ceph/src/boost/libs/multiprecision/doc/html/boost_multiprecision/ref/number.html

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  27 <a name="boost_multiprecision.ref.number"></a><a class="link" href="number.html" title="number">number</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="boost_multiprecision.ref.number.h0"></a>
  31         <span class="phrase"><a name="boost_multiprecision.ref.number.synopsis"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.synopsis">Synopsis</a>
  32       </h5>
  33 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">multiprecision</span><span class="special">{</span>
  34
  35 <span class="keyword">enum</span> <span class="identifier">expression_template_option</span> <span class="special">{</span> <span class="identifier">et_on</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">et_off</span> <span class="special">=</span> <span class="number">0</span> <span class="special">};</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">&gt;</span> <span class="keyword">struct</span> <span class="identifier">expression_template_default</span>
  38 <span class="special">{</span> <span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">expression_template_option</span> <span class="identifier">value</span> <span class="special">=</span> <span class="identifier">et_on</span><span class="special">;</span> <span class="special">};</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span> <span class="special">=</span> <span class="identifier">expression_template_default</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">&gt;::</span><span class="identifier">value</span><span class="special">&gt;</span>
  41 <span class="keyword">class</span> <span class="identifier">number</span>
  42 <span class="special">{</span>
  43    <span class="identifier">number</span><span class="special">();</span>
  44    <span class="identifier">number</span><span class="special">(</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">);</span>
  45    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">=(</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">);</span>
  46    <span class="identifier">number</span><span class="special">&amp;</span> <span class="identifier">assign</span><span class="special">(</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">);</span>
  47
  48    <span class="comment">// Member operators</span>
  49    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  50    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">-=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  51    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">*=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  52    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">/=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  53    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">++();</span>
  54    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">--();</span>
  55    <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">++(</span><span class="keyword">int</span><span class="special">);</span>
  56    <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">--(</span><span class="keyword">int</span><span class="special">);</span>
  57
  58    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">%=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  59    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">&amp;=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  60    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">|=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  61    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">^=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  62    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">&lt;&lt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
  63    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">&gt;&gt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
  64
  65    <span class="comment">// Use in Boolean context:</span>
  66    <span class="keyword">operator</span> <span class="emphasis"><em>convertible-to-bool-type</em></span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  67    <span class="comment">// swap:</span>
  68    <span class="keyword">void</span> <span class="identifier">swap</span><span class="special">(</span><span class="identifier">number</span><span class="special">&amp;</span> <span class="identifier">other</span><span class="special">);</span>
  69    <span class="comment">// Sign:</span>
  70    <span class="keyword">bool</span> <span class="identifier">is_zero</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  71    <span class="keyword">int</span> <span class="identifier">sign</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  72    <span class="comment">// string conversion:</span>
  73    <span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">str</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  74    <span class="comment">// Generic conversion mechanism</span>
  75    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  76    <span class="identifier">T</span> <span class="identifier">convert_to</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  77    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  78    <span class="keyword">explicit</span> <span class="keyword">operator</span> <span class="identifier">T</span> <span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  79    <span class="comment">// precision control:</span>
  80    <span class="keyword">static</span> <span class="keyword">unsigned</span> <span class="identifier">default_precision</span><span class="special">();</span>
  81    <span class="keyword">static</span> <span class="keyword">void</span> <span class="identifier">default_precision</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">);</span>
  82    <span class="keyword">unsigned</span> <span class="identifier">precision</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  83    <span class="keyword">void</span> <span class="identifier">precision</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">);</span>
  84    <span class="comment">// Comparison:</span>
  85    <span class="keyword">int</span> <span class="identifier">compare</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">&gt;&amp;</span> <span class="identifier">o</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  86    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  87    <span class="keyword">typename</span> <span class="identifier">enable_if</span><span class="special">&lt;</span><span class="identifier">is_convertible</span><span class="special">&lt;</span><span class="identifier">V</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="special">&gt;,</span> <span class="keyword">int</span><span class="special">&gt;::</span><span class="identifier">type</span>
  88       <span class="identifier">compare</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">o</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  89    <span class="comment">// Access to the underlying implementation:</span>
  90    <span class="identifier">Backend</span><span class="special">&amp;</span> <span class="identifier">backend</span><span class="special">();</span>
  91    <span class="keyword">const</span> <span class="identifier">Backend</span><span class="special">&amp;</span> <span class="identifier">backend</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  92 <span class="special">};</span>
  93
  94 <span class="comment">// Non member operators:</span>
  95 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">+(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  96 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">-(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  97 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">+(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  98 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">-(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
  99 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">*(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 100 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">/(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 101 <span class="comment">// Integer only operations:</span>
 102 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">%(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 103 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">&amp;(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 104 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">|(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 105 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">^(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 106 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">&lt;&lt;(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
 107 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">&gt;&gt;(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
 108 <span class="comment">// Comparison operators:</span>
 109 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">==(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 110 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">!=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 111 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&lt;</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 112 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&gt;</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 113 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&lt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 114 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&gt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 115
 116 <span class="comment">// Swap:</span>
 117 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 118 <span class="keyword">void</span> <span class="identifier">swap</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 119
 120 <span class="comment">// iostream support:</span>
 121 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 122 <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">r</span><span class="special">);</span>
 123 <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>unmentionable-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">r</span><span class="special">);</span>
 124 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 125 <span class="identifier">std</span><span class="special">::</span><span class="identifier">istream</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">&gt;&gt;</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">istream</span><span class="special">&amp;</span> <span class="identifier">is</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">r</span><span class="special">);</span>
 126
 127 <span class="comment">// Arithmetic with a higher precision result:</span>
 128 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">&gt;</span>
 129 <span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">add</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 130 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">&gt;</span>
 131 <span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">subtract</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 132 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">&gt;</span>
 133 <span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">multiply</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 134
 135 <span class="comment">// min and max overloads:</span>
 136 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">min</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 137 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">max</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 138
 139 <span class="comment">// C99 Non-member function standard library support:</span>
 140 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">abs</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 141 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">acos</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 142 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">acosh</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 143 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">asin</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 144 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">asinh</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 145 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">atan</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 146 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">atan2</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 147 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">atanh</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 148 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">cbrt</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 149 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">ceil</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 150 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">copysign</span>   <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 151 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">cos</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 152 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">cosh</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 153 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">erf</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 154 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">erfc</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 155 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">exp</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 156 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">exp2</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 157 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">expm1</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 158 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fabs</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 159 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fdim</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 160 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">floor</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 161 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fma</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 162 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fmin</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 163 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fmax</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 164 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fmod</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 165 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">frexp</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">*);</span>
 166 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">hypot</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 167 <span class="emphasis"><em>integer-type</em></span>                              <span class="identifier">ilogb</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 168 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">ldexp</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">);</span>
 169 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">lgamma</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 170 <span class="keyword">long</span> <span class="keyword">long</span>                                 <span class="identifier">llrint</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 171 <span class="keyword">long</span> <span class="keyword">long</span>                                 <span class="identifier">llround</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 172 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">log</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 173 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">log2</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 174 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">log10</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 175 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">log1p</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 176 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">logb</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 177 <span class="keyword">long</span>                                      <span class="identifier">lrint</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 178 <span class="keyword">long</span>                                      <span class="identifier">lround</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 179 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">modf</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 180 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">nearbyint</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 181 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">nextafter</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 182 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">nexttoward</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 183 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">pow</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 184 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">remainder</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 185 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">remquo</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">*);</span>
 186 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">rint</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 187 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">round</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 188 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">scalbn</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">);</span>
 189 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">scalbln</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">);</span>
 190 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sin</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 191 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sinh</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 192 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sqrt</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 193 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">tan</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 194 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">tanh</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 195 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">tgamma</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 196 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">trunc</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 197
 198 <span class="keyword">int</span>                                       <span class="identifier">fpclassify</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 199 <span class="keyword">bool</span>                                      <span class="identifier">isfinite</span>   <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 200 <span class="keyword">bool</span>                                      <span class="identifier">isinf</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 201 <span class="keyword">bool</span>                                      <span class="identifier">isnan</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 202 <span class="keyword">bool</span>                                      <span class="identifier">isnormal</span>   <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 203 <span class="keyword">int</span>                                       <span class="identifier">signbit</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 204
 205 <span class="keyword">bool</span>                                      <span class="identifier">isgreater</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 206 <span class="keyword">bool</span>                                      <span class="identifier">isgreaterequal</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 207 <span class="keyword">bool</span>                                      <span class="identifier">isless</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 208 <span class="keyword">bool</span>                                      <span class="identifier">islessequal</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 209 <span class="keyword">bool</span>                                      <span class="identifier">islessgreater</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 210 <span class="keyword">bool</span>                                      <span class="identifier">isunordered</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 211 <span class="comment">// Misc other common C library functions:</span>
 212 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">itrunc</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 213 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">ltrunc</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 214 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">lltrunc</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 215 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">iround</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 216 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">changesign</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 217 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">copysign</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 218
 219 <span class="comment">// Traits support:</span>
 220 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 221 <span class="keyword">struct</span> <span class="identifier">component_type</span><span class="special">;</span>
 222 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 223 <span class="keyword">struct</span> <span class="identifier">number_category</span><span class="special">;</span>
 224 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 225 <span class="keyword">struct</span> <span class="identifier">is_number</span><span class="special">;</span>
 226 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 227 <span class="keyword">struct</span> <span class="identifier">is_number_expression</span><span class="special">;</span>
 228
 229 <span class="comment">// Integer specific functions:</span>
 230 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">gcd</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 231 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">lcm</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 232 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">pow</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">unsigned</span><span class="special">);</span>
 233 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">powm</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">m</span><span class="special">);</span>
 234 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 235 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 236 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">EXpressionTemplates</span><span class="special">&gt;</span>      <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">EXpressionTemplates</span><span class="special">&gt;&amp;);</span>
 237 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 238 <span class="keyword">void</span> <span class="identifier">divide_qr</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">y</span><span class="special">,</span>
 239                <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">r</span><span class="special">);</span>
 240 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
 241 <span class="identifier">Integer</span> <span class="identifier">integer_modulus</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Integer</span> <span class="identifier">val</span><span class="special">);</span>
 242 <span class="keyword">unsigned</span> <span class="identifier">lsb</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
 243 <span class="keyword">unsigned</span> <span class="identifier">msb</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
 244 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 245 <span class="keyword">bool</span> <span class="identifier">bit_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
 246 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 247 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">bit_set</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
 248 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 249 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">bit_unset</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
 250 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 251 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">bit_flip</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
 252 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Engine</span><span class="special">&gt;</span>
 253 <span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="identifier">Engine</span><span class="special">&amp;</span> <span class="identifier">gen</span><span class="special">);</span>
 254 <span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">);</span>
 255
 256 <span class="comment">// Rational number support:</span>
 257 <span class="keyword">typename</span> <span class="identifier">component_type</span><span class="special">&lt;</span><span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&gt;::</span><span class="identifier">type</span> <span class="identifier">numerator</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 258 <span class="keyword">typename</span> <span class="identifier">component_type</span><span class="special">&lt;</span><span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&gt;::</span><span class="identifier">type</span> <span class="identifier">denominator</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 259
 260 <span class="special">}}</span> <span class="comment">// namespaces</span>
 261
 262 <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
 263
 264 <span class="comment">// Boost.Math interoperability functions:</span>
 265 <span class="keyword">int</span>                                              <span class="identifier">fpclassify</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
 266 <span class="keyword">bool</span>                                             <span class="identifier">isfinite</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
 267 <span class="keyword">bool</span>                                             <span class="identifier">isnan</span>          <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
 268 <span class="keyword">bool</span>                                             <span class="identifier">isinf</span>          <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
 269 <span class="keyword">bool</span>                                             <span class="identifier">isnormal</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
 270
 271 <span class="special">}}</span> <span class="comment">// namespaces</span>
 272
 273 <span class="comment">// numeric_limits support:</span>
 274 <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">{</span>
 275
 276 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 277 <span class="keyword">struct</span> <span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="special">&gt;</span>
 278 <span class="special">{</span>
 279    <span class="comment">/* Usual members here */</span>
 280 <span class="special">};</span>
 281
 282 <span class="special">}</span>
 283 </pre>
 284 <h5>
 285 <a name="boost_multiprecision.ref.number.h1"></a>
 286         <span class="phrase"><a name="boost_multiprecision.ref.number.description"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.description">Description</a>
 287       </h5>
 288 <pre class="programlisting"><span class="keyword">enum</span> <span class="identifier">expression_template_option</span> <span class="special">{</span> <span class="identifier">et_on</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">et_off</span> <span class="special">=</span> <span class="number">0</span> <span class="special">};</span>
 289 </pre>
 290 <p>
 291         This enumerated type is used to specify whether expression templates are
 292         turned on (et_on) or turned off (et_off).
 293       </p>
 294 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">&gt;</span> <span class="keyword">struct</span> <span class="identifier">expression_template_default</span>
 295 <span class="special">{</span> <span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">expression_template_option</span> <span class="identifier">value</span> <span class="special">=</span> <span class="identifier">et_on</span><span class="special">;</span> <span class="special">};</span>
 296 </pre>
 297 <p>
 298         This traits class specifies the default expression template option to be
 299         used with a particular Backend type. It defaults to <code class="computeroutput"><span class="identifier">et_on</span></code>.
 300       </p>
 301 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span> <span class="special">=</span> <span class="identifier">expression_template_default</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">&gt;::</span><span class="identifier">value</span><span class="special">&gt;</span>
 302 <span class="keyword">class</span> <span class="identifier">number</span><span class="special">;</span>
 303 </pre>
 304 <p>
 305         Class <code class="computeroutput"><span class="identifier">number</span></code> has two template
 306         arguments:
 307       </p>
 308 <div class="variablelist">
 309 <p class="title"><b></b></p>
 310 <dl class="variablelist">
 311 <dt><span class="term">Backend</span></dt>
 312 <dd><p>
 313               The actual arithmetic back-end that does all the work.
 314             </p></dd>
 315 <dt><span class="term">ExpressionTemplates</span></dt>
 316 <dd><p>
 317               A Boolean value: when <code class="computeroutput"><span class="identifier">et_on</span></code>,
 318               then expression templates are enabled, otherwise when set to <code class="computeroutput"><span class="identifier">et_off</span></code> they are disabled. The default
 319               for this parameter is computed via the traits class <code class="computeroutput"><span class="identifier">expression_template_default</span></code>
 320               whose member <code class="computeroutput"><span class="identifier">value</span></code>
 321               defaults to <code class="computeroutput"><span class="identifier">et_on</span></code> unless
 322               the traits class is specialized for a particular backend.
 323             </p></dd>
 324 </dl>
 325 </div>
 326 <pre class="programlisting"><span class="identifier">number</span><span class="special">();</span>
 327 <span class="identifier">number</span><span class="special">(</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">);</span>
 328 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">=(</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">);</span>
 329 <span class="identifier">number</span><span class="special">&amp;</span> <span class="identifier">assign</span><span class="special">(</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">);</span>
 330 </pre>
 331 <p>
 332         Type <code class="computeroutput"><span class="identifier">number</span></code> is default constructible,
 333         and both copy constructible and assignable from:
 334       </p>
 335 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 336 <li class="listitem">
 337             Itself.
 338           </li>
 339 <li class="listitem">
 340             An expression template which is the result of one of the arithmetic operators.
 341           </li>
 342 <li class="listitem">
 343             Any builtin arithmetic type, as long as the result would not be lossy
 344             (for example float to integer conversion).
 345           </li>
 346 <li class="listitem">
 347             Any type that the Backend is implicitly constructible or assignable from.
 348           </li>
 349 <li class="listitem">
 350             An rvalue reference to another <code class="computeroutput"><span class="identifier">number</span></code>.
 351             Move-semantics are used for construction if the backend also supports
 352             rvalue reference construction. In the case of assignment, move semantics
 353             are always supported when the argument is an rvalue reference irrespective
 354             of the backend.
 355           </li>
 356 <li class="listitem">
 357             Any type in the same family, as long as no loss of precision is involved.
 358             For example from <code class="computeroutput"><span class="identifier">int128_t</span></code>
 359             to <code class="computeroutput"><span class="identifier">int256_t</span></code>, or <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code> to <code class="computeroutput"><span class="identifier">cpp_dec_float_100</span></code>.
 360           </li>
 361 </ul></div>
 362 <p>
 363         Type <code class="computeroutput"><span class="identifier">number</span></code> is explicitly
 364         constructible from:
 365       </p>
 366 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 367 <li class="listitem">
 368             Any type mentioned above.
 369           </li>
 370 <li class="listitem">
 371             A <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span></code> or any type which is convertible
 372             to <code class="computeroutput"><span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span></code>.
 373           </li>
 374 <li class="listitem">
 375             Any arithmetic type (including those that would result in lossy conversions).
 376           </li>
 377 <li class="listitem">
 378             Any type in the same family, including those that result in loss of precision.
 379           </li>
 380 <li class="listitem">
 381             Any type that the Backend is explicitly constructible from.
 382           </li>
 383 <li class="listitem">
 384             Any pair of types for which a generic interconversion exists: that is
 385             from integer to integer, integer to rational, integer to float, rational
 386             to rational, rational to float, or float to float.
 387           </li>
 388 </ul></div>
 389 <p>
 390         The assign member function is available for any type for which an explicit
 391         converting constructor exists. It is intended to be used where a temporary
 392         generated from an explicit assignment would be expensive, for example:
 393       </p>
 394 <pre class="programlisting"><span class="identifier">mpfr_float_50</span>    <span class="identifier">f50</span><span class="special">;</span>
 395 <span class="identifier">mpfr_float_100</span>   <span class="identifier">f100</span><span class="special">;</span>
 396
 397 <span class="identifier">f50</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">mpfr_float_50</span><span class="special">&gt;(</span><span class="identifier">f100</span><span class="special">);</span>  <span class="comment">// explicit cast create a temporary</span>
 398 <span class="identifier">f50</span><span class="special">.</span><span class="identifier">assign</span><span class="special">(</span><span class="identifier">f100</span><span class="special">);</span>                        <span class="comment">// explicit call to assign create no temporary</span>
 399 </pre>
 400 <p>
 401         In addition, if the type has multiple components (for example rational or
 402         complex number types), then there is a two argument constructor:
 403       </p>
 404 <pre class="programlisting"><span class="identifier">number</span><span class="special">(</span><span class="identifier">arg1</span><span class="special">,</span> <span class="identifier">arg2</span><span class="special">);</span>
 405 </pre>
 406 <p>
 407         Where the two args must either be arithmetic types, or types that are convertible
 408         to the two components of <code class="computeroutput"><span class="keyword">this</span></code>.
 409       </p>
 410 <pre class="programlisting"><span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 411 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">-=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 412 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">*=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 413 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">/=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 414 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">++();</span>
 415 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">--();</span>
 416 <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">++(</span><span class="keyword">int</span><span class="special">);</span>
 417 <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">--(</span><span class="keyword">int</span><span class="special">);</span>
 418 <span class="comment">// Integer only operations:</span>
 419 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">%=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 420 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">&amp;=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 421 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">|=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 422 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">^=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 423 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">&lt;&lt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
 424 <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">&gt;&gt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
 425 </pre>
 426 <p>
 427         These operators all take their usual arithmetic meanings.
 428       </p>
 429 <p>
 430         The arguments to these operators is either:
 431       </p>
 432 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 433 <li class="listitem">
 434             Another <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span>
 435             <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span></code>.
 436           </li>
 437 <li class="listitem">
 438             An expression template derived from <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">&gt;</span></code>.
 439           </li>
 440 <li class="listitem">
 441             Any type implicitly convertible to <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span></code>, including some other instance of
 442             class <code class="computeroutput"><span class="identifier">number</span></code>.
 443           </li>
 444 </ul></div>
 445 <p>
 446         For the left and right shift operations, the argument must be a builtin integer
 447         type with a positive value (negative values result in a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">runtime_error</span></code>
 448         being thrown).
 449       </p>
 450 <pre class="programlisting"><span class="keyword">operator</span> <span class="emphasis"><em>convertible-to-bool-type</em></span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 451 </pre>
 452 <p>
 453         Returns an <span class="emphasis"><em>unmentionable-type</em></span> that is usable in Boolean
 454         contexts (this allows <code class="computeroutput"><span class="identifier">number</span></code>
 455         to be used in any Boolean context - if statements, conditional statements,
 456         or as an argument to a logical operator - without type <code class="computeroutput"><span class="identifier">number</span></code>
 457         being convertible to type <code class="computeroutput"><span class="keyword">bool</span></code>.
 458       </p>
 459 <p>
 460         This operator also enables the use of <code class="computeroutput"><span class="identifier">number</span></code>
 461         with any of the following operators: <code class="computeroutput"><span class="special">!</span></code>,
 462         <code class="computeroutput"><span class="special">||</span></code>, <code class="computeroutput"><span class="special">&amp;&amp;</span></code>
 463         and <code class="computeroutput"><span class="special">?:</span></code>.
 464       </p>
 465 <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">swap</span><span class="special">(</span><span class="identifier">number</span><span class="special">&amp;</span> <span class="identifier">other</span><span class="special">);</span>
 466 </pre>
 467 <p>
 468         Swaps <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
 469         with <code class="computeroutput"><span class="identifier">other</span></code>.
 470       </p>
 471 <pre class="programlisting"><span class="keyword">bool</span> <span class="identifier">is_zero</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 472 </pre>
 473 <p>
 474         Returns <code class="computeroutput"><span class="keyword">true</span></code> is <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code> is zero,
 475         otherwise <code class="computeroutput"><span class="keyword">false</span></code>.
 476       </p>
 477 <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">sign</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 478 </pre>
 479 <p>
 480         Returns a value less than zero if <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code> is negative, a value greater than zero
 481         if <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
 482         is positive, and zero if <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
 483         is zero.
 484       </p>
 485 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">str</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">precision</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">scientific</span> <span class="special">=</span> <span class="keyword">true</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
 486 </pre>
 487 <p>
 488         Returns the number formatted as a string, with at least <span class="emphasis"><em>precision</em></span>
 489         digits, and in scientific format if <span class="emphasis"><em>scientific</em></span> is true.
 490       </p>
 491 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 492 <span class="identifier">T</span> <span class="identifier">convert_to</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 493
 494 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 495 <span class="keyword">explicit</span> <span class="keyword">operator</span> <span class="identifier">T</span> <span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 496 </pre>
 497 <p>
 498         Provides a generic conversion mechanism to convert <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code> to type <code class="computeroutput"><span class="identifier">T</span></code>.
 499         Type <code class="computeroutput"><span class="identifier">T</span></code> may be any arithmetic
 500         type. Optionally other types may also be supported by specific <code class="computeroutput"><span class="identifier">Backend</span></code> types.
 501       </p>
 502 <pre class="programlisting"><span class="keyword">static</span> <span class="keyword">unsigned</span> <span class="identifier">default_precision</span><span class="special">();</span>
 503 <span class="keyword">static</span> <span class="keyword">void</span> <span class="identifier">default_precision</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">);</span>
 504 <span class="keyword">unsigned</span> <span class="identifier">precision</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 505 <span class="keyword">void</span> <span class="identifier">precision</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">);</span>
 506 </pre>
 507 <p>
 508         These functions are only available if the Backend template parameter supports
 509         runtime changes to precision. They get and set the default precision and
 510         the precision of <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
 511         respectively.
 512       </p>
 513 <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">compare</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">o</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
 514 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 515 <span class="keyword">typename</span> <span class="identifier">enable_if</span><span class="special">&lt;</span><span class="identifier">is_convertible</span><span class="special">&lt;</span><span class="identifier">V</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="special">&gt;,</span> <span class="keyword">int</span><span class="special">&gt;::</span><span class="identifier">type</span>
 516    <span class="identifier">compare</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">other</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
 517 </pre>
 518 <p>
 519         Returns:
 520       </p>
 521 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 522 <li class="listitem">
 523             A value less that 0 for <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span> <span class="special">&lt;</span> <span class="identifier">other</span></code>
 524           </li>
 525 <li class="listitem">
 526             A value greater that 0 for <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span> <span class="special">&gt;</span> <span class="identifier">other</span></code>
 527           </li>
 528 <li class="listitem">
 529 <p class="simpara">
 530             Zero for <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span>
 531             <span class="special">==</span> <span class="identifier">other</span></code>
 532           </p>
 533 <pre class="programlisting"><span class="identifier">Backend</span><span class="special">&amp;</span> <span class="identifier">backend</span><span class="special">();</span>
 534 <span class="keyword">const</span> <span class="identifier">Backend</span><span class="special">&amp;</span> <span class="identifier">backend</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 535 </pre>
 536 </li>
 537 </ul></div>
 538 <p>
 539         Returns the underlying back-end instance used by <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>.
 540       </p>
 541 <h5>
 542 <a name="boost_multiprecision.ref.number.h2"></a>
 543         <span class="phrase"><a name="boost_multiprecision.ref.number.non_member_operators"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.non_member_operators">Non-member
 544         operators</a>
 545       </h5>
 546 <pre class="programlisting"><span class="comment">// Non member operators:</span>
 547 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">+(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 548 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">-(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 549 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">+(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 550 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">-(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 551 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">*(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 552 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">/(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 553 <span class="comment">// Integer only operations:</span>
 554 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">%(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 555 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">&amp;(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 556 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">|(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 557 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">^(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 558 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">&lt;&lt;(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
 559 <span class="emphasis"><em>unmentionable-expression-template-type</em></span> <span class="keyword">operator</span><span class="special">&gt;&gt;(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>integer-type</em></span><span class="special">&amp;);</span>
 560 <span class="comment">// Comparison operators:</span>
 561 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">==(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 562 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">!=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 563 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&lt;</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 564 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&gt;</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 565 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&lt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 566 <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">&gt;=(</span><span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>see-below</em></span><span class="special">&amp;);</span>
 567 </pre>
 568 <p>
 569         These operators all take their usual arithmetic meanings.
 570       </p>
 571 <p>
 572         The arguments to these functions must contain at least one of the following:
 573       </p>
 574 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 575 <li class="listitem">
 576             A <code class="computeroutput"><span class="identifier">number</span></code>.
 577           </li>
 578 <li class="listitem">
 579             An expression template type derived from <code class="computeroutput"><span class="identifier">number</span></code>.
 580           </li>
 581 <li class="listitem">
 582             Any type for which <code class="computeroutput"><span class="identifier">number</span></code>
 583             has an implicit constructor - for example a builtin arithmetic type.
 584           </li>
 585 </ul></div>
 586 <p>
 587         The return type of these operators is either:
 588       </p>
 589 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 590 <li class="listitem">
 591             An <span class="emphasis"><em>unmentionable-type</em></span> expression template type when
 592             <code class="computeroutput"><span class="identifier">ExpressionTemplates</span></code> is
 593             <code class="computeroutput"><span class="keyword">true</span></code>.
 594           </li>
 595 <li class="listitem">
 596             Type <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span>
 597             <span class="identifier">et_off</span><span class="special">&gt;</span></code>
 598             when <code class="computeroutput"><span class="identifier">ExpressionTemplates</span></code>
 599             is <code class="computeroutput"><span class="keyword">false</span></code>.
 600           </li>
 601 <li class="listitem">
 602             Type <code class="computeroutput"><span class="keyword">bool</span></code> if the operator
 603             is a comparison operator.
 604           </li>
 605 </ul></div>
 606 <p>
 607         Finally note that the second argument to the left and right shift operations
 608         must be a builtin integer type, and that the argument must be positive (negative
 609         arguments result in a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">runtime_error</span></code>
 610         being thrown).
 611       </p>
 612 <h5>
 613 <a name="boost_multiprecision.ref.number.h3"></a>
 614         <span class="phrase"><a name="boost_multiprecision.ref.number.swap"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.swap">swap</a>
 615       </h5>
 616 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 617 <span class="keyword">void</span> <span class="identifier">swap</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 618 </pre>
 619 <p>
 620         Swaps <code class="computeroutput"><span class="identifier">a</span></code> and <code class="computeroutput"><span class="identifier">b</span></code>.
 621       </p>
 622 <h5>
 623 <a name="boost_multiprecision.ref.number.h4"></a>
 624         <span class="phrase"><a name="boost_multiprecision.ref.number.iostream_support"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.iostream_support">Iostream
 625         Support</a>
 626       </h5>
 627 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 628 <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">r</span><span class="special">);</span>
 629 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Unspecified</span><span class="special">...&gt;</span>
 630 <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">unmentionable</span><span class="special">-</span><span class="identifier">expression</span><span class="special">-</span><span class="keyword">template</span><span class="special">&amp;</span> <span class="identifier">r</span><span class="special">);</span>
 631 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 632 <span class="keyword">inline</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">istream</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">&gt;&gt;</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">istream</span><span class="special">&amp;</span> <span class="identifier">is</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">r</span><span class="special">)</span>
 633 </pre>
 634 <p>
 635         These operators provided formatted input-output operations on <code class="computeroutput"><span class="identifier">number</span></code> types, and expression templates
 636         derived from them.
 637       </p>
 638 <p>
 639         It's down to the back-end type to actually implement string conversion. However,
 640         the back-ends provided with this library support all of the iostream formatting
 641         flags, field width and precision settings.
 642       </p>
 643 <h5>
 644 <a name="boost_multiprecision.ref.number.h5"></a>
 645         <span class="phrase"><a name="boost_multiprecision.ref.number.arithmetic_with_a_higher_precisi"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.arithmetic_with_a_higher_precisi">Arithmetic
 646         with a higher precision result</a>
 647       </h5>
 648 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">&gt;</span>
 649 <span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">add</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 650
 651 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">&gt;</span>
 652 <span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">subtract</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 653
 654 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">&gt;</span>
 655 <span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">multiply</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&amp;</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 656 </pre>
 657 <p>
 658         These functions apply the named operator to the arguments <span class="emphasis"><em>a</em></span>
 659         and <span class="emphasis"><em>b</em></span> and store the result in <span class="emphasis"><em>result</em></span>,
 660         returning <span class="emphasis"><em>result</em></span>. In all cases they behave "as
 661         if" arguments <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> were
 662         first promoted to type <code class="computeroutput"><span class="identifier">ResultType</span></code>
 663         before applying the operator, though particular backends may well avoid that
 664         step by way of an optimization.
 665       </p>
 666 <p>
 667         The type <code class="computeroutput"><span class="identifier">ResultType</span></code> must
 668         be an instance of class <code class="computeroutput"><span class="identifier">number</span></code>,
 669         and the types <code class="computeroutput"><span class="identifier">Source1</span></code> and
 670         <code class="computeroutput"><span class="identifier">Source2</span></code> may be either instances
 671         of class <code class="computeroutput"><span class="identifier">number</span></code> or native
 672         integer types. The latter is an optimization that allows arithmetic to be
 673         performed on native integer types producing an extended precision result.
 674       </p>
 675 <h5>
 676 <a name="boost_multiprecision.ref.number.h6"></a>
 677         <span class="phrase"><a name="boost_multiprecision.ref.number.non_member_standard_library_func"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.non_member_standard_library_func">Non-member
 678         standard library function support</a>
 679       </h5>
 680 <pre class="programlisting"><span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">abs</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 681 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">acos</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 682 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">acosh</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 683 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">asin</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 684 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">asinh</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 685 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">atan</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 686 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">atan2</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 687 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">atanh</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 688 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">cbrt</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 689 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">ceil</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 690 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">copysign</span>   <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 691 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">cos</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 692 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">cosh</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 693 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">erf</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 694 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">erfc</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 695 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">exp</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 696 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">exp2</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 697 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">expm1</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 698 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fabs</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 699 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fdim</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 700 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">floor</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 701 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fma</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 702 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fmin</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 703 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fmax</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 704 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">fmod</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 705 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">frexp</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">*);</span>
 706 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">hypot</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 707 <span class="emphasis"><em>integer-type</em></span>                              <span class="identifier">ilogb</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 708 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">ldexp</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">);</span>
 709 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">lgamma</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 710 <span class="keyword">long</span> <span class="keyword">long</span>                                 <span class="identifier">llrint</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 711 <span class="keyword">long</span> <span class="keyword">long</span>                                 <span class="identifier">llround</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 712 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">log</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 713 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">log2</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 714 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">log10</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 715 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">log1p</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 716 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">logb</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 717 <span class="keyword">long</span>                                      <span class="identifier">lrint</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 718 <span class="keyword">long</span>                                      <span class="identifier">lround</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 719 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">modf</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 720 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">nearbyint</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 721 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">nextafter</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 722 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">nexttoward</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 723 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">pow</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 724 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">remainder</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 725 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">remquo</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">*);</span>
 726 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">rint</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 727 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">round</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 728 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">scalbn</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">);</span>
 729 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">scalbln</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="emphasis"><em>integer-type</em></span><span class="special">);</span>
 730 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sin</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 731 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sinh</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 732 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sqrt</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 733 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">tan</span>        <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 734 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">tanh</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 735 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">tgamma</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 736 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">trunc</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 737
 738 <span class="keyword">int</span>                                       <span class="identifier">fpclassify</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 739 <span class="keyword">bool</span>                                      <span class="identifier">isfinite</span>   <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 740 <span class="keyword">bool</span>                                      <span class="identifier">isinf</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 741 <span class="keyword">bool</span>                                      <span class="identifier">isnan</span>      <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 742 <span class="keyword">bool</span>                                      <span class="identifier">isnormal</span>   <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 743 <span class="keyword">int</span>                                       <span class="identifier">signbit</span>    <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 744
 745 <span class="keyword">bool</span>                                      <span class="identifier">isgreater</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 746 <span class="keyword">bool</span>                                      <span class="identifier">isgreaterequal</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 747 <span class="keyword">bool</span>                                      <span class="identifier">isless</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 748 <span class="keyword">bool</span>                                      <span class="identifier">islessequal</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 749 <span class="keyword">bool</span>                                      <span class="identifier">islessgreater</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 750 <span class="keyword">bool</span>                                      <span class="identifier">isunordered</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 751 </pre>
 752 <p>
 753         These functions all behave exactly as their standard library C++11 counterparts
 754         do: their argument is either an instance of <code class="computeroutput"><span class="identifier">number</span></code>
 755         or an expression template derived from it; If the argument is of type <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">et_off</span><span class="special">&gt;</span></code>
 756         then that is also the return type, otherwise the return type is an expression
 757         template unless otherwise stated.
 758       </p>
 759 <p>
 760         The integer type arguments to <code class="computeroutput"><span class="identifier">ldexp</span></code>,
 761         <code class="computeroutput"><span class="identifier">frexp</span></code>, <code class="computeroutput"><span class="identifier">scalbn</span></code>
 762         and <code class="computeroutput"><span class="identifier">ilogb</span></code> may be either type
 763         <code class="computeroutput"><span class="keyword">int</span></code>, or the actual type of the
 764         exponent of the number type.
 765       </p>
 766 <p>
 767         There are also some functions implemented for compatibility with the Boost.Math
 768         functions of the same name:
 769       </p>
 770 <pre class="programlisting"><span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">itrunc</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 771 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">ltrunc</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 772 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">lltrunc</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 773 <span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">iround</span> <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 774 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">changesign</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 775 <span class="emphasis"><em>number</em></span>                                    <span class="identifier">copysign</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
 776 </pre>
 777 <p>
 778         All these functions are normally implemented by the Backend type. However,
 779         default versions are provided for Backend types that don't have native support
 780         for these functions. Please note however, that this default support requires
 781         the precision of the type to be a compile time constant - this means for
 782         example that the <a href="http://gmplib.org" target="_top">GMP</a> MPF Backend will
 783         not work with these functions when that type is used at variable precision.
 784       </p>
 785 <p>
 786         Also note that with the exception of <code class="computeroutput"><span class="identifier">abs</span></code>
 787         that these functions can only be used with floating-point Backend types (if
 788         any other types such as fixed precision or complex types are added to the
 789         library later, then these functions may be extended to support those number
 790         types).
 791       </p>
 792 <p>
 793         The precision of these functions is generally determined by the backend implementation.
 794         For example the precision of these functions when used with <a class="link" href="../tut/floats/mpfr_float.html" title="mpfr_float">mpfr_float</a>
 795         is determined entirely by <a href="http://www.mpfr.org" target="_top">MPFR</a>.
 796         When these functions use our own implementations, the accuracy of the transcendental
 797         functions is generally a few epsilon. Note however, that the trigonometrical
 798         functions incur the usual accuracy loss when reducing arguments by large
 799         multiples of &#960;. Also note that both <a class="link" href="../tut/floats/gmp_float.html" title="gmp_float">gmp_float</a>
 800         and <a class="link" href="../tut/floats/cpp_dec_float.html" title="cpp_dec_float">cpp_dec_float</a>
 801         have a number of guard digits beyond their stated precision, so the error
 802         rates listed for these are in some sense artificially low.
 803       </p>
 804 <p>
 805         The following table shows the error rates we observe for these functions
 806         with various backend types, functions not listed here are exact (tested on
 807         Win32 with VC++10, MPFR-3.0.0, MPIR-2.1.1):
 808       </p>
 809 <div class="informaltable"><table class="table">
 810 <colgroup>
 811 <col>
 812 <col>
 813 <col>
 814 <col>
 815 </colgroup>
 816 <thead><tr>
 817 <th>
 818                 <p>
 819                   Function
 820                 </p>
 821               </th>
 822 <th>
 823                 <p>
 824                   mpfr_float_50
 825                 </p>
 826               </th>
 827 <th>
 828                 <p>
 829                   mpf_float_50
 830                 </p>
 831               </th>
 832 <th>
 833                 <p>
 834                   cpp_dec_float_50
 835                 </p>
 836               </th>
 837 </tr></thead>
 838 <tbody>
 839 <tr>
 840 <td>
 841                 <p>
 842                   sqrt
 843                 </p>
 844               </td>
 845 <td>
 846                 <p>
 847                   1eps
 848                 </p>
 849               </td>
 850 <td>
 851                 <p>
 852                   0eps
 853                 </p>
 854               </td>
 855 <td>
 856                 <p>
 857                   0eps
 858                 </p>
 859               </td>
 860 </tr>
 861 <tr>
 862 <td>
 863                 <p>
 864                   exp
 865                 </p>
 866               </td>
 867 <td>
 868                 <p>
 869                   1eps
 870                 </p>
 871               </td>
 872 <td>
 873                 <p>
 874                   0eps
 875                 </p>
 876               </td>
 877 <td>
 878                 <p>
 879                   0eps
 880                 </p>
 881               </td>
 882 </tr>
 883 <tr>
 884 <td>
 885                 <p>
 886                   log
 887                 </p>
 888               </td>
 889 <td>
 890                 <p>
 891                   1eps
 892                 </p>
 893               </td>
 894 <td>
 895                 <p>
 896                   0eps
 897                 </p>
 898               </td>
 899 <td>
 900                 <p>
 901                   0eps
 902                 </p>
 903               </td>
 904 </tr>
 905 <tr>
 906 <td>
 907                 <p>
 908                   log10
 909                 </p>
 910               </td>
 911 <td>
 912                 <p>
 913                   1eps
 914                 </p>
 915               </td>
 916 <td>
 917                 <p>
 918                   0eps
 919                 </p>
 920               </td>
 921 <td>
 922                 <p>
 923                   0eps
 924                 </p>
 925               </td>
 926 </tr>
 927 <tr>
 928 <td>
 929                 <p>
 930                   cos
 931                 </p>
 932               </td>
 933 <td>
 934                 <p>
 935                   700eps
 936                 </p>
 937               </td>
 938 <td>
 939                 <p>
 940                   0eps
 941                 </p>
 942               </td>
 943 <td>
 944                 <p>
 945                   0eps
 946                 </p>
 947               </td>
 948 </tr>
 949 <tr>
 950 <td>
 951                 <p>
 952                   sin
 953                 </p>
 954               </td>
 955 <td>
 956                 <p>
 957                   1eps
 958                 </p>
 959               </td>
 960 <td>
 961                 <p>
 962                   0eps
 963                 </p>
 964               </td>
 965 <td>
 966                 <p>
 967                   0eps
 968                 </p>
 969               </td>
 970 </tr>
 971 <tr>
 972 <td>
 973                 <p>
 974                   tan
 975                 </p>
 976               </td>
 977 <td>
 978                 <p>
 979                   0eps
 980                 </p>
 981               </td>
 982 <td>
 983                 <p>
 984                   0eps
 985                 </p>
 986               </td>
 987 <td>
 988                 <p>
 989                   0eps
 990                 </p>
 991               </td>
 992 </tr>
 993 <tr>
 994 <td>
 995                 <p>
 996                   acos
 997                 </p>
 998               </td>
 999 <td>
1000                 <p>
1001                   0eps
1002                 </p>
1003               </td>
1004 <td>
1005                 <p>
1006                   0eps
1007                 </p>
1008               </td>
1009 <td>
1010                 <p>
1011                   0eps
1012                 </p>
1013               </td>
1014 </tr>
1015 <tr>
1016 <td>
1017                 <p>
1018                   asin
1019                 </p>
1020               </td>
1021 <td>
1022                 <p>
1023                   0eps
1024                 </p>
1025               </td>
1026 <td>
1027                 <p>
1028                   0eps
1029                 </p>
1030               </td>
1031 <td>
1032                 <p>
1033                   0eps
1034                 </p>
1035               </td>
1036 </tr>
1037 <tr>
1038 <td>
1039                 <p>
1040                   atan
1041                 </p>
1042               </td>
1043 <td>
1044                 <p>
1045                   1eps
1046                 </p>
1047               </td>
1048 <td>
1049                 <p>
1050                   0eps
1051                 </p>
1052               </td>
1053 <td>
1054                 <p>
1055                   0eps
1056                 </p>
1057               </td>
1058 </tr>
1059 <tr>
1060 <td>
1061                 <p>
1062                   cosh
1063                 </p>
1064               </td>
1065 <td>
1066                 <p>
1067                   1045eps<a href="#ftn.boost_multiprecision.ref.number.f0" class="footnote"><sup class="footnote"><a name="boost_multiprecision.ref.number.f0"></a>[1]</sup></a>
1068                 </p>
1069               </td>
1070 <td>
1071                 <p>
1072                   0eps
1073                 </p>
1074               </td>
1075 <td>
1076                 <p>
1077                   0eps
1078                 </p>
1079               </td>
1080 </tr>
1081 <tr>
1082 <td>
1083                 <p>
1084                   sinh
1085                 </p>
1086               </td>
1087 <td>
1088                 <p>
1089                   2eps
1090                 </p>
1091               </td>
1092 <td>
1093                 <p>
1094                   0eps
1095                 </p>
1096               </td>
1097 <td>
1098                 <p>
1099                   0eps
1100                 </p>
1101               </td>
1102 </tr>
1103 <tr>
1104 <td>
1105                 <p>
1106                   tanh
1107                 </p>
1108               </td>
1109 <td>
1110                 <p>
1111                   1eps
1112                 </p>
1113               </td>
1114 <td>
1115                 <p>
1116                   0eps
1117                 </p>
1118               </td>
1119 <td>
1120                 <p>
1121                   0eps
1122                 </p>
1123               </td>
1124 </tr>
1125 <tr>
1126 <td>
1127                 <p>
1128                   pow
1129                 </p>
1130               </td>
1131 <td>
1132                 <p>
1133                   0eps
1134                 </p>
1135               </td>
1136 <td>
1137                 <p>
1138                   4eps
1139                 </p>
1140               </td>
1141 <td>
1142                 <p>
1143                   3eps
1144                 </p>
1145               </td>
1146 </tr>
1147 <tr>
1148 <td>
1149                 <p>
1150                   atan2
1151                 </p>
1152               </td>
1153 <td>
1154                 <p>
1155                   1eps
1156                 </p>
1157               </td>
1158 <td>
1159                 <p>
1160                   0eps
1161                 </p>
1162               </td>
1163 <td>
1164                 <p>
1165                   0eps
1166                 </p>
1167               </td>
1168 </tr>
1169 </tbody>
1170 <tbody class="footnotes"><tr><td colspan="4"><div id="ftn.boost_multiprecision.ref.number.f0" class="footnote"><p><a href="#boost_multiprecision.ref.number.f0" class="para"><sup class="para">[1] </sup></a>
1171                     It's likely that the inherent error in the input values to our
1172                     test cases are to blame here.
1173                   </p></div></td></tr></tbody>
1174 </table></div>
1175 <h5>
1176 <a name="boost_multiprecision.ref.number.h7"></a>
1177         <span class="phrase"><a name="boost_multiprecision.ref.number.traits_class_support"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.traits_class_support">Traits Class
1178         Support</a>
1179       </h5>
1180 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
1181 <span class="keyword">struct</span> <span class="identifier">component_type</span><span class="special">;</span>
1182 </pre>
1183 <p>
1184         If this is a type with multiple components (for example rational or complex
1185         types), then this trait has a single member <code class="computeroutput"><span class="identifier">type</span></code>
1186         that is the type of those components.
1187       </p>
1188 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
1189 <span class="keyword">struct</span> <span class="identifier">number_category</span><span class="special">;</span>
1190 </pre>
1191 <p>
1192         A traits class that inherits from <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">int_</span><span class="special">&lt;</span><span class="identifier">N</span><span class="special">&gt;</span></code>
1193         where <code class="computeroutput"><span class="identifier">N</span></code> is one of the enumerated
1194         values <code class="computeroutput"><span class="identifier">number_kind_integer</span></code>,
1195         <code class="computeroutput"><span class="identifier">number_kind_floating_point</span></code>,
1196         <code class="computeroutput"><span class="identifier">number_kind_rational</span></code>, <code class="computeroutput"><span class="identifier">number_kind_fixed_point</span></code>, or <code class="computeroutput"><span class="identifier">number_kind_unknown</span></code>. This traits class
1197         is specialized for any type that has <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code>
1198         support as well as for classes in this library: which means it can be used
1199         for generic code that must work with built in arithmetic types as well as
1200         multiprecision ones.
1201       </p>
1202 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
1203 <span class="keyword">struct</span> <span class="identifier">is_number</span><span class="special">;</span>
1204 </pre>
1205 <p>
1206         A traits class that inherits from <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">true_</span></code>
1207         if T is an instance of <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;&gt;</span></code>, otherwise from <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">false_</span></code>.
1208       </p>
1209 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
1210 <span class="keyword">struct</span> <span class="identifier">is_number_expression</span><span class="special">;</span>
1211 </pre>
1212 <p>
1213         A traits class that inherits from <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">true_</span></code>
1214         if T is an expression template type derived from <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;&gt;</span></code>, otherwise from <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">false_</span></code>.
1215       </p>
1216 <h5>
1217 <a name="boost_multiprecision.ref.number.h8"></a>
1218         <span class="phrase"><a name="boost_multiprecision.ref.number.integer_functions"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.integer_functions">Integer
1219         functions</a>
1220       </h5>
1221 <p>
1222         In addition to functioning with types from this library, these functions
1223         are also overloaded for built in integer types if you include <code class="computeroutput"><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>.
1224         Further, when used with fixed precision types (whether built in integers
1225         or multiprecision ones), the functions will promote to a wider type internally
1226         when the algorithm requires it. Versions overloaded for built in integer
1227         types return that integer type rather than an expression template.
1228       </p>
1229 <pre class="programlisting"><span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">gcd</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
1230 </pre>
1231 <p>
1232         Returns the largest integer <code class="computeroutput"><span class="identifier">x</span></code>
1233         that divides both <code class="computeroutput"><span class="identifier">a</span></code> and
1234         <code class="computeroutput"><span class="identifier">b</span></code>.
1235       </p>
1236 <pre class="programlisting"><span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">lcm</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
1237 </pre>
1238 <p>
1239         Returns the smallest integer <code class="computeroutput"><span class="identifier">x</span></code>
1240         that is divisible by both <code class="computeroutput"><span class="identifier">a</span></code>
1241         and <code class="computeroutput"><span class="identifier">b</span></code>.
1242       </p>
1243 <pre class="programlisting"><span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">pow</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">p</span><span class="special">);</span>
1244 </pre>
1245 <p>
1246         Returns <span class="emphasis"><em>b<sup>p</sup></em></span> as an expression template. Note that this
1247         function should be used with extreme care as the result can grow so large
1248         as to take "effectively forever" to compute, or else simply run
1249         the host machine out of memory. This is the one function in this category
1250         that is not overloaded for built in integer types, further, it's probably
1251         not a good idea to use it with fixed precision <code class="computeroutput"><span class="identifier">cpp_int</span></code>'s
1252         either.
1253       </p>
1254 <pre class="programlisting"><span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">powm</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">m</span><span class="special">);</span>
1255 </pre>
1256 <p>
1257         Returns <span class="emphasis"><em>b<sup>p</sup> mod m</em></span> as an expression template. Fixed precision
1258         types are promoted internally to ensure accuracy.
1259       </p>
1260 <pre class="programlisting"><span class="emphasis"><em>unmentionable-expression-template-type</em></span>    <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">);</span>
1261 </pre>
1262 <p>
1263         Returns the largest integer <code class="computeroutput"><span class="identifier">x</span></code>
1264         such that <code class="computeroutput"><span class="identifier">x</span> <span class="special">*</span>
1265         <span class="identifier">x</span> <span class="special">&lt;</span>
1266         <span class="identifier">a</span></code>.
1267       </p>
1268 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
1269 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">EXpressionTemplates</span><span class="special">&gt;</span>      <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">EXpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">r</span><span class="special">);</span>
1270 </pre>
1271 <p>
1272         Returns the largest integer <code class="computeroutput"><span class="identifier">x</span></code>
1273         such that <code class="computeroutput"><span class="identifier">x</span> <span class="special">*</span>
1274         <span class="identifier">x</span> <span class="special">&lt;</span>
1275         <span class="identifier">a</span></code>, and sets the remainder <code class="computeroutput"><span class="identifier">r</span></code> such that <code class="computeroutput"><span class="identifier">r</span>
1276         <span class="special">=</span> <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">x</span> <span class="special">*</span>
1277         <span class="identifier">x</span></code>.
1278       </p>
1279 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
1280 <span class="keyword">void</span> <span class="identifier">divide_qr</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">y</span><span class="special">,</span>
1281                <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">r</span><span class="special">);</span>
1282 </pre>
1283 <p>
1284         Divides x by y and returns both the quotient and remainder. After the call
1285         <code class="computeroutput"><span class="identifier">q</span> <span class="special">=</span>
1286         <span class="identifier">x</span> <span class="special">/</span> <span class="identifier">y</span></code> and <code class="computeroutput"><span class="identifier">r</span>
1287         <span class="special">=</span> <span class="identifier">x</span> <span class="special">%</span> <span class="identifier">y</span></code>.
1288       </p>
1289 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Integer</span><span class="special">&gt;</span>
1290 <span class="identifier">Integer</span> <span class="identifier">integer_modulus</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Integer</span> <span class="identifier">val</span><span class="special">);</span>
1291 </pre>
1292 <p>
1293         Returns the absolute value of <code class="computeroutput"><span class="identifier">x</span>
1294         <span class="special">%</span> <span class="identifier">val</span></code>.
1295       </p>
1296 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">lsb</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
1297 </pre>
1298 <p>
1299         Returns the (zero-based) index of the least significant bit that is set to
1300         1.
1301       </p>
1302 <p>
1303         Throws a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">range_error</span></code> if the argument is &lt;= 0.
1304       </p>
1305 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">msb</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
1306 </pre>
1307 <p>
1308         Returns the (zero-based) index of the most significant bit.
1309       </p>
1310 <p>
1311         Throws a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">range_error</span></code> if the argument is &lt;= 0.
1312       </p>
1313 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
1314 <span class="keyword">bool</span> <span class="identifier">bit_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
1315 </pre>
1316 <p>
1317         Returns <code class="computeroutput"><span class="keyword">true</span></code> if the bit at
1318         <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span> is set.
1319       </p>
1320 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
1321 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">bit_set</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
1322 </pre>
1323 <p>
1324         Sets the bit at <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span>, and
1325         returns <span class="emphasis"><em>val</em></span>.
1326       </p>
1327 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
1328 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">bit_unset</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
1329 </pre>
1330 <p>
1331         Unsets the bit at <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span>,
1332         and returns <span class="emphasis"><em>val</em></span>.
1333       </p>
1334 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
1335 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">bit_flip</span><span class="special">(</span><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">index</span><span class="special">);</span>
1336 </pre>
1337 <p>
1338         Flips the bit at <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span>,
1339         and returns <span class="emphasis"><em>val</em></span>.
1340       </p>
1341 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Engine</span><span class="special">&gt;</span>
1342 <span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="identifier">Engine</span><span class="special">&amp;</span> <span class="identifier">gen</span><span class="special">);</span>
1343 <span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">);</span>
1344 </pre>
1345 <p>
1346         Tests to see if the number <span class="emphasis"><em>n</em></span> is probably prime - the
1347         test excludes the vast majority of composite numbers by excluding small prime
1348         factors and performing a single Fermat test. Then performs <span class="emphasis"><em>trials</em></span>
1349         Miller-Rabin tests. Returns <code class="computeroutput"><span class="keyword">false</span></code>
1350         if <span class="emphasis"><em>n</em></span> is definitely composite, or <code class="computeroutput"><span class="keyword">true</span></code>
1351         if <span class="emphasis"><em>n</em></span> is probably prime with the probability of it being
1352         composite less than 0.25^trials. Fixed precision types are promoted internally
1353         to ensure accuracy.
1354       </p>
1355 <h5>
1356 <a name="boost_multiprecision.ref.number.h9"></a>
1357         <span class="phrase"><a name="boost_multiprecision.ref.number.rational_number_functions"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.rational_number_functions">Rational
1358         Number Functions</a>
1359       </h5>
1360 <pre class="programlisting"><span class="keyword">typename</span> <span class="identifier">component_type</span><span class="special">&lt;</span><span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&gt;::</span><span class="identifier">type</span> <span class="identifier">numerator</span>  <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
1361 <span class="keyword">typename</span> <span class="identifier">component_type</span><span class="special">&lt;</span><span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&gt;::</span><span class="identifier">type</span> <span class="identifier">denominator</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;);</span>
1362 </pre>
1363 <p>
1364         These functions return the numerator and denominator of a rational number
1365         respectively.
1366       </p>
1367 <h5>
1368 <a name="boost_multiprecision.ref.number.h10"></a>
1369         <span class="phrase"><a name="boost_multiprecision.ref.number.boost_math_interoperability_supp"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.boost_math_interoperability_supp">Boost.Math
1370         Interoperability Support</a>
1371       </h5>
1372 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
1373
1374 <span class="keyword">int</span>  <span class="identifier">fpclassify</span>     <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
1375 <span class="keyword">bool</span> <span class="identifier">isfinite</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
1376 <span class="keyword">bool</span> <span class="identifier">isnan</span>          <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
1377 <span class="keyword">bool</span> <span class="identifier">isinf</span>          <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
1378 <span class="keyword">bool</span> <span class="identifier">isnormal</span>       <span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>number-or-expression-template-type</em></span><span class="special">&amp;,</span> <span class="keyword">int</span><span class="special">);</span>
1379
1380 <span class="special">}}</span> <span class="comment">// namespaces</span>
1381 </pre>
1382 <p>
1383         These floating-point classification functions behave exactly as their Boost.Math
1384         equivalents.
1385       </p>
1386 <p>
1387         Other Boost.Math functions and templates may also be specialized or overloaded
1388         to ensure interoperability.
1389       </p>
1390 <h5>
1391 <a name="boost_multiprecision.ref.number.h11"></a>
1392         <span class="phrase"><a name="boost_multiprecision.ref.number.std_numeric_limits_support"></a></span><a class="link" href="number.html#boost_multiprecision.ref.number.std_numeric_limits_support">std::numeric_limits
1393         support</a>
1394       </h5>
1395 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">{</span>
1396
1397 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
1398 <span class="keyword">struct</span> <span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="special">&gt;</span>
1399 <span class="special">{</span>
1400    <span class="comment">/* Usual members here */</span>
1401 <span class="special">};</span>
1402
1403 <span class="special">}</span>
1404 </pre>
1405 <p>
1406         Class template <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code> is specialized for all instantiations
1407         of <code class="computeroutput"><span class="identifier">number</span></code> whose precision
1408         is known at compile time, plus those types whose precision is unlimited (though
1409         it is much less useful in those cases). It is not specialized for types whose
1410         precision can vary at compile time (such as <code class="computeroutput"><span class="identifier">mpf_float</span></code>).
1411       </p>
1412 </div>
1413 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1414 <td align="left"></td>
1415 <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2013 John Maddock and Christopher Kormanyos<p>
1416         Distributed under the Boost Software License, Version 1.0. (See accompanying
1417         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>)
1418       </p>
1419 </div></td>
1420 </tr></table>
1421 <hr>
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