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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Error due to subtraction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Consider computing\n",
      "$$\n",
      "x=10^{-8}, \\qquad y = 1 - \\cos(x)\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from numpy import sin,cos"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = 1.0e-8\n",
      "y = 1.0 - cos(x)\n",
      "print \"%24.14e\" % y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "    0.00000000000000e+00\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Because of roundoff error, the result is zero. But mathematically we have the equivalent expression\n",
      "$$\n",
      "z = 2 \\sin^2(x/2)\n",
      "$$\n",
      "which does not involve subtraction of nearly equal quantities."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "z = 2.0*sin(0.5*x)**2\n",
      "print \"%24.14e\" % z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "    5.00000000000000e-17\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}