{
 "cells": [
  {
   "cell_type": "markdown",
   "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",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    0.0000000000e+00\n"
     ]
    }
   ],
   "source": [
    "from math import sqrt\n",
    "x = 1.0e-8\n",
    "y = sqrt(1.0 + x**2) - 1.0\n",
    "print(\"%20.10e\" % y)"
   ]
  },
  {
   "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",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    5.0000000000e-17\n"
     ]
    }
   ],
   "source": [
    "y = x**2/(sqrt(1.0+x**2) + 1.0)\n",
    "print(\"%20.10e\" % y)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
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 },
 "nbformat": 4,
 "nbformat_minor": 1
}