1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Relative Error</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../backgrounders.html" title="Chapter 23. Backgrounders"> 9<link rel="prev" href="special_tut/special_tut_test.html" title="Testing"> 10<link rel="next" href="lanczos.html" title="The Lanczos Approximation"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="special_tut/special_tut_test.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="lanczos.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error">Relative Error</a> 28</h2></div></div></div> 29<p> 30 Given an actual value <span class="emphasis"><em>a</em></span> and a found value <span class="emphasis"><em>v</em></span> 31 the relative error can be calculated from: 32 </p> 33<div class="blockquote"><blockquote class="blockquote"><p> 34 <span class="inlinemediaobject"><img src="../../equations/error2.svg"></span> 35 36 </p></blockquote></div> 37<p> 38 However the test programs in the library use the symmetrical form: 39 </p> 40<div class="blockquote"><blockquote class="blockquote"><p> 41 <span class="inlinemediaobject"><img src="../../equations/error1.svg"></span> 42 43 </p></blockquote></div> 44<p> 45 which measures <span class="emphasis"><em>relative difference</em></span> and happens to be less 46 error prone in use since we don't have to worry which value is the "true" 47 result, and which is the experimental one. It guarantees to return a value 48 at least as large as the relative error. 49 </p> 50<p> 51 Special care needs to be taken when one value is zero: we could either take 52 the absolute error in this case (but that's cheating as the absolute error 53 is likely to be very small), or we could assign a value of either 1 or infinity 54 to the relative error in this special case. In the test cases for the special 55 functions in this library, everything below a threshold is regarded as "effectively 56 zero", otherwise the relative error is assigned the value of 1 if only 57 one of the terms is zero. The threshold is currently set at <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><>::</span><span class="identifier">min</span><span class="special">()</span></code>: in other words all denormalised numbers 58 are regarded as a zero. 59 </p> 60<p> 61 All the test programs calculate <span class="emphasis"><em>quantized relative error</em></span>, 62 whereas the graphs in this manual are produced with the <span class="emphasis"><em>actual error</em></span>. 63 The difference is as follows: in the test programs, the test data is rounded 64 to the target real type under test when the program is compiled, so the error 65 observed will then be a whole number of <span class="emphasis"><em>units in the last place</em></span> 66 either rounded up from the actual error, or rounded down (possibly to zero). 67 In contrast the <span class="emphasis"><em>true error</em></span> is obtained by extending the 68 precision of the calculated value, and then comparing to the actual value: 69 in this case the calculated error may be some fraction of <span class="emphasis"><em>units in 70 the last place</em></span>. 71 </p> 72<p> 73 Note that throughout this manual and the test programs the relative error is 74 usually quoted in units of epsilon. However, remember that <span class="emphasis"><em>units 75 in the last place</em></span> more accurately reflect the number of contaminated 76 digits, and that relative error can <span class="emphasis"><em>"wobble"</em></span> 77 by a factor of 2 compared to <span class="emphasis"><em>units in the last place</em></span>. 78 In other words: two implementations of the same function, whose maximum relative 79 errors differ by a factor of 2, can actually be accurate to the same number 80 of binary digits. You have been warned! 81 </p> 82<h5> 83<a name="math_toolkit.relative_error.h0"></a> 84 <span class="phrase"><a name="math_toolkit.relative_error.zero_error"></a></span><a class="link" href="relative_error.html#math_toolkit.relative_error.zero_error">The 85 Impossibility of Zero Error</a> 86 </h5> 87<p> 88 For many of the functions in this library, it is assumed that the error is 89 "effectively zero" if the computation can be done with a number of 90 guard digits. However it should be remembered that if the result is a <span class="emphasis"><em>transcendental 91 number</em></span> then as a point of principle we can never be sure that the 92 result is accurate to more than 1 ulp. This is an example of what <a href="http://en.wikipedia.org/wiki/William_Kahan" target="_top">http://en.wikipedia.org/wiki/William_Kahan</a> 93 called <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma</a>: 94 consider what happens if the first guard digit is a one, and the remaining 95 guard digits are all zero. Do we have a tie or not? Since the only thing we 96 can tell about a transcendental number is that its digits have no particular 97 pattern, we can never tell if we have a tie, no matter how many guard digits 98 we have. Therefore, we can never be completely sure that the result has been 99 rounded in the right direction. Of course, transcendental numbers that just 100 happen to be a tie - for however many guard digits we have - are extremely 101 rare, and get rarer the more guard digits we have, but even so.... 102 </p> 103<p> 104 Refer to the classic text <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What 105 Every Computer Scientist Should Know About Floating-Point Arithmetic</a> 106 for more information. 107 </p> 108</div> 109<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 110<td align="left"></td> 111<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 112 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 113 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 114 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 115 Daryle Walker and Xiaogang Zhang<p> 116 Distributed under the Boost Software License, Version 1.0. (See accompanying 117 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 118 </p> 119</div></td> 120</tr></table> 121<hr> 122<div class="spirit-nav"> 123<a accesskey="p" href="special_tut/special_tut_test.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="lanczos.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 124</div> 125</body> 126</html> 127
