1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Floating-point Comparison</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="../utils.html" title="Chapter 2. Floating Point Utilities"> 9<link rel="prev" href="next_float/ulp.html" title="Obtaining the Size of a Unit In the Last Place - ULP"> 10<link rel="next" href="cond.html" title="Condition Numbers"> 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="next_float/ulp.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../utils.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="cond.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.float_comparison"></a><a class="link" href="float_comparison.html" title="Floating-point Comparison">Floating-point Comparison</a> 28</h2></div></div></div> 29<p> 30 Comparison of floating-point values has always been a source of endless difficulty 31 and confusion. 32 </p> 33<p> 34 Unlike integral values that are exact, all floating-point operations will potentially 35 produce an inexact result that will be rounded to the nearest available binary 36 representation. Even apparently innocuous operations such as assigning 0.1 37 to a double produces an inexact result (as this decimal number has no exact 38 binary representation). 39 </p> 40<p> 41 Floating-point computations also involve rounding so that some 'computational 42 noise' is added, and hence results are also not exact (although repeatable, 43 at least under identical platforms and compile options). 44 </p> 45<p> 46 Sadly, this conflicts with the expectation of most users, as many articles 47 and innumerable cries for help show all too well. 48 </p> 49<p> 50 Some background reading is: 51 </p> 52<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 53<li class="listitem"> 54 Knuth D.E. The art of computer programming, vol II, section 4.2, especially 55 Floating-Point Comparison 4.2.2, pages 198-220. 56 </li> 57<li class="listitem"> 58 <a href="http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html" target="_top">David 59 Goldberg, "What Every Computer Scientist Should Know About Floating-Point 60 Arithmetic"</a> 61 </li> 62<li class="listitem"> 63 <a href="http://adtmag.com/articles/2000/03/16/comparing-floatshow-to-determine-if-floating-quantities-are-close-enough-once-a-tolerance-has-been-r.aspx" target="_top">Alberto 64 Squassabia, Comparing floats listing</a> 65 </li> 66<li class="listitem"> 67 <a href="https://code.google.com/p/googletest/wiki/AdvancedGuide#Floating-Point_Comparison" target="_top">Google 68 Floating-Point_Comparison guide</a> 69 </li> 70<li class="listitem"> 71 <a href="https://www.boost.org/doc/libs/release/libs/test/doc/html/boost_test/testing_tools/extended_comparison/floating_point.html" target="_top">Boost.Test 72 Floating-Point_Comparison</a> 73 </li> 74</ul></div> 75<p> 76 Boost provides a number of ways to compare floating-point values to see if 77 they are tolerably close enough to each other, but first we must decide what 78 kind of comparison we require: 79 </p> 80<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 81<li class="listitem"> 82 Absolute difference/error: the absolute difference between two values 83 <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> is simply <code class="computeroutput"><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">a</span><span class="special">-</span><span class="identifier">b</span><span class="special">)</span></code>. This is the only meaningful comparison 84 to make if we know that the result may have cancellation error (see below). 85 </li> 86<li class="listitem"> 87 The edit distance between the two values: i.e. how many (binary) floating-point 88 values are between two values <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>? 89 This is provided by the function <a href="../../../../../libs/math/doc/html/math_toolkit/next_float/float_distance.html" target="_top">Boost.Math 90 float_distance</a>, but is probably only useful when you know that 91 the distance should be very small. This function is somewhat difficult 92 to compute, and doesn't scale to values that are very far apart. In other 93 words, use with care. 94 </li> 95<li class="listitem"> 96 The relative distance/error between two values. This is quick and easy 97 to compute, and is generally the method of choice when checking that your 98 results are "tolerably close" to one another. However, it is 99 not as exact as the edit distance when dealing with small differences, 100 and due to the way floating-point values are encoded can "wobble" 101 by a factor of 2 compared to the "true" edit distance. This is 102 the method documented below: if <code class="computeroutput"><span class="identifier">float_distance</span></code> 103 is a surgeon's scalpel, then <code class="computeroutput"><span class="identifier">relative_difference</span></code> 104 is more like a Swiss army knife: both have important but different use 105 cases. 106 </li> 107</ul></div> 108<h6> 109<a name="math_toolkit.float_comparison.h0"></a> 110 <span class="phrase"><a name="math_toolkit.float_comparison.fp_relative"></a></span><a class="link" href="float_comparison.html#math_toolkit.float_comparison.fp_relative">Relative 111 Comparison of Floating-point Values</a> 112 </h6> 113<p> 114 <code class="computeroutput"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">relative_difference</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code> 115 </p> 116<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> 117<a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">b</span><span class="special">);</span> 118 119<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> 120<a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">epsilon_difference</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">b</span><span class="special">);</span> 121</pre> 122<p> 123 The function <code class="computeroutput"><span class="identifier">relative_difference</span></code> 124 returns the relative distance/error <span class="emphasis"><em>E</em></span> between two values 125 as defined by: 126 </p> 127<div class="blockquote"><blockquote class="blockquote"><p> 128 <span class="serif_italic">E = fabs((a - b) / min(a,b))</span> 129 </p></blockquote></div> 130<p> 131 The function <code class="computeroutput"><span class="identifier">epsilon_difference</span></code> 132 is a convenience function that returns <code class="computeroutput"><span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> 133 <span class="identifier">b</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">eps</span></code> where 134 <code class="computeroutput"><span class="identifier">eps</span></code> is the machine epsilon 135 for the result type. 136 </p> 137<p> 138 The following special cases are handled as follows: 139 </p> 140<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 141<li class="listitem"> 142 If either of <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> is a NaN, 143 then returns the largest representable value for T: for example for type 144 <code class="computeroutput"><span class="keyword">double</span></code>, this is <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code> 145 which is the same as <code class="computeroutput"><span class="identifier">DBL_MAX</span></code> 146 or <code class="computeroutput"><span class="number">1.7976931348623157e+308</span></code>. 147 </li> 148<li class="listitem"> 149 If <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> differ in sign then 150 returns the largest representable value for T. 151 </li> 152<li class="listitem"> 153 If both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are both infinities 154 (of the same sign), then returns zero. 155 </li> 156<li class="listitem"> 157 If just one of <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> is an 158 infinity, then returns the largest representable value for T. 159 </li> 160<li class="listitem"> 161 If both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are zero then 162 returns zero. 163 </li> 164<li class="listitem"> 165 If just one of <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> is a zero 166 or a denormalized value, then it is treated as if it were the smallest 167 (non-denormalized) value representable in T for the purposes of the above 168 calculation. 169 </li> 170</ul></div> 171<p> 172 These rules were primarily designed to assist with our own test suite, they 173 are designed to be robust enough that the function can in most cases be used 174 blindly, including in cases where the expected result is actually too small 175 to represent in type T and underflows to zero. 176 </p> 177<h6> 178<a name="math_toolkit.float_comparison.h1"></a> 179 <span class="phrase"><a name="math_toolkit.float_comparison.examples"></a></span><a class="link" href="float_comparison.html#math_toolkit.float_comparison.examples">Examples</a> 180 </h6> 181<p> 182 Some using statements will ensure that the functions we need are accessible. 183 </p> 184<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span> 185</pre> 186<p> 187 or 188 </p> 189<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">relative_difference</span><span class="special">;</span> 190<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">epsilon_difference</span><span class="special">;</span> 191<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">float_next</span><span class="special">;</span> 192<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">float_prior</span><span class="special">;</span> 193</pre> 194<p> 195 The following examples display values with all possibly significant digits. 196 Newer compilers should provide <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">FPT</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> for this purpose, and here we 197 use <code class="computeroutput"><span class="keyword">float</span></code> precision where <code class="computeroutput"><span class="identifier">max_digits10</span></code> = 9 to avoid displaying a distracting 198 number of decimal digits. 199 </p> 200<div class="note"><table border="0" summary="Note"> 201<tr> 202<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td> 203<th align="left">Note</th> 204</tr> 205<tr><td align="left" valign="top"><p> 206 Older compilers can use this formula to calculate <code class="computeroutput"><span class="identifier">max_digits10</span></code> 207 from <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">FPT</span><span class="special">>::</span><span class="identifier">digits10</span></code>: <code class="computeroutput"><span class="keyword">int</span> 208 <span class="identifier">max_digits10</span> <span class="special">=</span> 209 <span class="number">2</span> <span class="special">+</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">FPT</span><span class="special">>::</span><span class="identifier">digits10</span> 210 <span class="special">*</span> <span class="number">3010</span><span class="special">/</span><span class="number">10000</span><span class="special">;</span></code> 211 </p></td></tr> 212</table></div> 213<p> 214 One can set the display including all trailing zeros (helpful for this example 215 to show all potentially significant digits), and also to display <code class="computeroutput"><span class="keyword">bool</span></code> values as words rather than integers: 216 </p> 217<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> 218<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">boolalpha</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 219</pre> 220<p> 221 When comparing values that are <span class="emphasis"><em>quite close</em></span> or <span class="emphasis"><em>approximately 222 equal</em></span>, we could use either <code class="computeroutput"><span class="identifier">float_distance</span></code> 223 or <code class="computeroutput"><span class="identifier">relative_difference</span></code>/<code class="computeroutput"><span class="identifier">epsilon_difference</span></code>, for example with type 224 <code class="computeroutput"><span class="keyword">float</span></code>, these two values are adjacent 225 to each other: 226 </p> 227<pre class="programlisting"><span class="keyword">float</span> <span class="identifier">a</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> 228<span class="keyword">float</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">1</span> <span class="special">+</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> 229<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a = "</span> <span class="special"><<</span> <span class="identifier">a</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 230<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"b = "</span> <span class="special"><<</span> <span class="identifier">b</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 231<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"float_distance = "</span> <span class="special"><<</span> <span class="identifier">float_distance</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 232<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 233<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"epsilon_difference = "</span> <span class="special"><<</span> <span class="identifier">epsilon_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 234</pre> 235<p> 236 Which produces the output: 237 </p> 238<pre class="programlisting">a = 1.00000000 239b = 1.00000012 240float_distance = 1.00000000 241relative_difference = 1.19209290e-007 242epsilon_difference = 1.00000000 243</pre> 244<p> 245 In the example above, it just so happens that the edit distance as measured 246 by <code class="computeroutput"><span class="identifier">float_distance</span></code>, and the 247 difference measured in units of epsilon were equal. However, due to the way 248 floating point values are represented, that is not always the case: 249 </p> 250<pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="number">2.0f</span> <span class="special">/</span> <span class="number">3.0f</span><span class="special">;</span> <span class="comment">// 2/3 inexactly represented as a float</span> 251<span class="identifier">b</span> <span class="special">=</span> <span class="identifier">float_next</span><span class="special">(</span><span class="identifier">float_next</span><span class="special">(</span><span class="identifier">float_next</span><span class="special">(</span><span class="identifier">a</span><span class="special">)));</span> <span class="comment">// 3 floating point values above a</span> 252<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a = "</span> <span class="special"><<</span> <span class="identifier">a</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 253<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"b = "</span> <span class="special"><<</span> <span class="identifier">b</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 254<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"float_distance = "</span> <span class="special"><<</span> <span class="identifier">float_distance</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 255<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 256<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"epsilon_difference = "</span> <span class="special"><<</span> <span class="identifier">epsilon_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 257</pre> 258<p> 259 Which produces the output: 260 </p> 261<pre class="programlisting">a = 0.666666687 262b = 0.666666865 263float_distance = 3.00000000 264relative_difference = 2.68220901e-007 265epsilon_difference = 2.25000000 266</pre> 267<p> 268 There is another important difference between <code class="computeroutput"><span class="identifier">float_distance</span></code> 269 and the <code class="computeroutput"><span class="identifier">relative_difference</span><span class="special">/</span><span class="identifier">epsilon_difference</span></code> 270 functions in that <code class="computeroutput"><span class="identifier">float_distance</span></code> 271 returns a signed result that reflects which argument is larger in magnitude, 272 where as <code class="computeroutput"><span class="identifier">relative_difference</span><span class="special">/</span><span class="identifier">epsilon_difference</span></code> 273 simply return an unsigned value that represents how far apart the values are. 274 For example if we swap the order of the arguments: 275 </p> 276<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"float_distance = "</span> <span class="special"><<</span> <span class="identifier">float_distance</span><span class="special">(</span><span class="identifier">b</span><span class="special">,</span> <span class="identifier">a</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 277<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">b</span><span class="special">,</span> <span class="identifier">a</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 278<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"epsilon_difference = "</span> <span class="special"><<</span> <span class="identifier">epsilon_difference</span><span class="special">(</span><span class="identifier">b</span><span class="special">,</span> <span class="identifier">a</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 279</pre> 280<p> 281 The output is now: 282 </p> 283<pre class="programlisting">float_distance = -3.00000000 284relative_difference = 2.68220901e-007 285epsilon_difference = 2.25000000 286</pre> 287<p> 288 Zeros are always treated as equal, as are infinities as long as they have the 289 same sign: 290 </p> 291<pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> 292<span class="identifier">b</span> <span class="special">=</span> <span class="special">-</span><span class="number">0</span><span class="special">;</span> <span class="comment">// signed zero</span> 293<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 294<span class="identifier">a</span> <span class="special">=</span> <span class="identifier">b</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">();</span> 295<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 296<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="special">-</span><span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 297</pre> 298<p> 299 Which produces the output: 300 </p> 301<pre class="programlisting">relative_difference = 0.000000000 302relative_difference = 0.000000000 303relative_difference = 3.40282347e+038 304</pre> 305<p> 306 Note that finite values are always infinitely far away from infinities even 307 if those finite values are very large: 308 </p> 309<pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)();</span> 310<span class="identifier">b</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">();</span> 311<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a = "</span> <span class="special"><<</span> <span class="identifier">a</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 312<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"b = "</span> <span class="special"><<</span> <span class="identifier">b</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 313<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 314<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"epsilon_difference = "</span> <span class="special"><<</span> <span class="identifier">epsilon_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 315</pre> 316<p> 317 Which produces the output: 318 </p> 319<pre class="programlisting">a = 3.40282347e+038 320b = 1.#INF0000 321relative_difference = 3.40282347e+038 322epsilon_difference = 3.40282347e+038 323</pre> 324<p> 325 Finally, all denormalized values and zeros are treated as being effectively 326 equal: 327 </p> 328<pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">denorm_min</span><span class="special">();</span> 329<span class="identifier">b</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">*</span> <span class="number">2</span><span class="special">;</span> 330<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a = "</span> <span class="special"><<</span> <span class="identifier">a</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 331<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"b = "</span> <span class="special"><<</span> <span class="identifier">b</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 332<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"float_distance = "</span> <span class="special"><<</span> <span class="identifier">float_distance</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 333<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 334<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"epsilon_difference = "</span> <span class="special"><<</span> <span class="identifier">epsilon_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 335<span class="identifier">a</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> 336<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a = "</span> <span class="special"><<</span> <span class="identifier">a</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 337<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"b = "</span> <span class="special"><<</span> <span class="identifier">b</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 338<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"float_distance = "</span> <span class="special"><<</span> <span class="identifier">float_distance</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 339<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"relative_difference = "</span> <span class="special"><<</span> <span class="identifier">relative_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 340<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"epsilon_difference = "</span> <span class="special"><<</span> <span class="identifier">epsilon_difference</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 341</pre> 342<p> 343 Which produces the output: 344 </p> 345<pre class="programlisting">a = 1.40129846e-045 346b = 2.80259693e-045 347float_distance = 1.00000000 348relative_difference = 0.000000000 349epsilon_difference = 0.000000000 350a = 0.000000000 351b = 2.80259693e-045 352float_distance = 2.00000000 353relative_difference = 0.000000000 354epsilon_difference = 0.000000000</pre> 355<p> 356 Notice how, in the above example, two denormalized values that are a factor 357 of 2 apart are none the less only one representation apart! 358 </p> 359<p> 360 All the above examples are contained in <a href="../../../example/float_comparison_example.cpp" target="_top">float_comparison_example.cpp</a>. 361 </p> 362<h6> 363<a name="math_toolkit.float_comparison.h2"></a> 364 <span class="phrase"><a name="math_toolkit.float_comparison.small"></a></span><a class="link" href="float_comparison.html#math_toolkit.float_comparison.small">Handling 365 Absolute Errors</a> 366 </h6> 367<p> 368 Imagine we're testing the following function: 369 </p> 370<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">myspecial</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> 371<span class="special">{</span> 372 <span class="keyword">return</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">sin</span><span class="special">(</span><span class="number">4</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">);</span> 373<span class="special">}</span> 374</pre> 375<p> 376 This function has multiple roots, some of which are quite predictable in that 377 both <code class="computeroutput"><span class="identifier">sin</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code> and <code class="computeroutput"><span class="identifier">sin</span><span class="special">(</span><span class="number">4</span><span class="identifier">x</span><span class="special">)</span></code> are zero 378 together. Others occur because the values returned from those two functions 379 precisely cancel out. At such points the relative difference between the true 380 value of the function and the actual value returned may be <span class="emphasis"><em>arbitrarily 381 large</em></span> due to <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation 382 error</a>. 383 </p> 384<p> 385 In such a case, testing the function above by requiring that the values returned 386 by <code class="computeroutput"><span class="identifier">relative_error</span></code> or <code class="computeroutput"><span class="identifier">epsilon_error</span></code> are below some threshold is 387 pointless: the best we can do is to verify that the <span class="emphasis"><em>absolute difference</em></span> 388 between the true and calculated values is below some threshold. 389 </p> 390<p> 391 Of course, determining what that threshold should be is often tricky, but a 392 good starting point would be machine epsilon multiplied by the largest of the 393 values being summed. In the example above, the largest value returned by <code class="computeroutput"><span class="identifier">sin</span><span class="special">(</span><span class="identifier">whatever</span><span class="special">)</span></code> is 1, so simply using machine epsilon as the 394 target for maximum absolute difference might be a good start (though in practice 395 we may need a slightly higher value - some trial and error will be necessary). 396 </p> 397</div> 398<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 399<td align="left"></td> 400<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 401 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 402 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 403 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 404 Daryle Walker and Xiaogang Zhang<p> 405 Distributed under the Boost Software License, Version 1.0. (See accompanying 406 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>) 407 </p> 408</div></td> 409</tr></table> 410<hr> 411<div class="spirit-nav"> 412<a accesskey="p" href="next_float/ulp.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../utils.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="cond.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 413</div> 414</body> 415</html> 416
