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  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&lt;&gt; functions"><code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;&gt;</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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</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 built-in types there is usually a corresponding MACRO value TYPE_MAX,
  40           where TYPE is CHAR, INT, FLOAT etc.
  41         </p>
  42 <p>
  43           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>,
  44           may provide a macro definition.
  45         </p>
  46 <p>
  47           To cater for situations where no <code class="computeroutput"><span class="identifier">numeric_limits</span></code>
  48           specialization is available (for example because the precision of the type
  49           varies at runtime), packaged versions of this (and other functions) are
  50           provided using
  51         </p>
  52 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
  53
  54 <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;();</span>
  55 </pre>
  56 <p>
  57           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">max</span><span class="special">)()</span></code>
  58           if available, but otherwise 'do something sensible'.
  59         </p>
  60 <h5>
  61 <a name="boost_multiprecision.tut.limits.functions.h1"></a>
  62           <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
  63           function</a>
  64         </h5>
  65 <p>
  66           Since C++11: <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">lowest</span><span class="special">()</span></code>
  67           is
  68         </p>
  69 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  70 <li class="listitem">
  71               For integral types, the same as function <code class="computeroutput"><span class="identifier">min</span><span class="special">()</span></code>.
  72             </li>
  73 <li class="listitem">
  74               For floating-point types, generally the negative of <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code> (but implementation-dependent).
  75             </li>
  76 </ul></div>
  77 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">lowest</span><span class="special">();</span>
  78 </pre>
  79 <h5>
  80 <a name="boost_multiprecision.tut.limits.functions.h2"></a>
  81           <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>
  82         </h5>
  83 <p>
  84           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">min</span><span class="special">)()</span></code> returns the minimum finite value that
  85           can be represented by the type T.
  86         </p>
  87 <p>
  88           For built-in types, there is usually a corresponding MACRO value TYPE_MIN,
  89           where TYPE is CHAR, INT, FLOAT etc.
  90         </p>
  91 <p>
  92           Other types, including those provided by a <code class="computeroutput"><span class="keyword">typedef</span></code>,
  93           for example, <code class="computeroutput"><span class="identifier">INT64_T_MIN</span></code>
  94           for <code class="computeroutput"><span class="identifier">int64_t</span></code>, may provide
  95           a macro definition.
  96         </p>
  97 <p>
  98           For floating-point types, it is more fully defined as the <span class="emphasis"><em>minimum
  99           positive normalized value</em></span>.
 100         </p>
 101 <p>
 102           See <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">denorm_min</span><span class="special">()</span></code>
 103           for the smallest denormalized value, provided
 104         </p>
 105 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</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>
 106 </pre>
 107 <p>
 108           To cater for situations where no <code class="computeroutput"><span class="identifier">numeric_limits</span></code>
 109           specialization is available (for example because the precision of the type
 110           varies at runtime), packaged versions of this (and other functions) are
 111           provided using
 112         </p>
 113 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
 114
 115 <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;();</span>
 116 </pre>
 117 <p>
 118           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">min</span><span class="special">()</span></code> if available.
 119         </p>
 120 <h5>
 121 <a name="boost_multiprecision.tut.limits.functions.h3"></a>
 122           <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
 123           function</a>
 124         </h5>
 125 <p>
 126           Function <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">denorm_min</span><span class="special">()</span></code>
 127           returns the smallest <a href="http://en.wikipedia.org/wiki/Denormal_number" target="_top">denormalized
 128           value</a>, provided
 129         </p>
 130 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</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>
 131 </pre>
 132 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span><span class="special">);</span>
 133 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</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>
 134 <span class="special">{</span>
 135   <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">denorm_min</span><span class="special">();</span>
 136
 137     <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">d</span> <span class="special">&lt;&lt;</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>
 138
 139     <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span>
 140
 141     <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">&amp;</span><span class="identifier">exponent</span><span class="special">);</span>
 142     <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"exponent = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special">&lt;&lt;</span> <span class="identifier">exponent</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">//  fffffbcf</span>
 143     <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"significand = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special">&lt;&lt;</span> <span class="identifier">significand</span> <span class="special">&lt;&lt;</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>
 144 <span class="special">}</span>
 145 <span class="keyword">else</span>
 146 <span class="special">{</span>
 147   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"No denormalization. "</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 148 <span class="special">}</span>
 149 </pre>
 150 <p>
 151           The exponent is effectively reduced from -308 to -324 (though it remains
 152           encoded as zero and leading zeros appear in the significand, thereby losing
 153           precision until the significand reaches zero).
 154         </p>
 155 <h5>
 156 <a name="boost_multiprecision.tut.limits.functions.h4"></a>
 157           <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>
 158         </h5>
 159 <p>
 160           Function <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">round_error</span><span class="special">()</span></code>
 161           returns the maximum error (in units of <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">ULP</a>)
 162           that can be caused by any basic arithmetic operation.
 163         </p>
 164 <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>
 165 </pre>
 166 <p>
 167           The rounding style is indeterminable at compile time.
 168         </p>
 169 <p>
 170           For floating-point types, when rounding is to nearest, only half a bit
 171           is lost by rounding, and <code class="computeroutput"><span class="identifier">round_error</span>
 172           <span class="special">==</span> <span class="number">0.5</span></code>.
 173           In contrast when rounding is towards zero, or plus/minus infinity, we can
 174           loose up to one bit from rounding, and <code class="computeroutput"><span class="identifier">round_error</span>
 175           <span class="special">==</span> <span class="number">1</span></code>.
 176         </p>
 177 <p>
 178           For integer types, rounding always to zero, so at worst almost one bit
 179           can be rounded, so <code class="computeroutput"><span class="identifier">round_error</span>
 180           <span class="special">==</span> <span class="number">1</span></code>.
 181         </p>
 182 <p>
 183           <code class="computeroutput"><span class="identifier">round_error</span><span class="special">()</span></code>
 184           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()</span></code>
 185           to estimate the maximum potential error caused by rounding. For typical
 186           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
 187           epsilon is the maximum potential error.
 188         </p>
 189 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()</span> <span class="comment">// 2.2204460492503131e-016</span>
 190                  <span class="special">*</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">round_error</span><span class="special">();</span> <span class="comment">// 1/2</span>
 191 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">round_err</span> <span class="special">&lt;&lt;</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>
 192 </pre>
 193 <p>
 194           There are, of course, many occasions when much bigger loss of precision
 195           occurs, for example, caused by <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">Loss
 196           of significance or cancellation error</a> or very many iterations.
 197         </p>
 198 <h5>
 199 <a name="boost_multiprecision.tut.limits.functions.h5"></a>
 200           <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>
 201         </h5>
 202 <p>
 203           Function <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()</span></code>
 204           is meaningful only for non-integral types.
 205         </p>
 206 <p>
 207           It returns the difference between <code class="computeroutput"><span class="number">1.0</span></code>
 208           and the next value representable by the floating-point type T. So it is
 209           a one least-significant-bit change in this floating-point value.
 210         </p>
 211 <p>
 212           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>
 213           showing all possibly significant 17 decimal digits.
 214         </p>
 215 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span><span class="special">);</span>
 216 <span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span>
 217 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">();</span>
 218 <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>
 219 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</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>
 220   <span class="special">&lt;&lt;</span> <span class="identifier">d</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span>           <span class="comment">// 1.0000000000000000</span>
 221   <span class="special">&lt;&lt;</span> <span class="identifier">dpeps</span> <span class="special">&lt;&lt;</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>
 222 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">dpeps</span> <span class="special">-</span> <span class="identifier">d</span>   <span class="comment">// 1.0000000000000002</span>
 223   <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 224 </pre>
 225 <p>
 226           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>
 227           and the result is the same as adding <code class="computeroutput"><span class="identifier">epsilon</span></code>.
 228         </p>
 229 <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>
 230 <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>
 231 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">nad</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span>  <span class="comment">//  1.0000000000000002</span>
 232   <span class="special">&lt;&lt;</span> <span class="identifier">nad</span> <span class="special">-</span> <span class="identifier">one</span> <span class="comment">// 2.2204460492503131e-016</span>
 233   <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 234 </pre>
 235 <p>
 236           Adding any smaller value, like half <code class="computeroutput"><span class="identifier">epsilon</span></code>,
 237           will have no effect on this value.
 238         </p>
 239 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span><span class="special">);</span>
 240 <span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span>
 241 <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">();</span>
 242 <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>
 243
 244 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</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>
 245   <span class="special">&lt;&lt;</span> <span class="identifier">dpeps</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span>       <span class="comment">// 1.0000000000000000</span>
 246   <span class="special">&lt;&lt;</span> <span class="identifier">eps</span><span class="special">/</span><span class="number">2</span> <span class="special">&lt;&lt;</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>
 247 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">dpeps</span> <span class="special">-</span> <span class="identifier">d</span>   <span class="comment">// 0.00000000000000000</span>
 248   <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 249 </pre>
 250 <p>
 251           So this cancellation error leaves the values equal, despite adding half
 252           <code class="computeroutput"><span class="identifier">epsilon</span></code>.
 253         </p>
 254 <p>
 255           To achieve greater portability over platform and floating-point type, Boost.Math
 256           and Boost.Multiprecision provide a package of functions that 'do something
 257           sensible' if the standard <code class="computeroutput"><span class="identifier">numeric_limits</span></code>
 258           is not available. To use these <code class="computeroutput"><span class="preprocessor">#include</span>
 259           <span class="special">&lt;</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">&gt;</span></code>.
 260         </p>
 261 <p>
 262           A tolerance might be defined using this version of epsilon thus:
 263         </p>
 264 <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">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;()</span> <span class="special">*</span> <span class="number">2</span><span class="special">;</span>
 265 </pre>
 266 <h6>
 267 <a name="boost_multiprecision.tut.limits.functions.h6"></a>
 268           <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
 269           for Floating-point Comparisons</a>
 270         </h6>
 271 <p>
 272           <a href="https://en.wikipedia.org/wiki/Machine_epsilon" target="_top">Machine epsilon
 273           &#949;</a> is very useful to compute a tolerance when comparing floating-point
 274           values, a much more difficult task than is commonly imagined.
 275         </p>
 276 <p>
 277           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">&lt;&gt;::</span><span class="identifier">epsilon</span><span class="special">()</span></code></a>
 278           and Boost.Multiprecision implements this (where possible) for its program-defined
 279           types analogous to the __fundamental floating-point types like <code class="computeroutput"><span class="keyword">double</span></code> <code class="computeroutput"><span class="keyword">float</span></code>.
 280         </p>
 281 <p>
 282           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
 283           Every Computer Scientist Should Know About Floating-Point Arithmetic</a>
 284         </p>
 285 <p>
 286           The naive test comparing the absolute difference between two values and
 287           a tolerance does not give useful results if the values are too large or
 288           too small.
 289         </p>
 290 <p>
 291           So Boost.Test uses an algorithm first devised by Knuth for reliably checking
 292           if floating-point values are close enough.
 293         </p>
 294 <p>
 295           See Donald. E. Knuth. The art of computer programming (vol II). Copyright
 296           1998 Addison-Wesley Longman, Inc., 0-201-89684-2. Addison-Wesley Professional;
 297           3rd edition. (The relevant equations are in paragraph 4.2.2, Eq. 36 and
 298           37.)
 299         </p>
 300 <p>
 301           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
 302           floating_point comparison</a> for more details.
 303         </p>
 304 <p>
 305           See also:
 306         </p>
 307 <p>
 308           <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
 309           Squassia, Comparing floats</a>
 310         </p>
 311 <p>
 312           <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
 313           Squassia, Comparing floats code</a>
 314         </p>
 315 <p>
 316           <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
 317           Floating-Point_Comparison</a>
 318         </p>
 319 <p>
 320           For example, if we want a tolerance that might suit about 9 arithmetical
 321           operations, say sqrt(9) = 3, we could define:
 322         </p>
 323 <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">();</span>
 324 </pre>
 325 <p>
 326           This is very widely used in Boost.Math testing with Boost.Test's macro
 327           <code class="computeroutput"><span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span></code>
 328         </p>
 329 <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>
 330 <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">();</span>
 331
 332 <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>
 333 </pre>
 334 <p>
 335           used thus:
 336         </p>
 337 <pre class="programlisting"><span class="identifier">cd</span> <span class="special">./</span><span class="identifier">test</span>
 338 <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>
 339 </pre>
 340 <p>
 341           (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
 342           version is simpler to use).
 343         </p>
 344 <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>
 345 <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>
 346 <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>
 347
 348 <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float</span><span class="special">&lt;</span><span class="number">50</span><span class="special">&gt;,</span> <span class="identifier">et_off</span> <span class="special">&gt;</span> <span class="identifier">cpp_dec_float_50</span><span class="special">;</span> <span class="comment">// 50 decimal digits.</span>
 349 </pre>
 350 <div class="note"><table border="0" summary="Note">
 351 <tr>
 352 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 353 <th align="left">Note</th>
 354 </tr>
 355 <tr><td align="left" valign="top"><p>
 356             that Boost.Test does not yet allow floating-point comparisons with expression
 357             templates on, so the default expression template parameter has been replaced
 358             by <code class="computeroutput"><span class="identifier">et_off</span></code>.
 359           </p></td></tr>
 360 </table></div>
 361 <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">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">();</span>
 362 <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">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;();</span>
 363 <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">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;();</span>
 364
 365 <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>
 366 </pre>
 367 <h5>
 368 <a name="boost_multiprecision.tut.limits.functions.h7"></a>
 369           <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 -
 370           positive and negative</a>
 371         </h5>
 372 <p>
 373           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">has_infinity</span>
 374           <span class="special">==</span> <span class="keyword">true</span></code>,
 375           function <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">infinity</span><span class="special">()</span></code>
 376           provides an implementation-defined representation for &#8734;.
 377         </p>
 378 <p>
 379           The 'representation' is a particular bit pattern reserved for infinity.
 380           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">is_iec559</span>
 381           <span class="special">==</span> <span class="keyword">true</span></code>)
 382           <a href="http://en.wikipedia.org/wiki/IEEE_754-1985#Positive_and_negative_infinity" target="_top">positive
 383           and negative infinity</a> are assigned bit patterns for all defined
 384           floating-point types.
 385         </p>
 386 <p>
 387           Confusingly, the string resulting from outputting this representation,
 388           is also implementation-defined. And the string that can be input to generate
 389           the representation is also implementation-defined.
 390         </p>
 391 <p>
 392           For example, the output is <code class="computeroutput"><span class="number">1.</span><span class="special">#</span><span class="identifier">INF</span></code>
 393           on Microsoft systems, but <code class="computeroutput"><span class="identifier">inf</span></code>
 394           on most *nix platforms.
 395         </p>
 396 <p>
 397           This implementation-defined-ness has hampered use of infinity (and NaNs)
 398           but Boost.Math and Boost.Multiprecision work hard to provide a sensible
 399           representation for <span class="bold"><strong>all</strong></span> floating-point
 400           types, not just the built-in types, which with the use of suitable facets
 401           to define the input and output strings, makes it possible to use these
 402           useful features portably and including Boost.Serialization.
 403         </p>
 404 <h5>
 405 <a name="boost_multiprecision.tut.limits.functions.h8"></a>
 406           <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
 407           NaN</a>
 408         </h5>
 409 <h6>
 410 <a name="boost_multiprecision.tut.limits.functions.h9"></a>
 411           <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>
 412         </h6>
 413 <p>
 414           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">has_quiet_NaN</span>
 415           <span class="special">==</span> <span class="keyword">true</span></code>,
 416           function <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">()</span></code>
 417           provides an implementation-defined representation for NaN.
 418         </p>
 419 <p>
 420           <a href="http://en.wikipedia.org/wiki/NaN" target="_top">NaNs</a> are values to
 421           indicate that the result of an assignment or computation is meaningless.
 422           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.
 423         </p>
 424 <p>
 425           NaNs may also be used, to represent missing values: for example, these
 426           could, by convention, be ignored in calculations of statistics like means.
 427         </p>
 428 <p>
 429           Many of the problems with a representation for <a href="http://en.wikipedia.org/wiki/NaN" target="_top">Not-A-Number</a>
 430           has hampered portable use, similar to those with infinity.
 431         </p>
 432 <p>
 433           NaN can be used with binary multiprecision types like <code class="computeroutput"><span class="identifier">cpp_bin_float_quad</span></code>:
 434         </p>
 435 <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>
 436
 437 <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">&lt;</span><span class="identifier">cpp_bin_float_quad</span><span class="special">&gt;::</span><span class="identifier">has_quiet_NaN</span> <span class="special">==</span> <span class="keyword">true</span><span class="special">)</span>
 438 <span class="special">{</span>
 439   <span class="identifier">cpp_bin_float_quad</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">&lt;</span><span class="identifier">cpp_bin_float_quad</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">();</span>
 440
 441   <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">&lt;</span><span class="identifier">cpp_bin_float_quad</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">();</span>
 442   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"cpp_bin_float_quad NaN is "</span>  <span class="special">&lt;&lt;</span> <span class="identifier">NaN</span> <span class="special">&lt;&lt;</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>
 443
 444   <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>
 445   <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>
 446   <span class="comment">// Comparisons of NaN's always fail:</span>
 447   <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>
 448   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">b</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 449   <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>
 450   <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>
 451 <span class="special">}</span>
 452 <span class="keyword">else</span>
 453 <span class="special">{</span>
 454   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Type "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" does not have NaNs!"</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 455 <span class="special">}</span>
 456 </pre>
 457 <p>
 458           But using Boost.Math and suitable facets can permit portable use of both
 459           NaNs and positive and negative infinity.
 460         </p>
 461 <p>
 462           See <a href="../../../../../../../libs/math/example/nonfinite_facet_sstream.cpp" target="_top">boost:/libs/math/example/nonfinite_facet_sstream.cpp</a>
 463           and we also need
 464         </p>
 465 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
 466 </pre>
 467 <p>
 468           Then we can equally well use a multiprecision type cpp_bin_float_quad:
 469         </p>
 470 <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>
 471
 472 <span class="keyword">typedef</span> <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">T</span><span class="special">;</span>
 473
 474 <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>
 475 <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>
 476 <span class="special">{</span>
 477   <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">old_locale</span><span class="special">;</span>
 478   <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">&lt;</span><span class="keyword">char</span><span class="special">&gt;);</span>
 479   <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">&lt;</span><span class="keyword">char</span><span class="special">&gt;);</span>
 480   <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span>
 481   <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>
 482   <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">infinity</span><span class="special">();</span>
 483   <span class="identifier">ss</span> <span class="special">&lt;&lt;</span> <span class="identifier">inf</span><span class="special">;</span> <span class="comment">// Write out.</span>
 484  <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>
 485   <span class="identifier">T</span> <span class="identifier">r</span><span class="special">;</span>
 486   <span class="identifier">ss</span> <span class="special">&gt;&gt;</span> <span class="identifier">r</span><span class="special">;</span> <span class="comment">// Read back in.</span>
 487   <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>
 488   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"infinity output was "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 489   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"infinity input was "</span> <span class="special">&lt;&lt;</span> <span class="identifier">r</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 490 <span class="special">}</span>
 491 </pre>
 492 <pre class="programlisting"><span class="identifier">infinity</span> <span class="identifier">output</span> <span class="identifier">was</span> <span class="identifier">inf</span>
 493 <span class="identifier">infinity</span> <span class="identifier">input</span> <span class="identifier">was</span> <span class="identifier">inf</span>
 494 </pre>
 495 <p>
 496           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
 497           NaN always return false).
 498         </p>
 499 <pre class="programlisting"><span class="special">{</span>
 500   <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">old_locale</span><span class="special">;</span>
 501   <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">&lt;</span><span class="keyword">char</span><span class="special">&gt;);</span>
 502   <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">&lt;</span><span class="keyword">char</span><span class="special">&gt;);</span>
 503   <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span>
 504   <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>
 505   <span class="identifier">T</span> <span class="identifier">n</span><span class="special">;</span>
 506   <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">();</span>
 507   <span class="identifier">ss</span> <span class="special">&lt;&lt;</span> <span class="identifier">NaN</span><span class="special">;</span> <span class="comment">// Write out.</span>
 508   <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>
 509   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"NaN output was "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 510   <span class="identifier">ss</span> <span class="special">&gt;&gt;</span> <span class="identifier">n</span><span class="special">;</span> <span class="comment">// Read back in.</span>
 511   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"NaN input was "</span> <span class="special">&lt;&lt;</span> <span class="identifier">n</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 512 <span class="special">}</span>
 513 </pre>
 514 <pre class="programlisting"><span class="identifier">NaN</span> <span class="identifier">output</span> <span class="identifier">was</span> <span class="identifier">nan</span>
 515 <span class="identifier">NaN</span> <span class="identifier">input</span> <span class="identifier">was</span> <span class="identifier">nan</span>
 516 </pre>
 517 <h6>
 518 <a name="boost_multiprecision.tut.limits.functions.h10"></a>
 519           <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
 520           NaN</a>
 521         </h6>
 522 <p>
 523           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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">has_signaling_NaN</span>
 524           <span class="special">==</span> <span class="keyword">true</span></code>,
 525           function <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">signaling_NaN</span><span class="special">()</span></code>
 526           provides an implementation-defined representation for NaN that causes a
 527           hardware trap. It should be noted however, that at least one implementation
 528           of this function causes a hardware trap to be triggered simply by calling
 529           <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">signaling_NaN</span><span class="special">()</span></code>,
 530           and not only by using the value returned.
 531         </p>
 532 </div>
 533 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 534 <td align="left"></td>
 535 <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2019 John Maddock
 536       and Christopher Kormanyos<p>
 537         Distributed under the Boost Software License, Version 1.0. (See accompanying
 538         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>)
 539       </p>
 540 </div></td>
 541 </tr></table>
 542 <hr>
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