex: function (symbol, option) {
            // it might be an array
            if(symbol.clone) {
                symbol = symbol.clone(); // leave original as-is
            }
            if(symbol instanceof _.classes.Collection)
                symbol = symbol.elements;

            if(isArray(symbol)) {
                var LaTeXArray = [];
                for(var i = 0; i < symbol.length; i++) {
                    var sym = symbol[i];
                    //This way I can generate LaTeX on an array of strings.
                    if(!isSymbol(sym))
                        sym = _.parse(sym);
                    LaTeXArray.push(this.latex(sym, option));
                }
                return this.brackets(LaTeXArray.join(', '), 'square');
            }

            else if(isMatrix(symbol)) {
                var TeX = '\\begin{pmatrix}\n';
                for(var i = 0; i < symbol.elements.length; i++) {
                    var rowTeX = [],
                            e = symbol.elements[i];
                    for(var j = 0; j < e.length; j++) {
                        rowTeX.push(this.latex(e[j], option));
                    }
                    TeX += rowTeX.join(' & ');
                    if(i < symbol.elements.length - 1) {
                        TeX += '\\\\\n';
                    }
                }
                TeX += '\\end{pmatrix}';
                return TeX;
            }

            else if(isVector(symbol)) {
                var TeX = '\\left[';
                for(var i = 0; i < symbol.elements.length; i++) {
                    TeX += this.latex(symbol.elements[i], option) + ' ' + (i !== symbol.elements.length - 1 ? ',\\,' : '');
                }
                TeX += '\\right]';
                return TeX;
            }

            else if(isSet(symbol)) {
                var TeX = '\\{';
                for(var i = 0; i < symbol.elements.length; i++) {
                    TeX += this.latex(symbol.elements[i], option) + ' ' + (i !== symbol.elements.length - 1 ? ',\\,' : '');
                }
                TeX += '\\}';
                return TeX;
            }

            symbol = symbol.clone();

            var decimal = (option === 'decimal' || option === 'decimals'),
                    power = symbol.power,
                    invert = isNegative(power),
                    negative = symbol.multiplier.lessThan(0);

            if(symbol.group === P && decimal) {
                return String(symbol.multiplier.toDecimal() * Math.pow(symbol.value, symbol.power.toDecimal()));
            }
            else {
                symbol.multiplier = symbol.multiplier.abs();

                // if the user wants the result in decimal format then return it as such by placing it at the top part
                var m_array;

                if(decimal) {
                    var m = String(symbol.multiplier.toDecimal());
                    // if(String(m) === '1' && !decimal) m = '';
                    m_array = [m, ''];
                }
                else {
                    m_array = [symbol.multiplier.num, symbol.multiplier.den];
                }
                // get the value as a two part array
                var v_array = this.value(symbol, invert, option, negative),
                        p;
                // make it all positive since we know whether to push the power to the numerator or denominator already.
                if(invert)
                    power.negate();
                // the power is simple since it requires no additional formatting. We can get it to a
                // string right away. pass in true to neglect unit powers
                if(decimal) {
                    p = isSymbol(power) ? LaTeX.latex(power, option) : String(power.toDecimal());
                    if(String(p) === '1')
                        p = '';
                }
                // get the latex representation
                else if(isSymbol(power))
                    p = this.latex(power, option);
                // get it as a fraction
                else
                    p = this.formatFrac(power, true);
                // use this array to specify if the power is getting attached to the top or the bottom
                var p_array = ['', ''],
                        // stick it to the top or the bottom. If it's negative then the power gets placed on the bottom
                        index = invert ? 1 : 0;
                p_array[index] = p;

                // special case group P and decimal
                var retval = (negative ? '-' : '') + this.set(m_array, v_array, p_array, symbol.group === CB);

                return retval.replace(/\+\-/gi, '-');
            }

        },
        // greek mapping
        greek: {
            alpha: '\\alpha',
            beta: '\\beta',
            gamma: '\\gamma',
            delta: '\\delta',
            epsilon: '\\epsilon',
            zeta: '\\zeta',
            eta: '\\eta',
            theta: '\\theta',
            iota: '\\iota',
            kappa: '\\kappa',
            lambda: '\\lambda',
            mu: '\\mu',
            nu: '\\nu',
            xi: '\\xi',
            omnikron: '\\omnikron',
            pi: '\\pi',
            rho: '\\rho',
            sigma: '\\sigma',
            tau: '\\tau',
            upsilon: '\\upsilon',
            phi: '\\phi',
            chi: '\\chi',
            psi: '\\psi',
            omega: '\\omega',
            Gamma: '\\Gamma',
            Delta: '\\Delta',
            Epsilon: '\\Epsilon',
            Theta: '\\Theta',
            Lambda: '\\Lambda',
            Xi: '\\Xi',
            Pi: '\\Pi',
            Sigma: '\\Sigma',
            Phi: '\\Phi',
            Psi: '\\Psi',
            Omega: '\\Omega'
        },
        symbols: {
            arccos: '\\arccos',
            cos: '\\cos',
            csc: '\\csc',
            exp: '\\exp',
            ker: '\\ker',
            limsup: '\\limsup',
            min: '\\min',
            sinh: '\\sinh',
            arcsin: '\\arcsin',
            cosh: '\\cosh',
            deg: '\\deg',
            gcd: '\\gcd',
            lg: '\\lg',
            ln: '\\ln',
            Pr: '\\Pr',
            sqrt: '\\sqrt',
            sup: '\\sup',
            arctan: '\\arctan',
            cot: '\\cot',
            det: '\\det',
            hom: '\\hom',
            lim: '\\lim',
            log: '\\log',
            LN: '\\LN',
            sec: '\\sec',
            tan: '\\tan',
            arg: '\\arg',
            coth: '\\coth',
            dim: '\\dim',
            inf: '\\inf',
            liminf: '\\liminf',
            max: '\\max',
            sin: '\\sin',
            tanh: '\\tanh'
        },
        // get the raw value of the symbol as an array
        value: function (symbol, inverted, option, negative) {
            var group = symbol.group,
                    previousGroup = symbol.previousGroup,
                    v = ['', ''],
                    index = inverted ? 1 : 0;
            /*if(group === N) // do nothing since we want to return top & bottom blank; */
            if(symbol.isInfinity) {
                v[index] = '\\infty';
            }
            else if(group === S || group === P || previousGroup === S || previousGroup === P || previousGroup === N) {
                var value = this.formatSubscripts(symbol.value);
                if(value.replace)
                    value = value.replace(/(.+)_$/, '$1\\_');
                // split it so we can check for instances of alpha as well as alpha_b
                var t_varray = String(value).split('_');
                var greek = this.greek[t_varray[0]];
                if(greek) {
                    t_varray[0] = greek;
                    value = t_varray.join('_');
                }
                var symbol = this.symbols[t_varray[0]];
                if(symbol) {
                    t_varray[0] = symbol;
                    value = t_varray.join('_');
                }
                v[index] = value;
            }
            else if(group === FN || previousGroup === FN) {
                var name,
                        input = [],
                        fname = symbol.fname;
                // collect the arguments
                for(var i = 0; i < symbol.args.length; i++) {
                    var arg = symbol.args[i], item;
                    if(typeof arg === 'string')
                        item = arg;
                    else {
                        item = this.latex(arg, option);
                    }
                    input.push(item);
                }

                if(fname === SQRT) {
                    v[index] = '\\sqrt' + this.braces(input.join(','));
                }
                else if(fname === ABS) {
                    v[index] = this.brackets(input.join(','), 'abs');
                }
                else if(fname === PARENTHESIS) {
                    v[index] = this.brackets(input.join(','), 'parens');
                }
                else if(fname === 'limit') {
                    v[index] = ' \\lim\\limits_{' + input[1] + ' \\to ' + input[2] + '} ' + input[0];
                }
                else if(fname === 'integrate') {
                    v[index] = '\\int' + this.braces(input[0]) + this.braces('d' + input[1]);
                }
                else if(fname === 'defint') {
                    v[index] = '\\int\\limits_' + this.braces(input[1]) + '^' + this.braces(input[2]) + ' ' + input[0] + ' d' + input[3];
                }
                else if(fname === FACTORIAL || fname === DOUBLEFACTORIAL) {
                    var arg = symbol.args[0];
                    if(arg.power.equals(1) && (arg.isComposite() || arg.isCombination())) {
                        input[0] = this.brackets(input[0]);
                    }
                    v[index] = input[0] + (fname === FACTORIAL ? '!' : '!!');
                }
                else if(fname === 'floor') {
                    v[index] = '\\left \\lfloor' + this.braces(input[0]) + '\\right \\rfloor';
                }
                else if(fname === 'ceil') {
                    v[index] = '\\left \\lceil' + this.braces(input[0]) + '\\right \\rceil';
                }
                // capture log(a, b)
                else if(fname === Settings.LOG && input.length > 1) {
                    v[index] = '\\mathrm' + this.braces(Settings.LOG) + '_' + this.braces(input[1]) + this.brackets(input[0]);
                }
                // capture log(a, b)
                else if(fname === Settings.LOG10) {
                    v[index] = '\\mathrm' + this.braces(Settings.LOG) + '_' + this.braces(10) + this.brackets(input[0]);
                }
                else if(fname === 'sum') {
                    var a = input[0],
                            b = input[1],
                            c = input[2],
                            d = input[3];
                    v[index] = '\\sum\\limits_{' + this.braces(b) + '=' + this.braces(c) + '}^' + this.braces(d) + ' ' + this.braces(a) + '';
                }
                else if(fname === 'product') {
                    var a = input[0],
                            b = input[1],
                            c = input[2],
                            d = input[3];
                    v[index] = '\\prod\\limits_{' + this.braces(b) + '=' + this.braces(c) + '}^' + this.braces(d) + ' ' + this.braces(a) + '';
                }
                else if(fname === 'nthroot') {
                    v[index] = '\\sqrt[' + input[1] + ']' + this.braces(input[0]);
                }
                else if(fname === 'mod') {
                    v[index] = input[0] + ' \\bmod ' + input[1];
                }
                else if(fname === 'realpart') {
                    v[index] = '\\operatorname{Re}' + this.brackets(input[0]);
                }
                else if(fname === 'imagpart') {
                    v[index] = '\\operatorname{Im}' + this.brackets(input[0]);
                }
                else {
                    var name = fname !== '' ? '\\mathrm' + this.braces(fname.replace(/_/g, '\\_')) : '';
                    if(symbol.isConversion)
                        v[index] = name + this.brackets(input.join(''), 'parens');
                    else
                        v[index] = name + this.brackets(input.join(','), 'parens');
                }
            }
            else if(symbol.isComposite()) {
                var collected = symbol.collectSymbols().sort(
                        group === CP || previousGroup === CP ?
                        function (a, b) {
                            return b.group - a.group;
                        } :
                        function (a, b) {
                            var x = isSymbol(a.power) ? -1 : a.power;
                            var y = isSymbol(b.power) ? -1 : b.power;
                            return y - x;
                        }
                ),
                        symbols = [],
                        l = collected.length;
                for(var i = 0; i < l; i++) {
                    symbols.push(LaTeX.latex(collected[i], option));
                }
                var value = symbols.join('+');

                v[index] = !(symbol.isLinear() && symbol.multiplier.equals(1)) || negative ? this.brackets(value, 'parens') : value;
            }
            else if(group === CB || previousGroup === EX || previousGroup === CB) {
                if(group === CB)
                    symbol.distributeExponent();
                // This almost feels a little like cheating but I need to know if I should be wrapping the symbol
                // in brackets or not. We'll do this by checking the value of the numerator and then comparing it
                // to whether the symbol value is "simple" or not.
                var denominator = [],
                        numerator = [];
                // Generate a profile
                var den_map = [], num_map = [], num_c = 0, den_c = 0;
                var setBrackets = function (container, map, counter) {
                    if(counter > 1 && map.length > 0) {
                        var l = map.length;
                        for(var i = 0; i < l; i++) {
                            var idx = map[i], item = container[idx];
                            if(!(/^\\left\(.+\\right\)\^\{.+\}$/g.test(item) || /^\\left\(.+\\right\)$/g.test(item))) {
                                container[idx] = LaTeX.brackets(item, 'parens');
                            }
                        }
                    }
                    return container;
                };

                // Generate latex for each of them
                symbol.each(function (x) {
                    var isDenom = isNegative(x.power),
                            laTex;

                    if(isDenom) {
                        laTex = LaTeX.latex(x.invert(), option);
                        den_c++;
                        if(x.isComposite()) {
                            if(symbol.multiplier.den != 1 && Math.abs(x.power) == 1)
                                laTex = LaTeX.brackets(laTex, 'parens');
                            den_map.push(denominator.length); // make a note of where the composite was found
                        }

                        denominator.push(laTex);
                    }
                    else {
                        laTex = LaTeX.latex(x, option);
                        num_c++;
                        if(x.isComposite()) {
                            if(symbol.multiplier.num != 1 && Math.abs(x.power) == 1)
                                laTex = LaTeX.brackets(laTex, 'parens');
                            num_map.push(numerator.length);   // make a note of where the composite was found
                        }
                        numerator.push(laTex);
                    }
                });

                // Apply brackets
                setBrackets(numerator, num_map, num_c);
                v[0] = numerator.join(this.dot); // collapse the numerator into one string

                setBrackets(denominator, den_map, den_c);
                v[1] = denominator.join(this.dot);
            }

            return v;
        },
        set: function (m, v, p, combine_power) {
            var isBracketed = function (v) {
                return /^\\left\(.+\\right\)$/.test(v);
            };
            // format the power if it exists
            if(p)
                p = this.formatP(p);
            // group CB will have to be wrapped since the power applies to both it's numerator and denominator
            if(combine_power) {
                // POSSIBLE BUG: If powers for group CB format wrong, investigate this since I might have overlooked something
                // the assumption is that in every case the denonimator should be empty when dealing with CB. I can't think
                // of a case where this isn't true
                var tp = p[0];
                p[0] = ''; // temporarily make p blank
            }

            // merge v and p. Not that v MUST be first since the order matters
            v = this.merge(v, p);
            var mn = m[0], md = m[1], vn = v[0], vd = v[1];
            // filters
            // if the top has a variable but the numerator is one drop it
            if(vn && Number(mn) === 1)
                mn = '';
            // if denominator is 1 drop it always
            if(Number(md) === 1)
                md = '';
            // prepare the top portion but check that it's not already bracketed. If it is then leave out the cdot
            var top = this.join(mn, vn, !isBracketed(vn) ? this.dot : '');

            // prepare the bottom portion but check that it's not already bracketed. If it is then leave out the cdot
            var bottom = this.join(md, vd, !isBracketed(vd) ? this.dot : '');
            // format the power if it exists
            // make it a fraction if both top and bottom exists
            if(top && bottom) {
                var frac = this.frac(top, bottom);
                if(combine_power && tp)
                    frac = this.brackets(frac) + tp;
                return frac;
            }
            // otherwise only the top exists so return that
            else
                return top;
        },
        merge: function (a, b) {
            var r = [];
            for(var i = 0; i < 2; i++)
                r[i] = a[i] + b[i];
            return r;
        },
        // joins together two strings if both exist
        join: function (n, d, glue) {
            if(!n && !d)
                return '';
            if(n && !d)
                return n;
            if(d && !n)
                return d;
            return n + glue + d;
        },
        /**
         * Places subscripts in braces for proper formatting
         * @param {String} v
         * @returns {String}
         */
        formatSubscripts: function (v) {
            // Split it at the underscore
            var arr = v.toString().split('_');

            var name = '';

            // Loop over all entries except the first one
            while(arr.length > 1) {
                // Wrap all in braces except for the last one
                if(arr.length > 0) {
                    name = '_' + this.braces(arr.pop() + name);
                }
            }

            return arr[0] + name;
        },
        formatP: function (p_array) {
            for(var i = 0; i < 2; i++) {
                var p = p_array[i];
                if(p)
                    p_array[i] = '^' + this.braces(p);
            }
            return p_array;
        },
        /**
         * formats the fractions accordingly.
         * @param {Frac} f
         * @param {bool} is_pow
         */
        formatFrac: function (f, is_pow) {
            var n = f.num.toString(),
                    d = f.den.toString();
            // no need to have x^1
            if(is_pow && n === '1' && d === '1')
                return '';
            // no need to have x/1
            if(d === '1')
                return n;
            return this.frac(n, d);
        },
        frac: function (n, d) {
            return '\\frac' + this.braces(n) + this.braces(d);
        },
        braces: function (e) {
            return '{' + e + '}';
        },
        brackets: function (e, typ) {
            typ = typ || 'parens';
            var bracketTypes = {
                parens: ['(', ')'],
                square: ['[', ']'],
                brace: ['{', '}'],
                abs: ['|', '|'],
                angle: ['\\langle', '\\rangle']
            };
            var bracket = bracketTypes[typ];
            return '\\left' + bracket[0] + e + '\\right' + bracket[1];
        },
        /**
         * Removes extreneous tokens
         * @param {Tokens[]} tokens
         * @returns {Tokens[]}
         */
        filterTokens: function (tokens) {
            var filtered = [];

            // Copy over the type of the scope
            if(isArray(tokens)) {
                filtered.type = tokens.type;
            }

            // the items that need to be disposed
            var d = ['\\', 'left', 'right', 'big', 'Big', 'large', 'Large'];
            for(var i = 0, l = tokens.length; i < l; i++) {
                var token = tokens[i];
                var next_token = tokens[i + 1];
                if(token.value === '\\' && next_token.value === '\\') {
                    filtered.push(token);
                }
                else if(isArray(token)) {
                    filtered.push(LaTeX.filterTokens(token));
                }
                else if(d.indexOf(token.value) === -1) {
                    filtered.push(token);
                }
            }
            return filtered;
        },
        /*
         * Parses tokens from LaTeX string. Does not do any error checking
         * @param {Tokens[]} rpn
         * @returns {String}
         */
        parse: function (raw_tokens) {
            var i, l;
            var retval = '';
            var tokens = this.filterTokens(raw_tokens);
            var replace = {
                'cdot': '',
                'times': '',
                'infty': 'Infinity'
            };
            // get the next token
            var next = function (n) {
                return tokens[(typeof n === 'undefined' ? ++i : i += n)];
            };
            var parse_next = function () {
                return LaTeX.parse(next());
            };
            var get = function (token) {
                if(token in replace) {
                    return replace[token];
                }
                // A quirk with implicit multiplication forces us to check for *
                if(token === '*' && tokens[i + 1].value === '&') {
                    next(2); // skip this and the &
                    return ',';
                }

                if(token === '&') {
                    next();
                    return ','; // Skip the *
                }
                // If it's the end of a row, return the row separator
                if(token === '\\') {
                    return '],[';
                }
                return token;
            };

            // start parsing the tokens
            for(i = 0, l = tokens.length; i < l; i++) {
                var token = tokens[i];
                // fractions
                if(token.value === 'frac') {
                    // parse and wrap it in brackets
                    var n = parse_next();
                    var d = parse_next();
                    retval += n + '/' + d;
                }
                else if(token.value in LaTeX.symbols) {
                    if(token.value === SQRT && tokens[i + 1].type === 'vector' && tokens[i + 2].type === 'Set') {
                        var base = parse_next();
                        var expr = parse_next();
                        retval += (expr + '^' + inBrackets('1/' + base));
                    }
                    else {
                        retval += token.value + parse_next();
                    }
                }
                else if(token.value === 'int') {
                    var f = parse_next();
                    // skip the comma
                    i++;
                    // get the variable of integration
                    var dx = next().value;
                    dx = get(dx.substring(1, dx.length));
                    retval += 'integrate' + inBrackets(f + ',' + dx);
                }
                else if(token.value === 'int_') {
                    var l = parse_next(); // lower
                    i++; // skip the ^
                    var u = next().value; // upper
                    // if it is in brackets
                    if (u === undefined) {
                        i--;
                        var u = parse_next();
                    }
                    var f = parse_next(); // function
                    
                    // get the variable of integration
                    var dx = next().value;
                    // skip the comma
                    if (dx === ',') {
                        var dx = next().value;
                    }
                    // if 'd', skip
                    if (dx === 'differentialD') {
                        // skip the *
                        i++;
                        var dx = next().value;
                    }
                    if (dx === 'mathrm') {
                        // skip the mathrm{d}
                        i++;
                        var dx = next().value;
                    }
                    retval += 'defint' + inBrackets(f + ',' + l + ',' + u + ',' + dx);
                }
                else if(token.value && token.value.startsWith('int_')) {
                    // var l = parse_next(); // lower
                    var l = token.value.replace('int_', '')
                    console.log('uppernow')
                    i++; // skip the ^
                    var u = next().value; // upper
                    // if it is in brackets
                    if (u === undefined) {
                        i--;
                        var u = parse_next();
                    }
                    var f = parse_next(); // function
                    
                    // get the variable of integration
                    var dx = next().value;
                    // skip the comma
                    if (dx === ',') {
                        var dx = next().value;
                    }
                    // if 'd', skip
                    if (dx === 'differentialD') {
                        // skip the *
                        i++;
                        var dx = next().value;
                    }
                    if (dx === 'mathrm') {
                        // skip the mathrm{d}
                        i++;
                        var dx = next().value;
                    }
                    retval += 'defint' + inBrackets(f + ',' + l + ',' + u + ',' + dx);
                }
                else if(token.value === 'mathrm') {
                    var f = tokens[++i][0].value;
                    retval += f + parse_next();
                }
                // sum and product
                else if(token.value === 'sum_' || token.value === 'prod_') {
                    var fn = token.value === 'sum_' ? 'sum' : 'product';
                    var nxt = next();
                    i++; // skip the caret
                    var end = parse_next();
                    var f = parse_next();
                    retval += fn + inBrackets([f, get(nxt[0]), get(nxt[2]), get(end)].join(','));
                }
                else if(token.value === 'lim_') {
                    var nxt = next();
                    retval += 'limit' + inBrackets([parse_next(), get(nxt[0]), get(nxt[2])].join(','));
                }
                else if(token.value === 'begin') {
                    var nxt = next();
                    if(Array.isArray(nxt)) {
                        var v = nxt[0].value;
                        if(v === 'matrix') {
                            // Start a matrix
                            retval += 'matrix([';
                        }
                    }
                }
                else if(token.value === 'end') {
                    var nxt = next();
                    if(Array.isArray(nxt)) {
                        var v = nxt[0].value;
                        if(v === 'matrix') {
                            // End a matrix
                            retval += '])';
                        }
                    }
                }
                else {
                    if(Array.isArray(token)) {
                        retval += get(LaTeX.parse(token));
                    }
                    else {
                        retval += get(token.value.toString());
                    }
                }
            }

            return inBrackets(retval);
        }
    };
//Vector =======================================================================
    function Vector(v) {
        if(isVector(v))
            this.elements = v.items.slice(0);
        else if(isArray(v))
            this.elements = v.slice(0);
        else
            this.elements = [].slice.call(arguments);
    }
    /*
     * Generates a pre-filled array
     * @param {type} n
     * @param {type} val
     * @returns {unresolved}
     */
    Vector.arrayPrefill = function (n, val) {
        var a = [];
        val = val || 0;
        for(var i = 0; i < n; i++)
            a[i] = val;
        return a;
    };
    /**
     * Generate a vector from and array
     * @param {type} a
     * @returns {unresolved}
     */
    Vector.fromArray = function (a) {
        var v = new Vector();
        v.elements = a;
        return v;
    };

    /**
     * Convert a Set to a Vector
     * @param {Set} set
     * @returns {Vector}
     */
    Vector.fromSet = function (set) {
        return Vector.fromArray(set.elements);
    };

    //Ported from Sylvester.js
    Vector.prototype = {
        custom: true,
        // Returns element i of the vector
        e: function (i) {
            return (i < 1 || i > this.elements.length) ? null : this.elements[i - 1];
        },

        set: function (i, val) {
            if(!isSymbol(val))
                val = new Symbol(val);
            this.elements[i] = val;
        },

        // Returns the number of elements the vector has
        dimensions: function () {
            return this.elements.length;
        },

        // Returns the modulus ('length') of the vector
        modulus: function () {
            return block('SAFE', function () {
                return _.pow((this.dot(this.clone())), new Symbol(0.5));
            }, undefined, this);
        },

        // Returns true iff the vector is equal to the argument
        eql: function (vector) {
            var n = this.elements.length;
            var V = vector.elements || vector;
            if(n !== V.length) {
                return false;
            }
            do {
                if(Math.abs(_.subtract(this.elements[n - 1], V[n - 1]).valueOf()) > PRECISION) {
                    return false;
                }
            }
            while(--n);
            return true;
        },

        // Returns a clone of the vector
        clone: function () {
            var V = new Vector(),
                    l = this.elements.length;
            for(var i = 0; i < l; i++) {
                //Rule: all items within the vector must have a clone method.
                V.elements.push(this.elements[i].clone());
            }
            if(this.getter) {
                V.getter = this.getter.clone();
            }
            return V;
        },

        // Maps the vector to another vector according to the given function
        map: function (fn) {
            var elements = [];
            this.each(function (x, i) {
                elements.push(fn(x, i));
            });

            return new Vector(elements);
        },

        // Calls the iterator for each element of the vector in turn
        each: function (fn) {
            var n = this.elements.length, k = n, i;
            do {
                i = k - n;
                fn(this.elements[i], i + 1);
            }
            while(--n);
        },

        // Returns a new vector created by normalizing the receiver
        toUnitVector: function () {
            return block('SAFE', function () {
                var r = this.modulus();
                if(r.valueOf() === 0) {
                    return this.clone();
                }
                return this.map(function (x) {
                    return _.divide(x, r);
                });
            }, undefined, this);
        },

        // Returns the angle between the vector and the argument (also a vector)
        angleFrom: function (vector) {
            return block('SAFE', function () {
                var V = vector.elements || vector;
                var n = this.elements.length;
                if(n !== V.length) {
                    return null;
                }
                var dot = new Symbol(0), mod1 = new Symbol(0), mod2 = new Symbol(0);
                // Work things out in parallel to save time
                this.each(function (x, i) {
                    dot = _.add(dot, _.multiply(x, V[i - 1]));
                    mod1 = _.add(mod1, _.multiply(x, x));// will not conflict in safe block
                    mod2 = _.add(mod2, _.multiply(V[i - 1], V[i - 1]));// will not conflict in safe block
                });
                mod1 = _.pow(mod1, new Symbol(0.5));
                mod2 = _.pow(mod2, new Symbol(0.5));
                var product = _.multiply(mod1, mod2);
                if(product.valueOf() === 0) {
                    return null;
                }
                var theta = _.divide(dot, product);
                var theta_val = theta.valueOf();
                if(theta_val < -1) {
                    theta = -1;
                }
                if(theta_val > 1) {
                    theta = 1;
                }
                return new Symbol(Math.acos(theta));
            }, undefined, this);
        },

        // Returns true iff the vector is parallel to the argument
        isParallelTo: function (vector) {
            var angle = this.angleFrom(vector).valueOf();
            return (angle === null) ? null : (angle <= PRECISION);
        },

        // Returns true iff the vector is antiparallel to the argument
        isAntiparallelTo: function (vector) {
            var angle = this.angleFrom(vector).valueOf();
            return (angle === null) ? null : (Math.abs(angle - Math.PI) <= PRECISION);
        },

        // Returns true iff the vector is perpendicular to the argument
        isPerpendicularTo: function (vector) {
            var dot = this.dot(vector);
            return (dot === null) ? null : (Math.abs(dot) <= PRECISION);
        },

        // Returns the result of adding the argument to the vector
        add: function (vector) {
            return block('SAFE', function () {
                var V = vector.elements || vector;
                if(this.elements.length !== V.length) {
                    return null;
                }
                return this.map(function (x, i) {
                    return _.add(x, V[i - 1]);
                });
            }, undefined, this);
        },

        // Returns the result of subtracting the argument from the vector
        subtract: function (vector) {
            return block('SAFE', function () {
                var V = vector.elements || vector;
                if(this.elements.length !== V.length) {
                    return null;
                }
                return this.map(function (x, i) {
                    return _.subtract(x, V[i - 1]);
                });
            }, undefined, this);
        },

        // Returns the result of multiplying the elements of the vector by the argument
        multiply: function (k) {
            return this.map(function (x) {
                return x.clone() * k.clone();
            });
        },

        x: function (k) {
            return this.multiply(k);
        },

        // Returns the scalar product of the vector with the argument
        // Both vectors must have equal dimensionality
        dot: function (vector) {
            return block('SAFE', function () {
                var V = vector.elements || vector;
                var product = new Symbol(0), n = this.elements.length;
                if(n !== V.length) {
                    return null;
                }
                do {
                    product = _.add(product, _.multiply(this.elements[n - 1], V[n - 1]));
                }
                while(--n);
                return product;
            }, undefined, this);
        },

        // Returns the vector product of the vector with the argument
        // Both vectors must have dimensionality 3
        cross: function (vector) {
            var B = vector.elements || vector;
            if(this.elements.length !== 3 || B.length !== 3) {
                return null;
            }
            var A = this.elements;
            return block('SAFE', function () {
                return new Vector([
                    _.subtract(_.multiply(A[1], B[2]), _.multiply(A[2], B[1])),
                    _.subtract(_.multiply(A[2], B[0]), _.multiply(A[0], B[2])),
                    _.subtract(_.multiply(A[0], B[1]), _.multiply(A[1], B[0]))
                ]);
            }, undefined, this);
        },

        // Returns the (absolute) largest element of the vector
        max: function () {
            var m = 0, n = this.elements.length, k = n, i;
            do {
                i = k - n;
                if(Math.abs(this.elements[i].valueOf()) > Math.abs(m.valueOf())) {
                    m = this.elements[i];
                }
            }
            while(--n);
            return m;
        },
        magnitude: function () {
            var magnitude = new Symbol(0);
            this.each(function (e) {
                magnitude = _.add(magnitude, _.pow(e, new Symbol(2)));
            });
            return _.sqrt(magnitude);
        },
        // Returns the index of the first match found
        indexOf: function (x) {
            var index = null, n = this.elements.length, k = n, i;
            do {
                i = k - n;
                if(index === null && this.elements[i].valueOf() === x.valueOf()) {
                    index = i + 1;
                }
            }
            while(--n);
            return index;
        },
        text: function (x) {
            return text(this);
        },
        toString: function () {
            return this.text();
        },
        latex: function (option) {
            var tex = [];
            for(var i = 0; i < this.elements.length; i++) {
                tex.push(LaTeX.latex.call(LaTeX, this.elements[i], option));
            }
            return '[' + tex.join(', ') + ']';
        }
    };

//Matrix =======================================================================
    function Matrix() {
        var m = arguments,
                l = m.length, i, el = [];
        if(isMatrix(m)) { // if it's a matrix then make a clone
            for(i = 0; i < l; i++) {
                el.push(m[i].slice(0));
            }
        }
        else {
            var row, lw, rl;
            for(i = 0; i < l; i++) {
                row = m[i];
                if(isVector(row))
                    row = row.elements;
                if(!isArray(row))
                    row = [row];
                rl = row.length;
                if(lw && lw !== rl)
                    err('Unable to create Matrix. Row dimensions do not match!');
                el.push(row);
                lw = rl;
            }
        }
        this.elements = el;
    }
    Matrix.identity = function (n) {
        var m = new Matrix();
        for(var i = 0; i < n; i++) {
            m.elements.push([]);
            for(var j = 0; j < n; j++) {
                m.set(i, j, i === j ? new Symbol(1) : new Symbol(0));
            }
        }
        return m;
    };
    Matrix.fromArray = function (arr) {
        function F(args) {
            return Matrix.apply(this, args);
        }
        F.prototype = Matrix.prototype;

        return new F(arr);
    };
    Matrix.zeroMatrix = function (rows, cols) {
        var m = new Matrix();
        for(var i = 0; i < rows; i++) {
            m.elements.push(Vector.arrayPrefill(cols, new Symbol(0)));
        }
        return m;
    };
    Matrix.prototype = {
        // needs be true to let the parser know not to try to cast it to a symbol
        custom: true,
        get: function (row, column) {
            if(!this.elements[row])
                return undefined;
            return this.elements[row][column];
        },
        map: function (f, raw_values) {
            var M = new Matrix();
            this.each(function (e, i, j) {
                M.set(i, j, f.call(M, e), raw_values);
            });
            return M;
        },
        set: function (row, column, value, raw) {
            if(!this.elements[row])
                this.elements[row] = [];
            this.elements[row][column] = raw ? value : (isSymbol(value) ? value : new Symbol(value));
        },
        cols: function () {
            return this.elements[0].length;
        },
        rows: function () {
            return this.elements.length;
        },
        row: function (n) {
            if(!n || n > this.cols())
                return [];
            return this.elements[n - 1];
        },
        col: function (n) {
            var nr = this.rows(),
                    col = [];
            if(n > this.cols() || !n)
                return col;
            for(var i = 0; i < nr; i++) {
                col.push(this.elements[i][n - 1]);
            }
            return col;
        },
        eachElement: function (fn) {
            var nr = this.rows(),
                    nc = this.cols(), i, j;
            for(i = 0; i < nr; i++) {
                for(j = 0; j < nc; j++) {
                    fn.call(this, this.elements[i][j], i, j);
                }
            }
        },
        // ported from Sylvester.js
        determinant: function () {
            if(!this.isSquare()) {
                return null;
            }
            var M = this.toRightTriangular();
            var det = M.elements[0][0], n = M.elements.length - 1, k = n, i;
            do {
                i = k - n + 1;
                det = _.multiply(det, M.elements[i][i]);
            }
            while(--n);
            return det;
        },
        isSquare: function () {
            return this.elements.length === this.elements[0].length;
        },
        isSingular: function () {
            return this.isSquare() && this.determinant() === 0;
        },
        augment: function (m) {
            var r = this.rows(), rr = m.rows();
            if(r !== rr)
                err("Cannot augment matrix. Rows don't match.");
            for(var i = 0; i < r; i++) {
                this.elements[i] = this.elements[i].concat(m.elements[i]);
            }

            return this;
        },
        clone: function () {
            var r = this.rows(), c = this.cols(),
                    m = new Matrix();
            for(var i = 0; i < r; i++) {
                m.elements[i] = [];
                for(var j = 0; j < c; j++) {
                    var symbol = this.elements[i][j];
                    m.elements[i][j] = isSymbol(symbol) ? symbol.clone() : symbol;
                }
            }
            return m;
        },
        // ported from Sylvester.js
        invert: function () {
            if(!this.isSquare())
                err('Matrix is not square!');
            return block('SAFE', function () {
                var ni = this.elements.length, ki = ni, i, j;
                var imatrix = Matrix.identity(ni);
                var M = this.augment(imatrix).toRightTriangular();
                var np, kp = M.elements[0].length, p, els, divisor;
                var inverse_elements = [], new_element;
                // Matrix is non-singular so there will be no zeros on the diagonal
                // Cycle through rows from last to first
                do {
                    i = ni - 1;
                    // First, normalise diagonal elements to 1
                    els = [];
                    np = kp;
                    inverse_elements[i] = [];
                    divisor = M.elements[i][i];
                    do {
                        p = kp - np;
                        new_element = _.divide(M.elements[i][p], divisor.clone());
                        els.push(new_element);
                        // Shuffle of the current row of the right hand side into the results
                        // array as it will not be modified by later runs through this loop
                        if(p >= ki) {
                            inverse_elements[i].push(new_element);
                        }
                    }
                    while(--np);
                    M.elements[i] = els;
                    // Then, subtract this row from those above it to
                    // give the identity matrix on the left hand side
                    for(j = 0; j < i; j++) {
                        els = [];
                        np = kp;
                        do {
                            p = kp - np;
                            els.push(_.subtract(M.elements[j][p].clone(), _.multiply(M.elements[i][p].clone(), M.elements[j][i].clone())));
                        }
                        while(--np);
                        M.elements[j] = els;
                    }
                }
                while(--ni);
                return Matrix.fromArray(inverse_elements);
            }, undefined, this);
        },
        // ported from Sylvester.js
        toRightTriangular: function () {
            return block('SAFE', function () {
                var M = this.clone(), els, fel, nel,
                        n = this.elements.length, k = n, i, np, kp = this.elements[0].length, p;
                do {
                    i = k - n;
                    fel = M.elements[i][i];
                    if(fel.valueOf() === 0) {
                        for(var j = i + 1; j < k; j++) {
                            nel = M.elements[j][i];
                            if(nel && nel.valueOf() !== 0) {
                                els = [];
                                np = kp;
                                do {
                                    p = kp - np;
                                    els.push(_.add(M.elements[i][p].clone(), M.elements[j][p].clone()));
                                }
                                while(--np);
                                M.elements[i] = els;
                                break;
                            }
                        }
                    }
                    var fel = M.elements[i][i];
                    if(fel.valueOf() !== 0) {
                        for(j = i + 1; j < k; j++) {
                            var multiplier = _.divide(M.elements[j][i].clone(), M.elements[i][i].clone());
                            els = [];
                            np = kp;
                            do {
                                p = kp - np;
                                // Elements with column numbers up to an including the number
                                // of the row that we're subtracting can safely be set straight to
                                // zero, since that's the point of this routine and it avoids having
                                // to loop over and correct rounding errors later
                                els.push(p <= i ? new Symbol(0) :
                                        _.subtract(M.elements[j][p].clone(), _.multiply(M.elements[i][p].clone(), multiplier.clone())));
                            }
                            while(--np);
                            M.elements[j] = els;
                        }
                    }
                }
                while(--n);

                return M;
            }, undefined, this);
        },
        transpose: function () {
            var rows = this.elements.length, cols = this.elements[0].length;
            var M = new Matrix(), ni = cols, i, nj, j;

            do {
                i = cols - ni;
                M.elements[i] = [];
                nj = rows;
                do {
                    j = rows - nj;
                    M.elements[i][j] = this.elements[j][i].clone();
                }
                while(--nj);
            }
            while(--ni);
            return M;
        },
        // Returns true if the matrix can multiply the argument from the left
        canMultiplyFromLeft: function (matrix) {
            var l = isMatrix(matrix) ? matrix.elements.length : matrix.length;
            // this.columns should equal matrix.rows
            return (this.elements[0].length === l);
        },
        sameSize: function (matrix) {
            return this.rows() === matrix.rows() && this.cols() === matrix.cols();
        },
        multiply: function (matrix) {
            return block('SAFE', function () {
                var M = matrix.elements || matrix;
                if(!this.canMultiplyFromLeft(M)) {
                    if(this.sameSize(matrix)) {
                        var MM = new Matrix();
                        var rows = this.rows();
                        for(var i = 0; i < rows; i++) {
                            var e = _.multiply(new Vector(this.elements[i]), new Vector(matrix.elements[i]));
                            MM.elements[i] = e.elements;
                        }
                        return MM;
                    }
                    return null;
                }
                var ni = this.elements.length, ki = ni, i, nj, kj = M[0].length, j;
                var cols = this.elements[0].length, elements = [], sum, nc, c;
                do {
                    i = ki - ni;
                    elements[i] = [];
                    nj = kj;
                    do {
                        j = kj - nj;
                        sum = new Symbol(0);
                        nc = cols;
                        do {
                            c = cols - nc;
                            sum = _.add(sum, _.multiply(this.elements[i][c], M[c][j]));
                        }
                        while(--nc);
                        elements[i][j] = sum;
                    }
                    while(--nj);
                }
                while(--ni);
                return Matrix.fromArray(elements);
            }, undefined, this);
        },
        add: function (matrix, callback) {
            var M = new Matrix();
            if(this.sameSize(matrix)) {
                this.eachElement(function (e, i, j) {
                    var result = _.add(e.clone(), matrix.elements[i][j].clone());
                    if(callback) {
                        result = callback.call(M, result, e, matrix.elements[i][j]);
                    }
                    M.set(i, j, result);
                });
            }
            return M;
        },
        subtract: function (matrix, callback) {
            var M = new Matrix();
            if(this.sameSize(matrix)) {
                this.eachElement(function (e, i, j) {
                    var result = _.subtract(e.clone(), matrix.elements[i][j].clone());
                    if(callback) {
                        result = callback.call(M, result, e, matrix.elements[i][j]);
                    }
                    M.set(i, j, result);
                });
            }
            return M;
        },
        negate: function () {
            this.each(function (e) {
                return e.negate();
            });
            return this;
        },
        toVector: function () {
            if(this.rows() === 1 || this.cols() === 1) {
                var v = new Vector();
                v.elements = this.elements;
                return v;
            }
            return this;
        },
        toString: function (newline, to_decimal) {
            var l = this.rows(),
                    s = [];
            newline = newline === undefined ? '\n' : newline;
            for(var i = 0; i < l; i++) {
                s.push('[' + this.elements[i].map(function (x) {
                    var v = to_decimal ? x.multiplier.toDecimal() : x.toString();
                    return x !== undefined ? v : '';
                }).join(',') + ']');
            }
            return 'matrix' + inBrackets(s.join(','));
        },
        text: function () {
            return 'matrix(' + this.elements.toString('') + ')';
        },
        latex: function (option) {
            var cols = this.cols(), elements = this.elements;
            return format('\\begin{vmatrix}{0}\\end{vmatrix}', function () {
                var tex = [];
                for(var row in elements) {
                    var row_tex = [];
                    for(var i = 0; i < cols; i++) {
                        row_tex.push(LaTeX.latex.call(LaTeX, elements[row][i], option));
                    }
                    tex.push(row_tex.join(' & '));
                }
                return tex.join(' \\cr ');
            });
        }
    };
    // aliases
    Matrix.prototype.each = Matrix.prototype.eachElement;


    function Set(set) {
        this.elements = [];
        // if the first object isn't an array, convert it to one.
        if(!isVector(set))
            set = Vector.fromArray(arguments);

        if(set) {
            var elements = set.elements;
            for(var i = 0, l = elements.length; i < l; i++) {
                this.add(elements[i]);
            }
        }
    }

    Set.fromArray = function (arr) {
        function F(args) {
            return Set.apply(this, args);
        }
        F.prototype = Set.prototype;

        return new F(arr);
    };

    Set.prototype = {
        add: function (x) {
            if(!this.contains(x))
                this.elements.push(x.clone());
        },
        contains: function (x) {
            for(var i = 0; i < this.elements.length; i++) {
                var e = this.elements[i];
                if(x.equals(e))
                    return true;
            }
            return false;
        },
        each: function (f) {
            var elements = this.elements;
            var set = new Set();
            for(var i = 0, l = elements.length; i < l; i++) {
                var e = elements[i];
                f.call(this, e, set, i);
            }
            return set;
        },
        clone: function () {
            var set = new Set();
            this.each(function (e) {
                set.add(e.clone());
            });
            return set;
        },
        union: function (set) {
            var _union = this.clone();
            set.each(function (e) {
                _union.add(e);
            });

            return _union;
        },
        difference: function (set) {
            var diff = this.clone();
            set.each(function (e) {
                diff.remove(e);
            });
            return diff;
        },
        remove: function (element) {
            for(var i = 0, l = this.elements.length; i < l; i++) {
                var e = this.elements[i];
                if(e.equals(element)) {
                    remove(this.elements, i);
                    return true;
                }
            }
            return false;
        },
        intersection: function (set) {
            var _intersection = new Set();
            var A = this;
            set.each(function (e) {
                if(A.contains(e)) {
                    _intersection.add(e);
                }
                ;
            });

            return _intersection;
        },
        intersects: function (set) {
            return this.intersection(set).elements.length > 0;
        },
        is_subset: function (set) {
            var elements = set.elements;
            for(var i = 0, l = elements.length; i < l; i++) {
                if(!this.contains(elements[i])) {
                    return false;
                }
            }
            return true;
        },
        toString: function () {
            return '{' + this.elements.join(',') + '}';
        }
    };

//build ========================================================================
    var Build = {
        dependencies: {
            _rename: {
                'Math2.factorial': 'factorial'
            },
            factorial: {
                'Math2.gamma': Math2.gamma
            },
            gamma_incomplete: {
                'Math2.factorial': Math2.factorial
            },
            Li: {
                'Math2.Ei': Math2.Ei,
                'Math2.bigLog': Math2.bigLog,
                'Frac': Frac
            },
            Ci: {
                'Math2.factorial': Math2.factorial
            },
            Ei: {
                'Math2.factorial': Math2.factorial
            },
            Si: {
                'Math2.factorial': Math2.factorial
            },
            Shi: {
                'Math2.factorial': Math2.factorial
            },
            Chi: {
                'isInt': isInt,
                'nround': nround,
                'Math2.num_integrate': Math2.num_integrate
            },
            factor: {
                'Math2.ifactor': Math2.ifactor,
                'Symbol': Symbol
            },
            num_integrate: {
                'Math2.simpson': Math2.simpson,
                'nround': nround
            },
            fib: {
                'even': even
            }
        },
        /* Some functions need to be made numeric safe. Build checks if there's a
         * reformat option and calls that instead when compiling the function string.
         */
        reformat: {
            // this simply extends the build function
            diff: function (symbol, deps) {
                var v = symbol.args[1].toString();
                var f = 'var f = ' + Build.build(symbol.args[0].toString(), [v]) + ';';
                deps[1] += 'var diff = ' + Math2.diff.toString() + ';';
                deps[1] += f;

                return ['diff(f)(' + v + ')', deps];
            }
        },
        getProperName: function (f) {
            var map = {
                continued_fraction: 'continuedFraction'
            };
            return map[f] || f;
        },
        // assumes that dependences are at max 2 levels
        compileDependencies: function (f, deps) {
            // grab the predefined dependiences
            var dependencies = Build.dependencies[f];

            // the dependency string
            var dep_string = deps && deps[1] ? deps[1] : '';

            // the functions to be replaced
            var replacements = deps && deps[0] ? deps[0] : {};

            // loop through them and add them to the list
            for(var x in dependencies) {
                if(typeof dependencies[x] === 'object')
                    continue; // skip object
                var components = x.split('.'); //Math.f becomes f
                // if the function isn't part of an object then reference the function itself
                dep_string += 'var ' + (components.length > 1 ? components[1] : components[0]) + '=' + dependencies[x] + ';';
                replacements[x] = components.pop();
            }

            return [replacements, dep_string];
        },
        getArgsDeps: function (symbol, dependencies) {
            var args = symbol.args;
            for(var i = 0; i < args.length; i++) {
                symbol.args[i].each(function (x) {
                    if(x.group === FN)
                        dependencies = Build.compileDependencies(x.fname, dependencies);
                });
            }
            return dependencies;
        },
        build: function (symbol, arg_array) {
            symbol = block('PARSE2NUMBER', function () {
                return _.parse(symbol);
            }, true);
            var args = variables(symbol);
            var supplements = [];
            var dependencies = [];
            var ftext = function (symbol, xports) {
                //Fix for #545 - Parentheses confuse build.
                if(symbol.fname === '') {
                    symbol = Symbol.unwrapPARENS(symbol);
                }
                xports = xports || [];
                var c = [],
                        group = symbol.group,
                        prefix = '';

                var ftext_complex = function (group) {
                    var d = group === CB ? '*' : '+',
                            cc = [];

                    for(var x in symbol.symbols) {
                        var sym = symbol.symbols[x],
                                ft = ftext(sym, xports)[0];
                        // wrap it in brackets if it's group PL or CP
                        if(sym.isComposite())
                            ft = inBrackets(ft);
                        cc.push(ft);
                    }
                    var retval = cc.join(d);
                    retval = retval && !symbol.multiplier.equals(1) ? inBrackets(retval) : retval;
                    return retval;
                },
                        ftext_function = function (bn) {
                            var retval;
                            if(bn in Math)
                                retval = 'Math.' + bn;
                            else {
                                bn = Build.getProperName(bn);
                                if(supplements.indexOf(bn) === -1) { // make sure you're not adding the function twice
                                    //Math2 functions aren't part of the standard javascript
                                    //Math library and must be exported.
                                    xports.push('var ' + bn + ' = ' + Math2[bn].toString() + '; ');
                                    supplements.push(bn);
                                }
                                retval = bn;
                            }
                            retval = retval + inBrackets(symbol.args.map(function (x) {
                                return ftext(x, xports)[0];
                            }).join(','));

                            return retval;
                        };

                // the multiplier
                if(group === N)
                    c.push(symbol.multiplier.toDecimal());
                else if(symbol.multiplier.equals(-1))
                    prefix = '-';
                else if(!symbol.multiplier.equals(1))
                    c.push(symbol.multiplier.toDecimal());
                // the value
                var value;

                if(group === S || group === P)
                    value = symbol.value;
                else if(group === FN) {
                    dependencies = Build.compileDependencies(symbol.fname, dependencies);
                    dependencies = Build.getArgsDeps(symbol, dependencies);
                    if(Build.reformat[symbol.fname]) {
                        var components = Build.reformat[symbol.fname](symbol, dependencies);
                        dependencies = components[1];
                        value = components[0];
                    }
                    else {
                        value = ftext_function(symbol.fname);
                    }

                }
                else if(group === EX) {
                    var pg = symbol.previousGroup;
                    if(pg === N || pg === S)
                        value = symbol.value;
                    else if(pg === FN) {
                        value = ftext_function(symbol.fname);
                        dependencies = Build.compileDependencies(symbol.fname, dependencies);
                        dependencies = Build.getArgsDeps(symbol, dependencies);
                    }
                    else
                        value = ftext_complex(symbol.previousGroup);
                }
                else {
                    value = ftext_complex(symbol.group);
                }

                if(symbol.group !== N && !symbol.power.equals(1)) {
                    var pow = ftext(_.parse(symbol.power));
                    xports.push(pow[1]);
                    value = 'Math.pow' + inBrackets(value + ',' + pow[0]);
                }

                if(value)
                    c.push(prefix + value);

                return [c.join('*'), xports.join('').replace(/\n+\s+/g, ' ')];
            };
            if(arg_array) {
                // Fix for issue #546
                // Disable argument checking since it's a bit presumptuous.
                // Consider f(x) = 5; If I explicitely pass in an argument array contain x 
                // this check will fail and complain since the function doesn't contain x.
                /*
                 for (var i = 0; i < args.length; i++) {
                 var arg = args[i];
                 if (arg_array.indexOf(arg) === -1)
                 err(arg + ' not found in argument array');
                 }
                 */
                args = arg_array;
            }

            var f_array = ftext(symbol);

            // make all the substitutions;
            for(var x in dependencies[0]) {
                var alias = dependencies[0][x];
                f_array[1] = f_array[1].replace(x, alias);
                dependencies[1] = dependencies[1].replace(x, alias);
            }

            var f = new Function(args, (dependencies[1] || '') + f_array[1] + ' return ' + f_array[0] + ';');

            return f;
        }
    };


//finalize =====================================================================
    /* FINALIZE */
    (function () {
        reserveNames(_.CONSTANTS);
        reserveNames(_.functions);
        _.initConstants();
        //bug fix for error but needs to be revisited
        if(!_.error)
            _.error = err;

        //Store the log and log10 functions
        Settings.LOG_FNS = {
            log: _.functions['log'],
            log10: _.functions['log10']
        };

    })();

    /* END FINALIZE */

//Core =========================================================================
    var Utils = {
        allSame: allSame,
        allNumeric: allNumeric,
        arguments2Array: arguments2Array,
        arrayAddSlices: arrayAddSlices,
        arrayClone: arrayClone,
        arrayMax: arrayMax,
        arrayMin: arrayMin,
        arrayEqual: arrayEqual,
        arrayUnique: arrayUnique,
        arrayGetVariables: arrayGetVariables,
        arraySum: arraySum,
        block: block,
        build: Build.build,
        clearU: clearU,
        comboSort: comboSort,
        compare: compare,
        convertToVector: convertToVector,
        customError: customError,
        customType: customType,
        decompose_fn: decompose_fn,
        each: each,
        evaluate: evaluate,
        even: even,
        evenFraction: evenFraction,
        fillHoles: fillHoles,
        firstObject: firstObject,
        format: format,
        generatePrimes: generatePrimes,
        getCoeffs: getCoeffs,
        getU: getU,
        importFunctions: importFunctions,
        inBrackets: inBrackets,
        isArray: isArray,
        isExpression: isExpression,
        isFraction: isFraction,
        isInt: isInt,
        isMatrix: isMatrix,
        isNegative: isNegative,
        isNumericSymbol: isNumericSymbol,
        isPrime: isPrime,
        isReserved: isReserved,
        isSymbol: isSymbol,
        isVariableSymbol: isVariableSymbol,
        isVector: isVector,
        keys: keys,
        knownVariable: knownVariable,
        nroots: nroots,
        remove: remove,
        reserveNames: reserveNames,
        range: range,
        round: nround,
        sameSign: sameSign,
        scientificToDecimal: scientificToDecimal,
        separate: separate,
        stringReplace: stringReplace,
        text: text,
        validateName: validateName,
        variables: variables,
        warn: warn
    };

    //This contains all the parts of nerdamer and enables nerdamer's internal functions
    //to be used.
    var C = {
        groups: Groups,
        Symbol: Symbol,
        Expression: Expression,
        Frac: Frac,
        Vector: Vector,
        Matrix: Matrix,
        Parser: Parser,
        Scientific: Scientific,
        Fraction: Fraction,
        Math2: Math2,
        LaTeX: LaTeX,
        Utils: Utils,
        PARSER: _,
        PARENTHESIS: PARENTHESIS,
        Settings: Settings,
        err: err,
        bigInt: bigInt,
        bigDec: bigDec,
        exceptions: exceptions
    };

//libExports ===================================================================
    /**
     *
     * @param {String} expression the expression to be evaluated
     * @param {Object} subs the object containing the variable values
     * @param {Integer} location a specific location in the equation list to
     * insert the evaluated expression
     * @param {String} option additional options
     * @returns {Expression}
     */
    var libExports = function (expression, subs, option, location) {
        // Initiate the numer flag
        var numer = false;

        // Is the user declaring a function?
        var fndec = /^([a-z_][a-z\d\_]*)\(([a-z_,\s]*)\):=(.+)$/gi.exec(expression);
        if(fndec)
            return nerdamer.setFunction(fndec[1], fndec[2].split(','), fndec[3]);

        // var variable, fn, args;
        // Convert any expression passed in to a string
        if(expression instanceof Expression)
            expression = expression.toString();

        // Convert it to an array for simplicity
        if(!isArray(option)) {
            option = typeof option === 'undefined' ? [] : [option];
        }

        option.forEach(function (o) {
            // Turn on the numer flag if requested
            if(o === 'numer') {
                numer = true;
                return;
            }
            // Wrap it in a function if requested. This only holds true for
            // functions that take a single argument which is the expression
            var f = _.functions[option];
            // If there's a function and it takes a single argument, then wrap
            // the expression in it
            if(f && f[1] === 1) {
                expression = `${o}(${expression})`;
            }
        });

        var e = block('PARSE2NUMBER', function () {
            return _.parse(expression, subs);
        }, numer || Settings.PARSE2NUMBER);

        if(location) {
            EXPRESSIONS[location - 1] = e;
        }
        else {
            EXPRESSIONS.push(e);
        }

        return new Expression(e);
    };
    /**
     * Converts expression into rpn form
     * @param {String} expression
     * @returns {Token[]}
     */
    libExports.rpn = function (expression) {
        return _.tokenize(_.toRPN(expression));
    };

    /**
     * Generates LaTeX from expression string
     * @param {String} e
     * @param {object} opt
     * @returns {String}
     */
    libExports.convertToLaTeX = function (e, opt) {
        return _.toTeX(e, opt);
    };

    /**
     * Converts latex to text - Very very very basic at the moment
     * @param {String} e
     * @returns {String}
     */
    libExports.convertFromLaTeX = function (e) {
        var txt = LaTeX.parse(_.tokenize(e));
        return new Expression(_.parse(txt));
    };

    /**
     * Get the version of nerdamer or a loaded add-on
     * @param {String} add_on - The add-on being checked
     * @returns {String} returns the version of nerdamer
     */
    libExports.version = function (add_on) {
        if(add_on) {
            try {
                return C[add_on].version;
            }
            catch(e) {
                return "No module named " + add_on + " found!";
            }
        }
        return version;
    };

    /**
     * Get nerdamer generated warnings
     * @returns {String[]}
     */
    libExports.getWarnings = function () {
        return WARNINGS;
    };

    /**
     *
     * @param {String} constant The name of the constant to be set
     * @param {mixed} value The value of the constant
     * @returns {Object} Returns the nerdamer object
     */
    libExports.setConstant = function (constant, value) {
        validateName(constant);
        if(!isReserved(constant)) {
            //fix for issue #127
            if(value === 'delete' || value === '') {
                delete _.CONSTANTS[constant];
            }
            else {
                if(isNaN(value))
                    throw new NerdamerTypeError('Constant must be a number!');
                _.CONSTANTS[constant] = value;
            }
        }
        return this;
    };

    /**
     * Returns the value of a previously set constant
     * @param {type} constant
     * @returns {String}
     */
    libExports.getConstant = function (constant) {
        return String(_.constant[constant]);
    };

    /**
     *
     * @param {String} name The name of the function
     * @param {Array} params_array A list containing the parameter name of the functions
     * @param {String} body The body of the function
     * @returns {Boolean} returns true if succeeded and falls on fail
     * @example nerdamer.setFunction('f',['x'], 'x^2+2');
     */
    libExports.setFunction = setFunction;

    /**
     *
     * @returns {C} Exports the nerdamer core functions and objects
     */
    libExports.getCore = function () {
        return C;
    };

    libExports.getExpression = libExports.getEquation = Expression.getExpression;

    /**
     *
     * @param {Boolean} asArray The returned names are returned as an array if this is set to true;
     * @returns {String|Array}
     */
    libExports.reserved = function (asArray) {
        if(asArray) {
            return RESERVED;
        }
        return RESERVED.join(', ');
    };

    /**
     *
     * @param {Integer} equation_number the number of the equation to clear.
     * If 'all' is supplied then all equations are cleared
     * @param {Boolean} keep_EXPRESSIONS_fixed use true if you don't want to keep EXPRESSIONS length fixed
     * @returns {Object} Returns the nerdamer object
     */
    libExports.clear = function (equation_number, keep_EXPRESSIONS_fixed) {
        if(equation_number === 'all') {
            EXPRESSIONS = [];
        }
        else if(equation_number === 'last') {
            EXPRESSIONS.pop();
        }
        else if(equation_number === 'first') {
            EXPRESSIONS.shift();
        }
        else {
            var index = !equation_number ? EXPRESSIONS.length : equation_number - 1;
            keep_EXPRESSIONS_fixed === true ? EXPRESSIONS[index] = undefined : remove(EXPRESSIONS, index);
        }
        return this;
    };

    /**
     * Alias for nerdamer.clear('all')
     */
    libExports.flush = function () {
        this.clear('all');
        return this;
    };

    /**
     *
     * @param {Boolean} asObject
     * @param {Boolean} asLaTeX
     * @param {String|String[]} option
     * @returns {Array}
     */
    libExports.expressions = function (asObject, asLaTeX, option) {
        var result = asObject ? {} : [];
        for(var i = 0; i < EXPRESSIONS.length; i++) {
            var eq = asLaTeX ? LaTeX.latex(EXPRESSIONS[i], option) : text(EXPRESSIONS[i], option);
            asObject ? result[i + 1] = eq : result.push(eq);
        }
        return result;
    };

    //the method for registering modules
    libExports.register = function (obj) {
        var core = this.getCore();

        if(isArray(obj)) {
            for(var i = 0; i < obj.length; i++) {
                if(obj)
                    this.register(obj[i]);
            }
        }
        else if(obj && Settings.exclude.indexOf(obj.name) === -1) {
            //make sure all the dependencies are available
            if(obj.dependencies) {
                for(var i = 0; i < obj.dependencies.length; i++)
                    if(!core[obj.dependencies[i]])
                        throw new Error(format('{0} requires {1} to be loaded!', obj.name, obj.dependencies[i]));
            }
            //if no parent object is provided then the function does not have an address and cannot be called directly
            var parent_obj = obj.parent,
                    fn = obj.build.call(core); //call constructor to get function
            if(parent_obj) {
                if(!core[parent_obj])
                    core[obj.parent] = {};

                var ref_obj = parent_obj === 'nerdamer' ? this : core[parent_obj];
                //attach the function to the core
                ref_obj[obj.name] = fn;
            }
            if(obj.visible)
                _.functions[obj.name] = [fn, obj.numargs]; //make the function available

        }
    };

    /**
     * @param {String} name variable name
     * @returns {boolean} validates if the profided string is a valid variable name
     */
    libExports.validateName = validateName;

    /**
     * @param {String} varname variable name
     * @returns {boolean} validates if the profided string is a valid variable name
     */
    libExports.validVarName = function (varname) {
        try {
            validateName(varname);
            return RESERVED.indexOf(varname) === -1;
        }
        catch(e) {
            return false;
        }
    };

    /**
     *
     * @returns {Array} Array of functions currently supported by nerdamer
     */
    libExports.supported = function () {
        return keys(_.functions);
    };

    /**
     *
     * @returns {Number} The number equations/expressions currently loaded
     */
    libExports.numEquations = libExports.numExpressions = function () {
        return EXPRESSIONS.length;
    };
    /* END EXPORTS */

    /**
     *
     * @param {String} v variable to be set
     * @param {String} val value of variable. This can be a variable expression or number
     * @returns {Object} Returns the nerdamer object
     */
    libExports.setVar = function (v, val) {
        validateName(v);
        //check if it's not already a constant
        if(v in _.CONSTANTS)
            err('Cannot set value for constant ' + v);
        if(val === 'delete' || val === '')
            delete VARS[v];
        else {
            VARS[v] = isSymbol(val) ? val : _.parse(val);
        }
        return this;
    };

    /**
     * Returns the value of a set variable
     * @param {type} v
     * @returns {varies}
     */
    libExports.getVar = function (v) {
        return VARS[v];
    };
    /**
     * Clear the variables from the VARS object
     * @returns {Object} Returns the nerdamer object
     */
    libExports.clearVars = function () {
        VARS = {};
        return this;
    };

    /**
     *
     * @param {Function} loader
     * @returns {nerdamer}
     */
    libExports.load = function (loader) {
        loader.call(this);
        return this;
    };

    /**
     * @param {String} output - output format. Can be 'object' (just returns the VARS object), 'text' or 'latex'. Default: 'text'
     * @param {String|String[]} option
     * @returns {Object} Returns an object with the variables
     */
    libExports.getVars = function (output, option) {
        output = output || 'text';
        var variables = {};
        if(output === 'object')
            variables = VARS;
        else {
            for(var v in VARS) {
                if(output === 'latex') {
                    variables[v] = VARS[v].latex(option);
                }
                else if(output === 'text') {
                    variables[v] = VARS[v].text(option);
                }
            }
        }
        return variables;
    };

    /**
     * Set the value of a setting
     * @param {String} setting The setting to be changed
     * @param {boolean} value
     */
    libExports.set = function (setting, value) {
        //current options:
        //PARSE2NUMBER, suppress_errors
        if(typeof setting === 'object')
            for(var x in setting) {
                libExports.set(x, setting[x]);
            }

        var disallowed = ['SAFE'];
        if(disallowed.indexOf(setting) !== -1)
            err('Cannot modify setting: ' + setting);

        if(setting === 'PRECISION') {
            bigDec.set({precision: value});
            Settings.PRECISION = value;

            // Avoid that nerdamer puts out garbage after 21 decimal place
            if(value > 21) {
                this.set('USE_BIG', true);
            }
        }
        else if(setting === 'USE_LN' && value === true) {
            //set log as LN
            Settings.LOG = 'LN';
            //set log10 as log
            Settings.LOG10 = 'log';
            //point the functions in the right direction
            _.functions['log'] = Settings.LOG_FNS.log10; //log is now log10
            //the log10 function must be explicitly set
            _.functions['log'][0] = function (x) {
                if(x.isConstant())
                    return new Symbol(Math.log10(x));
                return _.symfunction(Settings.LOG10, [x]);
            };
            _.functions['LN'] = Settings.LOG_FNS.log; //LN is now log

            //remove log10
            delete _.functions['log10'];
        }
        else
            Settings[setting] = value;
    };

    /**
     * Get the value of a setting
     * @param {type} setting
     * @returns {undefined}
     */
    libExports.get = function (setting) {
        return Settings[setting];
    };

    /**
     * This functions makes internal functions available externally
     * @param {bool} override Override the functions when calling updateAPI if it exists
     */
    libExports.updateAPI = function (override) {
        //Map internal functions to external ones
        var linker = function (fname) {
            return function () {
                var args = [].slice.call(arguments);
                for(var i = 0; i < args.length; i++)
                    args[i] = _.parse(args[i]);
                return new Expression(block('PARSE2NUMBER', function () {
                    return _.callfunction(fname, args);
                }));
            };
        };
        //perform the mapping
        for(var x in _.functions)
            if(!(x in libExports) || override)
                libExports[x] = linker(x);
    };

    libExports.replaceFunction = function (name, fn, num_args) {
        var existing = _.functions[name];
        var new_num_args = typeof num_args === 'undefined' ? existing[1] : num_args;
        _.functions[name] = [fn.call(undefined, existing[0], C), new_num_args];
    };

    libExports.setOperator = function (operator, shift) {
        _.setOperator(operator, shift);
    };

    libExports.getOperator = function (operator) {
        return _.getOperator(operator);
    };

    libExports.aliasOperator = function (operator, withOperator) {
        _.aliasOperator(operator, withOperator);
    };

    libExports.tree = function (expression) {
        return _.tree(_.toRPN(_.tokenize(expression)));
    };

    libExports.htmlTree = function (expression, indent) {
        var tree = this.tree(expression);

        return '<div class="tree">\n' +
                '    <ul>\n' +
                '        <li>\n' +
                tree.toHTML(3, indent) + '\n' +
                '        </li>\n' +
                '    </ul>\n' +
                '</div>';
    };

    libExports.addPeeker = function (name, f) {
        if(_.peekers[name])
            _.peekers[name].push(f);
    };

    libExports.removePeeker = function (name, f) {
        remove(_.peekers[name], f);
    };

    libExports.parse = function (e) {
        return String(e).split(';').map(function (x) {
            return _.parse(x);
        });
    };

    libExports.updateAPI();

    return libExports; //Done
//imports ======================================================================
})({
    //https://github.com/peterolson/BigInteger.js
    bigInt: (function(){
        var bigInt=function(undefined){"use strict";var BASE=1e7,LOG_BASE=7,MAX_INT=9007199254740992,MAX_INT_ARR=smallToArray(MAX_INT),LOG_MAX_INT=Math.log(MAX_INT);function Integer(v,radix){if(typeof v==="undefined")return Integer[0];if(typeof radix!=="undefined")return+radix===10?parseValue(v):parseBase(v,radix);return parseValue(v)}function BigInteger(value,sign){this.value=value;this.sign=sign;this.isSmall=false}BigInteger.prototype=Object.create(Integer.prototype);function SmallInteger(value){this.value=value;this.sign=value<0;this.isSmall=true}SmallInteger.prototype=Object.create(Integer.prototype);function isPrecise(n){return-MAX_INT<n&&n<MAX_INT}function smallToArray(n){if(n<1e7)return[n];if(n<1e14)return[n%1e7,Math.floor(n/1e7)];return[n%1e7,Math.floor(n/1e7)%1e7,Math.floor(n/1e14)]}function arrayToSmall(arr){trim(arr);var length=arr.length;if(length<4&&compareAbs(arr,MAX_INT_ARR)<0){switch(length){case 0:return 0;case 1:return arr[0];case 2:return arr[0]+arr[1]*BASE;default:return arr[0]+(arr[1]+arr[2]*BASE)*BASE}}return arr}function trim(v){var i=v.length;while(v[--i]===0);v.length=i+1}function createArray(length){var x=new Array(length);var i=-1;while(++i<length){x[i]=0}return x}function truncate(n){if(n>0)return Math.floor(n);return Math.ceil(n)}function add(a,b){var l_a=a.length,l_b=b.length,r=new Array(l_a),carry=0,base=BASE,sum,i;for(i=0;i<l_b;i++){sum=a[i]+b[i]+carry;carry=sum>=base?1:0;r[i]=sum-carry*base}while(i<l_a){sum=a[i]+carry;carry=sum===base?1:0;r[i++]=sum-carry*base}if(carry>0)r.push(carry);return r}function addAny(a,b){if(a.length>=b.length)return add(a,b);return add(b,a)}function addSmall(a,carry){var l=a.length,r=new Array(l),base=BASE,sum,i;for(i=0;i<l;i++){sum=a[i]-base+carry;carry=Math.floor(sum/base);r[i]=sum-carry*base;carry+=1}while(carry>0){r[i++]=carry%base;carry=Math.floor(carry/base)}return r}BigInteger.prototype.add=function(v){var n=parseValue(v);if(this.sign!==n.sign){return this.subtract(n.negate())}var a=this.value,b=n.value;if(n.isSmall){return new BigInteger(addSmall(a,Math.abs(b)),this.sign)}return new BigInteger(addAny(a,b),this.sign)};BigInteger.prototype.plus=BigInteger.prototype.add;SmallInteger.prototype.add=function(v){var n=parseValue(v);var a=this.value;if(a<0!==n.sign){return this.subtract(n.negate())}var b=n.value;if(n.isSmall){if(isPrecise(a+b))return new SmallInteger(a+b);b=smallToArray(Math.abs(b))}return new BigInteger(addSmall(b,Math.abs(a)),a<0)};SmallInteger.prototype.plus=SmallInteger.prototype.add;function subtract(a,b){var a_l=a.length,b_l=b.length,r=new Array(a_l),borrow=0,base=BASE,i,difference;for(i=0;i<b_l;i++){difference=a[i]-borrow-b[i];if(difference<0){difference+=base;borrow=1}else borrow=0;r[i]=difference}for(i=b_l;i<a_l;i++){difference=a[i]-borrow;if(difference<0)difference+=base;else{r[i++]=difference;break}r[i]=difference}for(;i<a_l;i++){r[i]=a[i]}trim(r);return r}function subtractAny(a,b,sign){var value;if(compareAbs(a,b)>=0){value=subtract(a,b)}else{value=subtract(b,a);sign=!sign}value=arrayToSmall(value);if(typeof value==="number"){if(sign)value=-value;return new SmallInteger(value)}return new BigInteger(value,sign)}function subtractSmall(a,b,sign){var l=a.length,r=new Array(l),carry=-b,base=BASE,i,difference;for(i=0;i<l;i++){difference=a[i]+carry;carry=Math.floor(difference/base);difference%=base;r[i]=difference<0?difference+base:difference}r=arrayToSmall(r);if(typeof r==="number"){if(sign)r=-r;return new SmallInteger(r)}return new BigInteger(r,sign)}BigInteger.prototype.subtract=function(v){var n=parseValue(v);if(this.sign!==n.sign){return this.add(n.negate())}var a=this.value,b=n.value;if(n.isSmall)return subtractSmall(a,Math.abs(b),this.sign);return subtractAny(a,b,this.sign)};BigInteger.prototype.minus=BigInteger.prototype.subtract;SmallInteger.prototype.subtract=function(v){var n=parseValue(v);var a=this.value;if(a<0!==n.sign){return this.add(n.negate())}var b=n.value;if(n.isSmall){return new SmallInteger(a-b)}return subtractSmall(b,Math.abs(a),a>=0)};SmallInteger.prototype.minus=SmallInteger.prototype.subtract;BigInteger.prototype.negate=function(){return new BigInteger(this.value,!this.sign)};SmallInteger.prototype.negate=function(){var sign=this.sign;var small=new SmallInteger(-this.value);small.sign=!sign;return small};BigInteger.prototype.abs=function(){return new BigInteger(this.value,false)};SmallInteger.prototype.abs=function(){return new SmallInteger(Math.abs(this.value))};function multiplyLong(a,b){var a_l=a.length,b_l=b.length,l=a_l+b_l,r=createArray(l),base=BASE,product,carry,i,a_i,b_j;for(i=0;i<a_l;++i){a_i=a[i];for(var j=0;j<b_l;++j){b_j=b[j];product=a_i*b_j+r[i+j];carry=Math.floor(product/base);r[i+j]=product-carry*base;r[i+j+1]+=carry}}trim(r);return r}function multiplySmall(a,b){var l=a.length,r=new Array(l),base=BASE,carry=0,product,i;for(i=0;i<l;i++){product=a[i]*b+carry;carry=Math.floor(product/base);r[i]=product-carry*base}while(carry>0){r[i++]=carry%base;carry=Math.floor(carry/base)}return r}function shiftLeft(x,n){var r=[];while(n-- >0)r.push(0);return r.concat(x)}function multiplyKaratsuba(x,y){var n=Math.max(x.length,y.length);if(n<=30)return multiplyLong(x,y);n=Math.ceil(n/2);var b=x.slice(n),a=x.slice(0,n),d=y.slice(n),c=y.slice(0,n);var ac=multiplyKaratsuba(a,c),bd=multiplyKaratsuba(b,d),abcd=multiplyKaratsuba(addAny(a,b),addAny(c,d));var product=addAny(addAny(ac,shiftLeft(subtract(subtract(abcd,ac),bd),n)),shiftLeft(bd,2*n));trim(product);return product}function useKaratsuba(l1,l2){return-.012*l1-.012*l2+15e-6*l1*l2>0}BigInteger.prototype.multiply=function(v){var n=parseValue(v),a=this.value,b=n.value,sign=this.sign!==n.sign,abs;if(n.isSmall){if(b===0)return Integer[0];if(b===1)return this;if(b===-1)return this.negate();abs=Math.abs(b);if(abs<BASE){return new BigInteger(multiplySmall(a,abs),sign)}b=smallToArray(abs)}if(useKaratsuba(a.length,b.length))return new BigInteger(multiplyKaratsuba(a,b),sign);return new BigInteger(multiplyLong(a,b),sign)};BigInteger.prototype.times=BigInteger.prototype.multiply;function multiplySmallAndArray(a,b,sign){if(a<BASE){return new BigInteger(multiplySmall(b,a),sign)}return new BigInteger(multiplyLong(b,smallToArray(a)),sign)}SmallInteger.prototype._multiplyBySmall=function(a){if(isPrecise(a.value*this.value)){return new SmallInteger(a.value*this.value)}return multiplySmallAndArray(Math.abs(a.value),smallToArray(Math.abs(this.value)),this.sign!==a.sign)};BigInteger.prototype._multiplyBySmall=function(a){if(a.value===0)return Integer[0];if(a.value===1)return this;if(a.value===-1)return this.negate();return multiplySmallAndArray(Math.abs(a.value),this.value,this.sign!==a.sign)};SmallInteger.prototype.multiply=function(v){return parseValue(v)._multiplyBySmall(this)};SmallInteger.prototype.times=SmallInteger.prototype.multiply;function square(a){var l=a.length,r=createArray(l+l),base=BASE,product,carry,i,a_i,a_j;for(i=0;i<l;i++){a_i=a[i];carry=0-a_i*a_i;for(var j=i;j<l;j++){a_j=a[j];product=2*(a_i*a_j)+r[i+j]+carry;carry=Math.floor(product/base);r[i+j]=product-carry*base}r[i+l]=carry}trim(r);return r}BigInteger.prototype.square=function(){return new BigInteger(square(this.value),false)};SmallInteger.prototype.square=function(){var value=this.value*this.value;if(isPrecise(value))return new SmallInteger(value);return new BigInteger(square(smallToArray(Math.abs(this.value))),false)};function divMod1(a,b){var a_l=a.length,b_l=b.length,base=BASE,result=createArray(b.length),divisorMostSignificantDigit=b[b_l-1],lambda=Math.ceil(base/(2*divisorMostSignificantDigit)),remainder=multiplySmall(a,lambda),divisor=multiplySmall(b,lambda),quotientDigit,shift,carry,borrow,i,l,q;if(remainder.length<=a_l)remainder.push(0);divisor.push(0);divisorMostSignificantDigit=divisor[b_l-1];for(shift=a_l-b_l;shift>=0;shift--){quotientDigit=base-1;if(remainder[shift+b_l]!==divisorMostSignificantDigit){quotientDigit=Math.floor((remainder[shift+b_l]*base+remainder[shift+b_l-1])/divisorMostSignificantDigit)}carry=0;borrow=0;l=divisor.length;for(i=0;i<l;i++){carry+=quotientDigit*divisor[i];q=Math.floor(carry/base);borrow+=remainder[shift+i]-(carry-q*base);carry=q;if(borrow<0){remainder[shift+i]=borrow+base;borrow=-1}else{remainder[shift+i]=borrow;borrow=0}}while(borrow!==0){quotientDigit-=1;carry=0;for(i=0;i<l;i++){carry+=remainder[shift+i]-base+divisor[i];if(carry<0){remainder[shift+i]=carry+base;carry=0}else{remainder[shift+i]=carry;carry=1}}borrow+=carry}result[shift]=quotientDigit}remainder=divModSmall(remainder,lambda)[0];return[arrayToSmall(result),arrayToSmall(remainder)]}function divMod2(a,b){var a_l=a.length,b_l=b.length,result=[],part=[],base=BASE,guess,xlen,highx,highy,check;while(a_l){part.unshift(a[--a_l]);trim(part);if(compareAbs(part,b)<0){result.push(0);continue}xlen=part.length;highx=part[xlen-1]*base+part[xlen-2];highy=b[b_l-1]*base+b[b_l-2];if(xlen>b_l){highx=(highx+1)*base}guess=Math.ceil(highx/highy);do{check=multiplySmall(b,guess);if(compareAbs(check,part)<=0)break;guess--}while(guess);result.push(guess);part=subtract(part,check)}result.reverse();return[arrayToSmall(result),arrayToSmall(part)]}function divModSmall(value,lambda){var length=value.length,quotient=createArray(length),base=BASE,i,q,remainder,divisor;remainder=0;for(i=length-1;i>=0;--i){divisor=remainder*base+value[i];q=truncate(divisor/lambda);remainder=divisor-q*lambda;quotient[i]=q|0}return[quotient,remainder|0]}function divModAny(self,v){var value,n=parseValue(v);var a=self.value,b=n.value;var quotient;if(b===0)throw new Error("Cannot divide by zero");if(self.isSmall){if(n.isSmall){return[new SmallInteger(truncate(a/b)),new SmallInteger(a%b)]}return[Integer[0],self]}if(n.isSmall){if(b===1)return[self,Integer[0]];if(b==-1)return[self.negate(),Integer[0]];var abs=Math.abs(b);if(abs<BASE){value=divModSmall(a,abs);quotient=arrayToSmall(value[0]);var remainder=value[1];if(self.sign)remainder=-remainder;if(typeof quotient==="number"){if(self.sign!==n.sign)quotient=-quotient;return[new SmallInteger(quotient),new SmallInteger(remainder)]}return[new BigInteger(quotient,self.sign!==n.sign),new SmallInteger(remainder)]}b=smallToArray(abs)}var comparison=compareAbs(a,b);if(comparison===-1)return[Integer[0],self];if(comparison===0)return[Integer[self.sign===n.sign?1:-1],Integer[0]];if(a.length+b.length<=200)value=divMod1(a,b);else value=divMod2(a,b);quotient=value[0];var qSign=self.sign!==n.sign,mod=value[1],mSign=self.sign;if(typeof quotient==="number"){if(qSign)quotient=-quotient;quotient=new SmallInteger(quotient)}else quotient=new BigInteger(quotient,qSign);if(typeof mod==="number"){if(mSign)mod=-mod;mod=new SmallInteger(mod)}else mod=new BigInteger(mod,mSign);return[quotient,mod]}BigInteger.prototype.divmod=function(v){var result=divModAny(this,v);return{quotient:result[0],remainder:result[1]}};SmallInteger.prototype.divmod=BigInteger.prototype.divmod;BigInteger.prototype.divide=function(v){return divModAny(this,v)[0]};SmallInteger.prototype.over=SmallInteger.prototype.divide=BigInteger.prototype.over=BigInteger.prototype.divide;BigInteger.prototype.mod=function(v){return divModAny(this,v)[1]};SmallInteger.prototype.remainder=SmallInteger.prototype.mod=BigInteger.prototype.remainder=BigInteger.prototype.mod;BigInteger.prototype.pow=function(v){var n=parseValue(v),a=this.value,b=n.value,value,x,y;if(b===0)return Integer[1];if(a===0)return Integer[0];if(a===1)return Integer[1];if(a===-1)return n.isEven()?Integer[1]:Integer[-1];if(n.sign){return Integer[0]}if(!n.isSmall)throw new Error("The exponent "+n.toString()+" is too large.");if(this.isSmall){if(isPrecise(value=Math.pow(a,b)))return new SmallInteger(truncate(value))}x=this;y=Integer[1];while(true){if(b&1===1){y=y.times(x);--b}if(b===0)break;b/=2;x=x.square()}return y};SmallInteger.prototype.pow=BigInteger.prototype.pow;BigInteger.prototype.modPow=function(exp,mod){exp=parseValue(exp);mod=parseValue(mod);if(mod.isZero())throw new Error("Cannot take modPow with modulus 0");var r=Integer[1],base=this.mod(mod);while(exp.isPositive()){if(base.isZero())return Integer[0];if(exp.isOdd())r=r.multiply(base).mod(mod);exp=exp.divide(2);base=base.square().mod(mod)}return r};SmallInteger.prototype.modPow=BigInteger.prototype.modPow;function compareAbs(a,b){if(a.length!==b.length){return a.length>b.length?1:-1}for(var i=a.length-1;i>=0;i--){if(a[i]!==b[i])return a[i]>b[i]?1:-1}return 0}BigInteger.prototype.compareAbs=function(v){var n=parseValue(v),a=this.value,b=n.value;if(n.isSmall)return 1;return compareAbs(a,b)};SmallInteger.prototype.compareAbs=function(v){var n=parseValue(v),a=Math.abs(this.value),b=n.value;if(n.isSmall){b=Math.abs(b);return a===b?0:a>b?1:-1}return-1};BigInteger.prototype.compare=function(v){if(v===Infinity){return-1}if(v===-Infinity){return 1}var n=parseValue(v),a=this.value,b=n.value;if(this.sign!==n.sign){return n.sign?1:-1}if(n.isSmall){return this.sign?-1:1}return compareAbs(a,b)*(this.sign?-1:1)};BigInteger.prototype.compareTo=BigInteger.prototype.compare;SmallInteger.prototype.compare=function(v){if(v===Infinity){return-1}if(v===-Infinity){return 1}var n=parseValue(v),a=this.value,b=n.value;if(n.isSmall){return a==b?0:a>b?1:-1}if(a<0!==n.sign){return a<0?-1:1}return a<0?1:-1};SmallInteger.prototype.compareTo=SmallInteger.prototype.compare;BigInteger.prototype.equals=function(v){return this.compare(v)===0};SmallInteger.prototype.eq=SmallInteger.prototype.equals=BigInteger.prototype.eq=BigInteger.prototype.equals;BigInteger.prototype.notEquals=function(v){return this.compare(v)!==0};SmallInteger.prototype.neq=SmallInteger.prototype.notEquals=BigInteger.prototype.neq=BigInteger.prototype.notEquals;BigInteger.prototype.greater=function(v){return this.compare(v)>0};SmallInteger.prototype.gt=SmallInteger.prototype.greater=BigInteger.prototype.gt=BigInteger.prototype.greater;BigInteger.prototype.lesser=function(v){return this.compare(v)<0};SmallInteger.prototype.lt=SmallInteger.prototype.lesser=BigInteger.prototype.lt=BigInteger.prototype.lesser;BigInteger.prototype.greaterOrEquals=function(v){return this.compare(v)>=0};SmallInteger.prototype.geq=SmallInteger.prototype.greaterOrEquals=BigInteger.prototype.geq=BigInteger.prototype.greaterOrEquals;BigInteger.prototype.lesserOrEquals=function(v){return this.compare(v)<=0};SmallInteger.prototype.leq=SmallInteger.prototype.lesserOrEquals=BigInteger.prototype.leq=BigInteger.prototype.lesserOrEquals;BigInteger.prototype.isEven=function(){return(this.value[0]&1)===0};SmallInteger.prototype.isEven=function(){return(this.value&1)===0};BigInteger.prototype.isOdd=function(){return(this.value[0]&1)===1};SmallInteger.prototype.isOdd=function(){return(this.value&1)===1};BigInteger.prototype.isPositive=function(){return!this.sign};SmallInteger.prototype.isPositive=function(){return this.value>0};BigInteger.prototype.isNegative=function(){return this.sign};SmallInteger.prototype.isNegative=function(){return this.value<0};BigInteger.prototype.isUnit=function(){return false};SmallInteger.prototype.isUnit=function(){return Math.abs(this.value)===1};BigInteger.prototype.isZero=function(){return false};SmallInteger.prototype.isZero=function(){return this.value===0};BigInteger.prototype.isDivisibleBy=function(v){var n=parseValue(v);var value=n.value;if(value===0)return false;if(value===1)return true;if(value===2)return this.isEven();return this.mod(n).equals(Integer[0])};SmallInteger.prototype.isDivisibleBy=BigInteger.prototype.isDivisibleBy;function isBasicPrime(v){var n=v.abs();if(n.isUnit())return false;if(n.equals(2)||n.equals(3)||n.equals(5))return true;if(n.isEven()||n.isDivisibleBy(3)||n.isDivisibleBy(5))return false;if(n.lesser(49))return true}function millerRabinTest(n,a){var nPrev=n.prev(),b=nPrev,r=0,d,t,i,x;while(b.isEven())b=b.divide(2),r++;next:for(i=0;i<a.length;i++){if(n.lesser(a[i]))continue;x=bigInt(a[i]).modPow(b,n);if(x.equals(Integer[1])||x.equals(nPrev))continue;for(d=r-1;d!=0;d--){x=x.square().mod(n);if(x.isUnit())return false;if(x.equals(nPrev))continue next}return false}return true}BigInteger.prototype.isPrime=function(strict){var isPrime=isBasicPrime(this);if(isPrime!==undefined)return isPrime;var n=this.abs();var bits=n.bitLength();if(bits<=64)return millerRabinTest(n,[2,325,9375,28178,450775,9780504,1795265022]);var logN=Math.log(2)*bits;var t=Math.ceil(strict===true?2*Math.pow(logN,2):logN);for(var a=[],i=0;i<t;i++){a.push(bigInt(i+2))}return millerRabinTest(n,a)};SmallInteger.prototype.isPrime=BigInteger.prototype.isPrime;BigInteger.prototype.isProbablePrime=function(iterations){var isPrime=isBasicPrime(this);if(isPrime!==undefined)return isPrime;var n=this.abs();var t=iterations===undefined?5:iterations;for(var a=[],i=0;i<t;i++){a.push(bigInt.randBetween(2,n.minus(2)))}return millerRabinTest(n,a)};SmallInteger.prototype.isProbablePrime=BigInteger.prototype.isProbablePrime;BigInteger.prototype.modInv=function(n){var t=bigInt.zero,newT=bigInt.one,r=parseValue(n),newR=this.abs(),q,lastT,lastR;while(!newR.equals(bigInt.zero)){q=r.divide(newR);lastT=t;lastR=r;t=newT;r=newR;newT=lastT.subtract(q.multiply(newT));newR=lastR.subtract(q.multiply(newR))}if(!r.equals(1))throw new Error(this.toString()+" and "+n.toString()+" are not co-prime");if(t.compare(0)===-1){t=t.add(n)}if(this.isNegative()){return t.negate()}return t};SmallInteger.prototype.modInv=BigInteger.prototype.modInv;BigInteger.prototype.next=function(){var value=this.value;if(this.sign){return subtractSmall(value,1,this.sign)}return new BigInteger(addSmall(value,1),this.sign)};SmallInteger.prototype.next=function(){var value=this.value;if(value+1<MAX_INT)return new SmallInteger(value+1);return new BigInteger(MAX_INT_ARR,false)};BigInteger.prototype.prev=function(){var value=this.value;if(this.sign){return new BigInteger(addSmall(value,1),true)}return subtractSmall(value,1,this.sign)};SmallInteger.prototype.prev=function(){var value=this.value;if(value-1>-MAX_INT)return new SmallInteger(value-1);return new BigInteger(MAX_INT_ARR,true)};var powersOfTwo=[1];while(2*powersOfTwo[powersOfTwo.length-1]<=BASE)powersOfTwo.push(2*powersOfTwo[powersOfTwo.length-1]);var powers2Length=powersOfTwo.length,highestPower2=powersOfTwo[powers2Length-1];function shift_isSmall(n){return(typeof n==="number"||typeof n==="string")&&+Math.abs(n)<=BASE||n instanceof BigInteger&&n.value.length<=1}BigInteger.prototype.shiftLeft=function(n){if(!shift_isSmall(n)){throw new Error(String(n)+" is too large for shifting.")}n=+n;if(n<0)return this.shiftRight(-n);var result=this;if(result.isZero())return result;while(n>=powers2Length){result=result.multiply(highestPower2);n-=powers2Length-1}return result.multiply(powersOfTwo[n])};SmallInteger.prototype.shiftLeft=BigInteger.prototype.shiftLeft;BigInteger.prototype.shiftRight=function(n){var remQuo;if(!shift_isSmall(n)){throw new Error(String(n)+" is too large for shifting.")}n=+n;if(n<0)return this.shiftLeft(-n);var result=this;while(n>=powers2Length){if(result.isZero()||result.isNegative()&&result.isUnit())return result;remQuo=divModAny(result,highestPower2);result=remQuo[1].isNegative()?remQuo[0].prev():remQuo[0];n-=powers2Length-1}remQuo=divModAny(result,powersOfTwo[n]);return remQuo[1].isNegative()?remQuo[0].prev():remQuo[0]};SmallInteger.prototype.shiftRight=BigInteger.prototype.shiftRight;function bitwise(x,y,fn){y=parseValue(y);var xSign=x.isNegative(),ySign=y.isNegative();var xRem=xSign?x.not():x,yRem=ySign?y.not():y;var xDigit=0,yDigit=0;var xDivMod=null,yDivMod=null;var result=[];while(!xRem.isZero()||!yRem.isZero()){xDivMod=divModAny(xRem,highestPower2);xDigit=xDivMod[1].toJSNumber();if(xSign){xDigit=highestPower2-1-xDigit}yDivMod=divModAny(yRem,highestPower2);yDigit=yDivMod[1].toJSNumber();if(ySign){yDigit=highestPower2-1-yDigit}xRem=xDivMod[0];yRem=yDivMod[0];result.push(fn(xDigit,yDigit))}var sum=fn(xSign?1:0,ySign?1:0)!==0?bigInt(-1):bigInt(0);for(var i=result.length-1;i>=0;i-=1){sum=sum.multiply(highestPower2).add(bigInt(result[i]))}return sum}BigInteger.prototype.not=function(){return this.negate().prev()};SmallInteger.prototype.not=BigInteger.prototype.not;BigInteger.prototype.and=function(n){return bitwise(this,n,function(a,b){return a&b})};SmallInteger.prototype.and=BigInteger.prototype.and;BigInteger.prototype.or=function(n){return bitwise(this,n,function(a,b){return a|b})};SmallInteger.prototype.or=BigInteger.prototype.or;BigInteger.prototype.xor=function(n){return bitwise(this,n,function(a,b){return a^b})};SmallInteger.prototype.xor=BigInteger.prototype.xor;var LOBMASK_I=1<<30,LOBMASK_BI=(BASE&-BASE)*(BASE&-BASE)|LOBMASK_I;function roughLOB(n){var v=n.value,x=typeof v==="number"?v|LOBMASK_I:v[0]+v[1]*BASE|LOBMASK_BI;return x&-x}function integerLogarithm(value,base){if(base.compareTo(value)<=0){var tmp=integerLogarithm(value,base.square(base));var p=tmp.p;var e=tmp.e;var t=p.multiply(base);return t.compareTo(value)<=0?{p:t,e:e*2+1}:{p:p,e:e*2}}return{p:bigInt(1),e:0}}BigInteger.prototype.bitLength=function(){var n=this;if(n.compareTo(bigInt(0))<0){n=n.negate().subtract(bigInt(1))}if(n.compareTo(bigInt(0))===0){return bigInt(0)}return bigInt(integerLogarithm(n,bigInt(2)).e).add(bigInt(1))};SmallInteger.prototype.bitLength=BigInteger.prototype.bitLength;function max(a,b){a=parseValue(a);b=parseValue(b);return a.greater(b)?a:b}function min(a,b){a=parseValue(a);b=parseValue(b);return a.lesser(b)?a:b}function gcd(a,b){a=parseValue(a).abs();b=parseValue(b).abs();if(a.equals(b))return a;if(a.isZero())return b;if(b.isZero())return a;var c=Integer[1],d,t;while(a.isEven()&&b.isEven()){d=Math.min(roughLOB(a),roughLOB(b));a=a.divide(d);b=b.divide(d);c=c.multiply(d)}while(a.isEven()){a=a.divide(roughLOB(a))}do{while(b.isEven()){b=b.divide(roughLOB(b))}if(a.greater(b)){t=b;b=a;a=t}b=b.subtract(a)}while(!b.isZero());return c.isUnit()?a:a.multiply(c)}function lcm(a,b){a=parseValue(a).abs();b=parseValue(b).abs();return a.divide(gcd(a,b)).multiply(b)}function randBetween(a,b){a=parseValue(a);b=parseValue(b);var low=min(a,b),high=max(a,b);var range=high.subtract(low).add(1);if(range.isSmall)return low.add(Math.floor(Math.random()*range));var length=range.value.length-1;var result=[],restricted=true;for(var i=length;i>=0;i--){var top=restricted?range.value[i]:BASE;var digit=truncate(Math.random()*top);result.unshift(digit);if(digit<top)restricted=false}result=arrayToSmall(result);return low.add(typeof result==="number"?new SmallInteger(result):new BigInteger(result,false))}var parseBase=function(text,base){var length=text.length;var i;var absBase=Math.abs(base);for(var i=0;i<length;i++){var c=text[i].toLowerCase();if(c==="-")continue;if(/[a-z0-9]/.test(c)){if(/[0-9]/.test(c)&&+c>=absBase){if(c==="1"&&absBase===1)continue;throw new Error(c+" is not a valid digit in base "+base+".")}else if(c.charCodeAt(0)-87>=absBase){throw new Error(c+" is not a valid digit in base "+base+".")}}}if(2<=base&&base<=36){if(length<=LOG_MAX_INT/Math.log(base)){var result=parseInt(text,base);if(isNaN(result)){throw new Error(c+" is not a valid digit in base "+base+".")}return new SmallInteger(parseInt(text,base))}}base=parseValue(base);var digits=[];var isNegative=text[0]==="-";for(i=isNegative?1:0;i<text.length;i++){var c=text[i].toLowerCase(),charCode=c.charCodeAt(0);if(48<=charCode&&charCode<=57)digits.push(parseValue(c));else if(97<=charCode&&charCode<=122)digits.push(parseValue(c.charCodeAt(0)-87));else if(c==="<"){var start=i;do{i++}while(text[i]!==">");digits.push(parseValue(text.slice(start+1,i)))}else throw new Error(c+" is not a valid character")}return parseBaseFromArray(digits,base,isNegative)};function parseBaseFromArray(digits,base,isNegative){var val=Integer[0],pow=Integer[1],i;for(i=digits.length-1;i>=0;i--){val=val.add(digits[i].times(pow));pow=pow.times(base)}return isNegative?val.negate():val}function stringify(digit){if(digit<=35){return"0123456789abcdefghijklmnopqrstuvwxyz".charAt(digit)}return"<"+digit+">"}function toBase(n,base){base=bigInt(base);if(base.isZero()){if(n.isZero())return{value:[0],isNegative:false};throw new Error("Cannot convert nonzero numbers to base 0.")}if(base.equals(-1)){if(n.isZero())return{value:[0],isNegative:false};if(n.isNegative())return{value:[].concat.apply([],Array.apply(null,Array(-n)).map(Array.prototype.valueOf,[1,0])),isNegative:false};var arr=Array.apply(null,Array(+n-1)).map(Array.prototype.valueOf,[0,1]);arr.unshift([1]);return{value:[].concat.apply([],arr),isNegative:false}}var neg=false;if(n.isNegative()&&base.isPositive()){neg=true;n=n.abs()}if(base.equals(1)){if(n.isZero())return{value:[0],isNegative:false};return{value:Array.apply(null,Array(+n)).map(Number.prototype.valueOf,1),isNegative:neg}}var out=[];var left=n,divmod;while(left.isNegative()||left.compareAbs(base)>=0){divmod=left.divmod(base);left=divmod.quotient;var digit=divmod.remainder;if(digit.isNegative()){digit=base.minus(digit).abs();left=left.next()}out.push(digit.toJSNumber())}out.push(left.toJSNumber());return{value:out.reverse(),isNegative:neg}}function toBaseString(n,base){var arr=toBase(n,base);return(arr.isNegative?"-":"")+arr.value.map(stringify).join("")}BigInteger.prototype.toArray=function(radix){return toBase(this,radix)};SmallInteger.prototype.toArray=function(radix){return toBase(this,radix)};BigInteger.prototype.toString=function(radix){if(radix===undefined)radix=10;if(radix!==10)return toBaseString(this,radix);var v=this.value,l=v.length,str=String(v[--l]),zeros="0000000",digit;while(--l>=0){digit=String(v[l]);str+=zeros.slice(digit.length)+digit}var sign=this.sign?"-":"";return sign+str};SmallInteger.prototype.toString=function(radix){if(radix===undefined)radix=10;if(radix!=10)return toBaseString(this,radix);return String(this.value)};BigInteger.prototype.toJSON=SmallInteger.prototype.toJSON=function(){return this.toString()};BigInteger.prototype.valueOf=function(){return parseInt(this.toString(),10)};BigInteger.prototype.toJSNumber=BigInteger.prototype.valueOf;SmallInteger.prototype.valueOf=function(){return this.value};SmallInteger.prototype.toJSNumber=SmallInteger.prototype.valueOf;function parseStringValue(v){if(isPrecise(+v)){var x=+v;if(x===truncate(x))return new SmallInteger(x);throw new Error("Invalid integer: "+v)}var sign=v[0]==="-";if(sign)v=v.slice(1);var split=v.split(/e/i);if(split.length>2)throw new Error("Invalid integer: "+split.join("e"));if(split.length===2){var exp=split[1];if(exp[0]==="+")exp=exp.slice(1);exp=+exp;if(exp!==truncate(exp)||!isPrecise(exp))throw new Error("Invalid integer: "+exp+" is not a valid exponent.");var text=split[0];var decimalPlace=text.indexOf(".");if(decimalPlace>=0){exp-=text.length-decimalPlace-1;text=text.slice(0,decimalPlace)+text.slice(decimalPlace+1)}if(exp<0)throw new Error("Cannot include negative exponent part for integers");text+=new Array(exp+1).join("0");v=text}var isValid=/^([0-9][0-9]*)$/.test(v);if(!isValid)throw new Error("Invalid integer: "+v);var r=[],max=v.length,l=LOG_BASE,min=max-l;while(max>0){r.push(+v.slice(min,max));min-=l;if(min<0)min=0;max-=l}trim(r);return new BigInteger(r,sign)}function parseNumberValue(v){if(isPrecise(v)){if(v!==truncate(v))throw new Error(v+" is not an integer.");return new SmallInteger(v)}return parseStringValue(v.toString())}function parseValue(v){if(typeof v==="number"){return parseNumberValue(v)}if(typeof v==="string"){return parseStringValue(v)}return v}for(var i=0;i<1e3;i++){Integer[i]=new SmallInteger(i);if(i>0)Integer[-i]=new SmallInteger(-i)}Integer.one=Integer[1];Integer.zero=Integer[0];Integer.minusOne=Integer[-1];Integer.max=max;Integer.min=min;Integer.gcd=gcd;Integer.lcm=lcm;Integer.isInstance=function(x){return x instanceof BigInteger||x instanceof SmallInteger};Integer.randBetween=randBetween;Integer.fromArray=function(digits,base,isNegative){return parseBaseFromArray(digits.map(parseValue),parseValue(base||10),isNegative)};return Integer}();if(typeof module!=="undefined"&&module.hasOwnProperty("exports")){module.exports=bigInt}if(typeof define==="function"&&define.amd){define("big-integer",[],function(){return bigInt})}
        return bigInt;
    })(),
    /*
     *  decimal.js v10.2.1
     *  An arbitrary-precision Decimal type for JavaScript.
     *  https://github.com/MikeMcl/decimal.js
     *  Copyright (c) 2020 Michael Mclaughlin <M8ch88l@gmail.com>
     *  MIT Licence
     */
    bigDec: function(n){"use strict";var e,i,t,r,s=9e15,o=1e9,u="0123456789abcdef",c="2.3025850929940456840179914546843642076011014886287729760333279009675726096773524802359972050895982983419677840422862486334095254650828067566662873690987816894829072083255546808437998948262331985283935053089653777326288461633662222876982198867465436674744042432743651550489343149393914796194044002221051017141748003688084012647080685567743216228355220114804663715659121373450747856947683463616792101806445070648000277502684916746550586856935673420670581136429224554405758925724208241314695689016758940256776311356919292033376587141660230105703089634572075440370847469940168269282808481184289314848524948644871927809676271275775397027668605952496716674183485704422507197965004714951050492214776567636938662976979522110718264549734772662425709429322582798502585509785265383207606726317164309505995087807523710333101197857547331541421808427543863591778117054309827482385045648019095610299291824318237525357709750539565187697510374970888692180205189339507238539205144634197265287286965110862571492198849978748873771345686209167058",f="3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632789",a={precision:20,rounding:4,modulo:1,toExpNeg:-7,toExpPos:21,minE:-s,maxE:s,crypto:!1},h=!0,d="[DecimalError] 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