| 1 | <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
|
|---|
| 2 | <html>
|
|---|
| 3 | <!-- Copyright (C) 2022 Richard Stallman and Free Software Foundation, Inc.
|
|---|
| 4 |
|
|---|
| 5 | (The work of Trevis Rothwell and Nelson Beebe has been assigned or
|
|---|
| 6 | licensed to the FSF.)
|
|---|
| 7 |
|
|---|
| 8 | Permission is granted to copy, distribute and/or modify this document
|
|---|
| 9 | under the terms of the GNU Free Documentation License, Version 1.3 or
|
|---|
| 10 | any later version published by the Free Software Foundation; with the
|
|---|
| 11 | Invariant Sections being "GNU General Public License," with the
|
|---|
| 12 | Front-Cover Texts being "A GNU Manual," and with the Back-Cover
|
|---|
| 13 | Texts as in (a) below. A copy of the license is included in the
|
|---|
| 14 | section entitled "GNU Free Documentation License."
|
|---|
| 15 |
|
|---|
| 16 | (a) The FSF's Back-Cover Text is: "You have the freedom to copy and
|
|---|
| 17 | modify this GNU manual. Buying copies from the FSF supports it in
|
|---|
| 18 | developing GNU and promoting software freedom." -->
|
|---|
| 19 | <!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
|
|---|
| 20 | <head>
|
|---|
| 21 | <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
|---|
| 22 | <title>Rounding Issues (GNU C Language Manual)</title>
|
|---|
| 23 |
|
|---|
| 24 | <meta name="description" content="Rounding Issues (GNU C Language Manual)">
|
|---|
| 25 | <meta name="keywords" content="Rounding Issues (GNU C Language Manual)">
|
|---|
| 26 | <meta name="resource-type" content="document">
|
|---|
| 27 | <meta name="distribution" content="global">
|
|---|
| 28 | <meta name="Generator" content="makeinfo">
|
|---|
| 29 | <link href="index.html" rel="start" title="Top">
|
|---|
| 30 | <link href="Symbol-Index.html" rel="index" title="Symbol Index">
|
|---|
| 31 | <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
|
|---|
| 32 | <link href="Floating-Point-in-Depth.html" rel="up" title="Floating Point in Depth">
|
|---|
| 33 | <link href="Significance-Loss.html" rel="next" title="Significance Loss">
|
|---|
| 34 | <link href="Rounding.html" rel="prev" title="Rounding">
|
|---|
| 35 | <style type="text/css">
|
|---|
| 36 | <!--
|
|---|
| 37 | a.summary-letter {text-decoration: none}
|
|---|
| 38 | blockquote.indentedblock {margin-right: 0em}
|
|---|
| 39 | div.display {margin-left: 3.2em}
|
|---|
| 40 | div.example {margin-left: 3.2em}
|
|---|
| 41 | div.lisp {margin-left: 3.2em}
|
|---|
| 42 | kbd {font-style: oblique}
|
|---|
| 43 | pre.display {font-family: inherit}
|
|---|
| 44 | pre.format {font-family: inherit}
|
|---|
| 45 | pre.menu-comment {font-family: serif}
|
|---|
| 46 | pre.menu-preformatted {font-family: serif}
|
|---|
| 47 | span.nolinebreak {white-space: nowrap}
|
|---|
| 48 | span.roman {font-family: initial; font-weight: normal}
|
|---|
| 49 | span.sansserif {font-family: sans-serif; font-weight: normal}
|
|---|
| 50 | ul.no-bullet {list-style: none}
|
|---|
| 51 | -->
|
|---|
| 52 | </style>
|
|---|
| 53 |
|
|---|
| 54 |
|
|---|
| 55 | </head>
|
|---|
| 56 |
|
|---|
| 57 | <body lang="en">
|
|---|
| 58 | <span id="Rounding-Issues"></span><div class="header">
|
|---|
| 59 | <p>
|
|---|
| 60 | Next: <a href="Significance-Loss.html" accesskey="n" rel="next">Significance Loss</a>, Previous: <a href="Rounding.html" accesskey="p" rel="prev">Rounding</a>, Up: <a href="Floating-Point-in-Depth.html" accesskey="u" rel="up">Floating Point in Depth</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Symbol-Index.html" title="Index" rel="index">Index</a>]</p>
|
|---|
| 61 | </div>
|
|---|
| 62 | <hr>
|
|---|
| 63 | <span id="Rounding-Issues-1"></span><h3 class="section">28.8 Rounding Issues</h3>
|
|---|
| 64 | <span id="index-rounding-issues-_0028floating-point_0029"></span>
|
|---|
| 65 | <span id="index-floating_002dpoint-rounding-issues"></span>
|
|---|
| 66 |
|
|---|
| 67 | <p>The default IEEE 754 rounding mode minimizes errors, and most
|
|---|
| 68 | normal computations should not suffer any serious accumulation of
|
|---|
| 69 | errors from rounding.
|
|---|
| 70 | </p>
|
|---|
| 71 | <p>Of course, you can contrive examples where that is not so. Here
|
|---|
| 72 | is one: iterate a square root, then attempt to recover the
|
|---|
| 73 | original value by repeated squaring.
|
|---|
| 74 | </p>
|
|---|
| 75 | <div class="example">
|
|---|
| 76 | <pre class="example">#include <stdio.h>
|
|---|
| 77 | #include <math.h>
|
|---|
| 78 |
|
|---|
| 79 | int main (void)
|
|---|
| 80 | {
|
|---|
| 81 | double x = 100.0;
|
|---|
| 82 | double y;
|
|---|
| 83 | for (n = 10; n <= 100; n += 10)
|
|---|
| 84 | {
|
|---|
| 85 | y = x;
|
|---|
| 86 | for (k = 0; k < n; ++k) y = sqrt (y);
|
|---|
| 87 | for (k = 0; k < n; ++k) y *= y;
|
|---|
| 88 | printf ("n = %3d; x = %.0f\ty = %.6f\n", n, x, y);
|
|---|
| 89 | }
|
|---|
| 90 | return 0;
|
|---|
| 91 | }
|
|---|
| 92 | </pre></div>
|
|---|
| 93 |
|
|---|
| 94 | <p>Here is the output:
|
|---|
| 95 | </p>
|
|---|
| 96 | <div class="example">
|
|---|
| 97 | <pre class="example">n = 10; x = 100 y = 100.000000
|
|---|
| 98 | n = 20; x = 100 y = 100.000000
|
|---|
| 99 | n = 30; x = 100 y = 99.999977
|
|---|
| 100 | n = 40; x = 100 y = 99.981025
|
|---|
| 101 | n = 50; x = 100 y = 90.017127
|
|---|
| 102 | n = 60; x = 100 y = 1.000000
|
|---|
| 103 | n = 70; x = 100 y = 1.000000
|
|---|
| 104 | n = 80; x = 100 y = 1.000000
|
|---|
| 105 | n = 90; x = 100 y = 1.000000
|
|---|
| 106 | n = 100; x = 100 y = 1.000000
|
|---|
| 107 | </pre></div>
|
|---|
| 108 |
|
|---|
| 109 | <p>After 50 iterations, <code>y</code> has barely one correct digit, and
|
|---|
| 110 | soon after, there are no correct digits.
|
|---|
| 111 | </p>
|
|---|
| 112 |
|
|---|
| 113 |
|
|---|
| 114 |
|
|---|
| 115 |
|
|---|
| 116 | </body>
|
|---|
| 117 | </html>
|
|---|
