{
 "metadata": {
  "name": "",
  "signature": "sha256:30d4983d328c7423d055ee8319184f3624ed813d076cebcb4ce513c81d8669d9"
 },
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Error due to subtraction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Consider computing\n",
      "$$\n",
      "x = 10^{-8}, \\qquad y = \\sqrt{1+x^2} - 1\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = 1.0e-8\n",
      "y = sqrt(1.0 + x**2) - 1.0\n",
      "print \"%20.10e\" % y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "    0.0000000000e+00\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The result is zero due to round-off error. An equivalent expression is\n",
      "$$\n",
      "y = \\frac{x^2}{\\sqrt{1 + x^2} + 1}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = x**2/(sqrt(1.0+x**2) + 1.0)\n",
      "print \"%20.10e\" % y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "    5.0000000000e-17\n"
       ]
      }
     ],
     "prompt_number": 3
    }
   ],
   "metadata": {}
  }
 ]
}