Apply BigNumber config to functions too (currently it only applies to eval)

[adamazing]

Hello there,
I was attacking Project Euler, problem 26:

"Find the value of d <1000 for which 1/d contains the longest recurring cycle in its decimal fraction part."

I was trying to find the length of the repetend for each 1/d by calculating the smallest k for which 10ᵏ mod d == 1 but was getting weird/wrong results, e.g.

var mathjs = require('mathjs');
mathjs.config({
  number: 'BigNumber',
  precision: 4096
});

var p = mathjs.chain(10)
                  .pow(58)
                  .mod(531)
                  .done();
console.log(mathjs.format(p));
/*
Expected answer: 1 
Actual answer:  0
*/

The expected answer is 1, per https://www.wolframalpha.com/input/?i=10%5E58+mod+531 but it's returning 0?

Reproduction with a more obviously wrong result:

 (...)
var p = mathjs.chain(10)
                  .pow(16)
                  .mod(3)
                  .done();
console.log(mathjs.format(p));
/*
Expected answer: 1 
Actual answer:  0
*/

The greatest power of 10 for which mod(3) gives the correct answer is 15.

Is there something I'm doing wrong here?

[josdejong]

Whilst you configure to use BigNumbers, in chain you use regular numbers (which have a precision in the order of 16 bits which can explain this limit of 10^16).

The following works as expected, the expression parser will create BigNumbers:

mathjs.config({
  number: 'BigNumber'
});

console.log(mathjs.format(mathjs.eval('(10^58) mod 531'))); // 1

or:

mathjs.config({
  number: 'BigNumber'
});

var p = mathjs.chain(math.bignumber(10))
                  .pow(math.bignumber(58))
                  .mod(math.bignumber(531))
                  .done();
console.log(mathjs.format(p)); // 1

The bignumber configuration is confusing, it's quite logical to expect that numbers are always turned into BigNumbers which is not the case. This has bitten other people as well. Maybe we should rethink this behavior and automatically turn all numbers into BigNumbers (or Fractions) when configured like that.

[adamazing]

Thanks for your quick answer Jos.
You're right, the current behaviour seems counterintuitive to me.

[ericman314]

I'm jumping in late in the discussion, but I agree that regular numbers should be converted to BigNumbers automatically. Is this something typed-function can do?

[josdejong]

I don't have a clear idea how we could best tackle this challenge, but I indeed think that this may be something we can solve by extending typed-function with capability to do preprocessing of inputs depending on dynamic configuration.

[josdejong]

I think it is still relevant, and not as a question but as an idea. I'll reopen it and turn it into a discussion topic.

[gwhitney]

Another point specific to the particular function that the original poster was interested in is that the value of 10^58 mod 531 should absolutely not be computed by raising 10 to the 58th power and then taking the remainder mod 531; the power should be computed by some squarings and multiplications, taking the remainder mod 531 each time. In other words, take 10^2 mod 531 (that's 100) and multiply it by 10 to get 10^3 and take the remainder mod 531 (that's 469), then square that (to get 10^6 mod 531) and take the remainder mod 531, then multiply that remainder by 10 (to get 10^7 mod 531) and take the remainder mod 531, then square that twice taking the remainder each time (to get 10^28 mod 531) and then multiply that by 10 (to get 10^29 mod 531) and take the remainder mod 531 and finally square that (to get 10^58) and taking the remainder mod 531. The repeated taking of remainders keeps the whole computation well within the range of ordinary JavaScript number, and using the square-and-multiply trick greatly reduces the total amount of multiplication.

Number theorists routinely use a "powmod" function when they want to get the remainder of a power of a number. Moreover, mathjs could just automatically when it is compiling expressions convert mod(pow(a,b), n) into a call to powmod(a,b,n).

[gwhitney]

This came up again recently in #1485. It seems as though it would be possible (with some enhancements to typed-function) when config.number is 'BigNumber' to reprioritize number->bignumber conversion over accepting a number argument as is, if that's the desired behavior.