<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Floating-point Comparison</title> <link rel="stylesheet" href="../math.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> <link rel="home" href="../index.html" title="Math Toolkit 2.5.1"> <link rel="up" href="../utils.html" title="Chapter 2. Floating Point Utilities"> <link rel="prev" href="next_float/ulp.html" title="Obtaining the Size of a Unit In the Last Place - ULP"> <link rel="next" href="../cstdfloat.html" title="Chapter 3. Specified-width floating-point typedefs"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> <td align="center"><a href="../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <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="../cstdfloat.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h2 class="title" style="clear: both"> <a name="math_toolkit.float_comparison"></a><a class="link" href="float_comparison.html" title="Floating-point Comparison">Floating-point Comparison</a> </h2></div></div></div> <p> Comparison of floating-point values has always been a source of endless difficulty and confusion. </p> <p> Unlike integral values that are exact, all floating-point operations will potentially produce an inexact result that will be rounded to the nearest available binary representation. Even apparently inocuous operations such as assigning 0.1 to a double produces an inexact result (as this decimal number has no exact binary representation). </p> <p> Floating-point computations also involve rounding so that some 'computational noise' is added, and hence results are also not exact (although repeatable, at least under identical platforms and compile options). </p> <p> Sadly, this conflicts with the expectation of most users, as many articles and innumerable cries for help show all too well. </p> <p> Some background reading is: </p> <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> <li class="listitem"> Knuth D.E. The art of computer programming, vol II, section 4.2, especially Floating-Point Comparison 4.2.2, pages 198-220. </li> <li class="listitem"> <a href="http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html" target="_top">David Goldberg, "What Every Computer Scientist Should Know About Floating-Point Arithmetic"</a> </li> <li class="listitem"> <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 Squassabia, Comparing floats listing</a> </li> <li class="listitem"> <a href="https://code.google.com/p/googletest/wiki/AdvancedGuide#Floating-Point_Comparison" target="_top">Google Floating-Point_Comparison guide</a> </li> <li class="listitem"> <a href="../../../../../libs/test/doc/html/boost_test/users_guide/testing_tools/testing_floating_points.html" target="_top">Boost.Test Floating-Point_Comparison</a> </li> </ul></div> <p> Boost provides a number of ways to compare floating-point values to see if they are tolerably close enough to each other, but first we must decide what kind of comparison we require: </p> <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> <li class="listitem"> Absolute difference/error: the absolute difference between two values <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 to make if we know that the result may have cancellation error (see below). </li> <li class="listitem"> The edit distance between the two values: i.e. how many (binary) floating-point values are between two values <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>? This is provided by the function <a href="../../../../../libs/math/doc/html/math_toolkit/next_float/float_distance.html" target="_top">Boost.Math float_distance</a>, but is probably only useful when you know that the distance should be very small. This function is somewhat difficult to compute, and doesn't scale to values that are very far apart. In other words, use with care. </li> <li class="listitem"> The relative distance/error between two values. This is quick and easy to compute, and is generally the method of choice when checking that your results are "tolerably close" to one another. However, it is not as exact as the edit distance when dealing with small differences, and due to the way floating-point values are encoded can "wobble" by a factor of 2 compared to the "true" edit distance. This is the method documented below: if <code class="computeroutput"><span class="identifier">float_distance</span></code> is a surgeon's scalpel, then <code class="computeroutput"><span class="identifier">relative_difference</span></code> is more like a Swiss army knife: both have important but different use cases. </li> </ul></div> <h6> <a name="math_toolkit.float_comparison.h0"></a> <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 Comparison of Floating-point Values</a> </h6> <p> <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> </p> <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> <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> <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> <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> </pre> <p> The function <code class="computeroutput"><span class="identifier">relative_difference</span></code> returns the relative distance/error <span class="emphasis"><em>E</em></span> between two values as defined by: </p> <pre class="programlisting">E = fabs((a - b) / min(a,b))</pre> <p> The function <code class="computeroutput"><span class="identifier">epsilon_difference</span></code> 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> <span class="identifier">b</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">eps</span></code> where <code class="computeroutput"><span class="identifier">eps</span></code> is the machine epsilon for the result type. </p> <p> The following special cases are handled as follows: </p> <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> <li class="listitem"> If either of <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> is a NaN, then returns the largest representable value for T: for example for type <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> which is the same as <code class="computeroutput"><span class="identifier">DBL_MAX</span></code> or <code class="computeroutput"><span class="number">1.7976931348623157e+308</span></code>. </li> <li class="listitem"> If <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> differ in sign then returns the largest representable value for T. </li> <li class="listitem"> If both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are both infinities (of the same sign), then returns zero. </li> <li class="listitem"> If just one of <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> is an infinity, then returns the largest representable value for T. </li> <li class="listitem"> If both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are zero then returns zero. </li> <li class="listitem"> If just one of <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> is a zero or a denormalized value, then it is treated as if it were the smallest (non-denormalized) value representable in T for the purposes of the above calculation. </li> </ul></div> <p> These rules were primarily designed to assist with our own test suite, they are designed to be robust enough that the function can in most cases be used blindly, including in cases where the expected result is actually too small to represent in type T and underflows to zero. </p> <h6> <a name="math_toolkit.float_comparison.h1"></a> <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> </h6> <p> Some using statements will ensure that the functions we need are accessible. </p> <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> </pre> <p> or </p> <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> <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> <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> <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> </pre> <p> The following examples display values with all possibly significant digits. Newer compilers should provide <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limitsFPT</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> for this purpose, and here we 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 number of decimal digits. </p> <div class="note"><table border="0" summary="Note"> <tr> <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td> <th align="left">Note</th> </tr> <tr><td align="left" valign="top"><p> Older compilers can use this formula to calculate <code class="computeroutput"><span class="identifier">max_digits10</span></code> 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>:<br>    <code class="computeroutput"><span class="keyword">int</span> <span class="identifier">max_digits10</span> <span class="special">=</span> <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> <span class="special">*</span> <span class="number">3010</span><span class="special">/</span><span class="number">10000</span><span class="special">;</span></code> </p></td></tr> </table></div> <p> One can set the display including all trailing zeros (helpful for this example 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: </p> <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> <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> </pre> <p> When comparing values that are <span class="emphasis"><em>quite close</em></span> or <span class="emphasis"><em>approximately equal</em></span>, we could use either <code class="computeroutput"><span class="identifier">float_distance</span></code> 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 <code class="computeroutput"><span class="keyword">float</span></code>, these two values are adjacent to each other: </p> <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> <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> <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> <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> <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> <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> <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> </pre> <p> Which produces the output: </p> <pre class="programlisting">a = 1.00000000 b = 1.00000012 float_distance = 1.00000000 relative_difference = 1.19209290e-007 epsilon_difference = 1.00000000 </pre> <p> In the example above, it just so happens that the edit distance as measured by <code class="computeroutput"><span class="identifier">float_distance</span></code>, and the difference measured in units of epsilon were equal. However, due to the way floating point values are represented, that is not always the case: </p> <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> <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> <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> <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> <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> <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> <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> </pre> <p> Which produces the output: </p> <pre class="programlisting">a = 0.666666687 b = 0.666666865 float_distance = 3.00000000 relative_difference = 2.68220901e-007 epsilon_difference = 2.25000000 </pre> <p> There is another important difference between <code class="computeroutput"><span class="identifier">float_distance</span></code> and the <code class="computeroutput"><span class="identifier">relative_difference</span><span class="special">/</span><span class="identifier">epsilon_difference</span></code> functions in that <code class="computeroutput"><span class="identifier">float_distance</span></code> returns a signed result that reflects which argument is larger in magnitude, where as <code class="computeroutput"><span class="identifier">relative_difference</span><span class="special">/</span><span class="identifier">epsilon_difference</span></code> simply return an unsigned value that represents how far apart the values are. For example if we swap the order of the arguments: </p> <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> <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> <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> </pre> <p> The output is now: </p> <pre class="programlisting">float_distance = -3.00000000 relative_difference = 2.68220901e-007 epsilon_difference = 2.25000000 </pre> <p> Zeros are always treated as equal, as are infinities as long as they have the same sign: </p> <pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <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> <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> <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> <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> <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> </pre> <p> Which produces the output: </p> <pre class="programlisting">relative_difference = 0.000000000 relative_difference = 0.000000000 relative_difference = 3.40282347e+038 </pre> <p> Note that finite values are always infinitely far away from infinities even if those finite values are very large: </p> <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> <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> <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> <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> <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> <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> </pre> <p> Which produces the output: </p> <pre class="programlisting">a = 3.40282347e+038 b = 1.#INF0000 relative_difference = 3.40282347e+038 epsilon_difference = 3.40282347e+038 </pre> <p> Finally, all denormalized values and zeros are treated as being effectively equal: </p> <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> <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> <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> <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> <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> <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> <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> <span class="identifier">a</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <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> <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> <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> <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> <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> </pre> <p> Which produces the output: </p> <pre class="programlisting">a = 1.40129846e-045 b = 2.80259693e-045 float_distance = 1.00000000 relative_difference = 0.000000000 epsilon_difference = 0.000000000 a = 0.000000000 b = 2.80259693e-045 float_distance = 2.00000000 relative_difference = 0.000000000 epsilon_difference = 0.000000000</pre> <p> Notice how, in the above example, two denormalized values that are a factor of 2 apart are none the less only one representation apart! </p> <p> All the above examples are contained in <a href="../../../example/float_comparison_example.cpp" target="_top">float_comparison_example.cpp</a>. </p> <h6> <a name="math_toolkit.float_comparison.h2"></a> <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 Absolute Errors</a> </h6> <p> Imagine we're testing the following function: </p> <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> <span class="special">{</span> <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> <span class="special">}</span> </pre> <p> This function has multiple roots, some of which are quite predicable in that 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 together. Others occur because the values returned from those two functions precisely cancel out. At such points the relative difference between the true value of the function and the actual value returned may be <span class="emphasis"><em>arbitrarily large</em></span> due to <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation error</a>. </p> <p> In such a case, testing the function above by requiring that the values returned 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 pointless: the best we can do is to verify that the <span class="emphasis"><em>absolute difference</em></span> between the true and calculated values is below some threshold. </p> <p> Of course, determining what that threshold should be is often tricky, but a good starting point would be machine epsilon multiplied by the largest of the 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 target for maximum absolute difference might be a good start (though in practice we may need a slightly higher value - some trial and error will be necessary). </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> Distributed under the Boost Software License, Version 1.0. (See accompanying 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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <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="../cstdfloat.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>