Lect2_1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Physics 404/604\n",
    "\n",
    "## Computational Physics (Spring 2024)\n",
    "\n",
    "## BPB-250\n",
    "\n",
    "| Instructor | Prof. Zhaohuan Zhu                 |\n",
    "| ---------- | :--------------------------------- |\n",
    "| Email      | zhaohuan.zhu@unlv.edu              |\n",
    "| Website    | http://www.physics.unlv.edu/~zhzhu |\n",
    "| Office     | BPB 245                            |\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# 1. How to represent numbers?\n",
    "\n",
    "## 1.1 Integers:   \n",
    "\n",
    " ($-2^{n-1}$, $2^{n-1}$) where n is the number of bits used to store the signed integer\n",
    "      \n",
    "      Note that in python2 integer is either 4 bytes or 8 bytes. In python 3, integer can be any length as long as the memory allows. On the otherhand, some python libraries (e.g. numpy) donot support these very long integers. Remember only 19 digits integer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "int64\n",
      "[1] 0b1\n",
      "[2] 0b10\n",
      "[9223372036854775807] 0b111111111111111111111111111111111111111111111111111111111111111\n",
      "[-9223372036854775808] -0b1000000000000000000000000000000000000000000000000000000000000000\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/9y/650d6syj7198jyl1pprv_s2h0000gn/T/ipykernel_41740/3303376609.py:14: RuntimeWarning: overflow encountered in long_scalars\n",
      "  print(a+1,bin(a[0]+1))\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "a=np.array([1])\n",
    "print(a.dtype)\n",
    "print(a,bin(a[0]))\n",
    "\n",
    "a=np.array([2])\n",
    "print(a,bin(a[0]))\n",
    "\n",
    "a=np.array([2**63-1]) # largest integer stored\n",
    "print(a,bin(a[0]))\n",
    "\n",
    "# Try the following and explain what is happening here\n",
    "# adding one overflows - this will go to the sign bit = sign flip\n",
    "print(a+1,bin(a[0]+1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "## 1.2 Floating point numbers:\n",
    "\n",
    "8 bytes  \n",
    "One number in the memory   \n",
    "ABBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC\n",
    "\n",
    "A: sign  \n",
    "B: e  \n",
    "C: $b_{52-i}$\n",
    "i is the bit of C; the i of the leftmost C is 1; the i of the rightmost C is 52.\n",
    "\n",
    "\\begin{equation}  \n",
    "x=(-1)^{sign}(1+\\sum_{i=1}^{52} b_{52-i}2^{-i})2^{e-1023}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "import numpy as np\n",
    "import struct \n",
    "\n",
    "def fextract(num):\n",
    "    return ''.join(bin(c).replace('0b', '').rjust(8, '0') for c in struct.pack('!d', num))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0000000000000000000000000000000  0 01111111111 0000000000000000000000000000000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "num=1.0\n",
    "print(\"%.31f  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5000000000000000000000000000000  0 01111111110 0000000000000000000000000000000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "num=0.5\n",
    "print(\"%.31f  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7500000000000000000000000000000  0 01111111110 1000000000000000000000000000000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "num=0.75\n",
    "print(\"%.31f  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2000000000000000111022302462516  0 01111111100 1001100110011001100110011001100110011001100110011010\n"
     ]
    }
   ],
   "source": [
    "num=0.2\n",
    "#ccan not be split into base 2\n",
    "# must be truncated somewhere (accurate only to 16 digits - max for mantissa\n",
    "\n",
    "print(\"%.31f  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.8000000000000000444089209850063  0 01111111111 1100110011001100110011001100110011001100110011001101\n",
      "1.8000000000000000444089209850063  0 01111111111 1100110011001100110011001100110011001100110011001101\n",
      "1.7999999999999998223643160599750  0 01111111111 1100110011001100110011001100110011001100110011001100\n"
     ]
    }
   ],
   "source": [
    "# Left association is important !!! Keep this in mind when you are dealing with round-off errors.\n",
    "\n",
    "num1=0.2+1.9-0.3\n",
    "print(\"%.31f  %s\"%(num1,fextract(num)[0]+' '+fextract(num1)[1:12]+' '+fextract(num1)[12:]))\n",
    "\n",
    "\n",
    "num2=1.9+0.2-0.3\n",
    "print(\"%.31f  %s\"%(num2,fextract(num2)[0]+' '+fextract(num2)[1:12]+' '+fextract(num2)[12:]))\n",
    "\n",
    "\n",
    "num3=0.2-0.3+1.9\n",
    "print(\"%.31f  %s\"%(num3,fextract(num3)[0]+' '+fextract(num3)[1:12]+' '+fextract(num3)[12:]))\n",
    "\n",
    "# A+B-C does not equal A-C+B\n",
    "# computer will not think num2 == num3 due to round off error "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Range \n",
    "\n",
    "[-$10^{308}$, -$10^{-324}$] [$10^{-324}$,10$^{308}$]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0000000000000000000e+00  0 00000000000 0000000000000000000000000000000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "num=1.e-325 #underflow sometimes is treated as 0\n",
    "print(\"%.19e  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.9406564584124654418e-324  0 00000000000 0000000000000000000000000000000000000000000000000001\n"
     ]
    }
   ],
   "source": [
    "num=3.e-324\n",
    "print(\"%.19e  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "inf  0 11111111111 0000000000000000000000000000000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "num=1.e309 #overflow generates infinity\n",
    "print(\"%.19e  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.7976931348623157081e+308  0 11111111110 1111111111111111111111111111111111111111111111111111\n"
     ]
    }
   ],
   "source": [
    "num=1.7976931348623157e308\n",
    "print(\"%.19e  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Loop number 0, under   5.00000000000000000e-01, over   2.00000000000000000e+00\n",
      "Loop number 1, under   2.50000000000000000e-01, over   4.00000000000000000e+00\n",
      "Loop number 2, under   1.25000000000000000e-01, over   8.00000000000000000e+00\n",
      "Loop number 3, under   6.25000000000000000e-02, over   1.60000000000000000e+01\n",
      "Loop number 4, under   3.12500000000000000e-02, over   3.20000000000000000e+01\n",
      "Loop number 5, under   1.56250000000000000e-02, over   6.40000000000000000e+01\n",
      "Loop number 6, under   7.81250000000000000e-03, over   1.28000000000000000e+02\n",
      "Loop number 7, under   3.90625000000000000e-03, over   2.56000000000000000e+02\n",
      "Loop number 8, under   1.95312500000000000e-03, over   5.12000000000000000e+02\n",
      "Loop number 9, under   9.76562500000000000e-04, over   1.02400000000000000e+03\n",
      "Loop number 10, under   4.88281250000000000e-04, over   2.04800000000000000e+03\n",
      "Loop number 11, under   2.44140625000000000e-04, over   4.09600000000000000e+03\n",
      "Loop number 12, under   1.22070312500000000e-04, over   8.19200000000000000e+03\n",
      "Loop number 13, under   6.10351562500000000e-05, over   1.63840000000000000e+04\n",
      "Loop number 14, under   3.05175781250000000e-05, over   3.27680000000000000e+04\n",
      "Loop number 15, under   1.52587890625000000e-05, over   6.55360000000000000e+04\n",
      "Loop number 16, under   7.62939453125000000e-06, over   1.31072000000000000e+05\n",
      "Loop number 17, under   3.81469726562500000e-06, over   2.62144000000000000e+05\n",
      "Loop number 18, under   1.90734863281250000e-06, over   5.24288000000000000e+05\n",
      "Loop number 19, under   9.53674316406250000e-07, over   1.04857600000000000e+06\n",
      "Loop number 20, under   4.76837158203125000e-07, over   2.09715200000000000e+06\n",
      "Loop number 21, under   2.38418579101562500e-07, over   4.19430400000000000e+06\n",
      "Loop number 22, under   1.19209289550781250e-07, over   8.38860800000000000e+06\n",
      "Loop number 23, under   5.96046447753906250e-08, over   1.67772160000000000e+07\n",
      "Loop number 24, under   2.98023223876953125e-08, over   3.35544320000000000e+07\n",
      "Loop number 25, under   1.49011611938476562e-08, over   6.71088640000000000e+07\n",
      "Loop number 26, under   7.45058059692382812e-09, over   1.34217728000000000e+08\n",
      "Loop number 27, under   3.72529029846191406e-09, over   2.68435456000000000e+08\n",
      "Loop number 28, under   1.86264514923095703e-09, over   5.36870912000000000e+08\n",
      "Loop number 29, under   9.31322574615478516e-10, over   1.07374182400000000e+09\n",
      "Loop number 30, under   4.65661287307739258e-10, over   2.14748364800000000e+09\n",
      "Loop number 31, under   2.32830643653869629e-10, over   4.29496729600000000e+09\n",
      "Loop number 32, under   1.16415321826934814e-10, over   8.58993459200000000e+09\n",
      "Loop number 33, under   5.82076609134674072e-11, over   1.71798691840000000e+10\n",
      "Loop number 34, under   2.91038304567337036e-11, over   3.43597383680000000e+10\n",
      "Loop number 35, under   1.45519152283668518e-11, over   6.87194767360000000e+10\n",
      "Loop number 36, under   7.27595761418342590e-12, over   1.37438953472000000e+11\n",
      "Loop number 37, under   3.63797880709171295e-12, over   2.74877906944000000e+11\n",
      "Loop number 38, under   1.81898940354585648e-12, over   5.49755813888000000e+11\n",
      "Loop number 39, under   9.09494701772928238e-13, over   1.09951162777600000e+12\n",
      "Loop number 40, under   4.54747350886464119e-13, over   2.19902325555200000e+12\n",
      "Loop number 41, under   2.27373675443232059e-13, over   4.39804651110400000e+12\n",
      "Loop number 42, under   1.13686837721616030e-13, over   8.79609302220800000e+12\n",
      "Loop number 43, under   5.68434188608080149e-14, over   1.75921860444160000e+13\n",
      "Loop number 44, under   2.84217094304040074e-14, over   3.51843720888320000e+13\n",
      "Loop number 45, under   1.42108547152020037e-14, over   7.03687441776640000e+13\n",
      "Loop number 46, under   7.10542735760100186e-15, over   1.40737488355328000e+14\n",
      "Loop number 47, under   3.55271367880050093e-15, over   2.81474976710656000e+14\n",
      "Loop number 48, under   1.77635683940025046e-15, over   5.62949953421312000e+14\n",
      "Loop number 49, under   8.88178419700125232e-16, over   1.12589990684262400e+15\n",
      "Loop number 50, under   4.44089209850062616e-16, over   2.25179981368524800e+15\n",
      "Loop number 51, under   2.22044604925031308e-16, over   4.50359962737049600e+15\n",
      "Loop number 52, under   1.11022302462515654e-16, over   9.00719925474099200e+15\n",
      "Loop number 53, under   5.55111512312578270e-17, over   1.80143985094819840e+16\n",
      "Loop number 54, under   2.77555756156289135e-17, over   3.60287970189639680e+16\n",
      "Loop number 55, under   1.38777878078144568e-17, over   7.20575940379279360e+16\n",
      "Loop number 56, under   6.93889390390722838e-18, over   1.44115188075855872e+17\n",
      "Loop number 57, under   3.46944695195361419e-18, over   2.88230376151711744e+17\n",
      "Loop number 58, under   1.73472347597680709e-18, over   5.76460752303423488e+17\n",
      "Loop number 59, under   8.67361737988403547e-19, over   1.15292150460684698e+18\n",
      "Loop number 60, under   4.33680868994201774e-19, over   2.30584300921369395e+18\n",
      "Loop number 61, under   2.16840434497100887e-19, over   4.61168601842738790e+18\n",
      "Loop number 62, under   1.08420217248550443e-19, over   9.22337203685477581e+18\n",
      "Loop number 63, under   5.42101086242752217e-20, over   1.84467440737095516e+19\n",
      "Loop number 64, under   2.71050543121376109e-20, over   3.68934881474191032e+19\n",
      "Loop number 65, under   1.35525271560688054e-20, over   7.37869762948382065e+19\n",
      "Loop number 66, under   6.77626357803440271e-21, over   1.47573952589676413e+20\n",
      "Loop number 67, under   3.38813178901720136e-21, over   2.95147905179352826e+20\n",
      "Loop number 68, under   1.69406589450860068e-21, over   5.90295810358705652e+20\n",
      "Loop number 69, under   8.47032947254300339e-22, over   1.18059162071741130e+21\n",
      "Loop number 70, under   4.23516473627150170e-22, over   2.36118324143482261e+21\n",
      "Loop number 71, under   2.11758236813575085e-22, over   4.72236648286964521e+21\n",
      "Loop number 72, under   1.05879118406787542e-22, over   9.44473296573929043e+21\n",
      "Loop number 73, under   5.29395592033937712e-23, over   1.88894659314785809e+22\n",
      "Loop number 74, under   2.64697796016968856e-23, over   3.77789318629571617e+22\n",
      "Loop number 75, under   1.32348898008484428e-23, over   7.55578637259143234e+22\n",
      "Loop number 76, under   6.61744490042422140e-24, over   1.51115727451828647e+23\n",
      "Loop number 77, under   3.30872245021211070e-24, over   3.02231454903657294e+23\n",
      "Loop number 78, under   1.65436122510605535e-24, over   6.04462909807314587e+23\n",
      "Loop number 79, under   8.27180612553027675e-25, over   1.20892581961462917e+24\n",
      "Loop number 80, under   4.13590306276513837e-25, over   2.41785163922925835e+24\n",
      "Loop number 81, under   2.06795153138256919e-25, over   4.83570327845851670e+24\n",
      "Loop number 82, under   1.03397576569128459e-25, over   9.67140655691703340e+24\n",
      "Loop number 83, under   5.16987882845642297e-26, over   1.93428131138340668e+25\n",
      "Loop number 84, under   2.58493941422821148e-26, over   3.86856262276681336e+25\n",
      "Loop number 85, under   1.29246970711410574e-26, over   7.73712524553362672e+25\n",
      "Loop number 86, under   6.46234853557052871e-27, over   1.54742504910672534e+26\n",
      "Loop number 87, under   3.23117426778526435e-27, over   3.09485009821345069e+26\n",
      "Loop number 88, under   1.61558713389263218e-27, over   6.18970019642690137e+26\n",
      "Loop number 89, under   8.07793566946316089e-28, over   1.23794003928538027e+27\n",
      "Loop number 90, under   4.03896783473158044e-28, over   2.47588007857076055e+27\n",
      "Loop number 91, under   2.01948391736579022e-28, over   4.95176015714152110e+27\n",
      "Loop number 92, under   1.00974195868289511e-28, over   9.90352031428304220e+27\n",
      "Loop number 93, under   5.04870979341447555e-29, over   1.98070406285660844e+28\n",
      "Loop number 94, under   2.52435489670723778e-29, over   3.96140812571321688e+28\n",
      "Loop number 95, under   1.26217744835361889e-29, over   7.92281625142643376e+28\n",
      "Loop number 96, under   6.31088724176809444e-30, over   1.58456325028528675e+29\n",
      "Loop number 97, under   3.15544362088404722e-30, over   3.16912650057057350e+29\n",
      "Loop number 98, under   1.57772181044202361e-30, over   6.33825300114114701e+29\n",
      "Loop number 99, under   7.88860905221011805e-31, over   1.26765060022822940e+30\n",
      "Loop number 100, under   3.94430452610505903e-31, over   2.53530120045645880e+30\n",
      "Loop number 101, under   1.97215226305252951e-31, over   5.07060240091291761e+30\n",
      "Loop number 102, under   9.86076131526264757e-32, over   1.01412048018258352e+31\n",
      "Loop number 103, under   4.93038065763132378e-32, over   2.02824096036516704e+31\n",
      "Loop number 104, under   2.46519032881566189e-32, over   4.05648192073033408e+31\n",
      "Loop number 105, under   1.23259516440783095e-32, over   8.11296384146066817e+31\n",
      "Loop number 106, under   6.16297582203915473e-33, over   1.62259276829213363e+32\n",
      "Loop number 107, under   3.08148791101957736e-33, over   3.24518553658426727e+32\n",
      "Loop number 108, under   1.54074395550978868e-33, over   6.49037107316853454e+32\n",
      "Loop number 109, under   7.70371977754894341e-34, over   1.29807421463370691e+33\n",
      "Loop number 110, under   3.85185988877447171e-34, over   2.59614842926741381e+33\n",
      "Loop number 111, under   1.92592994438723585e-34, over   5.19229685853482763e+33\n",
      "Loop number 112, under   9.62964972193617927e-35, over   1.03845937170696553e+34\n",
      "Loop number 113, under   4.81482486096808963e-35, over   2.07691874341393105e+34\n",
      "Loop number 114, under   2.40741243048404482e-35, over   4.15383748682786210e+34\n",
      "Loop number 115, under   1.20370621524202241e-35, over   8.30767497365572421e+34\n",
      "Loop number 116, under   6.01853107621011204e-36, over   1.66153499473114484e+35\n",
      "Loop number 117, under   3.00926553810505602e-36, over   3.32306998946228968e+35\n",
      "Loop number 118, under   1.50463276905252801e-36, over   6.64613997892457936e+35\n",
      "Loop number 119, under   7.52316384526264005e-37, over   1.32922799578491587e+36\n",
      "Loop number 120, under   3.76158192263132003e-37, over   2.65845599156983175e+36\n",
      "Loop number 121, under   1.88079096131566001e-37, over   5.31691198313966349e+36\n",
      "Loop number 122, under   9.40395480657830006e-38, over   1.06338239662793270e+37\n",
      "Loop number 123, under   4.70197740328915003e-38, over   2.12676479325586540e+37\n",
      "Loop number 124, under   2.35098870164457502e-38, over   4.25352958651173079e+37\n",
      "Loop number 125, under   1.17549435082228751e-38, over   8.50705917302346159e+37\n",
      "Loop number 126, under   5.87747175411143754e-39, over   1.70141183460469232e+38\n",
      "Loop number 127, under   2.93873587705571877e-39, over   3.40282366920938463e+38\n",
      "Loop number 128, under   1.46936793852785938e-39, over   6.80564733841876927e+38\n",
      "Loop number 129, under   7.34683969263929692e-40, over   1.36112946768375385e+39\n",
      "Loop number 130, under   3.67341984631964846e-40, over   2.72225893536750771e+39\n",
      "Loop number 131, under   1.83670992315982423e-40, over   5.44451787073501542e+39\n",
      "Loop number 132, under   9.18354961579912116e-41, over   1.08890357414700308e+40\n",
      "Loop number 133, under   4.59177480789956058e-41, over   2.17780714829400617e+40\n",
      "Loop number 134, under   2.29588740394978029e-41, over   4.35561429658801233e+40\n",
      "Loop number 135, under   1.14794370197489014e-41, over   8.71122859317602466e+40\n",
      "Loop number 136, under   5.73971850987445072e-42, over   1.74224571863520493e+41\n",
      "Loop number 137, under   2.86985925493722536e-42, over   3.48449143727040987e+41\n",
      "Loop number 138, under   1.43492962746861268e-42, over   6.96898287454081973e+41\n",
      "Loop number 139, under   7.17464813734306340e-43, over   1.39379657490816395e+42\n",
      "Loop number 140, under   3.58732406867153170e-43, over   2.78759314981632789e+42\n",
      "Loop number 141, under   1.79366203433576585e-43, over   5.57518629963265579e+42\n",
      "Loop number 142, under   8.96831017167882925e-44, over   1.11503725992653116e+43\n",
      "Loop number 143, under   4.48415508583941463e-44, over   2.23007451985306231e+43\n",
      "Loop number 144, under   2.24207754291970731e-44, over   4.46014903970612463e+43\n",
      "Loop number 145, under   1.12103877145985366e-44, over   8.92029807941224926e+43\n",
      "Loop number 146, under   5.60519385729926828e-45, over   1.78405961588244985e+44\n",
      "Loop number 147, under   2.80259692864963414e-45, over   3.56811923176489970e+44\n",
      "Loop number 148, under   1.40129846432481707e-45, over   7.13623846352979941e+44\n",
      "Loop number 149, under   7.00649232162408535e-46, over   1.42724769270595988e+45\n",
      "Loop number 150, under   3.50324616081204268e-46, over   2.85449538541191976e+45\n",
      "Loop number 151, under   1.75162308040602134e-46, over   5.70899077082383952e+45\n",
      "Loop number 152, under   8.75811540203010669e-47, over   1.14179815416476790e+46\n",
      "Loop number 153, under   4.37905770101505335e-47, over   2.28359630832953581e+46\n",
      "Loop number 154, under   2.18952885050752667e-47, over   4.56719261665907162e+46\n",
      "Loop number 155, under   1.09476442525376334e-47, over   9.13438523331814324e+46\n",
      "Loop number 156, under   5.47382212626881668e-48, over   1.82687704666362865e+47\n",
      "Loop number 157, under   2.73691106313440834e-48, over   3.65375409332725730e+47\n",
      "Loop number 158, under   1.36845553156720417e-48, over   7.30750818665451459e+47\n",
      "Loop number 159, under   6.84227765783602085e-49, over   1.46150163733090292e+48\n",
      "Loop number 160, under   3.42113882891801043e-49, over   2.92300327466180584e+48\n",
      "Loop number 161, under   1.71056941445900521e-49, over   5.84600654932361167e+48\n",
      "Loop number 162, under   8.55284707229502607e-50, over   1.16920130986472233e+49\n",
      "Loop number 163, under   4.27642353614751303e-50, over   2.33840261972944467e+49\n",
      "Loop number 164, under   2.13821176807375652e-50, over   4.67680523945888934e+49\n",
      "Loop number 165, under   1.06910588403687826e-50, over   9.35361047891777868e+49\n",
      "Loop number 166, under   5.34552942018439129e-51, over   1.87072209578355574e+50\n",
      "Loop number 167, under   2.67276471009219565e-51, over   3.74144419156711147e+50\n",
      "Loop number 168, under   1.33638235504609782e-51, over   7.48288838313422294e+50\n",
      "Loop number 169, under   6.68191177523048912e-52, over   1.49657767662684459e+51\n",
      "Loop number 170, under   3.34095588761524456e-52, over   2.99315535325368918e+51\n",
      "Loop number 171, under   1.67047794380762228e-52, over   5.98631070650737835e+51\n",
      "Loop number 172, under   8.35238971903811139e-53, over   1.19726214130147567e+52\n",
      "Loop number 173, under   4.17619485951905570e-53, over   2.39452428260295134e+52\n",
      "Loop number 174, under   2.08809742975952785e-53, over   4.78904856520590268e+52\n",
      "Loop number 175, under   1.04404871487976392e-53, over   9.57809713041180536e+52\n",
      "Loop number 176, under   5.22024357439881962e-54, over   1.91561942608236107e+53\n",
      "Loop number 177, under   2.61012178719940981e-54, over   3.83123885216472215e+53\n",
      "Loop number 178, under   1.30506089359970491e-54, over   7.66247770432944429e+53\n",
      "Loop number 179, under   6.52530446799852453e-55, over   1.53249554086588886e+54\n",
      "Loop number 180, under   3.26265223399926226e-55, over   3.06499108173177772e+54\n",
      "Loop number 181, under   1.63132611699963113e-55, over   6.12998216346355543e+54\n",
      "Loop number 182, under   8.15663058499815566e-56, over   1.22599643269271109e+55\n",
      "Loop number 183, under   4.07831529249907783e-56, over   2.45199286538542217e+55\n",
      "Loop number 184, under   2.03915764624953891e-56, over   4.90398573077084435e+55\n",
      "Loop number 185, under   1.01957882312476946e-56, over   9.80797146154168869e+55\n",
      "Loop number 186, under   5.09789411562384729e-57, over   1.96159429230833774e+56\n",
      "Loop number 187, under   2.54894705781192364e-57, over   3.92318858461667548e+56\n",
      "Loop number 188, under   1.27447352890596182e-57, over   7.84637716923335095e+56\n",
      "Loop number 189, under   6.37236764452980911e-58, over   1.56927543384667019e+57\n",
      "Loop number 190, under   3.18618382226490455e-58, over   3.13855086769334038e+57\n",
      "Loop number 191, under   1.59309191113245228e-58, over   6.27710173538668076e+57\n",
      "Loop number 192, under   7.96545955566226139e-59, over   1.25542034707733615e+58\n",
      "Loop number 193, under   3.98272977783113069e-59, over   2.51084069415467231e+58\n",
      "Loop number 194, under   1.99136488891556535e-59, over   5.02168138830934461e+58\n",
      "Loop number 195, under   9.95682444457782673e-60, over   1.00433627766186892e+59\n",
      "Loop number 196, under   4.97841222228891337e-60, over   2.00867255532373784e+59\n",
      "Loop number 197, under   2.48920611114445668e-60, over   4.01734511064747569e+59\n",
      "Loop number 198, under   1.24460305557222834e-60, over   8.03469022129495138e+59\n",
      "Loop number 199, under   6.22301527786114171e-61, over   1.60693804425899028e+60\n",
      "Loop number 200, under   3.11150763893057085e-61, over   3.21387608851798055e+60\n",
      "Loop number 201, under   1.55575381946528543e-61, over   6.42775217703596110e+60\n",
      "Loop number 202, under   7.77876909732642713e-62, over   1.28555043540719222e+61\n",
      "Loop number 203, under   3.88938454866321357e-62, over   2.57110087081438444e+61\n",
      "Loop number 204, under   1.94469227433160678e-62, over   5.14220174162876888e+61\n",
      "Loop number 205, under   9.72346137165803392e-63, over   1.02844034832575378e+62\n",
      "Loop number 206, under   4.86173068582901696e-63, over   2.05688069665150755e+62\n",
      "Loop number 207, under   2.43086534291450848e-63, over   4.11376139330301511e+62\n",
      "Loop number 208, under   1.21543267145725424e-63, over   8.22752278660603021e+62\n",
      "Loop number 209, under   6.07716335728627120e-64, over   1.64550455732120604e+63\n",
      "Loop number 210, under   3.03858167864313560e-64, over   3.29100911464241208e+63\n",
      "Loop number 211, under   1.51929083932156780e-64, over   6.58201822928482417e+63\n",
      "Loop number 212, under   7.59645419660783900e-65, over   1.31640364585696483e+64\n",
      "Loop number 213, under   3.79822709830391950e-65, over   2.63280729171392967e+64\n",
      "Loop number 214, under   1.89911354915195975e-65, over   5.26561458342785933e+64\n",
      "Loop number 215, under   9.49556774575979875e-66, over   1.05312291668557187e+65\n",
      "Loop number 216, under   4.74778387287989937e-66, over   2.10624583337114373e+65\n",
      "Loop number 217, under   2.37389193643994969e-66, over   4.21249166674228747e+65\n",
      "Loop number 218, under   1.18694596821997484e-66, over   8.42498333348457494e+65\n",
      "Loop number 219, under   5.93472984109987422e-67, over   1.68499666669691499e+66\n",
      "Loop number 220, under   2.96736492054993711e-67, over   3.36999333339382997e+66\n",
      "Loop number 221, under   1.48368246027496855e-67, over   6.73998666678765995e+66\n",
      "Loop number 222, under   7.41841230137484277e-68, over   1.34799733335753199e+67\n",
      "Loop number 223, under   3.70920615068742139e-68, over   2.69599466671506398e+67\n",
      "Loop number 224, under   1.85460307534371069e-68, over   5.39198933343012796e+67\n",
      "Loop number 225, under   9.27301537671855346e-69, over   1.07839786668602559e+68\n",
      "Loop number 226, under   4.63650768835927673e-69, over   2.15679573337205118e+68\n",
      "Loop number 227, under   2.31825384417963837e-69, over   4.31359146674410237e+68\n",
      "Loop number 228, under   1.15912692208981918e-69, over   8.62718293348820473e+68\n",
      "Loop number 229, under   5.79563461044909592e-70, over   1.72543658669764095e+69\n",
      "Loop number 230, under   2.89781730522454796e-70, over   3.45087317339528189e+69\n",
      "Loop number 231, under   1.44890865261227398e-70, over   6.90174634679056379e+69\n",
      "Loop number 232, under   7.24454326306136989e-71, over   1.38034926935811276e+70\n",
      "Loop number 233, under   3.62227163153068495e-71, over   2.76069853871622551e+70\n",
      "Loop number 234, under   1.81113581576534247e-71, over   5.52139707743245103e+70\n",
      "Loop number 235, under   9.05567907882671237e-72, over   1.10427941548649021e+71\n",
      "Loop number 236, under   4.52783953941335618e-72, over   2.20855883097298041e+71\n",
      "Loop number 237, under   2.26391976970667809e-72, over   4.41711766194596082e+71\n",
      "Loop number 238, under   1.13195988485333905e-72, over   8.83423532389192165e+71\n",
      "Loop number 239, under   5.65979942426669523e-73, over   1.76684706477838433e+72\n",
      "Loop number 240, under   2.82989971213334761e-73, over   3.53369412955676866e+72\n",
      "Loop number 241, under   1.41494985606667381e-73, over   7.06738825911353732e+72\n",
      "Loop number 242, under   7.07474928033336904e-74, over   1.41347765182270746e+73\n",
      "Loop number 243, under   3.53737464016668452e-74, over   2.82695530364541493e+73\n",
      "Loop number 244, under   1.76868732008334226e-74, over   5.65391060729082985e+73\n",
      "Loop number 245, under   8.84343660041671130e-75, over   1.13078212145816597e+74\n",
      "Loop number 246, under   4.42171830020835565e-75, over   2.26156424291633194e+74\n",
      "Loop number 247, under   2.21085915010417782e-75, over   4.52312848583266388e+74\n",
      "Loop number 248, under   1.10542957505208891e-75, over   9.04625697166532777e+74\n",
      "Loop number 249, under   5.52714787526044456e-76, over   1.80925139433306555e+75\n",
      "Loop number 250, under   2.76357393763022228e-76, over   3.61850278866613111e+75\n",
      "Loop number 251, under   1.38178696881511114e-76, over   7.23700557733226221e+75\n",
      "Loop number 252, under   6.90893484407555570e-77, over   1.44740111546645244e+76\n",
      "Loop number 253, under   3.45446742203777785e-77, over   2.89480223093290489e+76\n",
      "Loop number 254, under   1.72723371101888893e-77, over   5.78960446186580977e+76\n",
      "Loop number 255, under   8.63616855509444463e-78, over   1.15792089237316195e+77\n",
      "Loop number 256, under   4.31808427754722231e-78, over   2.31584178474632391e+77\n",
      "Loop number 257, under   2.15904213877361116e-78, over   4.63168356949264782e+77\n",
      "Loop number 258, under   1.07952106938680558e-78, over   9.26336713898529563e+77\n",
      "Loop number 259, under   5.39760534693402789e-79, over   1.85267342779705913e+78\n",
      "Loop number 260, under   2.69880267346701395e-79, over   3.70534685559411825e+78\n",
      "Loop number 261, under   1.34940133673350697e-79, over   7.41069371118823651e+78\n",
      "Loop number 262, under   6.74700668366753486e-80, over   1.48213874223764730e+79\n",
      "Loop number 263, under   3.37350334183376743e-80, over   2.96427748447529460e+79\n",
      "Loop number 264, under   1.68675167091688372e-80, over   5.92855496895058921e+79\n",
      "Loop number 265, under   8.43375835458441858e-81, over   1.18571099379011784e+80\n",
      "Loop number 266, under   4.21687917729220929e-81, over   2.37142198758023568e+80\n",
      "Loop number 267, under   2.10843958864610464e-81, over   4.74284397516047136e+80\n",
      "Loop number 268, under   1.05421979432305232e-81, over   9.48568795032094273e+80\n",
      "Loop number 269, under   5.27109897161526161e-82, over   1.89713759006418855e+81\n",
      "Loop number 270, under   2.63554948580763081e-82, over   3.79427518012837709e+81\n",
      "Loop number 271, under   1.31777474290381540e-82, over   7.58855036025675418e+81\n",
      "Loop number 272, under   6.58887371451907702e-83, over   1.51771007205135084e+82\n",
      "Loop number 273, under   3.29443685725953851e-83, over   3.03542014410270167e+82\n",
      "Loop number 274, under   1.64721842862976925e-83, over   6.07084028820540335e+82\n",
      "Loop number 275, under   8.23609214314884627e-84, over   1.21416805764108067e+83\n",
      "Loop number 276, under   4.11804607157442313e-84, over   2.42833611528216134e+83\n",
      "Loop number 277, under   2.05902303578721157e-84, over   4.85667223056432268e+83\n",
      "Loop number 278, under   1.02951151789360578e-84, over   9.71334446112864535e+83\n",
      "Loop number 279, under   5.14755758946802892e-85, over   1.94266889222572907e+84\n",
      "Loop number 280, under   2.57377879473401446e-85, over   3.88533778445145814e+84\n",
      "Loop number 281, under   1.28688939736700723e-85, over   7.77067556890291628e+84\n",
      "Loop number 282, under   6.43444698683503615e-86, over   1.55413511378058326e+85\n",
      "Loop number 283, under   3.21722349341751807e-86, over   3.10827022756116651e+85\n",
      "Loop number 284, under   1.60861174670875904e-86, over   6.21654045512233303e+85\n",
      "Loop number 285, under   8.04305873354379518e-87, over   1.24330809102446661e+86\n",
      "Loop number 286, under   4.02152936677189759e-87, over   2.48661618204893321e+86\n",
      "Loop number 287, under   2.01076468338594880e-87, over   4.97323236409786642e+86\n",
      "Loop number 288, under   1.00538234169297440e-87, over   9.94646472819573284e+86\n",
      "Loop number 289, under   5.02691170846487199e-88, over   1.98929294563914657e+87\n",
      "Loop number 290, under   2.51345585423243600e-88, over   3.97858589127829314e+87\n",
      "Loop number 291, under   1.25672792711621800e-88, over   7.95717178255658627e+87\n",
      "Loop number 292, under   6.28363963558108999e-89, over   1.59143435651131725e+88\n",
      "Loop number 293, under   3.14181981779054499e-89, over   3.18286871302263451e+88\n",
      "Loop number 294, under   1.57090990889527250e-89, over   6.36573742604526902e+88\n",
      "Loop number 295, under   7.85454954447636248e-90, over   1.27314748520905380e+89\n",
      "Loop number 296, under   3.92727477223818124e-90, over   2.54629497041810761e+89\n",
      "Loop number 297, under   1.96363738611909062e-90, over   5.09258994083621522e+89\n",
      "Loop number 298, under   9.81818693059545311e-91, over   1.01851798816724304e+90\n",
      "Loop number 299, under   4.90909346529772655e-91, over   2.03703597633448609e+90\n",
      "Loop number 300, under   2.45454673264886328e-91, over   4.07407195266897217e+90\n",
      "Loop number 301, under   1.22727336632443164e-91, over   8.14814390533794435e+90\n",
      "Loop number 302, under   6.13636683162215819e-92, over   1.62962878106758887e+91\n",
      "Loop number 303, under   3.06818341581107910e-92, over   3.25925756213517774e+91\n",
      "Loop number 304, under   1.53409170790553955e-92, over   6.51851512427035548e+91\n",
      "Loop number 305, under   7.67045853952769774e-93, over   1.30370302485407110e+92\n",
      "Loop number 306, under   3.83522926976384887e-93, over   2.60740604970814219e+92\n",
      "Loop number 307, under   1.91761463488192443e-93, over   5.21481209941628438e+92\n",
      "Loop number 308, under   9.58807317440962217e-94, over   1.04296241988325688e+93\n",
      "Loop number 309, under   4.79403658720481109e-94, over   2.08592483976651375e+93\n",
      "Loop number 310, under   2.39701829360240554e-94, over   4.17184967953302750e+93\n",
      "Loop number 311, under   1.19850914680120277e-94, over   8.34369935906605501e+93\n",
      "Loop number 312, under   5.99254573400601386e-95, over   1.66873987181321100e+94\n",
      "Loop number 313, under   2.99627286700300693e-95, over   3.33747974362642200e+94\n",
      "Loop number 314, under   1.49813643350150346e-95, over   6.67495948725284401e+94\n",
      "Loop number 315, under   7.49068216750751732e-96, over   1.33499189745056880e+95\n",
      "Loop number 316, under   3.74534108375375866e-96, over   2.66998379490113760e+95\n",
      "Loop number 317, under   1.87267054187687933e-96, over   5.33996758980227521e+95\n",
      "Loop number 318, under   9.36335270938439665e-97, over   1.06799351796045504e+96\n",
      "Loop number 319, under   4.68167635469219833e-97, over   2.13598703592091008e+96\n",
      "Loop number 320, under   2.34083817734609916e-97, over   4.27197407184182016e+96\n",
      "Loop number 321, under   1.17041908867304958e-97, over   8.54394814368364033e+96\n",
      "Loop number 322, under   5.85209544336524791e-98, over   1.70878962873672807e+97\n",
      "Loop number 323, under   2.92604772168262395e-98, over   3.41757925747345613e+97\n",
      "Loop number 324, under   1.46302386084131198e-98, over   6.83515851494691226e+97\n",
      "Loop number 325, under   7.31511930420655989e-99, over   1.36703170298938245e+98\n",
      "Loop number 326, under   3.65755965210327994e-99, over   2.73406340597876491e+98\n",
      "Loop number 327, under   1.82877982605163997e-99, over   5.46812681195752981e+98\n",
      "Loop number 328, under  9.14389913025819986e-100, over   1.09362536239150596e+99\n",
      "Loop number 329, under  4.57194956512909993e-100, over   2.18725072478301192e+99\n",
      "Loop number 330, under  2.28597478256454996e-100, over   4.37450144956602385e+99\n",
      "Loop number 331, under  1.14298739128227498e-100, over   8.74900289913204770e+99\n",
      "Loop number 332, under  5.71493695641137491e-101, over  1.74980057982640954e+100\n",
      "Loop number 333, under  2.85746847820568746e-101, over  3.49960115965281908e+100\n",
      "Loop number 334, under  1.42873423910284373e-101, over  6.99920231930563816e+100\n",
      "Loop number 335, under  7.14367119551421864e-102, over  1.39984046386112763e+101\n",
      "Loop number 336, under  3.57183559775710932e-102, over  2.79968092772225526e+101\n",
      "Loop number 337, under  1.78591779887855466e-102, over  5.59936185544451053e+101\n",
      "Loop number 338, under  8.92958899439277330e-103, over  1.11987237108890211e+102\n",
      "Loop number 339, under  4.46479449719638665e-103, over  2.23974474217780421e+102\n",
      "Loop number 340, under  2.23239724859819332e-103, over  4.47948948435560842e+102\n",
      "Loop number 341, under  1.11619862429909666e-103, over  8.95897896871121684e+102\n",
      "Loop number 342, under  5.58099312149548331e-104, over  1.79179579374224337e+103\n",
      "Loop number 343, under  2.79049656074774166e-104, over  3.58359158748448674e+103\n",
      "Loop number 344, under  1.39524828037387083e-104, over  7.16718317496897347e+103\n",
      "Loop number 345, under  6.97624140186935414e-105, over  1.43343663499379469e+104\n",
      "Loop number 346, under  3.48812070093467707e-105, over  2.86687326998758939e+104\n",
      "Loop number 347, under  1.74406035046733853e-105, over  5.73374653997517878e+104\n",
      "Loop number 348, under  8.72030175233669267e-106, over  1.14674930799503576e+105\n",
      "Loop number 349, under  4.36015087616834634e-106, over  2.29349861599007151e+105\n",
      "Loop number 350, under  2.18007543808417317e-106, over  4.58699723198014302e+105\n",
      "Loop number 351, under  1.09003771904208658e-106, over  9.17399446396028605e+105\n",
      "Loop number 352, under  5.45018859521043292e-107, over  1.83479889279205721e+106\n",
      "Loop number 353, under  2.72509429760521646e-107, over  3.66959778558411442e+106\n",
      "Loop number 354, under  1.36254714880260823e-107, over  7.33919557116822884e+106\n",
      "Loop number 355, under  6.81273574401304115e-108, over  1.46783911423364577e+107\n",
      "Loop number 356, under  3.40636787200652058e-108, over  2.93567822846729153e+107\n",
      "Loop number 357, under  1.70318393600326029e-108, over  5.87135645693458307e+107\n",
      "Loop number 358, under  8.51591968001630144e-109, over  1.17427129138691661e+108\n",
      "Loop number 359, under  4.25795984000815072e-109, over  2.34854258277383323e+108\n",
      "Loop number 360, under  2.12897992000407536e-109, over  4.69708516554766646e+108\n",
      "Loop number 361, under  1.06448996000203768e-109, over  9.39417033109533291e+108\n",
      "Loop number 362, under  5.32244980001018840e-110, over  1.87883406621906658e+109\n",
      "Loop number 363, under  2.66122490000509420e-110, over  3.75766813243813316e+109\n",
      "Loop number 364, under  1.33061245000254710e-110, over  7.51533626487626633e+109\n",
      "Loop number 365, under  6.65306225001273550e-111, over  1.50306725297525327e+110\n",
      "Loop number 366, under  3.32653112500636775e-111, over  3.00613450595050653e+110\n",
      "Loop number 367, under  1.66326556250318387e-111, over  6.01226901190101306e+110\n",
      "Loop number 368, under  8.31632781251591937e-112, over  1.20245380238020261e+111\n",
      "Loop number 369, under  4.15816390625795969e-112, over  2.40490760476040523e+111\n",
      "Loop number 370, under  2.07908195312897984e-112, over  4.80981520952081045e+111\n",
      "Loop number 371, under  1.03954097656448992e-112, over  9.61963041904162090e+111\n",
      "Loop number 372, under  5.19770488282244961e-113, over  1.92392608380832418e+112\n",
      "Loop number 373, under  2.59885244141122480e-113, over  3.84785216761664836e+112\n",
      "Loop number 374, under  1.29942622070561240e-113, over  7.69570433523329672e+112\n",
      "Loop number 375, under  6.49713110352806201e-114, over  1.53914086704665934e+113\n",
      "Loop number 376, under  3.24856555176403101e-114, over  3.07828173409331869e+113\n",
      "Loop number 377, under  1.62428277588201550e-114, over  6.15656346818663738e+113\n",
      "Loop number 378, under  8.12141387941007751e-115, over  1.23131269363732748e+114\n",
      "Loop number 379, under  4.06070693970503876e-115, over  2.46262538727465495e+114\n",
      "Loop number 380, under  2.03035346985251938e-115, over  4.92525077454930990e+114\n",
      "Loop number 381, under  1.01517673492625969e-115, over  9.85050154909861980e+114\n",
      "Loop number 382, under  5.07588367463129845e-116, over  1.97010030981972396e+115\n",
      "Loop number 383, under  2.53794183731564922e-116, over  3.94020061963944792e+115\n",
      "Loop number 384, under  1.26897091865782461e-116, over  7.88040123927889584e+115\n",
      "Loop number 385, under  6.34485459328912306e-117, over  1.57608024785577917e+116\n",
      "Loop number 386, under  3.17242729664456153e-117, over  3.15216049571155834e+116\n",
      "Loop number 387, under  1.58621364832228076e-117, over  6.30432099142311667e+116\n",
      "Loop number 388, under  7.93106824161140382e-118, over  1.26086419828462333e+117\n",
      "Loop number 389, under  3.96553412080570191e-118, over  2.52172839656924667e+117\n",
      "Loop number 390, under  1.98276706040285096e-118, over  5.04345679313849334e+117\n",
      "Loop number 391, under  9.91383530201425478e-119, over  1.00869135862769867e+118\n",
      "Loop number 392, under  4.95691765100712739e-119, over  2.01738271725539734e+118\n",
      "Loop number 393, under  2.47845882550356369e-119, over  4.03476543451079467e+118\n",
      "Loop number 394, under  1.23922941275178185e-119, over  8.06953086902158934e+118\n",
      "Loop number 395, under  6.19614706375890924e-120, over  1.61390617380431787e+119\n",
      "Loop number 396, under  3.09807353187945462e-120, over  3.22781234760863574e+119\n",
      "Loop number 397, under  1.54903676593972731e-120, over  6.45562469521727147e+119\n",
      "Loop number 398, under  7.74518382969863655e-121, over  1.29112493904345429e+120\n",
      "Loop number 399, under  3.87259191484931827e-121, over  2.58224987808690859e+120\n",
      "Loop number 400, under  1.93629595742465914e-121, over  5.16449975617381718e+120\n",
      "Loop number 401, under  9.68147978712329568e-122, over  1.03289995123476344e+121\n",
      "Loop number 402, under  4.84073989356164784e-122, over  2.06579990246952687e+121\n",
      "Loop number 403, under  2.42036994678082392e-122, over  4.13159980493905374e+121\n",
      "Loop number 404, under  1.21018497339041196e-122, over  8.26319960987810749e+121\n",
      "Loop number 405, under  6.05092486695205980e-123, over  1.65263992197562150e+122\n",
      "Loop number 406, under  3.02546243347602990e-123, over  3.30527984395124299e+122\n",
      "Loop number 407, under  1.51273121673801495e-123, over  6.61055968790248599e+122\n",
      "Loop number 408, under  7.56365608369007475e-124, over  1.32211193758049720e+123\n",
      "Loop number 409, under  3.78182804184503738e-124, over  2.64422387516099440e+123\n",
      "Loop number 410, under  1.89091402092251869e-124, over  5.28844775032198879e+123\n",
      "Loop number 411, under  9.45457010461259344e-125, over  1.05768955006439776e+124\n",
      "Loop number 412, under  4.72728505230629672e-125, over  2.11537910012879552e+124\n",
      "Loop number 413, under  2.36364252615314836e-125, over  4.23075820025759103e+124\n",
      "Loop number 414, under  1.18182126307657418e-125, over  8.46151640051518207e+124\n",
      "Loop number 415, under  5.90910631538287090e-126, over  1.69230328010303641e+125\n",
      "Loop number 416, under  2.95455315769143545e-126, over  3.38460656020607283e+125\n",
      "Loop number 417, under  1.47727657884571772e-126, over  6.76921312041214565e+125\n",
      "Loop number 418, under  7.38638289422858862e-127, over  1.35384262408242913e+126\n",
      "Loop number 419, under  3.69319144711429431e-127, over  2.70768524816485826e+126\n",
      "Loop number 420, under  1.84659572355714716e-127, over  5.41537049632971652e+126\n",
      "Loop number 421, under  9.23297861778573578e-128, over  1.08307409926594330e+127\n",
      "Loop number 422, under  4.61648930889286789e-128, over  2.16614819853188661e+127\n",
      "Loop number 423, under  2.30824465444643395e-128, over  4.33229639706377322e+127\n",
      "Loop number 424, under  1.15412232722321697e-128, over  8.66459279412754644e+127\n",
      "Loop number 425, under  5.77061163611608486e-129, over  1.73291855882550929e+128\n",
      "Loop number 426, under  2.88530581805804243e-129, over  3.46583711765101857e+128\n",
      "Loop number 427, under  1.44265290902902122e-129, over  6.93167423530203715e+128\n",
      "Loop number 428, under  7.21326454514510608e-130, over  1.38633484706040743e+129\n",
      "Loop number 429, under  3.60663227257255304e-130, over  2.77266969412081486e+129\n",
      "Loop number 430, under  1.80331613628627652e-130, over  5.54533938824162972e+129\n",
      "Loop number 431, under  9.01658068143138260e-131, over  1.10906787764832594e+130\n",
      "Loop number 432, under  4.50829034071569130e-131, over  2.21813575529665189e+130\n",
      "Loop number 433, under  2.25414517035784565e-131, over  4.43627151059330378e+130\n",
      "Loop number 434, under  1.12707258517892282e-131, over  8.87254302118660755e+130\n",
      "Loop number 435, under  5.63536292589461412e-132, over  1.77450860423732151e+131\n",
      "Loop number 436, under  2.81768146294730706e-132, over  3.54901720847464302e+131\n",
      "Loop number 437, under  1.40884073147365353e-132, over  7.09803441694928604e+131\n",
      "Loop number 438, under  7.04420365736826765e-133, over  1.41960688338985721e+132\n",
      "Loop number 439, under  3.52210182868413383e-133, over  2.83921376677971442e+132\n",
      "Loop number 440, under  1.76105091434206691e-133, over  5.67842753355942883e+132\n",
      "Loop number 441, under  8.80525457171033457e-134, over  1.13568550671188577e+133\n",
      "Loop number 442, under  4.40262728585516728e-134, over  2.27137101342377153e+133\n",
      "Loop number 443, under  2.20131364292758364e-134, over  4.54274202684754307e+133\n",
      "Loop number 444, under  1.10065682146379182e-134, over  9.08548405369508613e+133\n",
      "Loop number 445, under  5.50328410731895911e-135, over  1.81709681073901723e+134\n",
      "Loop number 446, under  2.75164205365947955e-135, over  3.63419362147803445e+134\n",
      "Loop number 447, under  1.37582102682973978e-135, over  7.26838724295606891e+134\n",
      "Loop number 448, under  6.87910513414869888e-136, over  1.45367744859121378e+135\n",
      "Loop number 449, under  3.43955256707434944e-136, over  2.90735489718242756e+135\n",
      "Loop number 450, under  1.71977628353717472e-136, over  5.81470979436485512e+135\n",
      "Loop number 451, under  8.59888141768587360e-137, over  1.16294195887297102e+136\n",
      "Loop number 452, under  4.29944070884293680e-137, over  2.32588391774594205e+136\n",
      "Loop number 453, under  2.14972035442146840e-137, over  4.65176783549188410e+136\n",
      "Loop number 454, under  1.07486017721073420e-137, over  9.30353567098376820e+136\n",
      "Loop number 455, under  5.37430088605367100e-138, over  1.86070713419675364e+137\n",
      "Loop number 456, under  2.68715044302683550e-138, over  3.72141426839350728e+137\n",
      "Loop number 457, under  1.34357522151341775e-138, over  7.44282853678701456e+137\n",
      "Loop number 458, under  6.71787610756708875e-139, over  1.48856570735740291e+138\n",
      "Loop number 459, under  3.35893805378354438e-139, over  2.97713141471480582e+138\n",
      "Loop number 460, under  1.67946902689177219e-139, over  5.95426282942961165e+138\n",
      "Loop number 461, under  8.39734513445886094e-140, over  1.19085256588592233e+139\n",
      "Loop number 462, under  4.19867256722943047e-140, over  2.38170513177184466e+139\n",
      "Loop number 463, under  2.09933628361471523e-140, over  4.76341026354368932e+139\n",
      "Loop number 464, under  1.04966814180735762e-140, over  9.52682052708737864e+139\n",
      "Loop number 465, under  5.24834070903678809e-141, over  1.90536410541747573e+140\n",
      "Loop number 466, under  2.62417035451839404e-141, over  3.81072821083495145e+140\n",
      "Loop number 467, under  1.31208517725919702e-141, over  7.62145642166990291e+140\n",
      "Loop number 468, under  6.56042588629598511e-142, over  1.52429128433398058e+141\n",
      "Loop number 469, under  3.28021294314799255e-142, over  3.04858256866796116e+141\n",
      "Loop number 470, under  1.64010647157399628e-142, over  6.09716513733592233e+141\n",
      "Loop number 471, under  8.20053235786998139e-143, over  1.21943302746718447e+142\n",
      "Loop number 472, under  4.10026617893499069e-143, over  2.43886605493436893e+142\n",
      "Loop number 473, under  2.05013308946749535e-143, over  4.87773210986873786e+142\n",
      "Loop number 474, under  1.02506654473374767e-143, over  9.75546421973747572e+142\n",
      "Loop number 475, under  5.12533272366873837e-144, over  1.95109284394749514e+143\n",
      "Loop number 476, under  2.56266636183436918e-144, over  3.90218568789499029e+143\n",
      "Loop number 477, under  1.28133318091718459e-144, over  7.80437137578998058e+143\n",
      "Loop number 478, under  6.40666590458592296e-145, over  1.56087427515799612e+144\n",
      "Loop number 479, under  3.20333295229296148e-145, over  3.12174855031599223e+144\n",
      "Loop number 480, under  1.60166647614648074e-145, over  6.24349710063198446e+144\n",
      "Loop number 481, under  8.00833238073240370e-146, over  1.24869942012639689e+145\n",
      "Loop number 482, under  4.00416619036620185e-146, over  2.49739884025279379e+145\n",
      "Loop number 483, under  2.00208309518310092e-146, over  4.99479768050558757e+145\n",
      "Loop number 484, under  1.00104154759155046e-146, over  9.98959536101117514e+145\n",
      "Loop number 485, under  5.00520773795775231e-147, over  1.99791907220223503e+146\n",
      "Loop number 486, under  2.50260386897887616e-147, over  3.99583814440447006e+146\n",
      "Loop number 487, under  1.25130193448943808e-147, over  7.99167628880894011e+146\n",
      "Loop number 488, under  6.25650967244719039e-148, over  1.59833525776178802e+147\n",
      "Loop number 489, under  3.12825483622359519e-148, over  3.19667051552357604e+147\n",
      "Loop number 490, under  1.56412741811179760e-148, over  6.39334103104715209e+147\n",
      "Loop number 491, under  7.82063709055898799e-149, over  1.27866820620943042e+148\n",
      "Loop number 492, under  3.91031854527949399e-149, over  2.55733641241886084e+148\n",
      "Loop number 493, under  1.95515927263974700e-149, over  5.11467282483772167e+148\n",
      "Loop number 494, under  9.77579636319873498e-150, over  1.02293456496754433e+149\n",
      "Loop number 495, under  4.88789818159936749e-150, over  2.04586912993508867e+149\n",
      "Loop number 496, under  2.44394909079968375e-150, over  4.09173825987017734e+149\n",
      "Loop number 497, under  1.22197454539984187e-150, over  8.18347651974035468e+149\n",
      "Loop number 498, under  6.10987272699920936e-151, over  1.63669530394807094e+150\n",
      "Loop number 499, under  3.05493636349960468e-151, over  3.27339060789614187e+150\n",
      "Loop number 500, under  1.52746818174980234e-151, over  6.54678121579228374e+150\n",
      "Loop number 501, under  7.63734090874901171e-152, over  1.30935624315845675e+151\n",
      "Loop number 502, under  3.81867045437450585e-152, over  2.61871248631691350e+151\n",
      "Loop number 503, under  1.90933522718725293e-152, over  5.23742497263382699e+151\n",
      "Loop number 504, under  9.54667613593626463e-153, over  1.04748499452676540e+152\n",
      "Loop number 505, under  4.77333806796813232e-153, over  2.09496998905353080e+152\n",
      "Loop number 506, under  2.38666903398406616e-153, over  4.18993997810706159e+152\n",
      "Loop number 507, under  1.19333451699203308e-153, over  8.37987995621412319e+152\n",
      "Loop number 508, under  5.96667258496016539e-154, over  1.67597599124282464e+153\n",
      "Loop number 509, under  2.98333629248008270e-154, over  3.35195198248564927e+153\n",
      "Loop number 510, under  1.49166814624004135e-154, over  6.70390396497129855e+153\n",
      "Loop number 511, under  7.45834073120020674e-155, over  1.34078079299425971e+154\n",
      "Loop number 512, under  3.72917036560010337e-155, over  2.68156158598851942e+154\n",
      "Loop number 513, under  1.86458518280005169e-155, over  5.36312317197703884e+154\n",
      "Loop number 514, under  9.32292591400025843e-156, over  1.07262463439540777e+155\n",
      "Loop number 515, under  4.66146295700012921e-156, over  2.14524926879081554e+155\n",
      "Loop number 516, under  2.33073147850006461e-156, over  4.29049853758163107e+155\n",
      "Loop number 517, under  1.16536573925003230e-156, over  8.58099707516326214e+155\n",
      "Loop number 518, under  5.82682869625016152e-157, over  1.71619941503265243e+156\n",
      "Loop number 519, under  2.91341434812508076e-157, over  3.43239883006530486e+156\n",
      "Loop number 520, under  1.45670717406254038e-157, over  6.86479766013060971e+156\n",
      "Loop number 521, under  7.28353587031270190e-158, over  1.37295953202612194e+157\n",
      "Loop number 522, under  3.64176793515635095e-158, over  2.74591906405224389e+157\n",
      "Loop number 523, under  1.82088396757817547e-158, over  5.49183812810448777e+157\n",
      "Loop number 524, under  9.10441983789087737e-159, over  1.09836762562089755e+158\n",
      "Loop number 525, under  4.55220991894543869e-159, over  2.19673525124179511e+158\n",
      "Loop number 526, under  2.27610495947271934e-159, over  4.39347050248359022e+158\n",
      "Loop number 527, under  1.13805247973635967e-159, over  8.78694100496718044e+158\n",
      "Loop number 528, under  5.69026239868179836e-160, over  1.75738820099343609e+159\n",
      "Loop number 529, under  2.84513119934089918e-160, over  3.51477640198687217e+159\n",
      "Loop number 530, under  1.42256559967044959e-160, over  7.02955280397374435e+159\n",
      "Loop number 531, under  7.11282799835224795e-161, over  1.40591056079474887e+160\n",
      "Loop number 532, under  3.55641399917612397e-161, over  2.81182112158949774e+160\n",
      "Loop number 533, under  1.77820699958806199e-161, over  5.62364224317899548e+160\n",
      "Loop number 534, under  8.89103499794030993e-162, over  1.12472844863579910e+161\n",
      "Loop number 535, under  4.44551749897015497e-162, over  2.24945689727159819e+161\n",
      "Loop number 536, under  2.22275874948507748e-162, over  4.49891379454319638e+161\n",
      "Loop number 537, under  1.11137937474253874e-162, over  8.99782758908639277e+161\n",
      "Loop number 538, under  5.55689687371269371e-163, over  1.79956551781727855e+162\n",
      "Loop number 539, under  2.77844843685634685e-163, over  3.59913103563455711e+162\n",
      "Loop number 540, under  1.38922421842817343e-163, over  7.19826207126911421e+162\n",
      "Loop number 541, under  6.94612109214086714e-164, over  1.43965241425382284e+163\n",
      "Loop number 542, under  3.47306054607043357e-164, over  2.87930482850764568e+163\n",
      "Loop number 543, under  1.73653027303521678e-164, over  5.75860965701529137e+163\n",
      "Loop number 544, under  8.68265136517608392e-165, over  1.15172193140305827e+164\n",
      "Loop number 545, under  4.34132568258804196e-165, over  2.30344386280611655e+164\n",
      "Loop number 546, under  2.17066284129402098e-165, over  4.60688772561223310e+164\n",
      "Loop number 547, under  1.08533142064701049e-165, over  9.21377545122446619e+164\n",
      "Loop number 548, under  5.42665710323505245e-166, over  1.84275509024489324e+165\n",
      "Loop number 549, under  2.71332855161752622e-166, over  3.68551018048978648e+165\n",
      "Loop number 550, under  1.35666427580876311e-166, over  7.37102036097957295e+165\n",
      "Loop number 551, under  6.78332137904381556e-167, over  1.47420407219591459e+166\n",
      "Loop number 552, under  3.39166068952190778e-167, over  2.94840814439182918e+166\n",
      "Loop number 553, under  1.69583034476095389e-167, over  5.89681628878365836e+166\n",
      "Loop number 554, under  8.47915172380476945e-168, over  1.17936325775673167e+167\n",
      "Loop number 555, under  4.23957586190238473e-168, over  2.35872651551346335e+167\n",
      "Loop number 556, under  2.11978793095119236e-168, over  4.71745303102692669e+167\n",
      "Loop number 557, under  1.05989396547559618e-168, over  9.43490606205385338e+167\n",
      "Loop number 558, under  5.29946982737798091e-169, over  1.88698121241077068e+168\n",
      "Loop number 559, under  2.64973491368899045e-169, over  3.77396242482154135e+168\n",
      "Loop number 560, under  1.32486745684449523e-169, over  7.54792484964308270e+168\n",
      "Loop number 561, under  6.62433728422247614e-170, over  1.50958496992861654e+169\n",
      "Loop number 562, under  3.31216864211123807e-170, over  3.01916993985723308e+169\n",
      "Loop number 563, under  1.65608432105561903e-170, over  6.03833987971446616e+169\n",
      "Loop number 564, under  8.28042160527809517e-171, over  1.20766797594289323e+170\n",
      "Loop number 565, under  4.14021080263904758e-171, over  2.41533595188578647e+170\n",
      "Loop number 566, under  2.07010540131952379e-171, over  4.83067190377157293e+170\n",
      "Loop number 567, under  1.03505270065976190e-171, over  9.66134380754314586e+170\n",
      "Loop number 568, under  5.17526350329880948e-172, over  1.93226876150862917e+171\n",
      "Loop number 569, under  2.58763175164940474e-172, over  3.86453752301725834e+171\n",
      "Loop number 570, under  1.29381587582470237e-172, over  7.72907504603451669e+171\n",
      "Loop number 571, under  6.46907937912351185e-173, over  1.54581500920690334e+172\n",
      "Loop number 572, under  3.23453968956175593e-173, over  3.09163001841380668e+172\n",
      "Loop number 573, under  1.61726984478087796e-173, over  6.18326003682761335e+172\n",
      "Loop number 574, under  8.08634922390438981e-174, over  1.23665200736552267e+173\n",
      "Loop number 575, under  4.04317461195219491e-174, over  2.47330401473104534e+173\n",
      "Loop number 576, under  2.02158730597609745e-174, over  4.94660802946209068e+173\n",
      "Loop number 577, under  1.01079365298804873e-174, over  9.89321605892418136e+173\n",
      "Loop number 578, under  5.05396826494024363e-175, over  1.97864321178483627e+174\n",
      "Loop number 579, under  2.52698413247012182e-175, over  3.95728642356967254e+174\n",
      "Loop number 580, under  1.26349206623506091e-175, over  7.91457284713934509e+174\n",
      "Loop number 581, under  6.31746033117530454e-176, over  1.58291456942786902e+175\n",
      "Loop number 582, under  3.15873016558765227e-176, over  3.16582913885573804e+175\n",
      "Loop number 583, under  1.57936508279382614e-176, over  6.33165827771147607e+175\n",
      "Loop number 584, under  7.89682541396913068e-177, over  1.26633165554229521e+176\n",
      "Loop number 585, under  3.94841270698456534e-177, over  2.53266331108459043e+176\n",
      "Loop number 586, under  1.97420635349228267e-177, over  5.06532662216918086e+176\n",
      "Loop number 587, under  9.87103176746141335e-178, over  1.01306532443383617e+177\n",
      "Loop number 588, under  4.93551588373070667e-178, over  2.02613064886767234e+177\n",
      "Loop number 589, under  2.46775794186535334e-178, over  4.05226129773534469e+177\n",
      "Loop number 590, under  1.23387897093267667e-178, over  8.10452259547068937e+177\n",
      "Loop number 591, under  6.16939485466338334e-179, over  1.62090451909413787e+178\n",
      "Loop number 592, under  3.08469742733169167e-179, over  3.24180903818827575e+178\n",
      "Loop number 593, under  1.54234871366584584e-179, over  6.48361807637655150e+178\n",
      "Loop number 594, under  7.71174356832922918e-180, over  1.29672361527531030e+179\n",
      "Loop number 595, under  3.85587178416461459e-180, over  2.59344723055062060e+179\n",
      "Loop number 596, under  1.92793589208230729e-180, over  5.18689446110124120e+179\n",
      "Loop number 597, under  9.63967946041153647e-181, over  1.03737889222024824e+180\n",
      "Loop number 598, under  4.81983973020576824e-181, over  2.07475778444049648e+180\n",
      "Loop number 599, under  2.40991986510288412e-181, over  4.14951556888099296e+180\n",
      "Loop number 600, under  1.20495993255144206e-181, over  8.29903113776198592e+180\n",
      "Loop number 601, under  6.02479966275721029e-182, over  1.65980622755239718e+181\n",
      "Loop number 602, under  3.01239983137860515e-182, over  3.31961245510479437e+181\n",
      "Loop number 603, under  1.50619991568930257e-182, over  6.63922491020958873e+181\n",
      "Loop number 604, under  7.53099957844651287e-183, over  1.32784498204191775e+182\n",
      "Loop number 605, under  3.76549978922325643e-183, over  2.65568996408383549e+182\n",
      "Loop number 606, under  1.88274989461162822e-183, over  5.31137992816767099e+182\n",
      "Loop number 607, under  9.41374947305814108e-184, over  1.06227598563353420e+183\n",
      "Loop number 608, under  4.70687473652907054e-184, over  2.12455197126706839e+183\n",
      "Loop number 609, under  2.35343736826453527e-184, over  4.24910394253413679e+183\n",
      "Loop number 610, under  1.17671868413226764e-184, over  8.49820788506827358e+183\n",
      "Loop number 611, under  5.88359342066133818e-185, over  1.69964157701365472e+184\n",
      "Loop number 612, under  2.94179671033066909e-185, over  3.39928315402730943e+184\n",
      "Loop number 613, under  1.47089835516533454e-185, over  6.79856630805461886e+184\n",
      "Loop number 614, under  7.35449177582667272e-186, over  1.35971326161092377e+185\n",
      "Loop number 615, under  3.67724588791333636e-186, over  2.71942652322184755e+185\n",
      "Loop number 616, under  1.83862294395666818e-186, over  5.43885304644369509e+185\n",
      "Loop number 617, under  9.19311471978334090e-187, over  1.08777060928873902e+186\n",
      "Loop number 618, under  4.59655735989167045e-187, over  2.17554121857747804e+186\n",
      "Loop number 619, under  2.29827867994583523e-187, over  4.35108243715495607e+186\n",
      "Loop number 620, under  1.14913933997291761e-187, over  8.70216487430991214e+186\n",
      "Loop number 621, under  5.74569669986458806e-188, over  1.74043297486198243e+187\n",
      "Loop number 622, under  2.87284834993229403e-188, over  3.48086594972396486e+187\n",
      "Loop number 623, under  1.43642417496614702e-188, over  6.96173189944792972e+187\n",
      "Loop number 624, under  7.18212087483073508e-189, over  1.39234637988958594e+188\n",
      "Loop number 625, under  3.59106043741536754e-189, over  2.78469275977917189e+188\n",
      "Loop number 626, under  1.79553021870768377e-189, over  5.56938551955834377e+188\n",
      "Loop number 627, under  8.97765109353841885e-190, over  1.11387710391166875e+189\n",
      "Loop number 628, under  4.48882554676920943e-190, over  2.22775420782333751e+189\n",
      "Loop number 629, under  2.24441277338460471e-190, over  4.45550841564667502e+189\n",
      "Loop number 630, under  1.12220638669230236e-190, over  8.91101683129335004e+189\n",
      "Loop number 631, under  5.61103193346151178e-191, over  1.78220336625867001e+190\n",
      "Loop number 632, under  2.80551596673075589e-191, over  3.56440673251734001e+190\n",
      "Loop number 633, under  1.40275798336537795e-191, over  7.12881346503468003e+190\n",
      "Loop number 634, under  7.01378991682688973e-192, over  1.42576269300693601e+191\n",
      "Loop number 635, under  3.50689495841344486e-192, over  2.85152538601387201e+191\n",
      "Loop number 636, under  1.75344747920672243e-192, over  5.70305077202774402e+191\n",
      "Loop number 637, under  8.76723739603361216e-193, over  1.14061015440554880e+192\n",
      "Loop number 638, under  4.38361869801680608e-193, over  2.28122030881109761e+192\n",
      "Loop number 639, under  2.19180934900840304e-193, over  4.56244061762219522e+192\n",
      "Loop number 640, under  1.09590467450420152e-193, over  9.12488123524439044e+192\n",
      "Loop number 641, under  5.47952337252100760e-194, over  1.82497624704887809e+193\n",
      "Loop number 642, under  2.73976168626050380e-194, over  3.64995249409775617e+193\n",
      "Loop number 643, under  1.36988084313025190e-194, over  7.29990498819551235e+193\n",
      "Loop number 644, under  6.84940421565125950e-195, over  1.45998099763910247e+194\n",
      "Loop number 645, under  3.42470210782562975e-195, over  2.91996199527820494e+194\n",
      "Loop number 646, under  1.71235105391281487e-195, over  5.83992399055640988e+194\n",
      "Loop number 647, under  8.56175526956407437e-196, over  1.16798479811128198e+195\n",
      "Loop number 648, under  4.28087763478203719e-196, over  2.33596959622256395e+195\n",
      "Loop number 649, under  2.14043881739101859e-196, over  4.67193919244512790e+195\n",
      "Loop number 650, under  1.07021940869550930e-196, over  9.34387838489025581e+195\n",
      "Loop number 651, under  5.35109704347754648e-197, over  1.86877567697805116e+196\n",
      "Loop number 652, under  2.67554852173877324e-197, over  3.73755135395610232e+196\n",
      "Loop number 653, under  1.33777426086938662e-197, over  7.47510270791220465e+196\n",
      "Loop number 654, under  6.68887130434693310e-198, over  1.49502054158244093e+197\n",
      "Loop number 655, under  3.34443565217346655e-198, over  2.99004108316488186e+197\n",
      "Loop number 656, under  1.67221782608673328e-198, over  5.98008216632976372e+197\n",
      "Loop number 657, under  8.36108913043366638e-199, over  1.19601643326595274e+198\n",
      "Loop number 658, under  4.18054456521683319e-199, over  2.39203286653190549e+198\n",
      "Loop number 659, under  2.09027228260841660e-199, over  4.78406573306381097e+198\n",
      "Loop number 660, under  1.04513614130420830e-199, over  9.56813146612762195e+198\n",
      "Loop number 661, under  5.22568070652104149e-200, over  1.91362629322552439e+199\n",
      "Loop number 662, under  2.61284035326052074e-200, over  3.82725258645104878e+199\n",
      "Loop number 663, under  1.30642017663026037e-200, over  7.65450517290209756e+199\n",
      "Loop number 664, under  6.53210088315130186e-201, over  1.53090103458041951e+200\n",
      "Loop number 665, under  3.26605044157565093e-201, over  3.06180206916083902e+200\n",
      "Loop number 666, under  1.63302522078782547e-201, over  6.12360413832167805e+200\n",
      "Loop number 667, under  8.16512610393912733e-202, over  1.22472082766433561e+201\n",
      "Loop number 668, under  4.08256305196956366e-202, over  2.44944165532867122e+201\n",
      "Loop number 669, under  2.04128152598478183e-202, over  4.89888331065734244e+201\n",
      "Loop number 670, under  1.02064076299239092e-202, over  9.79776662131468487e+201\n",
      "Loop number 671, under  5.10320381496195458e-203, over  1.95955332426293697e+202\n",
      "Loop number 672, under  2.55160190748097729e-203, over  3.91910664852587395e+202\n",
      "Loop number 673, under  1.27580095374048864e-203, over  7.83821329705174790e+202\n",
      "Loop number 674, under  6.37900476870244322e-204, over  1.56764265941034958e+203\n",
      "Loop number 675, under  3.18950238435122161e-204, over  3.13528531882069916e+203\n",
      "Loop number 676, under  1.59475119217561081e-204, over  6.27057063764139832e+203\n",
      "Loop number 677, under  7.97375596087805403e-205, over  1.25411412752827966e+204\n",
      "Loop number 678, under  3.98687798043902701e-205, over  2.50822825505655933e+204\n",
      "Loop number 679, under  1.99343899021951351e-205, over  5.01645651011311866e+204\n",
      "Loop number 680, under  9.96719495109756754e-206, over  1.00329130202262373e+205\n",
      "Loop number 681, under  4.98359747554878377e-206, over  2.00658260404524746e+205\n",
      "Loop number 682, under  2.49179873777439188e-206, over  4.01316520809049492e+205\n",
      "Loop number 683, under  1.24589936888719594e-206, over  8.02633041618098985e+205\n",
      "Loop number 684, under  6.22949684443597971e-207, over  1.60526608323619797e+206\n",
      "Loop number 685, under  3.11474842221798985e-207, over  3.21053216647239594e+206\n",
      "Loop number 686, under  1.55737421110899493e-207, over  6.42106433294479188e+206\n",
      "Loop number 687, under  7.78687105554497464e-208, over  1.28421286658895838e+207\n",
      "Loop number 688, under  3.89343552777248732e-208, over  2.56842573317791675e+207\n",
      "Loop number 689, under  1.94671776388624366e-208, over  5.13685146635583350e+207\n",
      "Loop number 690, under  9.73358881943121830e-209, over  1.02737029327116670e+208\n",
      "Loop number 691, under  4.86679440971560915e-209, over  2.05474058654233340e+208\n",
      "Loop number 692, under  2.43339720485780457e-209, over  4.10948117308466680e+208\n",
      "Loop number 693, under  1.21669860242890229e-209, over  8.21896234616933361e+208\n",
      "Loop number 694, under  6.08349301214451144e-210, over  1.64379246923386672e+209\n",
      "Loop number 695, under  3.04174650607225572e-210, over  3.28758493846773344e+209\n",
      "Loop number 696, under  1.52087325303612786e-210, over  6.57516987693546688e+209\n",
      "Loop number 697, under  7.60436626518063929e-211, over  1.31503397538709338e+210\n",
      "Loop number 698, under  3.80218313259031965e-211, over  2.63006795077418675e+210\n",
      "Loop number 699, under  1.90109156629515982e-211, over  5.26013590154837351e+210\n",
      "Loop number 700, under  9.50545783147579912e-212, over  1.05202718030967470e+211\n",
      "Loop number 701, under  4.75272891573789956e-212, over  2.10405436061934940e+211\n",
      "Loop number 702, under  2.37636445786894978e-212, over  4.20810872123869881e+211\n",
      "Loop number 703, under  1.18818222893447489e-212, over  8.41621744247739761e+211\n",
      "Loop number 704, under  5.94091114467237445e-213, over  1.68324348849547952e+212\n",
      "Loop number 705, under  2.97045557233618722e-213, over  3.36648697699095904e+212\n",
      "Loop number 706, under  1.48522778616809361e-213, over  6.73297395398191809e+212\n",
      "Loop number 707, under  7.42613893084046806e-214, over  1.34659479079638362e+213\n",
      "Loop number 708, under  3.71306946542023403e-214, over  2.69318958159276724e+213\n",
      "Loop number 709, under  1.85653473271011702e-214, over  5.38637916318553447e+213\n",
      "Loop number 710, under  9.28267366355058508e-215, over  1.07727583263710689e+214\n",
      "Loop number 711, under  4.64133683177529254e-215, over  2.15455166527421379e+214\n",
      "Loop number 712, under  2.32066841588764627e-215, over  4.30910333054842758e+214\n",
      "Loop number 713, under  1.16033420794382313e-215, over  8.61820666109685515e+214\n",
      "Loop number 714, under  5.80167103971911567e-216, over  1.72364133221937103e+215\n",
      "Loop number 715, under  2.90083551985955784e-216, over  3.44728266443874206e+215\n",
      "Loop number 716, under  1.45041775992977892e-216, over  6.89456532887748412e+215\n",
      "Loop number 717, under  7.25208879964889459e-217, over  1.37891306577549682e+216\n",
      "Loop number 718, under  3.62604439982444730e-217, over  2.75782613155099365e+216\n",
      "Loop number 719, under  1.81302219991222365e-217, over  5.51565226310198730e+216\n",
      "Loop number 720, under  9.06511099956111824e-218, over  1.10313045262039746e+217\n",
      "Loop number 721, under  4.53255549978055912e-218, over  2.20626090524079492e+217\n",
      "Loop number 722, under  2.26627774989027956e-218, over  4.41252181048158984e+217\n",
      "Loop number 723, under  1.13313887494513978e-218, over  8.82504362096317968e+217\n",
      "Loop number 724, under  5.66569437472569890e-219, over  1.76500872419263594e+218\n",
      "Loop number 725, under  2.83284718736284945e-219, over  3.53001744838527187e+218\n",
      "Loop number 726, under  1.41642359368142472e-219, over  7.06003489677054374e+218\n",
      "Loop number 727, under  7.08211796840712362e-220, over  1.41200697935410875e+219\n",
      "Loop number 728, under  3.54105898420356181e-220, over  2.82401395870821750e+219\n",
      "Loop number 729, under  1.77052949210178091e-220, over  5.64802791741643499e+219\n",
      "Loop number 730, under  8.85264746050890453e-221, over  1.12960558348328700e+220\n",
      "Loop number 731, under  4.42632373025445226e-221, over  2.25921116696657400e+220\n",
      "Loop number 732, under  2.21316186512722613e-221, over  4.51842233393314800e+220\n",
      "Loop number 733, under  1.10658093256361307e-221, over  9.03684466786629599e+220\n",
      "Loop number 734, under  5.53290466281806533e-222, over  1.80736893357325920e+221\n",
      "Loop number 735, under  2.76645233140903267e-222, over  3.61473786714651840e+221\n",
      "Loop number 736, under  1.38322616570451633e-222, over  7.22947573429303679e+221\n",
      "Loop number 737, under  6.91613082852258166e-223, over  1.44589514685860736e+222\n",
      "Loop number 738, under  3.45806541426129083e-223, over  2.89179029371721472e+222\n",
      "Loop number 739, under  1.72903270713064542e-223, over  5.78358058743442943e+222\n",
      "Loop number 740, under  8.64516353565322708e-224, over  1.15671611748688589e+223\n",
      "Loop number 741, under  4.32258176782661354e-224, over  2.31343223497377177e+223\n",
      "Loop number 742, under  2.16129088391330677e-224, over  4.62686446994754355e+223\n",
      "Loop number 743, under  1.08064544195665338e-224, over  9.25372893989508709e+223\n",
      "Loop number 744, under  5.40322720978326692e-225, over  1.85074578797901742e+224\n",
      "Loop number 745, under  2.70161360489163346e-225, over  3.70149157595803484e+224\n",
      "Loop number 746, under  1.35080680244581673e-225, over  7.40298315191606968e+224\n",
      "Loop number 747, under  6.75403401222908366e-226, over  1.48059663038321394e+225\n",
      "Loop number 748, under  3.37701700611454183e-226, over  2.96119326076642787e+225\n",
      "Loop number 749, under  1.68850850305727091e-226, over  5.92238652153285574e+225\n",
      "Loop number 750, under  8.44254251528635457e-227, over  1.18447730430657115e+226\n",
      "Loop number 751, under  4.22127125764317728e-227, over  2.36895460861314230e+226\n",
      "Loop number 752, under  2.11063562882158864e-227, over  4.73790921722628459e+226\n",
      "Loop number 753, under  1.05531781441079432e-227, over  9.47581843445256918e+226\n",
      "Loop number 754, under  5.27658907205397161e-228, over  1.89516368689051384e+227\n",
      "Loop number 755, under  2.63829453602698580e-228, over  3.79032737378102767e+227\n",
      "Loop number 756, under  1.31914726801349290e-228, over  7.58065474756205535e+227\n",
      "Loop number 757, under  6.59573634006746451e-229, over  1.51613094951241107e+228\n",
      "Loop number 758, under  3.29786817003373225e-229, over  3.03226189902482214e+228\n",
      "Loop number 759, under  1.64893408501686613e-229, over  6.06452379804964428e+228\n",
      "Loop number 760, under  8.24467042508433063e-230, over  1.21290475960992886e+229\n",
      "Loop number 761, under  4.12233521254216532e-230, over  2.42580951921985771e+229\n",
      "Loop number 762, under  2.06116760627108266e-230, over  4.85161903843971542e+229\n",
      "Loop number 763, under  1.03058380313554133e-230, over  9.70323807687943084e+229\n",
      "Loop number 764, under  5.15291901567770665e-231, over  1.94064761537588617e+230\n",
      "Loop number 765, under  2.57645950783885332e-231, over  3.88129523075177234e+230\n",
      "Loop number 766, under  1.28822975391942666e-231, over  7.76259046150354468e+230\n",
      "Loop number 767, under  6.44114876959713331e-232, over  1.55251809230070894e+231\n",
      "Loop number 768, under  3.22057438479856665e-232, over  3.10503618460141787e+231\n",
      "Loop number 769, under  1.61028719239928333e-232, over  6.21007236920283574e+231\n",
      "Loop number 770, under  8.05143596199641664e-233, over  1.24201447384056715e+232\n",
      "Loop number 771, under  4.02571798099820832e-233, over  2.48402894768113430e+232\n",
      "Loop number 772, under  2.01285899049910416e-233, over  4.96805789536226859e+232\n",
      "Loop number 773, under  1.00642949524955208e-233, over  9.93611579072453718e+232\n",
      "Loop number 774, under  5.03214747624776040e-234, over  1.98722315814490744e+233\n",
      "Loop number 775, under  2.51607373812388020e-234, over  3.97444631628981487e+233\n",
      "Loop number 776, under  1.25803686906194010e-234, over  7.94889263257962975e+233\n",
      "Loop number 777, under  6.29018434530970050e-235, over  1.58977852651592595e+234\n",
      "Loop number 778, under  3.14509217265485025e-235, over  3.17955705303185190e+234\n",
      "Loop number 779, under  1.57254608632742512e-235, over  6.35911410606370380e+234\n",
      "Loop number 780, under  7.86273043163712562e-236, over  1.27182282121274076e+235\n",
      "Loop number 781, under  3.93136521581856281e-236, over  2.54364564242548152e+235\n",
      "Loop number 782, under  1.96568260790928141e-236, over  5.08729128485096304e+235\n",
      "Loop number 783, under  9.82841303954640703e-237, over  1.01745825697019261e+236\n",
      "Loop number 784, under  4.91420651977320351e-237, over  2.03491651394038522e+236\n",
      "Loop number 785, under  2.45710325988660176e-237, over  4.06983302788077043e+236\n",
      "Loop number 786, under  1.22855162994330088e-237, over  8.13966605576154086e+236\n",
      "Loop number 787, under  6.14275814971650439e-238, over  1.62793321115230817e+237\n",
      "Loop number 788, under  3.07137907485825220e-238, over  3.25586642230461634e+237\n",
      "Loop number 789, under  1.53568953742912610e-238, over  6.51173284460923269e+237\n",
      "Loop number 790, under  7.67844768714563049e-239, over  1.30234656892184654e+238\n",
      "Loop number 791, under  3.83922384357281524e-239, over  2.60469313784369308e+238\n",
      "Loop number 792, under  1.91961192178640762e-239, over  5.20938627568738615e+238\n",
      "Loop number 793, under  9.59805960893203811e-240, over  1.04187725513747723e+239\n",
      "Loop number 794, under  4.79902980446601906e-240, over  2.08375451027495446e+239\n",
      "Loop number 795, under  2.39951490223300953e-240, over  4.16750902054990892e+239\n",
      "Loop number 796, under  1.19975745111650476e-240, over  8.33501804109981784e+239\n",
      "Loop number 797, under  5.99878725558252382e-241, over  1.66700360821996357e+240\n",
      "Loop number 798, under  2.99939362779126191e-241, over  3.33400721643992714e+240\n",
      "Loop number 799, under  1.49969681389563095e-241, over  6.66801443287985427e+240\n",
      "Loop number 800, under  7.49848406947815477e-242, over  1.33360288657597085e+241\n",
      "Loop number 801, under  3.74924203473907739e-242, over  2.66720577315194171e+241\n",
      "Loop number 802, under  1.87462101736953869e-242, over  5.33441154630388342e+241\n",
      "Loop number 803, under  9.37310508684769347e-243, over  1.06688230926077668e+242\n",
      "Loop number 804, under  4.68655254342384673e-243, over  2.13376461852155337e+242\n",
      "Loop number 805, under  2.34327627171192337e-243, over  4.26752923704310674e+242\n",
      "Loop number 806, under  1.17163813585596168e-243, over  8.53505847408621347e+242\n",
      "Loop number 807, under  5.85819067927980842e-244, over  1.70701169481724269e+243\n",
      "Loop number 808, under  2.92909533963990421e-244, over  3.41402338963448539e+243\n",
      "Loop number 809, under  1.46454766981995210e-244, over  6.82804677926897078e+243\n",
      "Loop number 810, under  7.32273834909976052e-245, over  1.36560935585379416e+244\n",
      "Loop number 811, under  3.66136917454988026e-245, over  2.73121871170758831e+244\n",
      "Loop number 812, under  1.83068458727494013e-245, over  5.46243742341517662e+244\n",
      "Loop number 813, under  9.15342293637470065e-246, over  1.09248748468303532e+245\n",
      "Loop number 814, under  4.57671146818735033e-246, over  2.18497496936607065e+245\n",
      "Loop number 815, under  2.28835573409367516e-246, over  4.36994993873214130e+245\n",
      "Loop number 816, under  1.14417786704683758e-246, over  8.73989987746428259e+245\n",
      "Loop number 817, under  5.72088933523418791e-247, over  1.74797997549285652e+246\n",
      "Loop number 818, under  2.86044466761709395e-247, over  3.49595995098571304e+246\n",
      "Loop number 819, under  1.43022233380854698e-247, over  6.99191990197142608e+246\n",
      "Loop number 820, under  7.15111166904273488e-248, over  1.39838398039428522e+247\n",
      "Loop number 821, under  3.57555583452136744e-248, over  2.79676796078857043e+247\n",
      "Loop number 822, under  1.78777791726068372e-248, over  5.59353592157714086e+247\n",
      "Loop number 823, under  8.93888958630341861e-249, over  1.11870718431542817e+248\n",
      "Loop number 824, under  4.46944479315170930e-249, over  2.23741436863085634e+248\n",
      "Loop number 825, under  2.23472239657585465e-249, over  4.47482873726171269e+248\n",
      "Loop number 826, under  1.11736119828792733e-249, over  8.94965747452342538e+248\n",
      "Loop number 827, under  5.58680599143963663e-250, over  1.78993149490468508e+249\n",
      "Loop number 828, under  2.79340299571981831e-250, over  3.57986298980937015e+249\n",
      "Loop number 829, under  1.39670149785990916e-250, over  7.15972597961874030e+249\n",
      "Loop number 830, under  6.98350748929954579e-251, over  1.43194519592374806e+250\n",
      "Loop number 831, under  3.49175374464977289e-251, over  2.86389039184749612e+250\n",
      "Loop number 832, under  1.74587687232488645e-251, over  5.72778078369499224e+250\n",
      "Loop number 833, under  8.72938436162443223e-252, over  1.14555615673899845e+251\n",
      "Loop number 834, under  4.36469218081221612e-252, over  2.29111231347799690e+251\n",
      "Loop number 835, under  2.18234609040610806e-252, over  4.58222462695599379e+251\n",
      "Loop number 836, under  1.09117304520305403e-252, over  9.16444925391198759e+251\n",
      "Loop number 837, under  5.45586522601527014e-253, over  1.83288985078239752e+252\n",
      "Loop number 838, under  2.72793261300763507e-253, over  3.66577970156479503e+252\n",
      "Loop number 839, under  1.36396630650381754e-253, over  7.33155940312959007e+252\n",
      "Loop number 840, under  6.81983153251908768e-254, over  1.46631188062591801e+253\n",
      "Loop number 841, under  3.40991576625954384e-254, over  2.93262376125183603e+253\n",
      "Loop number 842, under  1.70495788312977192e-254, over  5.86524752250367205e+253\n",
      "Loop number 843, under  8.52478941564885960e-255, over  1.17304950450073441e+254\n",
      "Loop number 844, under  4.26239470782442980e-255, over  2.34609900900146882e+254\n",
      "Loop number 845, under  2.13119735391221490e-255, over  4.69219801800293764e+254\n",
      "Loop number 846, under  1.06559867695610745e-255, over  9.38439603600587529e+254\n",
      "Loop number 847, under  5.32799338478053725e-256, over  1.87687920720117506e+255\n",
      "Loop number 848, under  2.66399669239026863e-256, over  3.75375841440235011e+255\n",
      "Loop number 849, under  1.33199834619513431e-256, over  7.50751682880470023e+255\n",
      "Loop number 850, under  6.65999173097567156e-257, over  1.50150336576094005e+256\n",
      "Loop number 851, under  3.32999586548783578e-257, over  3.00300673152188009e+256\n",
      "Loop number 852, under  1.66499793274391789e-257, over  6.00601346304376018e+256\n",
      "Loop number 853, under  8.32498966371958945e-258, over  1.20120269260875204e+257\n",
      "Loop number 854, under  4.16249483185979473e-258, over  2.40240538521750407e+257\n",
      "Loop number 855, under  2.08124741592989736e-258, over  4.80481077043500815e+257\n",
      "Loop number 856, under  1.04062370796494868e-258, over  9.60962154087001629e+257\n",
      "Loop number 857, under  5.20311853982474341e-259, over  1.92192430817400326e+258\n",
      "Loop number 858, under  2.60155926991237170e-259, over  3.84384861634800652e+258\n",
      "Loop number 859, under  1.30077963495618585e-259, over  7.68769723269601304e+258\n",
      "Loop number 860, under  6.50389817478092926e-260, over  1.53753944653920261e+259\n",
      "Loop number 861, under  3.25194908739046463e-260, over  3.07507889307840521e+259\n",
      "Loop number 862, under  1.62597454369523232e-260, over  6.15015778615681043e+259\n",
      "Loop number 863, under  8.12987271847616158e-261, over  1.23003155723136209e+260\n",
      "Loop number 864, under  4.06493635923808079e-261, over  2.46006311446272417e+260\n",
      "Loop number 865, under  2.03246817961904039e-261, over  4.92012622892544834e+260\n",
      "Loop number 866, under  1.01623408980952020e-261, over  9.84025245785089669e+260\n",
      "Loop number 867, under  5.08117044904760099e-262, over  1.96805049157017934e+261\n",
      "Loop number 868, under  2.54058522452380049e-262, over  3.93610098314035867e+261\n",
      "Loop number 869, under  1.27029261226190025e-262, over  7.87220196628071735e+261\n",
      "Loop number 870, under  6.35146306130950123e-263, over  1.57444039325614347e+262\n",
      "Loop number 871, under  3.17573153065475062e-263, over  3.14888078651228694e+262\n",
      "Loop number 872, under  1.58786576532737531e-263, over  6.29776157302457388e+262\n",
      "Loop number 873, under  7.93932882663687654e-264, over  1.25955231460491478e+263\n",
      "Loop number 874, under  3.96966441331843827e-264, over  2.51910462920982955e+263\n",
      "Loop number 875, under  1.98483220665921913e-264, over  5.03820925841965910e+263\n",
      "Loop number 876, under  9.92416103329609567e-265, over  1.00764185168393182e+264\n",
      "Loop number 877, under  4.96208051664804784e-265, over  2.01528370336786364e+264\n",
      "Loop number 878, under  2.48104025832402392e-265, over  4.03056740673572728e+264\n",
      "Loop number 879, under  1.24052012916201196e-265, over  8.06113481347145456e+264\n",
      "Loop number 880, under  6.20260064581005980e-266, over  1.61222696269429091e+265\n",
      "Loop number 881, under  3.10130032290502990e-266, over  3.22445392538858183e+265\n",
      "Loop number 882, under  1.55065016145251495e-266, over  6.44890785077716365e+265\n",
      "Loop number 883, under  7.75325080726257475e-267, over  1.28978157015543273e+266\n",
      "Loop number 884, under  3.87662540363128737e-267, over  2.57956314031086546e+266\n",
      "Loop number 885, under  1.93831270181564369e-267, over  5.15912628062173092e+266\n",
      "Loop number 886, under  9.69156350907821843e-268, over  1.03182525612434618e+267\n",
      "Loop number 887, under  4.84578175453910922e-268, over  2.06365051224869237e+267\n",
      "Loop number 888, under  2.42289087726955461e-268, over  4.12730102449738474e+267\n",
      "Loop number 889, under  1.21144543863477730e-268, over  8.25460204899476947e+267\n",
      "Loop number 890, under  6.05722719317388652e-269, over  1.65092040979895389e+268\n",
      "Loop number 891, under  3.02861359658694326e-269, over  3.30184081959790779e+268\n",
      "Loop number 892, under  1.51430679829347163e-269, over  6.60368163919581558e+268\n",
      "Loop number 893, under  7.57153399146735815e-270, over  1.32073632783916312e+269\n",
      "Loop number 894, under  3.78576699573367908e-270, over  2.64147265567832623e+269\n",
      "Loop number 895, under  1.89288349786683954e-270, over  5.28294531135665246e+269\n",
      "Loop number 896, under  9.46441748933419769e-271, over  1.05658906227133049e+270\n",
      "Loop number 897, under  4.73220874466709884e-271, over  2.11317812454266099e+270\n",
      "Loop number 898, under  2.36610437233354942e-271, over  4.22635624908532197e+270\n",
      "Loop number 899, under  1.18305218616677471e-271, over  8.45271249817064394e+270\n",
      "Loop number 900, under  5.91526093083387355e-272, over  1.69054249963412879e+271\n",
      "Loop number 901, under  2.95763046541693678e-272, over  3.38108499926825758e+271\n",
      "Loop number 902, under  1.47881523270846839e-272, over  6.76216999853651515e+271\n",
      "Loop number 903, under  7.39407616354234194e-273, over  1.35243399970730303e+272\n",
      "Loop number 904, under  3.69703808177117097e-273, over  2.70486799941460606e+272\n",
      "Loop number 905, under  1.84851904088558549e-273, over  5.40973599882921212e+272\n",
      "Loop number 906, under  9.24259520442792743e-274, over  1.08194719976584242e+273\n",
      "Loop number 907, under  4.62129760221396371e-274, over  2.16389439953168485e+273\n",
      "Loop number 908, under  2.31064880110698186e-274, over  4.32778879906336970e+273\n",
      "Loop number 909, under  1.15532440055349093e-274, over  8.65557759812673940e+273\n",
      "Loop number 910, under  5.77662200276745464e-275, over  1.73111551962534788e+274\n",
      "Loop number 911, under  2.88831100138372732e-275, over  3.46223103925069576e+274\n",
      "Loop number 912, under  1.44415550069186366e-275, over  6.92446207850139152e+274\n",
      "Loop number 913, under  7.22077750345931830e-276, over  1.38489241570027830e+275\n",
      "Loop number 914, under  3.61038875172965915e-276, over  2.76978483140055661e+275\n",
      "Loop number 915, under  1.80519437586482958e-276, over  5.53956966280111321e+275\n",
      "Loop number 916, under  9.02597187932414788e-277, over  1.10791393256022264e+276\n",
      "Loop number 917, under  4.51298593966207394e-277, over  2.21582786512044529e+276\n",
      "Loop number 918, under  2.25649296983103697e-277, over  4.43165573024089057e+276\n",
      "Loop number 919, under  1.12824648491551849e-277, over  8.86331146048178114e+276\n",
      "Loop number 920, under  5.64123242457759243e-278, over  1.77266229209635623e+277\n",
      "Loop number 921, under  2.82061621228879621e-278, over  3.54532458419271246e+277\n",
      "Loop number 922, under  1.41030810614439811e-278, over  7.09064916838542491e+277\n",
      "Loop number 923, under  7.05154053072199053e-279, over  1.41812983367708498e+278\n",
      "Loop number 924, under  3.52577026536099527e-279, over  2.83625966735416997e+278\n",
      "Loop number 925, under  1.76288513268049763e-279, over  5.67251933470833993e+278\n",
      "Loop number 926, under  8.81442566340248816e-280, over  1.13450386694166799e+279\n",
      "Loop number 927, under  4.40721283170124408e-280, over  2.26900773388333597e+279\n",
      "Loop number 928, under  2.20360641585062204e-280, over  4.53801546776667194e+279\n",
      "Loop number 929, under  1.10180320792531102e-280, over  9.07603093553334389e+279\n",
      "Loop number 930, under  5.50901603962655510e-281, over  1.81520618710666878e+280\n",
      "Loop number 931, under  2.75450801981327755e-281, over  3.63041237421333756e+280\n",
      "Loop number 932, under  1.37725400990663878e-281, over  7.26082474842667511e+280\n",
      "Loop number 933, under  6.88627004953319388e-282, over  1.45216494968533502e+281\n",
      "Loop number 934, under  3.44313502476659694e-282, over  2.90432989937067004e+281\n",
      "Loop number 935, under  1.72156751238329847e-282, over  5.80865979874134009e+281\n",
      "Loop number 936, under  8.60783756191649235e-283, over  1.16173195974826802e+282\n",
      "Loop number 937, under  4.30391878095824617e-283, over  2.32346391949653604e+282\n",
      "Loop number 938, under  2.15195939047912309e-283, over  4.64692783899307207e+282\n",
      "Loop number 939, under  1.07597969523956154e-283, over  9.29385567798614414e+282\n",
      "Loop number 940, under  5.37989847619780772e-284, over  1.85877113559722883e+283\n",
      "Loop number 941, under  2.68994923809890386e-284, over  3.71754227119445766e+283\n",
      "Loop number 942, under  1.34497461904945193e-284, over  7.43508454238891531e+283\n",
      "Loop number 943, under  6.72487309524725965e-285, over  1.48701690847778306e+284\n",
      "Loop number 944, under  3.36243654762362982e-285, over  2.97403381695556613e+284\n",
      "Loop number 945, under  1.68121827381181491e-285, over  5.94806763391113225e+284\n",
      "Loop number 946, under  8.40609136905907456e-286, over  1.18961352678222645e+285\n",
      "Loop number 947, under  4.20304568452953728e-286, over  2.37922705356445290e+285\n",
      "Loop number 948, under  2.10152284226476864e-286, over  4.75845410712890580e+285\n",
      "Loop number 949, under  1.05076142113238432e-286, over  9.51690821425781160e+285\n",
      "Loop number 950, under  5.25380710566192160e-287, over  1.90338164285156232e+286\n",
      "Loop number 951, under  2.62690355283096080e-287, over  3.80676328570312464e+286\n",
      "Loop number 952, under  1.31345177641548040e-287, over  7.61352657140624928e+286\n",
      "Loop number 953, under  6.56725888207740200e-288, over  1.52270531428124986e+287\n",
      "Loop number 954, under  3.28362944103870100e-288, over  3.04541062856249971e+287\n",
      "Loop number 955, under  1.64181472051935050e-288, over  6.09082125712499943e+287\n",
      "Loop number 956, under  8.20907360259675250e-289, over  1.21816425142499989e+288\n",
      "Loop number 957, under  4.10453680129837625e-289, over  2.43632850284999977e+288\n",
      "Loop number 958, under  2.05226840064918812e-289, over  4.87265700569999954e+288\n",
      "Loop number 959, under  1.02613420032459406e-289, over  9.74531401139999908e+288\n",
      "Loop number 960, under  5.13067100162297031e-290, over  1.94906280227999982e+289\n",
      "Loop number 961, under  2.56533550081148516e-290, over  3.89812560455999963e+289\n",
      "Loop number 962, under  1.28266775040574258e-290, over  7.79625120911999926e+289\n",
      "Loop number 963, under  6.41333875202871289e-291, over  1.55925024182399985e+290\n",
      "Loop number 964, under  3.20666937601435644e-291, over  3.11850048364799971e+290\n",
      "Loop number 965, under  1.60333468800717822e-291, over  6.23700096729599941e+290\n",
      "Loop number 966, under  8.01667344003589111e-292, over  1.24740019345919988e+291\n",
      "Loop number 967, under  4.00833672001794556e-292, over  2.49480038691839976e+291\n",
      "Loop number 968, under  2.00416836000897278e-292, over  4.98960077383679953e+291\n",
      "Loop number 969, under  1.00208418000448639e-292, over  9.97920154767359906e+291\n",
      "Loop number 970, under  5.01042090002243194e-293, over  1.99584030953471981e+292\n",
      "Loop number 971, under  2.50521045001121597e-293, over  3.99168061906943962e+292\n",
      "Loop number 972, under  1.25260522500560799e-293, over  7.98336123813887925e+292\n",
      "Loop number 973, under  6.26302612502803993e-294, over  1.59667224762777585e+293\n",
      "Loop number 974, under  3.13151306251401997e-294, over  3.19334449525555170e+293\n",
      "Loop number 975, under  1.56575653125700998e-294, over  6.38668899051110340e+293\n",
      "Loop number 976, under  7.82878265628504991e-295, over  1.27733779810222068e+294\n",
      "Loop number 977, under  3.91439132814252496e-295, over  2.55467559620444136e+294\n",
      "Loop number 978, under  1.95719566407126248e-295, over  5.10935119240888272e+294\n",
      "Loop number 979, under  9.78597832035631239e-296, over  1.02187023848177654e+295\n",
      "Loop number 980, under  4.89298916017815620e-296, over  2.04374047696355309e+295\n",
      "Loop number 981, under  2.44649458008907810e-296, over  4.08748095392710617e+295\n",
      "Loop number 982, under  1.22324729004453905e-296, over  8.17496190785421235e+295\n",
      "Loop number 983, under  6.11623645022269525e-297, over  1.63499238157084247e+296\n",
      "Loop number 984, under  3.05811822511134762e-297, over  3.26998476314168494e+296\n",
      "Loop number 985, under  1.52905911255567381e-297, over  6.53996952628336988e+296\n",
      "Loop number 986, under  7.64529556277836906e-298, over  1.30799390525667398e+297\n",
      "Loop number 987, under  3.82264778138918453e-298, over  2.61598781051334795e+297\n",
      "Loop number 988, under  1.91132389069459226e-298, over  5.23197562102669590e+297\n",
      "Loop number 989, under  9.55661945347296132e-299, over  1.04639512420533918e+298\n",
      "Loop number 990, under  4.77830972673648066e-299, over  2.09279024841067836e+298\n",
      "Loop number 991, under  2.38915486336824033e-299, over  4.18558049682135672e+298\n",
      "Loop number 992, under  1.19457743168412017e-299, over  8.37116099364271344e+298\n",
      "Loop number 993, under  5.97288715842060083e-300, over  1.67423219872854269e+299\n",
      "Loop number 994, under  2.98644357921030041e-300, over  3.34846439745708538e+299\n",
      "Loop number 995, under  1.49322178960515021e-300, over  6.69692879491417076e+299\n",
      "Loop number 996, under  7.46610894802575103e-301, over  1.33938575898283415e+300\n",
      "Loop number 997, under  3.73305447401287552e-301, over  2.67877151796566830e+300\n",
      "Loop number 998, under  1.86652723700643776e-301, over  5.35754303593133660e+300\n",
      "Loop number 999, under  9.33263618503218879e-302, over  1.07150860718626732e+301\n",
      "Loop number 1000, under  4.66631809251609439e-302, over  2.14301721437253464e+301\n",
      "Loop number 1001, under  2.33315904625804720e-302, over  4.28603442874506928e+301\n",
      "Loop number 1002, under  1.16657952312902360e-302, over  8.57206885749013857e+301\n",
      "Loop number 1003, under  5.83289761564511799e-303, over  1.71441377149802771e+302\n",
      "Loop number 1004, under  2.91644880782255900e-303, over  3.42882754299605543e+302\n",
      "Loop number 1005, under  1.45822440391127950e-303, over  6.85765508599211085e+302\n",
      "Loop number 1006, under  7.29112201955639749e-304, over  1.37153101719842217e+303\n",
      "Loop number 1007, under  3.64556100977819875e-304, over  2.74306203439684434e+303\n",
      "Loop number 1008, under  1.82278050488909937e-304, over  5.48612406879368868e+303\n",
      "Loop number 1009, under  9.11390252444549687e-305, over  1.09722481375873774e+304\n",
      "Loop number 1010, under  4.55695126222274843e-305, over  2.19444962751747547e+304\n",
      "Loop number 1011, under  2.27847563111137422e-305, over  4.38889925503495095e+304\n",
      "Loop number 1012, under  1.13923781555568711e-305, over  8.77779851006990189e+304\n",
      "Loop number 1013, under  5.69618907777843554e-306, over  1.75555970201398038e+305\n",
      "Loop number 1014, under  2.84809453888921777e-306, over  3.51111940402796076e+305\n",
      "Loop number 1015, under  1.42404726944460889e-306, over  7.02223880805592151e+305\n",
      "Loop number 1016, under  7.12023634722304443e-307, over  1.40444776161118430e+306\n",
      "Loop number 1017, under  3.56011817361152221e-307, over  2.80889552322236861e+306\n",
      "Loop number 1018, under  1.78005908680576111e-307, over  5.61779104644473721e+306\n",
      "Loop number 1019, under  8.90029543402880553e-308, over  1.12355820928894744e+307\n",
      "Loop number 1020, under  4.45014771701440277e-308, over  2.24711641857789488e+307\n",
      "Loop number 1021, under  2.22507385850720138e-308, over  4.49423283715578977e+307\n",
      "Loop number 1022, under  1.11253692925360069e-308, over  8.98846567431157954e+307\n",
      "Loop number 1023, under  5.56268464626800346e-309, over                       inf\n",
      "Loop number 1024, under  2.78134232313400173e-309, over                       inf\n",
      "Loop number 1025, under  1.39067116156700086e-309, over                       inf\n",
      "Loop number 1026, under  6.95335580783500432e-310, over                       inf\n",
      "Loop number 1027, under  3.47667790391750216e-310, over                       inf\n",
      "Loop number 1028, under  1.73833895195875108e-310, over                       inf\n",
      "Loop number 1029, under  8.69169475979375540e-311, over                       inf\n",
      "Loop number 1030, under  4.34584737989687770e-311, over                       inf\n",
      "Loop number 1031, under  2.17292368994843885e-311, over                       inf\n",
      "Loop number 1032, under  1.08646184497421943e-311, over                       inf\n",
      "Loop number 1033, under  5.43230922487109713e-312, over                       inf\n",
      "Loop number 1034, under  2.71615461243554856e-312, over                       inf\n",
      "Loop number 1035, under  1.35807730621777428e-312, over                       inf\n",
      "Loop number 1036, under  6.79038653108887141e-313, over                       inf\n",
      "Loop number 1037, under  3.39519326554443570e-313, over                       inf\n",
      "Loop number 1038, under  1.69759663277221785e-313, over                       inf\n",
      "Loop number 1039, under  8.48798316386108926e-314, over                       inf\n",
      "Loop number 1040, under  4.24399158193054463e-314, over                       inf\n",
      "Loop number 1041, under  2.12199579096527232e-314, over                       inf\n",
      "Loop number 1042, under  1.06099789548263616e-314, over                       inf\n",
      "Loop number 1043, under  5.30498947741318079e-315, over                       inf\n",
      "Loop number 1044, under  2.65249473870659039e-315, over                       inf\n",
      "Loop number 1045, under  1.32624736935329520e-315, over                       inf\n",
      "Loop number 1046, under  6.63123684676647598e-316, over                       inf\n",
      "Loop number 1047, under  3.31561842338323799e-316, over                       inf\n",
      "Loop number 1048, under  1.65780921169161900e-316, over                       inf\n",
      "Loop number 1049, under  8.28904605845809498e-317, over                       inf\n",
      "Loop number 1050, under  4.14452302922904749e-317, over                       inf\n",
      "Loop number 1051, under  2.07226151461452375e-317, over                       inf\n",
      "Loop number 1052, under  1.03613075730726187e-317, over                       inf\n",
      "Loop number 1053, under  5.18065378653630936e-318, over                       inf\n",
      "Loop number 1054, under  2.59032689326815468e-318, over                       inf\n",
      "Loop number 1055, under  1.29516344663407734e-318, over                       inf\n",
      "Loop number 1056, under  6.47581723317038670e-319, over                       inf\n",
      "Loop number 1057, under  3.23790861658519335e-319, over                       inf\n",
      "Loop number 1058, under  1.61895430829259668e-319, over                       inf\n",
      "Loop number 1059, under  8.09477154146298338e-320, over                       inf\n",
      "Loop number 1060, under  4.04738577073149169e-320, over                       inf\n",
      "Loop number 1061, under  2.02369288536574584e-320, over                       inf\n",
      "Loop number 1062, under  1.01184644268287292e-320, over                       inf\n",
      "Loop number 1063, under  5.05923221341436461e-321, over                       inf\n",
      "Loop number 1064, under  2.52961610670718231e-321, over                       inf\n",
      "Loop number 1065, under  1.26480805335359115e-321, over                       inf\n",
      "Loop number 1066, under  6.32404026676795577e-322, over                       inf\n",
      "Loop number 1067, under  3.16202013338397788e-322, over                       inf\n",
      "Loop number 1068, under  1.58101006669198894e-322, over                       inf\n",
      "Loop number 1069, under  7.90505033345994471e-323, over                       inf\n",
      "Loop number 1070, under  3.95252516672997235e-323, over                       inf\n",
      "Loop number 1071, under  1.97626258336498618e-323, over                       inf\n",
      "Loop number 1072, under  9.88131291682493088e-324, over                       inf\n",
      "Loop number 1073, under  4.94065645841246544e-324, over                       inf\n",
      "Loop number 1074, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1075, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1076, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1077, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1078, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1079, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1080, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1081, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1082, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1083, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1084, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1085, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1086, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1087, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1088, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1089, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1090, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1091, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1092, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1093, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1094, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1095, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1096, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1097, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1098, under   0.00000000000000000e+00, over                       inf\n",
      "Loop number 1099, under   0.00000000000000000e+00, over                       inf\n"
     ]
    }
   ],
   "source": [
    "# try it: Write a code to determine the overflow and underflow limit to be accurate by a factor 2\n",
    "# at some point will become 0 and inf, the number just before this limit will be the limits\n",
    "\n",
    "under =1.\n",
    "over =1.\n",
    "for i in range(1100):\n",
    "    under = under / 2\n",
    "    over = over * 2\n",
    "    print(\"Loop number %d, under %25.17e, over %25.17e\"%(i,under,over))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Biggest and smallest number (factor of 2) the computer can store: \n",
    "\n",
    "final under is : 4.94065645841246544e-324 \n",
    "\n",
    "final over is: 8.98846567431157954e+307"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Machine Precision: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0000000000000000100000000000000  0 01111000110 0111000011101111010101000110010001101101010010010111\n"
     ]
    }
   ],
   "source": [
    "num=1.0e-17  # this is the relative error \n",
    "print(\"%.31f  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0000000000000000000000000000000  0 01111111111 0000000000000000000000000000000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "num=1.0+1.0e-17\n",
    "print(\"%.31f  %s\"%(num,fextract(num)[0]+' '+fextract(num)[1:12]+' '+fextract(num)[12:]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100.0999999999999943156581139191985 100.0999999999999943156581139191985\n"
     ]
    }
   ],
   "source": [
    "num1 = 100.1\n",
    "num2= 100.1+100.1*1.0e-17\n",
    "print(\"%.31f %.31f\"%(num1,num2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eps= 0.5  one_plus_eps 1.5\n",
      "eps= 0.25  one_plus_eps 1.25\n",
      "eps= 0.125  one_plus_eps 1.125\n",
      "eps= 0.0625  one_plus_eps 1.0625\n",
      "eps= 0.03125  one_plus_eps 1.03125\n",
      "eps= 0.015625  one_plus_eps 1.015625\n",
      "eps= 0.0078125  one_plus_eps 1.0078125\n",
      "eps= 0.00390625  one_plus_eps 1.00390625\n",
      "eps= 0.001953125  one_plus_eps 1.001953125\n",
      "eps= 0.0009765625  one_plus_eps 1.0009765625\n",
      "eps= 0.00048828125  one_plus_eps 1.00048828125\n",
      "eps= 0.000244140625  one_plus_eps 1.000244140625\n",
      "eps= 0.0001220703125  one_plus_eps 1.0001220703125\n",
      "eps= 6.103515625e-05  one_plus_eps 1.00006103515625\n",
      "eps= 3.0517578125e-05  one_plus_eps 1.000030517578125\n",
      "eps= 1.52587890625e-05  one_plus_eps 1.0000152587890625\n",
      "eps= 7.62939453125e-06  one_plus_eps 1.0000076293945312\n",
      "eps= 3.814697265625e-06  one_plus_eps 1.0000038146972656\n",
      "eps= 1.9073486328125e-06  one_plus_eps 1.0000019073486328\n",
      "eps= 9.5367431640625e-07  one_plus_eps 1.0000009536743164\n",
      "eps= 4.76837158203125e-07  one_plus_eps 1.0000004768371582\n",
      "eps= 2.384185791015625e-07  one_plus_eps 1.000000238418579\n",
      "eps= 1.1920928955078125e-07  one_plus_eps 1.0000001192092896\n",
      "eps= 5.960464477539063e-08  one_plus_eps 1.0000000596046448\n",
      "eps= 2.9802322387695312e-08  one_plus_eps 1.0000000298023224\n",
      "eps= 1.4901161193847656e-08  one_plus_eps 1.0000000149011612\n",
      "eps= 7.450580596923828e-09  one_plus_eps 1.0000000074505806\n",
      "eps= 3.725290298461914e-09  one_plus_eps 1.0000000037252903\n",
      "eps= 1.862645149230957e-09  one_plus_eps 1.0000000018626451\n",
      "eps= 9.313225746154785e-10  one_plus_eps 1.0000000009313226\n",
      "eps= 4.656612873077393e-10  one_plus_eps 1.0000000004656613\n",
      "eps= 2.3283064365386963e-10  one_plus_eps 1.0000000002328306\n",
      "eps= 1.1641532182693481e-10  one_plus_eps 1.0000000001164153\n",
      "eps= 5.820766091346741e-11  one_plus_eps 1.0000000000582077\n",
      "eps= 2.9103830456733704e-11  one_plus_eps 1.0000000000291038\n",
      "eps= 1.4551915228366852e-11  one_plus_eps 1.000000000014552\n",
      "eps= 7.275957614183426e-12  one_plus_eps 1.000000000007276\n",
      "eps= 3.637978807091713e-12  one_plus_eps 1.000000000003638\n",
      "eps= 1.8189894035458565e-12  one_plus_eps 1.000000000001819\n",
      "eps= 9.094947017729282e-13  one_plus_eps 1.0000000000009095\n",
      "eps= 4.547473508864641e-13  one_plus_eps 1.0000000000004547\n",
      "eps= 2.2737367544323206e-13  one_plus_eps 1.0000000000002274\n",
      "eps= 1.1368683772161603e-13  one_plus_eps 1.0000000000001137\n",
      "eps= 5.684341886080802e-14  one_plus_eps 1.0000000000000568\n",
      "eps= 2.842170943040401e-14  one_plus_eps 1.0000000000000284\n",
      "eps= 1.4210854715202004e-14  one_plus_eps 1.0000000000000142\n",
      "eps= 7.105427357601002e-15  one_plus_eps 1.000000000000007\n",
      "eps= 3.552713678800501e-15  one_plus_eps 1.0000000000000036\n",
      "eps= 1.7763568394002505e-15  one_plus_eps 1.0000000000000018\n",
      "eps= 8.881784197001252e-16  one_plus_eps 1.0000000000000009\n",
      "eps= 4.440892098500626e-16  one_plus_eps 1.0000000000000004\n",
      "eps= 2.220446049250313e-16  one_plus_eps 1.0000000000000002\n",
      "eps= 1.1102230246251565e-16  one_plus_eps 1.0\n",
      "eps= 5.551115123125783e-17  one_plus_eps 1.0\n",
      "eps= 2.7755575615628914e-17  one_plus_eps 1.0\n",
      "eps= 1.3877787807814457e-17  one_plus_eps 1.0\n",
      "eps= 6.938893903907228e-18  one_plus_eps 1.0\n",
      "eps= 3.469446951953614e-18  one_plus_eps 1.0\n",
      "eps= 1.734723475976807e-18  one_plus_eps 1.0\n",
      "eps= 8.673617379884035e-19  one_plus_eps 1.0\n",
      "eps= 4.336808689942018e-19  one_plus_eps 1.0\n",
      "eps= 2.168404344971009e-19  one_plus_eps 1.0\n",
      "eps= 1.0842021724855044e-19  one_plus_eps 1.0\n",
      "eps= 5.421010862427522e-20  one_plus_eps 1.0\n",
      "eps= 2.710505431213761e-20  one_plus_eps 1.0\n",
      "eps= 1.3552527156068805e-20  one_plus_eps 1.0\n",
      "eps= 6.776263578034403e-21  one_plus_eps 1.0\n",
      "eps= 3.3881317890172014e-21  one_plus_eps 1.0\n",
      "eps= 1.6940658945086007e-21  one_plus_eps 1.0\n",
      "eps= 8.470329472543003e-22  one_plus_eps 1.0\n",
      "eps= 4.235164736271502e-22  one_plus_eps 1.0\n",
      "eps= 2.117582368135751e-22  one_plus_eps 1.0\n",
      "eps= 1.0587911840678754e-22  one_plus_eps 1.0\n",
      "eps= 5.293955920339377e-23  one_plus_eps 1.0\n",
      "eps= 2.6469779601696886e-23  one_plus_eps 1.0\n",
      "eps= 1.3234889800848443e-23  one_plus_eps 1.0\n",
      "eps= 6.617444900424222e-24  one_plus_eps 1.0\n",
      "eps= 3.308722450212111e-24  one_plus_eps 1.0\n",
      "eps= 1.6543612251060553e-24  one_plus_eps 1.0\n",
      "eps= 8.271806125530277e-25  one_plus_eps 1.0\n",
      "eps= 4.1359030627651384e-25  one_plus_eps 1.0\n",
      "eps= 2.0679515313825692e-25  one_plus_eps 1.0\n",
      "eps= 1.0339757656912846e-25  one_plus_eps 1.0\n",
      "eps= 5.169878828456423e-26  one_plus_eps 1.0\n",
      "eps= 2.5849394142282115e-26  one_plus_eps 1.0\n",
      "eps= 1.2924697071141057e-26  one_plus_eps 1.0\n",
      "eps= 6.462348535570529e-27  one_plus_eps 1.0\n",
      "eps= 3.2311742677852644e-27  one_plus_eps 1.0\n",
      "eps= 1.6155871338926322e-27  one_plus_eps 1.0\n",
      "eps= 8.077935669463161e-28  one_plus_eps 1.0\n",
      "eps= 4.0389678347315804e-28  one_plus_eps 1.0\n",
      "eps= 2.0194839173657902e-28  one_plus_eps 1.0\n",
      "eps= 1.0097419586828951e-28  one_plus_eps 1.0\n",
      "eps= 5.048709793414476e-29  one_plus_eps 1.0\n",
      "eps= 2.524354896707238e-29  one_plus_eps 1.0\n",
      "eps= 1.262177448353619e-29  one_plus_eps 1.0\n",
      "eps= 6.310887241768095e-30  one_plus_eps 1.0\n",
      "eps= 3.1554436208840472e-30  one_plus_eps 1.0\n",
      "eps= 1.5777218104420236e-30  one_plus_eps 1.0\n",
      "eps= 7.888609052210118e-31  one_plus_eps 1.0\n"
     ]
    }
   ],
   "source": [
    "# try it: determine your machine's precision\n",
    "eps=1\n",
    "for i in range(100):\n",
    "    eps= eps/2\n",
    "    # add one line of code here to calculate eps \n",
    "    print('eps=',eps,' one_plus_eps',1+eps)\n",
    "    \n",
    "# when 1+eps = 1, we have reached the machien precision - my computer relative error"
   ]
  }
 ],
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