1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>std::numeric_limits<> functions</title> 5<link rel="stylesheet" href="../../../multiprecision.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Chapter 1. Boost.Multiprecision"> 8<link rel="up" href="../limits.html" title="Numeric Limits"> 9<link rel="prev" href="constants.html" title="std::numeric_limits<> constants"> 10<link rel="next" href="limits32.html" title="Numeric limits for 32-bit platform"> 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="constants.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../limits.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="limits32.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h4 class="title"> 27<a name="boost_multiprecision.tut.limits.functions"></a><a class="link" href="functions.html" title="std::numeric_limits<> functions"><code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><></span></code> functions</a> 28</h4></div></div></div> 29<h5> 30<a name="boost_multiprecision.tut.limits.functions.h0"></a> 31 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.max_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.max_function"><code class="computeroutput"><span class="identifier">max</span></code> function</a> 32 </h5> 33<p> 34 Function <code class="computeroutput"><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">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)()</span></code> returns the largest finite value that 35 can be represented by the type T. If there is no such value (and <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">bounded</span></code> is <code class="computeroutput"><span class="keyword">false</span></code>) 36 then returns <code class="computeroutput"><span class="identifier">T</span><span class="special">()</span></code>. 37 </p> 38<p> 39 For <a href="https://en.cppreference.com/w/cpp/language/types" target="_top">fundamental 40 (built-in)</a> types there is usually a corresponding MACRO value TYPE_MAX, 41 where TYPE is CHAR, INT, FLOAT etc. 42 </p> 43<p> 44 Other types, including those provided by a typedef, for example <code class="computeroutput"><span class="identifier">INT64_T_MAX</span></code> for <code class="computeroutput"><span class="identifier">int64_t</span></code>, 45 may provide a macro definition. 46 </p> 47<p> 48 To cater for situations where no <code class="computeroutput"><span class="identifier">numeric_limits</span></code> 49 specialization is available (for example because the precision of the type 50 varies at runtime), packaged versions of this (and other functions) are 51 provided using 52 </p> 53<pre class="programlisting"><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">tools</span><span class="special">/</span><span class="identifier">precision</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 54 55<span class="identifier">T</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">tools</span><span class="special">::</span><span class="identifier">max_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>();</span> 56</pre> 57<p> 58 Of course, these simply use <code class="computeroutput"><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">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)()</span></code> 59 if available, but otherwise 'do something sensible'. 60 </p> 61<h5> 62<a name="boost_multiprecision.tut.limits.functions.h1"></a> 63 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.lowest_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.lowest_function">lowest 64 function</a> 65 </h5> 66<p> 67 Since C++11: <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">T</span><span class="special">>::</span><span class="identifier">lowest</span><span class="special">()</span></code> 68 is 69 </p> 70<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 71<li class="listitem"> 72 For integral types, the same as function <code class="computeroutput"><span class="identifier">min</span><span class="special">()</span></code>. 73 </li> 74<li class="listitem"> 75 For floating-point types, generally the negative of <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code> (but implementation-dependent). 76 </li> 77</ul></div> 78<pre class="programlisting"><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">double</span><span class="special">>::</span><span class="identifier">max</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">double</span><span class="special">>::</span><span class="identifier">lowest</span><span class="special">();</span> 79</pre> 80<h5> 81<a name="boost_multiprecision.tut.limits.functions.h2"></a> 82 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.min_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.min_function"><code class="computeroutput"><span class="identifier">min</span></code> function</a> 83 </h5> 84<p> 85 Function <code class="computeroutput"><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">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">)()</span></code> returns the minimum finite value that 86 can be represented by the type T. 87 </p> 88<p> 89 For <a href="https://en.cppreference.com/w/cpp/language/types" target="_top">fundamental 90 (built-in)</a> types, there is usually a corresponding MACRO value 91 TYPE_MIN, where TYPE is CHAR, INT, FLOAT etc. 92 </p> 93<p> 94 Other types, including those provided by a <code class="computeroutput"><span class="keyword">typedef</span></code>, 95 for example, <code class="computeroutput"><span class="identifier">INT64_T_MIN</span></code> 96 for <code class="computeroutput"><span class="identifier">int64_t</span></code>, may provide 97 a macro definition. 98 </p> 99<p> 100 For floating-point types, it is more fully defined as the <span class="emphasis"><em>minimum 101 positive normalized value</em></span>. 102 </p> 103<p> 104 See <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">T</span><span class="special">>::</span><span class="identifier">denorm_min</span><span class="special">()</span></code> 105 for the smallest denormalized value, provided 106 </p> 107<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">has_denorm</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span> 108</pre> 109<p> 110 To cater for situations where no <code class="computeroutput"><span class="identifier">numeric_limits</span></code> 111 specialization is available (for example because the precision of the type 112 varies at runtime), packaged versions of this (and other functions) are 113 provided using 114 </p> 115<pre class="programlisting"><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">tools</span><span class="special">/</span><span class="identifier">precision</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 116 117<span class="identifier">T</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">tools</span><span class="special">::</span><span class="identifier">min_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>();</span> 118</pre> 119<p> 120 Of course, these simply use <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">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span></code> if available. 121 </p> 122<h5> 123<a name="boost_multiprecision.tut.limits.functions.h3"></a> 124 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.denorm_min_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.denorm_min_function">denorm_min 125 function</a> 126 </h5> 127<p> 128 Function <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">T</span><span class="special">>::</span><span class="identifier">denorm_min</span><span class="special">()</span></code> 129 returns the smallest <a href="http://en.wikipedia.org/wiki/Denormal_number" target="_top">denormalized 130 value</a>, provided 131 </p> 132<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">has_denorm</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span> 133</pre> 134<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">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> 135<span class="keyword">if</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">double</span><span class="special">>::</span><span class="identifier">has_denorm</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span><span class="special">)</span> 136<span class="special">{</span> 137 <span class="keyword">double</span> <span class="identifier">d</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">double</span><span class="special">>::</span><span class="identifier">denorm_min</span><span class="special">();</span> 138 139 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">d</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="comment">// 4.9406564584124654e-324</span> 140 141 <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span> 142 143 <span class="keyword">double</span> <span class="identifier">significand</span> <span class="special">=</span> <span class="identifier">frexp</span><span class="special">(</span><span class="identifier">d</span><span class="special">,</span> <span class="special">&</span><span class="identifier">exponent</span><span class="special">);</span> 144 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"exponent = "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">exponent</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="comment">// fffffbcf</span> 145 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"significand = "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">significand</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="comment">// 0.50000000000000000</span> 146<span class="special">}</span> 147<span class="keyword">else</span> 148<span class="special">{</span> 149 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"No denormalization. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 150<span class="special">}</span> 151</pre> 152<p> 153 The exponent is effectively reduced from -308 to -324 (though it remains 154 encoded as zero and leading zeros appear in the significand, thereby losing 155 precision until the significand reaches zero). 156 </p> 157<h5> 158<a name="boost_multiprecision.tut.limits.functions.h4"></a> 159 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.round_error"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.round_error">round_error</a> 160 </h5> 161<p> 162 Function <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">T</span><span class="special">>::</span><span class="identifier">round_error</span><span class="special">()</span></code> 163 returns the maximum error (in units of <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit 164 in the last place (ULP)</a>) that can be caused by any basic arithmetic 165 operation. 166 </p> 167<pre class="programlisting"><span class="identifier">round_style</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round_indeterminate</span><span class="special">;</span> 168</pre> 169<p> 170 The rounding style is indeterminable at compile time. 171 </p> 172<p> 173 For floating-point types, when rounding is to nearest, only half a bit 174 is lost by rounding, and <code class="computeroutput"><span class="identifier">round_error</span> 175 <span class="special">==</span> <span class="number">0.5</span></code>. 176 In contrast when rounding is towards zero, or plus/minus infinity, we can 177 loose up to one bit from rounding, and <code class="computeroutput"><span class="identifier">round_error</span> 178 <span class="special">==</span> <span class="number">1</span></code>. 179 </p> 180<p> 181 For integer types, rounding always to zero, so at worst almost one bit 182 can be rounded, so <code class="computeroutput"><span class="identifier">round_error</span> 183 <span class="special">==</span> <span class="number">1</span></code>. 184 </p> 185<p> 186 <code class="computeroutput"><span class="identifier">round_error</span><span class="special">()</span></code> 187 can be used with <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">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code> 188 to estimate the maximum potential error caused by rounding. For typical 189 floating-point types, <code class="computeroutput"><span class="identifier">round_error</span><span class="special">()</span> <span class="special">=</span> <span class="number">1</span><span class="special">/</span><span class="number">2</span></code>, so half 190 epsilon is the maximum potential error. 191 </p> 192<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">round_err</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">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span> <span class="comment">// 2.2204460492503131e-016</span> 193 <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">double</span><span class="special">>::</span><span class="identifier">round_error</span><span class="special">();</span> <span class="comment">// 1/2</span> 194<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">round_err</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="comment">// 1.1102230246251565e-016</span> 195</pre> 196<p> 197 There are, of course, many occasions when much bigger loss of precision 198 occurs, for example, caused by <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">Loss 199 of significance or cancellation error</a> or very many iterations. 200 </p> 201<h5> 202<a name="boost_multiprecision.tut.limits.functions.h5"></a> 203 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.epsilon"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.epsilon">epsilon</a> 204 </h5> 205<p> 206 Function <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">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code> 207 is meaningful only for non-integral types. 208 </p> 209<p> 210 It returns the difference between <code class="computeroutput"><span class="number">1.0</span></code> 211 and the next value representable by the floating-point type T. So it is 212 a one least-significant-bit change in this floating-point value. 213 </p> 214<p> 215 For <code class="computeroutput"><span class="keyword">double</span></code> (<code class="computeroutput"><span class="identifier">float_64t</span></code>) it is <code class="computeroutput"><span class="number">2.2204460492503131e-016</span></code> 216 showing all possibly significant 17 decimal digits. 217 </p> 218<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">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> 219<span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> 220<span class="keyword">double</span> <span class="identifier">eps</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">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> 221<span class="keyword">double</span> <span class="identifier">dpeps</span> <span class="special">=</span> <span class="identifier">d</span><span class="special">+</span><span class="identifier">eps</span><span class="special">;</span> 222<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">showpoint</span> <span class="comment">// Ensure all trailing zeros are shown.</span> 223 <span class="special"><<</span> <span class="identifier">d</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="comment">// 1.0000000000000000</span> 224 <span class="special"><<</span> <span class="identifier">dpeps</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="comment">// 2.2204460492503131e-016</span> 225<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dpeps</span> <span class="special">-</span> <span class="identifier">d</span> <span class="comment">// 1.0000000000000002</span> 226 <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 227</pre> 228<p> 229 We can explicitly increment by one bit using the function <code class="computeroutput"><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></code> 230 and the result is the same as adding <code class="computeroutput"><span class="identifier">epsilon</span></code>. 231 </p> 232<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">one</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> 233<span class="keyword">double</span> <span class="identifier">nad</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">float_next</span><span class="special">(</span><span class="identifier">one</span><span class="special">);</span> 234<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">nad</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="comment">// 1.0000000000000002</span> 235 <span class="special"><<</span> <span class="identifier">nad</span> <span class="special">-</span> <span class="identifier">one</span> <span class="comment">// 2.2204460492503131e-016</span> 236 <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 237</pre> 238<p> 239 Adding any smaller value, like half <code class="computeroutput"><span class="identifier">epsilon</span></code>, 240 will have no effect on this value. 241 </p> 242<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">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> 243<span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> 244<span class="keyword">double</span> <span class="identifier">eps</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">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> 245<span class="keyword">double</span> <span class="identifier">dpeps</span> <span class="special">=</span> <span class="identifier">d</span> <span class="special">+</span> <span class="identifier">eps</span><span class="special">/</span><span class="number">2</span><span class="special">;</span> 246 247<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">showpoint</span> <span class="comment">// Ensure all trailing zeros are shown.</span> 248 <span class="special"><<</span> <span class="identifier">dpeps</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="comment">// 1.0000000000000000</span> 249 <span class="special"><<</span> <span class="identifier">eps</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">endl</span><span class="special">;</span> <span class="comment">// 1.1102230246251565e-016</span> 250<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dpeps</span> <span class="special">-</span> <span class="identifier">d</span> <span class="comment">// 0.00000000000000000</span> 251 <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 252</pre> 253<p> 254 So this cancellation error leaves the values equal, despite adding half 255 <code class="computeroutput"><span class="identifier">epsilon</span></code>. 256 </p> 257<p> 258 To achieve greater portability over platform and floating-point type, Boost.Math 259 and Boost.Multiprecision provide a package of functions that 'do something 260 sensible' if the standard <code class="computeroutput"><span class="identifier">numeric_limits</span></code> 261 is not available. To use these <code class="computeroutput"><span class="preprocessor">#include</span> 262 <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">tools</span><span class="special">/</span><span class="identifier">precision</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code>. 263 </p> 264<p> 265 A tolerance might be defined using this version of epsilon thus: 266 </p> 267<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">tolerance</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">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span> <span class="special">*</span> <span class="number">2</span><span class="special">;</span> 268</pre> 269<h6> 270<a name="boost_multiprecision.tut.limits.functions.h6"></a> 271 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.FP_tolerance"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.FP_tolerance">Tolerance 272 for Floating-point Comparisons</a> 273 </h6> 274<p> 275 <a href="https://en.wikipedia.org/wiki/Machine_epsilon" target="_top">Machine epsilon 276 ε</a> is very useful to compute a tolerance when comparing floating-point 277 values, a much more difficult task than is commonly imagined. 278 </p> 279<p> 280 The C++ standard specifies <a href="https://en.cppreference.com/w/cpp/types/numeric_limits/epsilon" target="_top"><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">epsilon</span><span class="special">()</span></code></a> 281 and Boost.Multiprecision implements this (where possible) for its program-defined 282 types analogous to the <a href="https://en.cppreference.com/w/cpp/language/types" target="_top">fundamental 283 (built-in)</a> floating-point types like <code class="computeroutput"><span class="keyword">double</span></code> 284 <code class="computeroutput"><span class="keyword">float</span></code>. 285 </p> 286<p> 287 For more information than you probably want (but still need) see <a href="http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html" target="_top">What 288 Every Computer Scientist Should Know About Floating-Point Arithmetic</a> 289 </p> 290<p> 291 The naive test comparing the absolute difference between two values and 292 a tolerance does not give useful results if the values are too large or 293 too small. 294 </p> 295<p> 296 So Boost.Test uses an algorithm first devised by Knuth for reliably checking 297 if floating-point values are close enough. 298 </p> 299<p> 300 See Donald. E. Knuth. The art of computer programming (vol II). Copyright 301 1998 Addison-Wesley Longman, Inc., 0-201-89684-2. Addison-Wesley Professional; 302 3rd edition. (The relevant equations are in paragraph 4.2.2, Eq. 36 and 303 37.) 304 </p> 305<p> 306 See <a href="https://www.boost.org/doc/libs/release/libs/test/doc/html/boost_test/testing_tools/extended_comparison/floating_point/floating_points_comparison_theory.html" target="_top">Boost.Math 307 floating_point comparison</a> for more details. 308 </p> 309<p> 310 See also: 311 </p> 312<p> 313 <a href="http://adtmag.com/articles/2000/03/15/comparing-floats-how-to-determine-if-floating-quantities-are-close-enough-once-a-tolerance-has-been.aspx" target="_top">Alberto 314 Squassia, Comparing floats</a> 315 </p> 316<p> 317 <a href="http://adtmag.com/articles/2000/03/16/comparing-floats-how-to-determine-if-floating-quantities-are-close-enough-once-a-tolerance-has-been.aspx" target="_top">Alberto 318 Squassia, Comparing floats code</a> 319 </p> 320<p> 321 <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 322 Floating-Point_Comparison</a> 323 </p> 324<p> 325 For example, if we want a tolerance that might suit about 9 arithmetical 326 operations, say sqrt(9) = 3, we could define: 327 </p> 328<pre class="programlisting"><span class="identifier">T</span> <span class="identifier">tolerance</span> <span class="special">=</span> <span class="number">3</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">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> 329</pre> 330<p> 331 This is very widely used in Boost.Math testing with Boost.Test's macro 332 <code class="computeroutput"><span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span></code> 333 </p> 334<pre class="programlisting"><span class="identifier">T</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> 335<span class="identifier">T</span> <span class="identifier">calculated</span> <span class="special">=</span> <span class="number">1.0</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">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> 336 337<span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span> 338</pre> 339<p> 340 used thus: 341 </p> 342<pre class="programlisting"><span class="identifier">cd</span> <span class="special">./</span><span class="identifier">test</span> 343<span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span> 344</pre> 345<p> 346 (There is also a version BOOST_CHECK_CLOSE using tolerance as a <span class="bold"><strong>percentage</strong></span> rather than a fraction; usually the fraction 347 version is simpler to use). 348 </p> 349<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special">;</span> 350<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float</span><span class="special">;</span> 351<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">et_off</span><span class="special">;</span> 352 353<span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">cpp_dec_float</span><span class="special"><</span><span class="number">50</span><span class="special">>,</span> <span class="identifier">et_off</span> <span class="special">></span> <span class="identifier">cpp_dec_float_50</span><span class="special">;</span> <span class="comment">// 50 decimal digits.</span> 354</pre> 355<div class="note"><table border="0" summary="Note"> 356<tr> 357<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> 358<th align="left">Note</th> 359</tr> 360<tr><td align="left" valign="top"><p> 361 that Boost.Test does not yet allow floating-point comparisons with expression 362 templates on, so the default expression template parameter has been replaced 363 by <code class="computeroutput"><span class="identifier">et_off</span></code>. 364 </p></td></tr> 365</table></div> 366<pre class="programlisting"><span class="identifier">cpp_dec_float_50</span> <span class="identifier">tolerance</span> <span class="special">=</span> <span class="number">3</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">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> 367<span class="identifier">cpp_dec_float_50</span> <span class="identifier">expected</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">constants</span><span class="special">::</span><span class="identifier">two_pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>();</span> 368<span class="identifier">cpp_dec_float_50</span> <span class="identifier">calculated</span> <span class="special">=</span> <span class="number">2</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">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>();</span> 369 370<span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span> 371</pre> 372<h5> 373<a name="boost_multiprecision.tut.limits.functions.h7"></a> 374 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.infinity"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.infinity">Infinity - 375 positive and negative</a> 376 </h5> 377<p> 378 For floating-point types only, for which <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">T</span><span class="special">>::</span><span class="identifier">has_infinity</span> 379 <span class="special">==</span> <span class="keyword">true</span></code>, 380 function <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">T</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">()</span></code> 381 provides an implementation-defined representation for ∞. 382 </p> 383<p> 384 The 'representation' is a particular bit pattern reserved for infinity. 385 For IEEE754 system (for which <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">T</span><span class="special">>::</span><span class="identifier">is_iec559</span> 386 <span class="special">==</span> <span class="keyword">true</span></code>) 387 <a href="http://en.wikipedia.org/wiki/IEEE_754-1985#Positive_and_negative_infinity" target="_top">positive 388 and negative infinity</a> are assigned bit patterns for all defined 389 floating-point types. 390 </p> 391<p> 392 Confusingly, the string resulting from outputting this representation, 393 is also implementation-defined. And the string that can be input to generate 394 the representation is also implementation-defined. 395 </p> 396<p> 397 For example, the output is <code class="computeroutput"><span class="number">1.</span><span class="special">#</span><span class="identifier">INF</span></code> 398 on Microsoft systems, but <code class="computeroutput"><span class="identifier">inf</span></code> 399 on most *nix platforms. 400 </p> 401<p> 402 This implementation-defined-ness has hampered use of infinity (and NaNs) 403 but <a href="https://www.boost.org/doc/libs/release/libs/math/doc/index.html" target="_top">Boost.Math</a> 404 and <a href="https://www.boost.org/doc/libs/release/libs/multiprecision/doc/index.html" target="_top">Boost.Multiprecision</a> 405 work hard to provide a sensible representation for <span class="bold"><strong>all</strong></span> 406 floating-point types, not just the <a href="https://en.cppreference.com/w/cpp/language/types" target="_top">fundamental 407 (built-in) types</a>, which with the use of suitable facets to define 408 the input and output strings, makes it possible to use these useful features 409 portably and including <a href="https://www.boost.org/doc/libs/release/libs/serialization/doc/index.html" target="_top">Boost.Serialization</a>. 410 </p> 411<h5> 412<a name="boost_multiprecision.tut.limits.functions.h8"></a> 413 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.not_a_number_nan"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.not_a_number_nan">Not-A-Number 414 NaN</a> 415 </h5> 416<h6> 417<a name="boost_multiprecision.tut.limits.functions.h9"></a> 418 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.quiet_nan"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.quiet_nan">Quiet_NaN</a> 419 </h6> 420<p> 421 For floating-point types only, for which <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">T</span><span class="special">>::</span><span class="identifier">has_quiet_NaN</span> 422 <span class="special">==</span> <span class="keyword">true</span></code>, 423 function <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">T</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">()</span></code> 424 provides an implementation-defined representation for NaN. 425 </p> 426<p> 427 <a href="http://en.wikipedia.org/wiki/NaN" target="_top">NaNs</a> are values to 428 indicate that the result of an assignment or computation is meaningless. 429 A typical example is <code class="computeroutput"><span class="number">0</span><span class="special">/</span><span class="number">0</span></code> but there are many others. 430 </p> 431<p> 432 NaNs may also be used, to represent missing values: for example, these 433 could, by convention, be ignored in calculations of statistics like means. 434 </p> 435<p> 436 Many of the problems with a representation for <a href="http://en.wikipedia.org/wiki/NaN" target="_top">Not-A-Number</a> 437 has hampered portable use, similar to those with infinity. 438 </p> 439<p> 440 NaN can be used with binary multiprecision types like <code class="computeroutput"><span class="identifier">cpp_bin_float_quad</span></code>: 441 </p> 442<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_quad</span><span class="special">;</span> 443 444<span class="keyword">if</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">cpp_bin_float_quad</span><span class="special">>::</span><span class="identifier">has_quiet_NaN</span> <span class="special">==</span> <span class="keyword">true</span><span class="special">)</span> 445<span class="special">{</span> 446 <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">NaN</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">cpp_bin_float_quad</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">();</span> 447 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cpp_bin_float_quad NaN is "</span> <span class="special"><<</span> <span class="identifier">NaN</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="comment">// cpp_bin_float_quad NaN is nan</span> 448 449 <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="identifier">NaN</span><span class="special">;</span> 450 <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">calculated</span> <span class="special">=</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">NaN</span><span class="special">;</span> 451 <span class="comment">// Comparisons of NaN's always fail:</span> 452 <span class="keyword">bool</span> <span class="identifier">b</span> <span class="special">=</span> <span class="identifier">expected</span> <span class="special">==</span> <span class="identifier">calculated</span><span class="special">;</span> 453 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</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> 454 <span class="identifier">BOOST_CHECK_NE</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">expected</span><span class="special">);</span> 455 <span class="identifier">BOOST_CHECK_NE</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">);</span> 456<span class="special">}</span> 457<span class="keyword">else</span> 458<span class="special">{</span> 459 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Type "</span> <span class="special"><<</span> <span class="keyword">typeid</span><span class="special">(</span><span class="identifier">cpp_bin_float_quad</span><span class="special">).</span><span class="identifier">name</span><span class="special">()</span> <span class="special"><<</span> <span class="string">" does not have NaNs!"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 460<span class="special">}</span> 461</pre> 462<p> 463 But using Boost.Math and suitable facets can permit portable use of both 464 NaNs and positive and negative infinity. 465 </p> 466<p> 467 See <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/doc/html/../../../../libs/math/example/nonfinite_facet_sstream.cpp" target="_top">boost:/libs/math/example/nonfinite_facet_sstream.cpp</a> 468 and we also need 469 </p> 470<pre class="programlisting"><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">nonfinite_num_facets</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 471</pre> 472<p> 473 Then we can equally well use a multiprecision type cpp_bin_float_quad: 474 </p> 475<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_quad</span><span class="special">;</span> 476 477<span class="keyword">typedef</span> <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">T</span><span class="special">;</span> 478 479<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">nonfinite_num_put</span><span class="special">;</span> 480<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">nonfinite_num_get</span><span class="special">;</span> 481<span class="special">{</span> 482 <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">old_locale</span><span class="special">;</span> 483 <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">tmp_locale</span><span class="special">(</span><span class="identifier">old_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_put</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> 484 <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">new_locale</span><span class="special">(</span><span class="identifier">tmp_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_get</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> 485 <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span> 486 <span class="identifier">ss</span><span class="special">.</span><span class="identifier">imbue</span><span class="special">(</span><span class="identifier">new_locale</span><span class="special">);</span> 487 <span class="identifier">T</span> <span class="identifier">inf</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">T</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">();</span> 488 <span class="identifier">ss</span> <span class="special"><<</span> <span class="identifier">inf</span><span class="special">;</span> <span class="comment">// Write out.</span> 489 <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special">==</span> <span class="string">"inf"</span><span class="special">);</span> 490 <span class="identifier">T</span> <span class="identifier">r</span><span class="special">;</span> 491 <span class="identifier">ss</span> <span class="special">>></span> <span class="identifier">r</span><span class="special">;</span> <span class="comment">// Read back in.</span> 492 <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">inf</span> <span class="special">==</span> <span class="identifier">r</span><span class="special">);</span> <span class="comment">// Confirms that the floating-point values really are identical.</span> 493 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity output was "</span> <span class="special"><<</span> <span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</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> 494 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity input was "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 495<span class="special">}</span> 496</pre> 497<pre class="programlisting"><span class="identifier">infinity</span> <span class="identifier">output</span> <span class="identifier">was</span> <span class="identifier">inf</span> 498<span class="identifier">infinity</span> <span class="identifier">input</span> <span class="identifier">was</span> <span class="identifier">inf</span> 499</pre> 500<p> 501 Similarly we can do the same with NaN (except that we cannot use <code class="computeroutput"><span class="identifier">assert</span></code> (because any comparisons with 502 NaN always return false). 503 </p> 504<pre class="programlisting"><span class="special">{</span> 505 <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">old_locale</span><span class="special">;</span> 506 <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">tmp_locale</span><span class="special">(</span><span class="identifier">old_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_put</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> 507 <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">new_locale</span><span class="special">(</span><span class="identifier">tmp_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_get</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> 508 <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span> 509 <span class="identifier">ss</span><span class="special">.</span><span class="identifier">imbue</span><span class="special">(</span><span class="identifier">new_locale</span><span class="special">);</span> 510 <span class="identifier">T</span> <span class="identifier">n</span><span class="special">;</span> 511 <span class="identifier">T</span> <span class="identifier">NaN</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">T</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">();</span> 512 <span class="identifier">ss</span> <span class="special"><<</span> <span class="identifier">NaN</span><span class="special">;</span> <span class="comment">// Write out.</span> 513 <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special">==</span> <span class="string">"nan"</span><span class="special">);</span> 514 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"NaN output was "</span> <span class="special"><<</span> <span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</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> 515 <span class="identifier">ss</span> <span class="special">>></span> <span class="identifier">n</span><span class="special">;</span> <span class="comment">// Read back in.</span> 516 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"NaN input was "</span> <span class="special"><<</span> <span class="identifier">n</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 517<span class="special">}</span> 518</pre> 519<pre class="programlisting"><span class="identifier">NaN</span> <span class="identifier">output</span> <span class="identifier">was</span> <span class="identifier">nan</span> 520<span class="identifier">NaN</span> <span class="identifier">input</span> <span class="identifier">was</span> <span class="identifier">nan</span> 521</pre> 522<h6> 523<a name="boost_multiprecision.tut.limits.functions.h10"></a> 524 <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.signaling_nan"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.signaling_nan">Signaling 525 NaN</a> 526 </h6> 527<p> 528 For floating-point types only, for which <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">T</span><span class="special">>::</span><span class="identifier">has_signaling_NaN</span> 529 <span class="special">==</span> <span class="keyword">true</span></code>, 530 function <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">T</span><span class="special">>::</span><span class="identifier">signaling_NaN</span><span class="special">()</span></code> 531 provides an implementation-defined representation for NaN that causes a 532 hardware trap. It should be noted however, that at least one implementation 533 of this function causes a hardware trap to be triggered simply by calling 534 <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">T</span><span class="special">>::</span><span class="identifier">signaling_NaN</span><span class="special">()</span></code>, 535 and not only by using the value returned. 536 </p> 537</div> 538<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 539<td align="left"></td> 540<td align="right"><div class="copyright-footer">Copyright © 2002-2020 John 541 Maddock and Christopher Kormanyos<p> 542 Distributed under the Boost Software License, Version 1.0. (See accompanying 543 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>) 544 </p> 545</div></td> 546</tr></table> 547<hr> 548<div class="spirit-nav"> 549<a accesskey="p" href="constants.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../limits.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="limits32.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 550</div> 551</body> 552</html> 553
