fun_with_ulp.ipynb

{
 "metadata": {
  "name": "",
  "signature": "sha256:b42dc62c52256a5ad10132c8c054384342edb41fd8f75a3785d5248935849627"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pandas as pd\n",
      "from cStringIO import StringIO\n",
      "import random\n",
      "from decimal import Decimal\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def xstrtod(s):\n",
      "    return pd.read_csv(StringIO(s), header=None)[0][0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def randnum():\n",
      "    s = '1.'  # Start with 1, otherwise you can randomly get very small numbers\n",
      "    for i in range(20):\n",
      "        s += random.choice('0123456789')\n",
      "    return s"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 117
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def ulp(s):\n",
      "    num = Decimal(s)\n",
      "    f = np.float64(s)\n",
      "    a = f.view((np.uint8, 8))\n",
      "    # Since this is uint8 make sure the result doesn't accidentally wrap\n",
      "    if a[0] == 0:\n",
      "        a[0] = 1\n",
      "    elif a[0] == 255:\n",
      "        a[0] = 254\n",
      "    elif Decimal(f) < num:\n",
      "        a[0] += 1\n",
      "    elif Decimal(f) > num:\n",
      "        a[0] -= 1\n",
      "    f2 = a.view(np.float64)[0]\n",
      "    return abs(f2 - f)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 118
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n_vals = 10000\n",
      "y = np.empty(n_vals)\n",
      "good_vals = 0\n",
      "ulp_vals = np.empty(n_vals)\n",
      "diffs = np.empty(n_vals)\n",
      "\n",
      "for i in range(n_vals):\n",
      "    val = randnum()\n",
      "    guess = xstrtod(val)\n",
      "    decimal_ulp_val = Decimal(ulp(val))\n",
      "    ulp_vals[i] = float(decimal_ulp_val / Decimal(1e-16))\n",
      "    decimal_diff = abs(Decimal(val) - Decimal(guess))\n",
      "    ulp_diff = decimal_diff / decimal_ulp_val\n",
      "    y[i] = float(ulp_diff)\n",
      "    diffs[i] = float(decimal_diff)\n",
      "    if float(ulp_diff) < 1.0:\n",
      "        good_vals += 1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 119
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print('{0}% of values within 1.0 ULP'.format(good_vals * 100.0 / 10000))\n",
      "plt.hist(y, bins=30, log=True)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "71.2% of values within 1.0 ULP\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x11028f2d0>"
       ]
      }
     ],
     "prompt_number": 120
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# OK, the ULP is always very near 2**-52 which is one bit in the\n",
      "# 52-bit mantissa (http://kipirvine.com/asm/workbook/floating_tut.htm)\n",
      "plt.hist(ulp_vals, bins=50)\n",
      "plt.xlabel('ULP / 1e-16')\n",
      "plt.grid()\n",
      "plt.show()\n",
      "np.min(ulp_vals)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10c8839d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 121,
       "text": [
        "2.2204460492503131"
       ]
      }
     ],
     "prompt_number": 121
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(diffs, bins=50, log=True)\n",
      "plt.xlabel('Diff')\n",
      "plt.grid()\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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hdQIZCakTyEhInUDRLPSS1HH26IsYw1jdiFVavnt/xh59GvboJUljEhf6AfbewDkYFVIn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\nSSn8P4GGBnniDU56AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10de77750>"
       ]
      }
     ],
     "prompt_number": 122
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.float64(1.23123123123123123123)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 123
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = x.view((np.uint8, 8))\n",
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 124,
       "text": [
        "array([ 50,  59, 255, 132,  31, 179, 243,  63], dtype=uint8)"
       ]
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y0 = a.view(np.float64)[0]\n",
      "a[0] += 1\n",
      "y1 = a.view(np.float64)[0]\n",
      "np.abs(y1 - y0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 125,
       "text": [
        "2.2204460492503131e-16"
       ]
      }
     ],
     "prompt_number": 125
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2**-52"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 126,
       "text": [
        "2.220446049250313e-16"
       ]
      }
     ],
     "prompt_number": 126
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}