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    25. 

  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h3 class="title">
    27. 

  27. <a name="boost_multiprecision.ref.number"></a><a class="link" href="number.html" title="number">number</a>
    28. 

  28. </h3></div></div></div>
    29. 

  29. <h5>
    30. 

  30. <a name="boost_multiprecision.ref.number.h0"></a>
    31. 

  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. 

  32.       </h5>
    33. 

  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. 

  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. 

  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. 

  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. 

  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. 

  41. <span class="keyword">class</span> <span class="identifier">number</span>
    42. 

  42. <span class="special">{</span>
    43. 

  43. <span class="keyword">public</span><span class="special">:</span>
    44. 

  44.    <span class="keyword">typedef</span>          <span class="identifier">Backend</span>                          <span class="identifier">backend_type</span><span class="special">;</span>
    45. 

  45.    <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">component_type</span><span class="special">&lt;</span><span class="identifier">self_type</span><span class="special">&gt;::</span><span class="identifier">type</span>  <span class="identifier">value_type</span><span class="special">;</span>
    46. 

  46. 

  47.    <span class="identifier">number</span><span class="special">();</span>
    48. 

  48.    <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>
    49. 

  49.    <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>
    50. 

  50.    <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>
    51. 

  51. 

  52.    <span class="comment">// Member operators</span>
    53. 

  53.    <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>
    54. 

  54.    <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>
    55. 

  55.    <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>
    56. 

  56.    <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>
    57. 

  57.    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">++();</span>
    58. 

  58.    <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">--();</span>
    59. 

  59.    <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">++(</span><span class="keyword">int</span><span class="special">);</span>
    60. 

  60.    <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">--(</span><span class="keyword">int</span><span class="special">);</span>
    61. 

  61. 

  62.    <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>
    63. 

  63.    <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>
    64. 

  64.    <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>
    65. 

  65.    <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>
    66. 

  66.    <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>
    67. 

  67.    <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>
    68. 

  68. 

  69.    <span class="comment">// Use in Boolean context:</span>
    70. 

  70.    <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>
    71. 

  71.    <span class="comment">// swap:</span>
    72. 

  72.    <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>
    73. 

  73.    <span class="comment">// Sign:</span>
    74. 

  74.    <span class="keyword">bool</span> <span class="identifier">is_zero</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    75. 

  75.    <span class="keyword">int</span> <span class="identifier">sign</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    76. 

  76.    <span class="comment">// string conversion:</span>
    77. 

  77.    <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>
    78. 

  78.    <span class="comment">// Generic conversion mechanism</span>
    79. 

  79.    <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>
    80. 

  80.    <span class="identifier">T</span> <span class="identifier">convert_to</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    81. 

  81.    <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>
    82. 

  82.    <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>
    83. 

  83.    <span class="comment">// precision control:</span>
    84. 

  84.    <span class="keyword">static</span> <span class="keyword">unsigned</span> <span class="identifier">default_precision</span><span class="special">();</span>
    85. 

  85.    <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>
    86. 

  86.    <span class="keyword">unsigned</span> <span class="identifier">precision</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    87. 

  87.    <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>
    88. 

  88.    <span class="comment">// Comparison:</span>
    89. 

  89.    <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>
    90. 

  90.    <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>
    91. 

  91.    <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>
    92. 

  92.       <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>
    93. 

  93.    <span class="comment">// real and imaginary parts:</span>
    94. 

  94.    <span class="identifier">value_type</span> <span class="identifier">real</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    95. 

  95.    <span class="identifier">value_type</span> <span class="identifier">imag</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    96. 

  96.    <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>
    97. 

  97.    <span class="keyword">void</span> <span class="identifier">real</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">);</span>
    98. 

  98.    <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>
    99. 

  99.    <span class="keyword">void</span> <span class="identifier">imag</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">);</span>
    100. 

  100.    <span class="comment">// Access to the underlying implementation:</span>
    101. 

  101.    <span class="identifier">Backend</span><span class="special">&amp;</span> <span class="identifier">backend</span><span class="special">();</span>
    102. 

  102.    <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>
    103. 

  103. <span class="special">};</span>
    104. 

  104. 

  105. <span class="comment">// Non member operators:</span>
    106. 

  106. <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>
    107. 

  107. <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>
    108. 

  108. <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>
    109. 

  109. <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>
    110. 

  110. <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>
    111. 

  111. <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>
    112. 

  112. <span class="comment">// Integer only operations:</span>
    113. 

  113. <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>
    114. 

  114. <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>
    115. 

  115. <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>
    116. 

  116. <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>
    117. 

  117. <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>
    118. 

  118. <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>
    119. 

  119. <span class="comment">// Comparison operators:</span>
    120. 

  120. <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>
    121. 

  121. <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>
    122. 

  122. <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>
    123. 

  123. <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>
    124. 

  124. <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>
    125. 

  125. <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>
    126. 

  126. 

  127. <span class="comment">// Swap:</span>
    128. 

  128. <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>
    129. 

  129. <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>
    130. 

  130. 

  131. <span class="comment">// iostream support:</span>
    132. 

  132. <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>
    133. 

  133. <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>
    134. 

  134. <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>
    135. 

  135. <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>
    136. 

  136. <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>
    137. 

  137. 

  138. <span class="comment">// Arithmetic with a higher precision result:</span>
    139. 

  139. <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>
    140. 

  140. <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>
    141. 

  141. <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>
    142. 

  142. <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>
    143. 

  143. <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>
    144. 

  144. <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>
    145. 

  145. 

  146. <span class="comment">// min and max overloads:</span>
    147. 

  147. <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>
    148. 

  148. <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>
    149. 

  149. 

  150. <span class="comment">// C99 Non-member function standard library support:</span>
    151. 

  151. <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>
    152. 

  152. <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>
    153. 

  153. <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>
    154. 

  154. <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>
    155. 

  155. <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>
    156. 

  156. <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>
    157. 

  157. <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>
    158. 

  158. <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>
    159. 

  159. <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>
    160. 

  160. <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>
    161. 

  161. <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>
    162. 

  162. <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>
    163. 

  163. <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>
    164. 

  164. <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>
    165. 

  165. <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>
    166. 

  166. <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>
    167. 

  167. <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>
    168. 

  168. <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>
    169. 

  169. <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>
    170. 

  170. <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>
    171. 

  171. <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>
    172. 

  172. <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>
    173. 

  173. <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>
    174. 

  174. <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>
    175. 

  175. <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>
    176. 

  176. <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>
    177. 

  177. <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>
    178. 

  178. <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>
    179. 

  179. <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>
    180. 

  180. <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>
    181. 

  181. <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>
    182. 

  182. <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>
    183. 

  183. <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>
    184. 

  184. <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>
    185. 

  185. <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>
    186. 

  186. <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>
    187. 

  187. <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>
    188. 

  188. <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>
    189. 

  189. <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>
    190. 

  190. <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>
    191. 

  191. <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>
    192. 

  192. <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>
    193. 

  193. <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>
    194. 

  194. <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>
    195. 

  195. <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>
    196. 

  196. <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>
    197. 

  197. <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>
    198. 

  198. <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>
    199. 

  199. <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>
    200. 

  200. <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>
    201. 

  201. <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>
    202. 

  202. <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>
    203. 

  203. <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>
    204. 

  204. <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>
    205. 

  205. <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>
    206. 

  206. <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>
    207. 

  207. <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>
    208. 

  208. 

  209. <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>
    210. 

  210. <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>
    211. 

  211. <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>
    212. 

  212. <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>
    213. 

  213. <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>
    214. 

  214. <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>
    215. 

  215. 

  216. <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>
    217. 

  217. <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>
    218. 

  218. <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>
    219. 

  219. <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>
    220. 

  220. <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>
    221. 

  221. <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>
    222. 

  222. <span class="comment">// Complex number functions:</span>
    223. 

  223. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">real</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>
    224. 

  224. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">imag</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>
    225. 

  225. <span class="emphasis"><em>number&lt;...&gt;::value_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>
    226. 

  226. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">arg</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>
    227. 

  227. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">norm</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>
    228. 

  228. <span class="emphasis"><em>number</em></span>                                    <span class="identifier">conj</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>
    229. 

  229. <span class="emphasis"><em>number</em></span>                                    <span class="identifier">proj</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>
    230. 

  230. <span class="emphasis"><em>number</em></span>                                    <span class="identifier">polar</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. 

  231. <span class="comment">// Misc other common C library functions:</span>
    232. 

  232. <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>
    233. 

  233. <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>
    234. 

  234. <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>
    235. 

  235. <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>
    236. 

  236. <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>
    237. 

  237. <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>
    238. 

  238. 

  239. <span class="comment">// Traits support:</span>
    240. 

  240. <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>
    241. 

  241. <span class="keyword">struct</span> <span class="identifier">component_type</span><span class="special">;</span>
    242. 

  242. <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>
    243. 

  243. <span class="keyword">struct</span> <span class="identifier">number_category</span><span class="special">;</span>
    244. 

  244. <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>
    245. 

  245. <span class="keyword">struct</span> <span class="identifier">is_number</span><span class="special">;</span>
    246. 

  246. <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>
    247. 

  247. <span class="keyword">struct</span> <span class="identifier">is_number_expression</span><span class="special">;</span>
    248. 

  248. 

  249. <span class="comment">// Integer specific functions:</span>
    250. 

  250. <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>
    251. 

  251. <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>
    252. 

  252. <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>
    253. 

  253. <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>
    254. 

  254. <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>
    255. 

  255. <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>
    256. 

  256. <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>
    257. 

  257. <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>
    258. 

  258. <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>
    259. 

  259.                <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>
    260. 

  260. <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>
    261. 

  261. <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>
    262. 

  262. <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>
    263. 

  263. <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>
    264. 

  264. <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>
    265. 

  265. <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>
    266. 

  266. <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>
    267. 

  267. <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>
    268. 

  268. <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>
    269. 

  269. <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>
    270. 

  270. <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>
    271. 

  271. <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>
    272. 

  272. <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>
    273. 

  273. <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>
    274. 

  274. <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>
    275. 

  275. 

  276. <span class="comment">// Rational number support:</span>
    277. 

  277. <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>
    278. 

  278. <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>
    279. 

  279. 

  280. <span class="special">}}</span> <span class="comment">// namespaces</span>
    281. 

  281. 

  282. <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>
    283. 

  283. 

  284. <span class="comment">// Boost.Math interoperability functions:</span>
    285. 

  285. <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>
    286. 

  286. <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>
    287. 

  287. <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>
    288. 

  288. <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>
    289. 

  289. <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>
    290. 

  290. 

  291. <span class="special">}}</span> <span class="comment">// namespaces</span>
    292. 

  292. 

  293. <span class="comment">// numeric_limits support:</span>
    294. 

  294. <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">{</span>
    295. 

  295. 

  296. <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>
    297. 

  297. <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>
    298. 

  298. <span class="special">{</span>
    299. 

  299.    <span class="comment">/* Usual members here */</span>
    300. 

  300. <span class="special">};</span>
    301. 

  301. 

  302. <span class="special">}</span>
    303. 

  303. </pre>
    304. 

  304. <h5>
    305. 

  305. <a name="boost_multiprecision.ref.number.h1"></a>
    306. 

  306.         <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>
    307. 

  307.       </h5>
    308. 

  308. <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>
    309. 

  309. </pre>
    310. 

  310. <p>
    311. 

  311.         This enumerated type is used to specify whether expression templates are
    312. 

  312.         turned on (et_on) or turned off (et_off).
    313. 

  313.       </p>
    314. 

  314. <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>
    315. 

  315. <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>
    316. 

  316. </pre>
    317. 

  317. <p>
    318. 

  318.         This traits class specifies the default expression template option to be
    319. 

  319.         used with a particular Backend type. It defaults to <code class="computeroutput"><span class="identifier">et_on</span></code>.
    320. 

  320.       </p>
    321. 

  321. <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>
    322. 

  322. <span class="keyword">class</span> <span class="identifier">number</span><span class="special">;</span>
    323. 

  323. </pre>
    324. 

  324. <p>
    325. 

  325.         Class <code class="computeroutput"><span class="identifier">number</span></code> has two template
    326. 

  326.         arguments:
    327. 

  327.       </p>
    328. 

  328. <div class="variablelist">
    329. 

  329. <p class="title"><b></b></p>
    330. 

  330. <dl class="variablelist">
    331. 

  331. <dt><span class="term">Backend</span></dt>
    332. 

  332. <dd><p>
    333. 

  333.               The actual arithmetic back-end that does all the work.
    334. 

  334.             </p></dd>
    335. 

  335. <dt><span class="term">ExpressionTemplates</span></dt>
    336. 

  336. <dd><p>
    337. 

  337.               A Boolean value: when <code class="computeroutput"><span class="identifier">et_on</span></code>,
    338. 

  338.               then expression templates are enabled, otherwise when set to <code class="computeroutput"><span class="identifier">et_off</span></code> they are disabled. The default
    339. 

  339.               for this parameter is computed via the traits class <code class="computeroutput"><span class="identifier">expression_template_default</span></code>
    340. 

  340.               whose member <code class="computeroutput"><span class="identifier">value</span></code>
    341. 

  341.               defaults to <code class="computeroutput"><span class="identifier">et_on</span></code> unless
    342. 

  342.               the traits class is specialized for a particular backend.
    343. 

  343.             </p></dd>
    344. 

  344. </dl>
    345. 

  345. </div>
    346. 

  346. <pre class="programlisting"><span class="identifier">number</span><span class="special">();</span>
    347. 

  347. <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>
    348. 

  348. <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>
    349. 

  349. <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>
    350. 

  350. </pre>
    351. 

  351. <p>
    352. 

  352.         Type <code class="computeroutput"><span class="identifier">number</span></code> is default constructible,
    353. 

  353.         and both copy constructible and assignable from:
    354. 

  354.       </p>
    355. 

  355. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    356. 

  356. <li class="listitem">
    357. 

  357.             Itself.
    358. 

  358.           </li>
    359. 

  359. <li class="listitem">
    360. 

  360.             An expression template which is the result of one of the arithmetic operators.
    361. 

  361.           </li>
    362. 

  362. <li class="listitem">
    363. 

  363.             Any builtin arithmetic type, as long as the result would not be lossy
    364. 

  364.             (for example float to integer conversion).
    365. 

  365.           </li>
    366. 

  366. <li class="listitem">
    367. 

  367.             Any type that the Backend is implicitly constructible or assignable from.
    368. 

  368.           </li>
    369. 

  369. <li class="listitem">
    370. 

  370.             An rvalue reference to another <code class="computeroutput"><span class="identifier">number</span></code>.
    371. 

  371.             Move-semantics are used for construction if the backend also supports
    372. 

  372.             rvalue reference construction. In the case of assignment, move semantics
    373. 

  373.             are always supported when the argument is an rvalue reference irrespective
    374. 

  374.             of the backend.
    375. 

  375.           </li>
    376. 

  376. <li class="listitem">
    377. 

  377.             Any type in the same family, as long as no loss of precision is involved.
    378. 

  378.             For example from <code class="computeroutput"><span class="identifier">int128_t</span></code>
    379. 

  379.             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>.
    380. 

  380.           </li>
    381. 

  381. </ul></div>
    382. 

  382. <p>
    383. 

  383.         Type <code class="computeroutput"><span class="identifier">number</span></code> is explicitly
    384. 

  384.         constructible from:
    385. 

  385.       </p>
    386. 

  386. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    387. 

  387. <li class="listitem">
    388. 

  388.             Any type mentioned above.
    389. 

  389.           </li>
    390. 

  390. <li class="listitem">
    391. 

  391.             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
    392. 

  392.             to <code class="computeroutput"><span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span></code>.
    393. 

  393.           </li>
    394. 

  394. <li class="listitem">
    395. 

  395.             Any arithmetic type (including those that would result in lossy conversions).
    396. 

  396.           </li>
    397. 

  397. <li class="listitem">
    398. 

  398.             Any type in the same family, including those that result in loss of precision.
    399. 

  399.           </li>
    400. 

  400. <li class="listitem">
    401. 

  401.             Any type that the Backend is explicitly constructible from.
    402. 

  402.           </li>
    403. 

  403. <li class="listitem">
    404. 

  404.             Any pair of types for which a generic interconversion exists: that is
    405. 

  405.             from integer to integer, integer to rational, integer to float, rational
    406. 

  406.             to rational, rational to float, or float to float.
    407. 

  407.           </li>
    408. 

  408. </ul></div>
    409. 

  409. <p>
    410. 

  410.         The assign member function is available for any type for which an explicit
    411. 

  411.         converting constructor exists. It is intended to be used where a temporary
    412. 

  412.         generated from an explicit assignment would be expensive, for example:
    413. 

  413.       </p>
    414. 

  414. <pre class="programlisting"><span class="identifier">mpfr_float_50</span>    <span class="identifier">f50</span><span class="special">;</span>
    415. 

  415. <span class="identifier">mpfr_float_100</span>   <span class="identifier">f100</span><span class="special">;</span>
    416. 

  416. 

  417. <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>
    418. 

  418. <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>
    419. 

  419. </pre>
    420. 

  420. <p>
    421. 

  421.         In addition, if the type has multiple components (for example rational or
    422. 

  422.         complex number types), then there is a two argument constructor:
    423. 

  423.       </p>
    424. 

  424. <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>
    425. 

  425. </pre>
    426. 

  426. <p>
    427. 

  427.         Where the two args must either be arithmetic types, or types that are convertible
    428. 

  428.         to the two components of <code class="computeroutput"><span class="keyword">this</span></code>.
    429. 

  429.       </p>
    430. 

  430. <p>
    431. 

  431.         Finally, when the type has a variable precision, then there are constructors:
    432. 

  432.       </p>
    433. 

  433. <pre class="programlisting"><span class="identifier">number</span><span class="special">(</span><span class="identifier">arg1</span><span class="special">,</span> <span class="identifier">precision</span><span class="special">);</span>
    434. 

  434. <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> <span class="identifier">precision</span><span class="special">);</span>
    435. 

  435. </pre>
    436. 

  436. <p>
    437. 

  437.         Where <code class="computeroutput"><span class="identifier">precision</span></code> is an unsigned
    438. 

  438.         value, the 2 arg version is active for scalar types and/or copy-construction
    439. 

  439.         with specific precision, and the 3-arg version for complex types.
    440. 

  440.       </p>
    441. 

  441. <p>
    442. 

  442.         Likewise <code class="computeroutput"><span class="identifier">assign</span></code> has a 2-arg
    443. 

  443.         overloaded, with the second argument being the precision.
    444. 

  444.       </p>
    445. 

  445. <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>
    446. 

  446. <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>
    447. 

  447. <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>
    448. 

  448. <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>
    449. 

  449. <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">++();</span>
    450. 

  450. <span class="identifier">number</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">--();</span>
    451. 

  451. <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">++(</span><span class="keyword">int</span><span class="special">);</span>
    452. 

  452. <span class="identifier">number</span>  <span class="keyword">operator</span><span class="special">--(</span><span class="keyword">int</span><span class="special">);</span>
    453. 

  453. <span class="comment">// Integer only operations:</span>
    454. 

  454. <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>
    455. 

  455. <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>
    456. 

  456. <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>
    457. 

  457. <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>
    458. 

  458. <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>
    459. 

  459. <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>
    460. 

  460. </pre>
    461. 

  461. <p>
    462. 

  462.         These operators all take their usual arithmetic meanings.
    463. 

  463.       </p>
    464. 

  464. <p>
    465. 

  465.         The arguments to these operators is either:
    466. 

  466.       </p>
    467. 

  467. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    468. 

  468. <li class="listitem">
    469. 

  469.             Another <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span>
    470. 

  470.             <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span></code>.
    471. 

  471.           </li>
    472. 

  472. <li class="listitem">
    473. 

  473.             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>.
    474. 

  474.           </li>
    475. 

  475. <li class="listitem">
    476. 

  476.             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
    477. 

  477.             class <code class="computeroutput"><span class="identifier">number</span></code>.
    478. 

  478.           </li>
    479. 

  479. </ul></div>
    480. 

  480. <p>
    481. 

  481.         For the left and right shift operations, the argument must be a builtin integer
    482. 

  482.         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>
    483. 

  483.         being thrown).
    484. 

  484.       </p>
    485. 

  485. <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>
    486. 

  486. </pre>
    487. 

  487. <p>
    488. 

  488.         Returns an <span class="emphasis"><em>unmentionable-type</em></span> that is usable in Boolean
    489. 

  489.         contexts (this allows <code class="computeroutput"><span class="identifier">number</span></code>
    490. 

  490.         to be used in any Boolean context - if statements, conditional statements,
    491. 

  491.         or as an argument to a logical operator - without type <code class="computeroutput"><span class="identifier">number</span></code>
    492. 

  492.         being convertible to type <code class="computeroutput"><span class="keyword">bool</span></code>.
    493. 

  493.       </p>
    494. 

  494. <p>
    495. 

  495.         This operator also enables the use of <code class="computeroutput"><span class="identifier">number</span></code>
    496. 

  496.         with any of the following operators: <code class="computeroutput"><span class="special">!</span></code>,
    497. 

  497.         <code class="computeroutput"><span class="special">||</span></code>, <code class="computeroutput"><span class="special">&amp;&amp;</span></code>
    498. 

  498.         and <code class="computeroutput"><span class="special">?:</span></code>.
    499. 

  499.       </p>
    500. 

  500. <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>
    501. 

  501. </pre>
    502. 

  502. <p>
    503. 

  503.         Swaps <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
    504. 

  504.         with <code class="computeroutput"><span class="identifier">other</span></code>.
    505. 

  505.       </p>
    506. 

  506. <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>
    507. 

  507. </pre>
    508. 

  508. <p>
    509. 

  509.         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,
    510. 

  510.         otherwise <code class="computeroutput"><span class="keyword">false</span></code>.
    511. 

  511.       </p>
    512. 

  512. <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>
    513. 

  513. </pre>
    514. 

  514. <p>
    515. 

  515.         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
    516. 

  516.         if <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
    517. 

  517.         is positive, and zero if <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
    518. 

  518.         is zero.
    519. 

  519.       </p>
    520. 

  520. <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>
    521. 

  521. </pre>
    522. 

  522. <p>
    523. 

  523.         Returns the number formatted as a string, with at least <span class="emphasis"><em>precision</em></span>
    524. 

  524.         digits, and in scientific format if <span class="emphasis"><em>scientific</em></span> is true.
    525. 

  525.       </p>
    526. 

  526. <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>
    527. 

  527. <span class="identifier">T</span> <span class="identifier">convert_to</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    528. 

  528. 

  529. <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>
    530. 

  530. <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>
    531. 

  531. </pre>
    532. 

  532. <p>
    533. 

  533.         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>.
    534. 

  534.         Type <code class="computeroutput"><span class="identifier">T</span></code> may be any arithmetic
    535. 

  535.         type. Optionally other types may also be supported by specific <code class="computeroutput"><span class="identifier">Backend</span></code> types.
    536. 

  536.       </p>
    537. 

  537. <pre class="programlisting"><span class="keyword">static</span> <span class="keyword">unsigned</span> <span class="identifier">default_precision</span><span class="special">();</span>
    538. 

  538. <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>
    539. 

  539. <span class="keyword">unsigned</span> <span class="identifier">precision</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    540. 

  540. <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>
    541. 

  541. </pre>
    542. 

  542. <p>
    543. 

  543.         These functions are only available if the Backend template parameter supports
    544. 

  544.         runtime changes to precision. They get and set the default precision and
    545. 

  545.         the precision of <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>
    546. 

  546.         respectively.
    547. 

  547.       </p>
    548. 

  548. <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>
    549. 

  549. <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>
    550. 

  550. <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>
    551. 

  551.    <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>
    552. 

  552. </pre>
    553. 

  553. <p>
    554. 

  554.         Returns:
    555. 

  555.       </p>
    556. 

  556. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    557. 

  557. <li class="listitem">
    558. 

  558.             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>
    559. 

  559.           </li>
    560. 

  560. <li class="listitem">
    561. 

  561.             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>
    562. 

  562.           </li>
    563. 

  563. <li class="listitem">
    564. 

  564. <p class="simpara">
    565. 

  565.             Zero for <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span>
    566. 

  566.             <span class="special">==</span> <span class="identifier">other</span></code>
    567. 

  567.           </p>
    568. 

  568. <pre class="programlisting"><span class="identifier">value_type</span> <span class="identifier">real</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    569. 

  569. <span class="identifier">value_type</span> <span class="identifier">imag</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
    570. 

  570. </pre>
    571. 

  571. </li>
    572. 

  572. </ul></div>
    573. 

  573. <p>
    574. 

  574.         These return the real and imaginary parts respectively. If the number is
    575. 

  575.         not a complex type, then the imaginary part is always zero.
    576. 

  576.       </p>
    577. 

  577. <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>
    578. 

  578. <span class="keyword">void</span> <span class="identifier">real</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">);</span>
    579. 

  579. <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>
    580. 

  580. <span class="keyword">void</span> <span class="identifier">imag</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">);</span>
    581. 

  581. </pre>
    582. 

  582. <p>
    583. 

  583.         These set the real and imaginary parts respectively of the number. If the
    584. 

  584.         number is not a complex type, then setting the real part is equivalent to
    585. 

  585.         assignment, and attempting to set the imaginary part will result in a compile
    586. 

  586.         time error.
    587. 

  587.       </p>
    588. 

  588. <pre class="programlisting"><span class="identifier">Backend</span><span class="special">&amp;</span> <span class="identifier">backend</span><span class="special">();</span>
    589. 

  589. <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>
    590. 

  590. </pre>
    591. 

  591. <p>
    592. 

  592.         Returns the underlying back-end instance used by <code class="computeroutput"><span class="special">*</span><span class="keyword">this</span></code>.
    593. 

  593.       </p>
    594. 

  594. <h5>
    595. 

  595. <a name="boost_multiprecision.ref.number.h2"></a>
    596. 

  596.         <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
    597. 

  597.         operators</a>
    598. 

  598.       </h5>
    599. 

  599. <pre class="programlisting"><span class="comment">// Non member operators:</span>
    600. 

  600. <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>
    601. 

  601. <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>
    602. 

  602. <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>
    603. 

  603. <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>
    604. 

  604. <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>
    605. 

  605. <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>
    606. 

  606. <span class="comment">// Integer only operations:</span>
    607. 

  607. <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>
    608. 

  608. <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>
    609. 

  609. <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>
    610. 

  610. <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>
    611. 

  611. <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>
    612. 

  612. <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>
    613. 

  613. <span class="comment">// Comparison operators:</span>
    614. 

  614. <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>
    615. 

  615. <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>
    616. 

  616. <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>
    617. 

  617. <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>
    618. 

  618. <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>
    619. 

  619. <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>
    620. 

  620. </pre>
    621. 

  621. <p>
    622. 

  622.         These operators all take their usual arithmetic meanings.
    623. 

  623.       </p>
    624. 

  624. <p>
    625. 

  625.         The arguments to these functions must contain at least one of the following:
    626. 

  626.       </p>
    627. 

  627. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    628. 

  628. <li class="listitem">
    629. 

  629.             A <code class="computeroutput"><span class="identifier">number</span></code>.
    630. 

  630.           </li>
    631. 

  631. <li class="listitem">
    632. 

  632.             An expression template type derived from <code class="computeroutput"><span class="identifier">number</span></code>.
    633. 

  633.           </li>
    634. 

  634. <li class="listitem">
    635. 

  635.             Any type for which <code class="computeroutput"><span class="identifier">number</span></code>
    636. 

  636.             has an implicit constructor - for example a builtin arithmetic type.
    637. 

  637.           </li>
    638. 

  638. </ul></div>
    639. 

  639. <p>
    640. 

  640.         The return type of these operators is either:
    641. 

  641.       </p>
    642. 

  642. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    643. 

  643. <li class="listitem">
    644. 

  644.             An <span class="emphasis"><em>unmentionable-type</em></span> expression template type when
    645. 

  645.             <code class="computeroutput"><span class="identifier">ExpressionTemplates</span></code> is
    646. 

  646.             <code class="computeroutput"><span class="keyword">true</span></code>.
    647. 

  647.           </li>
    648. 

  648. <li class="listitem">
    649. 

  649.             Type <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">Backend</span><span class="special">,</span>
    650. 

  650.             <span class="identifier">et_off</span><span class="special">&gt;</span></code>
    651. 

  651.             when <code class="computeroutput"><span class="identifier">ExpressionTemplates</span></code>
    652. 

  652.             is <code class="computeroutput"><span class="keyword">false</span></code>.
    653. 

  653.           </li>
    654. 

  654. <li class="listitem">
    655. 

  655.             Type <code class="computeroutput"><span class="keyword">bool</span></code> if the operator
    656. 

  656.             is a comparison operator.
    657. 

  657.           </li>
    658. 

  658. </ul></div>
    659. 

  659. <p>
    660. 

  660.         Finally note that the second argument to the left and right shift operations
    661. 

  661.         must be a builtin integer type, and that the argument must be positive (negative
    662. 

  662.         arguments result in a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">runtime_error</span></code>
    663. 

  663.         being thrown).
    664. 

  664.       </p>
    665. 

  665. <h5>
    666. 

  666. <a name="boost_multiprecision.ref.number.h3"></a>
    667. 

  667.         <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>
    668. 

  668.       </h5>
    669. 

  669. <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>
    670. 

  670. <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>
    671. 

  671. </pre>
    672. 

  672. <p>
    673. 

  673.         Swaps <code class="computeroutput"><span class="identifier">a</span></code> and <code class="computeroutput"><span class="identifier">b</span></code>.
    674. 

  674.       </p>
    675. 

  675. <h5>
    676. 

  676. <a name="boost_multiprecision.ref.number.h4"></a>
    677. 

  677.         <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
    678. 

  678.         Support</a>
    679. 

  679.       </h5>
    680. 

  680. <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>
    681. 

  681. <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>
    682. 

  682. <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>
    683. 

  683. <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>
    684. 

  684. <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>
    685. 

  685. <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>
    686. 

  686. </pre>
    687. 

  687. <p>
    688. 

  688.         These operators provided formatted input-output operations on <code class="computeroutput"><span class="identifier">number</span></code> types, and expression templates
    689. 

  689.         derived from them.
    690. 

  690.       </p>
    691. 

  691. <p>
    692. 

  692.         It's down to the back-end type to actually implement string conversion. However,
    693. 

  693.         the back-ends provided with this library support all of the iostream formatting
    694. 

  694.         flags, field width and precision settings.
    695. 

  695.       </p>
    696. 

  696. <h5>
    697. 

  697. <a name="boost_multiprecision.ref.number.h5"></a>
    698. 

  698.         <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
    699. 

  699.         with a higher precision result</a>
    700. 

  700.       </h5>
    701. 

  701. <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>
    702. 

  702. <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>
    703. 

  703. 

  704. <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>
    705. 

  705. <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>
    706. 

  706. 

  707. <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>
    708. 

  708. <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>
    709. 

  709. </pre>
    710. 

  710. <p>
    711. 

  711.         These functions apply the named operator to the arguments <span class="emphasis"><em>a</em></span>
    712. 

  712.         and <span class="emphasis"><em>b</em></span> and store the result in <span class="emphasis"><em>result</em></span>,
    713. 

  713.         returning <span class="emphasis"><em>result</em></span>. In all cases they behave "as
    714. 

  714.         if" arguments <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> were
    715. 

  715.         first promoted to type <code class="computeroutput"><span class="identifier">ResultType</span></code>
    716. 

  716.         before applying the operator, though particular backends may well avoid that
    717. 

  717.         step by way of an optimization.
    718. 

  718.       </p>
    719. 

  719. <p>
    720. 

  720.         The type <code class="computeroutput"><span class="identifier">ResultType</span></code> must
    721. 

  721.         be an instance of class <code class="computeroutput"><span class="identifier">number</span></code>,
    722. 

  722.         and the types <code class="computeroutput"><span class="identifier">Source1</span></code> and
    723. 

  723.         <code class="computeroutput"><span class="identifier">Source2</span></code> may be either instances
    724. 

  724.         of class <code class="computeroutput"><span class="identifier">number</span></code> or native
    725. 

  725.         integer types. The latter is an optimization that allows arithmetic to be
    726. 

  726.         performed on native integer types producing an extended precision result.
    727. 

  727.       </p>
    728. 

  728. <h5>
    729. 

  729. <a name="boost_multiprecision.ref.number.h6"></a>
    730. 

  730.         <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
    731. 

  731.         standard library function support</a>
    732. 

  732.       </h5>
    733. 

  733. <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>
    734. 

  734. <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>
    735. 

  735. <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>
    736. 

  736. <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>
    737. 

  737. <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>
    738. 

  738. <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>
    739. 

  739. <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>
    740. 

  740. <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>
    741. 

  741. <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>
    742. 

  742. <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>
    743. 

  743. <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>
    744. 

  744. <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>
    745. 

  745. <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>
    746. 

  746. <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>
    747. 

  747. <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>
    748. 

  748. <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>
    749. 

  749. <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>
    750. 

  750. <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>
    751. 

  751. <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>
    752. 

  752. <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>
    753. 

  753. <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>
    754. 

  754. <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>
    755. 

  755. <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>
    756. 

  756. <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>
    757. 

  757. <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>
    758. 

  758. <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>
    759. 

  759. <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>
    760. 

  760. <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>
    761. 

  761. <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>
    762. 

  762. <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>
    763. 

  763. <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>
    764. 

  764. <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>
    765. 

  765. <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>
    766. 

  766. <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>
    767. 

  767. <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>
    768. 

  768. <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>
    769. 

  769. <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>
    770. 

  770. <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>
    771. 

  771. <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>
    772. 

  772. <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>
    773. 

  773. <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>
    774. 

  774. <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>
    775. 

  775. <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>
    776. 

  776. <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>
    777. 

  777. <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>
    778. 

  778. <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>
    779. 

  779. <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>
    780. 

  780. <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>
    781. 

  781. <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>
    782. 

  782. <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>
    783. 

  783. <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>
    784. 

  784. <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>
    785. 

  785. <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>
    786. 

  786. <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>
    787. 

  787. <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>
    788. 

  788. <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>
    789. 

  789. <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>
    790. 

  790. 

  791. <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>
    792. 

  792. <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>
    793. 

  793. <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>
    794. 

  794. <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>
    795. 

  795. <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>
    796. 

  796. <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>
    797. 

  797. 

  798. <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>
    799. 

  799. <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>
    800. 

  800. <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>
    801. 

  801. <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>
    802. 

  802. <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>
    803. 

  803. <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>
    804. 

  804. </pre>
    805. 

  805. <p>
    806. 

  806.         These functions all behave exactly as their standard library C++11 counterparts
    807. 

  807.         do: their argument is either an instance of <code class="computeroutput"><span class="identifier">number</span></code>
    808. 

  808.         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>
    809. 

  809.         then that is also the return type, otherwise the return type is an expression
    810. 

  810.         template unless otherwise stated.
    811. 

  811.       </p>
    812. 

  812. <p>
    813. 

  813.         The integer type arguments to <code class="computeroutput"><span class="identifier">ldexp</span></code>,
    814. 

  814.         <code class="computeroutput"><span class="identifier">frexp</span></code>, <code class="computeroutput"><span class="identifier">scalbn</span></code>
    815. 

  815.         and <code class="computeroutput"><span class="identifier">ilogb</span></code> may be either type
    816. 

  816.         <code class="computeroutput"><span class="keyword">int</span></code>, or the actual type of the
    817. 

  817.         exponent of the number type.
    818. 

  818.       </p>
    819. 

  819. <p>
    820. 

  820.         Complex number types support the following functions:
    821. 

  821.       </p>
    822. 

  822. <pre class="programlisting"><span class="comment">// Complex number functions:</span>
    823. 

  823. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">real</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>
    824. 

  824. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">imag</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>
    825. 

  825. <span class="emphasis"><em>number&lt;...&gt;::value_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>
    826. 

  826. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">arg</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>
    827. 

  827. <span class="emphasis"><em>number&lt;...&gt;::value_type</em></span>                   <span class="identifier">norm</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>
    828. 

  828. <span class="emphasis"><em>number</em></span>                                    <span class="identifier">conj</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>
    829. 

  829. <span class="emphasis"><em>number</em></span>                                    <span class="identifier">proj</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>
    830. 

  830. <span class="emphasis"><em>number</em></span>                                    <span class="identifier">polar</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>
    831. 

  831. </pre>
    832. 

  832. <p>
    833. 

  833.         In addition the functions <code class="computeroutput"><span class="identifier">real</span></code>,
    834. 

  834.         <code class="computeroutput"><span class="identifier">imag</span></code>, <code class="computeroutput"><span class="identifier">arg</span></code>,
    835. 

  835.         <code class="computeroutput"><span class="identifier">norm</span></code>, <code class="computeroutput"><span class="identifier">conj</span></code>
    836. 

  836.         and <code class="computeroutput"><span class="identifier">proj</span></code> are overloaded for
    837. 

  837.         scalar (ie non-complex) types in the same manner as <code class="computeroutput"><span class="special">&lt;</span><span class="identifier">complex</span><span class="special">&gt;</span></code>
    838. 

  838.         and treat the argument as a value whose imaginary part is zero.
    839. 

  839.       </p>
    840. 

  840. <p>
    841. 

  841.         There are also some functions implemented for compatibility with the Boost.Math
    842. 

  842.         functions of the same name:
    843. 

  843.       </p>
    844. 

  844. <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>
    845. 

  845. <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>
    846. 

  846. <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>
    847. 

  847. <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>
    848. 

  848. <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>
    849. 

  849. <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>
    850. 

  850. </pre>
    851. 

  851. <p>
    852. 

  852.         All these functions are normally implemented by the Backend type. However,
    853. 

  853.         default versions are provided for Backend types that don't have native support
    854. 

  854.         for these functions. Please note however, that this default support requires
    855. 

  855.         the precision of the type to be a compile time constant - this means for
    856. 

  856.         example that the <a href="http://gmplib.org" target="_top">GMP</a> MPF Backend will
    857. 

  857.         not work with these functions when that type is used at variable precision.
    858. 

  858.       </p>
    859. 

  859. <p>
    860. 

  860.         Also note that with the exception of <code class="computeroutput"><span class="identifier">abs</span></code>
    861. 

  861.         that these functions can only be used with floating-point Backend types (if
    862. 

  862.         any other types such as fixed precision or complex types are added to the
    863. 

  863.         library later, then these functions may be extended to support those number
    864. 

  864.         types).
    865. 

  865.       </p>
    866. 

  866. <p>
    867. 

  867.         The precision of these functions is generally determined by the backend implementation.
    868. 

  868.         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>
    869. 

  869.         is determined entirely by <a href="http://www.mpfr.org" target="_top">MPFR</a>.
    870. 

  870.         When these functions use our own implementations, the accuracy of the transcendental
    871. 

  871.         functions is generally a few epsilon. Note however, that the trigonometrical
    872. 

  872.         functions incur the usual accuracy loss when reducing arguments by large
    873. 

  873.         multiples of &#960;. Also note that both <a class="link" href="../tut/floats/gmp_float.html" title="gmp_float">gmp_float</a>
    874. 

  874.         and <a class="link" href="../tut/floats/cpp_dec_float.html" title="cpp_dec_float">cpp_dec_float</a>
    875. 

  875.         have a number of guard digits beyond their stated precision, so the error
    876. 

  876.         rates listed for these are in some sense artificially low.
    877. 

  877.       </p>
    878. 

  878. <p>
    879. 

  879.         The following table shows the error rates we observe for these functions
    880. 

  880.         with various backend types, functions not listed here are exact (tested on
    881. 

  881.         Win32 with VC++10, MPFR-3.0.0, MPIR-2.1.1):
    882. 

  882.       </p>
    883. 

  883. <div class="informaltable"><table class="table">
    884. 

  884. <colgroup>
    885. 

  885. <col>
    886. 

  886. <col>
    887. 

  887. <col>
    888. 

  888. <col>
    889. 

  889. </colgroup>
    890. 

  890. <thead><tr>
    891. 

  891. <th>
    892. 

  892.                 <p>
    893. 

  893.                   Function
    894. 

  894.                 </p>
    895. 

  895.               </th>
    896. 

  896. <th>
    897. 

  897.                 <p>
    898. 

  898.                   mpfr_float_50
    899. 

  899.                 </p>
    900. 

  900.               </th>
    901. 

  901. <th>
    902. 

  902.                 <p>
    903. 

  903.                   mpf_float_50
    904. 

  904.                 </p>
    905. 

  905.               </th>
    906. 

  906. <th>
    907. 

  907.                 <p>
    908. 

  908.                   cpp_dec_float_50
    909. 

  909.                 </p>
    910. 

  910.               </th>
    911. 

  911. </tr></thead>
    912. 

  912. <tbody>
    913. 

  913. <tr>
    914. 

  914. <td>
    915. 

  915.                 <p>
    916. 

  916.                   sqrt
    917. 

  917.                 </p>
    918. 

  918.               </td>
    919. 

  919. <td>
    920. 

  920.                 <p>
    921. 

  921.                   1eps
    922. 

  922.                 </p>
    923. 

  923.               </td>
    924. 

  924. <td>
    925. 

  925.                 <p>
    926. 

  926.                   0eps
    927. 

  927.                 </p>
    928. 

  928.               </td>
    929. 

  929. <td>
    930. 

  930.                 <p>
    931. 

  931.                   0eps
    932. 

  932.                 </p>
    933. 

  933.               </td>
    934. 

  934. </tr>
    935. 

  935. <tr>
    936. 

  936. <td>
    937. 

  937.                 <p>
    938. 

  938.                   exp
    939. 

  939.                 </p>
    940. 

  940.               </td>
    941. 

  941. <td>
    942. 

  942.                 <p>
    943. 

  943.                   1eps
    944. 

  944.                 </p>
    945. 

  945.               </td>
    946. 

  946. <td>
    947. 

  947.                 <p>
    948. 

  948.                   0eps
    949. 

  949.                 </p>
    950. 

  950.               </td>
    951. 

  951. <td>
    952. 

  952.                 <p>
    953. 

  953.                   0eps
    954. 

  954.                 </p>
    955. 

  955.               </td>
    956. 

  956. </tr>
    957. 

  957. <tr>
    958. 

  958. <td>
    959. 

  959.                 <p>
    960. 

  960.                   log
    961. 

  961.                 </p>
    962. 

  962.               </td>
    963. 

  963. <td>
    964. 

  964.                 <p>
    965. 

  965.                   1eps
    966. 

  966.                 </p>
    967. 

  967.               </td>
    968. 

  968. <td>
    969. 

  969.                 <p>
    970. 

  970.                   0eps
    971. 

  971.                 </p>
    972. 

  972.               </td>
    973. 

  973. <td>
    974. 

  974.                 <p>
    975. 

  975.                   0eps
    976. 

  976.                 </p>
    977. 

  977.               </td>
    978. 

  978. </tr>
    979. 

  979. <tr>
    980. 

  980. <td>
    981. 

  981.                 <p>
    982. 

  982.                   log10
    983. 

  983.                 </p>
    984. 

  984.               </td>
    985. 

  985. <td>
    986. 

  986.                 <p>
    987. 

  987.                   1eps
    988. 

  988.                 </p>
    989. 

  989.               </td>
    990. 

  990. <td>
    991. 

  991.                 <p>
    992. 

  992.                   0eps
    993. 

  993.                 </p>
    994. 

  994.               </td>
    995. 

  995. <td>
    996. 

  996.                 <p>
    997. 

  997.                   0eps
    998. 

  998.                 </p>
    999. 

  999.               </td>
    1000. 

  1000. </tr>
    1001. 

  1001. <tr>
    1002. 

  1002. <td>
    1003. 

  1003.                 <p>
    1004. 

  1004.                   cos
    1005. 

  1005.                 </p>
    1006. 

  1006.               </td>
    1007. 

  1007. <td>
    1008. 

  1008.                 <p>
    1009. 

  1009.                   700eps
    1010. 

  1010.                 </p>
    1011. 

  1011.               </td>
    1012. 

  1012. <td>
    1013. 

  1013.                 <p>
    1014. 

  1014.                   0eps
    1015. 

  1015.                 </p>
    1016. 

  1016.               </td>
    1017. 

  1017. <td>
    1018. 

  1018.                 <p>
    1019. 

  1019.                   0eps
    1020. 

  1020.                 </p>
    1021. 

  1021.               </td>
    1022. 

  1022. </tr>
    1023. 

  1023. <tr>
    1024. 

  1024. <td>
    1025. 

  1025.                 <p>
    1026. 

  1026.                   sin
    1027. 

  1027.                 </p>
    1028. 

  1028.               </td>
    1029. 

  1029. <td>
    1030. 

  1030.                 <p>
    1031. 

  1031.                   1eps
    1032. 

  1032.                 </p>
    1033. 

  1033.               </td>
    1034. 

  1034. <td>
    1035. 

  1035.                 <p>
    1036. 

  1036.                   0eps
    1037. 

  1037.                 </p>
    1038. 

  1038.               </td>
    1039. 

  1039. <td>
    1040. 

  1040.                 <p>
    1041. 

  1041.                   0eps
    1042. 

  1042.                 </p>
    1043. 

  1043.               </td>
    1044. 

  1044. </tr>
    1045. 

  1045. <tr>
    1046. 

  1046. <td>
    1047. 

  1047.                 <p>
    1048. 

  1048.                   tan
    1049. 

  1049.                 </p>
    1050. 

  1050.               </td>
    1051. 

  1051. <td>
    1052. 

  1052.                 <p>
    1053. 

  1053.                   0eps
    1054. 

  1054.                 </p>
    1055. 

  1055.               </td>
    1056. 

  1056. <td>
    1057. 

  1057.                 <p>
    1058. 

  1058.                   0eps
    1059. 

  1059.                 </p>
    1060. 

  1060.               </td>
    1061. 

  1061. <td>
    1062. 

  1062.                 <p>
    1063. 

  1063.                   0eps
    1064. 

  1064.                 </p>
    1065. 

  1065.               </td>
    1066. 

  1066. </tr>
    1067. 

  1067. <tr>
    1068. 

  1068. <td>
    1069. 

  1069.                 <p>
    1070. 

  1070.                   acos
    1071. 

  1071.                 </p>
    1072. 

  1072.               </td>
    1073. 

  1073. <td>
    1074. 

  1074.                 <p>
    1075. 

  1075.                   0eps
    1076. 

  1076.                 </p>
    1077. 

  1077.               </td>
    1078. 

  1078. <td>
    1079. 

  1079.                 <p>
    1080. 

  1080.                   0eps
    1081. 

  1081.                 </p>
    1082. 

  1082.               </td>
    1083. 

  1083. <td>
    1084. 

  1084.                 <p>
    1085. 

  1085.                   0eps
    1086. 

  1086.                 </p>
    1087. 

  1087.               </td>
    1088. 

  1088. </tr>
    1089. 

  1089. <tr>
    1090. 

  1090. <td>
    1091. 

  1091.                 <p>
    1092. 

  1092.                   asin
    1093. 

  1093.                 </p>
    1094. 

  1094.               </td>
    1095. 

  1095. <td>
    1096. 

  1096.                 <p>
    1097. 

  1097.                   0eps
    1098. 

  1098.                 </p>
    1099. 

  1099.               </td>
    1100. 

  1100. <td>
    1101. 

  1101.                 <p>
    1102. 

  1102.                   0eps
    1103. 

  1103.                 </p>
    1104. 

  1104.               </td>
    1105. 

  1105. <td>
    1106. 

  1106.                 <p>
    1107. 

  1107.                   0eps
    1108. 

  1108.                 </p>
    1109. 

  1109.               </td>
    1110. 

  1110. </tr>
    1111. 

  1111. <tr>
    1112. 

  1112. <td>
    1113. 

  1113.                 <p>
    1114. 

  1114.                   atan
    1115. 

  1115.                 </p>
    1116. 

  1116.               </td>
    1117. 

  1117. <td>
    1118. 

  1118.                 <p>
    1119. 

  1119.                   1eps
    1120. 

  1120.                 </p>
    1121. 

  1121.               </td>
    1122. 

  1122. <td>
    1123. 

  1123.                 <p>
    1124. 

  1124.                   0eps
    1125. 

  1125.                 </p>
    1126. 

  1126.               </td>
    1127. 

  1127. <td>
    1128. 

  1128.                 <p>
    1129. 

  1129.                   0eps
    1130. 

  1130.                 </p>
    1131. 

  1131.               </td>
    1132. 

  1132. </tr>
    1133. 

  1133. <tr>
    1134. 

  1134. <td>
    1135. 

  1135.                 <p>
    1136. 

  1136.                   cosh
    1137. 

  1137.                 </p>
    1138. 

  1138.               </td>
    1139. 

  1139. <td>
    1140. 

  1140.                 <p>
    1141. 

  1141.                   1045eps<a href="#ftn.boost_multiprecision.ref.number.f0" class="footnote" name="boost_multiprecision.ref.number.f0"><sup class="footnote">[1]</sup></a>
    1142. 

  1142.                 </p>
    1143. 

  1143.               </td>
    1144. 

  1144. <td>
    1145. 

  1145.                 <p>
    1146. 

  1146.                   0eps
    1147. 

  1147.                 </p>
    1148. 

  1148.               </td>
    1149. 

  1149. <td>
    1150. 

  1150.                 <p>
    1151. 

  1151.                   0eps
    1152. 

  1152.                 </p>
    1153. 

  1153.               </td>
    1154. 

  1154. </tr>
    1155. 

  1155. <tr>
    1156. 

  1156. <td>
    1157. 

  1157.                 <p>
    1158. 

  1158.                   sinh
    1159. 

  1159.                 </p>
    1160. 

  1160.               </td>
    1161. 

  1161. <td>
    1162. 

  1162.                 <p>
    1163. 

  1163.                   2eps
    1164. 

  1164.                 </p>
    1165. 

  1165.               </td>
    1166. 

  1166. <td>
    1167. 

  1167.                 <p>
    1168. 

  1168.                   0eps
    1169. 

  1169.                 </p>
    1170. 

  1170.               </td>
    1171. 

  1171. <td>
    1172. 

  1172.                 <p>
    1173. 

  1173.                   0eps
    1174. 

  1174.                 </p>
    1175. 

  1175.               </td>
    1176. 

  1176. </tr>
    1177. 

  1177. <tr>
    1178. 

  1178. <td>
    1179. 

  1179.                 <p>
    1180. 

  1180.                   tanh
    1181. 

  1181.                 </p>
    1182. 

  1182.               </td>
    1183. 

  1183. <td>
    1184. 

  1184.                 <p>
    1185. 

  1185.                   1eps
    1186. 

  1186.                 </p>
    1187. 

  1187.               </td>
    1188. 

  1188. <td>
    1189. 

  1189.                 <p>
    1190. 

  1190.                   0eps
    1191. 

  1191.                 </p>
    1192. 

  1192.               </td>
    1193. 

  1193. <td>
    1194. 

  1194.                 <p>
    1195. 

  1195.                   0eps
    1196. 

  1196.                 </p>
    1197. 

  1197.               </td>
    1198. 

  1198. </tr>
    1199. 

  1199. <tr>
    1200. 

  1200. <td>
    1201. 

  1201.                 <p>
    1202. 

  1202.                   pow
    1203. 

  1203.                 </p>
    1204. 

  1204.               </td>
    1205. 

  1205. <td>
    1206. 

  1206.                 <p>
    1207. 

  1207.                   0eps
    1208. 

  1208.                 </p>
    1209. 

  1209.               </td>
    1210. 

  1210. <td>
    1211. 

  1211.                 <p>
    1212. 

  1212.                   4eps
    1213. 

  1213.                 </p>
    1214. 

  1214.               </td>
    1215. 

  1215. <td>
    1216. 

  1216.                 <p>
    1217. 

  1217.                   3eps
    1218. 

  1218.                 </p>
    1219. 

  1219.               </td>
    1220. 

  1220. </tr>
    1221. 

  1221. <tr>
    1222. 

  1222. <td>
    1223. 

  1223.                 <p>
    1224. 

  1224.                   atan2
    1225. 

  1225.                 </p>
    1226. 

  1226.               </td>
    1227. 

  1227. <td>
    1228. 

  1228.                 <p>
    1229. 

  1229.                   1eps
    1230. 

  1230.                 </p>
    1231. 

  1231.               </td>
    1232. 

  1232. <td>
    1233. 

  1233.                 <p>
    1234. 

  1234.                   0eps
    1235. 

  1235.                 </p>
    1236. 

  1236.               </td>
    1237. 

  1237. <td>
    1238. 

  1238.                 <p>
    1239. 

  1239.                   0eps
    1240. 

  1240.                 </p>
    1241. 

  1241.               </td>
    1242. 

  1242. </tr>
    1243. 

  1243. </tbody>
    1244. 

  1244. <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>
    1245. 

  1245.                     It's likely that the inherent error in the input values to our
    1246. 

  1246.                     test cases are to blame here.
    1247. 

  1247.                   </p></div></td></tr></tbody>
    1248. 

  1248. </table></div>
    1249. 

  1249. <h5>
    1250. 

  1250. <a name="boost_multiprecision.ref.number.h7"></a>
    1251. 

  1251.         <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
    1252. 

  1252.         Support</a>
    1253. 

  1253.       </h5>
    1254. 

  1254. <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>
    1255. 

  1255. <span class="keyword">struct</span> <span class="identifier">component_type</span><span class="special">;</span>
    1256. 

  1256. </pre>
    1257. 

  1257. <p>
    1258. 

  1258.         If this is a type with multiple components (for example rational or complex
    1259. 

  1259.         types), then this trait has a single member <code class="computeroutput"><span class="identifier">type</span></code>
    1260. 

  1260.         that is the type of those components.
    1261. 

  1261.       </p>
    1262. 

  1262. <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>
    1263. 

  1263. <span class="keyword">struct</span> <span class="identifier">number_category</span><span class="special">;</span>
    1264. 

  1264. </pre>
    1265. 

  1265. <p>
    1266. 

  1266.         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>
    1267. 

  1267.         where <code class="computeroutput"><span class="identifier">N</span></code> is one of the enumerated
    1268. 

  1268.         values <code class="computeroutput"><span class="identifier">number_kind_integer</span></code>,
    1269. 

  1269.         <code class="computeroutput"><span class="identifier">number_kind_floating_point</span></code>,
    1270. 

  1270.         <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
    1271. 

  1271.         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>
    1272. 

  1272.         support as well as for classes in this library: which means it can be used
    1273. 

  1273.         for generic code that must work with built in arithmetic types as well as
    1274. 

  1274.         multiprecision ones.
    1275. 

  1275.       </p>
    1276. 

  1276. <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>
    1277. 

  1277. <span class="keyword">struct</span> <span class="identifier">is_number</span><span class="special">;</span>
    1278. 

  1278. </pre>
    1279. 

  1279. <p>
    1280. 

  1280.         A traits class that inherits from <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">true_</span></code>
    1281. 

  1281.         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>.
    1282. 

  1282.       </p>
    1283. 

  1283. <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>
    1284. 

  1284. <span class="keyword">struct</span> <span class="identifier">is_number_expression</span><span class="special">;</span>
    1285. 

  1285. </pre>
    1286. 

  1286. <p>
    1287. 

  1287.         A traits class that inherits from <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">true_</span></code>
    1288. 

  1288.         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>.
    1289. 

  1289.       </p>
    1290. 

  1290. <h5>
    1291. 

  1291. <a name="boost_multiprecision.ref.number.h8"></a>
    1292. 

  1292.         <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
    1293. 

  1293.         functions</a>
    1294. 

  1294.       </h5>
    1295. 

  1295. <p>
    1296. 

  1296.         In addition to functioning with types from this library, these functions
    1297. 

  1297.         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>.
    1298. 

  1298.         Further, when used with fixed precision types (whether built in integers
    1299. 

  1299.         or multiprecision ones), the functions will promote to a wider type internally
    1300. 

  1300.         when the algorithm requires it. Versions overloaded for built in integer
    1301. 

  1301.         types return that integer type rather than an expression template.
    1302. 

  1302.       </p>
    1303. 

  1303. <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>
    1304. 

  1304. </pre>
    1305. 

  1305. <p>
    1306. 

  1306.         Returns the largest integer <code class="computeroutput"><span class="identifier">x</span></code>
    1307. 

  1307.         that divides both <code class="computeroutput"><span class="identifier">a</span></code> and
    1308. 

  1308.         <code class="computeroutput"><span class="identifier">b</span></code>.
    1309. 

  1309.       </p>
    1310. 

  1310. <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>
    1311. 

  1311. </pre>
    1312. 

  1312. <p>
    1313. 

  1313.         Returns the smallest integer <code class="computeroutput"><span class="identifier">x</span></code>
    1314. 

  1314.         that is divisible by both <code class="computeroutput"><span class="identifier">a</span></code>
    1315. 

  1315.         and <code class="computeroutput"><span class="identifier">b</span></code>.
    1316. 

  1316.       </p>
    1317. 

  1317. <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>
    1318. 

  1318. </pre>
    1319. 

  1319. <p>
    1320. 

  1320.         Returns <span class="emphasis"><em>b<sup>p</sup></em></span> as an expression template. Note that this
    1321. 

  1321.         function should be used with extreme care as the result can grow so large
    1322. 

  1322.         as to take "effectively forever" to compute, or else simply run
    1323. 

  1323.         the host machine out of memory. This is the one function in this category
    1324. 

  1324.         that is not overloaded for built in integer types, further, it's probably
    1325. 

  1325.         not a good idea to use it with fixed precision <code class="computeroutput"><span class="identifier">cpp_int</span></code>'s
    1326. 

  1326.         either.
    1327. 

  1327.       </p>
    1328. 

  1328. <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>
    1329. 

  1329. </pre>
    1330. 

  1330. <p>
    1331. 

  1331.         Returns <span class="emphasis"><em>b<sup>p</sup> mod m</em></span> as an expression template. Fixed precision
    1332. 

  1332.         types are promoted internally to ensure accuracy.
    1333. 

  1333.       </p>
    1334. 

  1334. <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>
    1335. 

  1335. </pre>
    1336. 

  1336. <p>
    1337. 

  1337.         Returns the largest integer <code class="computeroutput"><span class="identifier">x</span></code>
    1338. 

  1338.         such that <code class="computeroutput"><span class="identifier">x</span> <span class="special">*</span>
    1339. 

  1339.         <span class="identifier">x</span> <span class="special">&lt;</span>
    1340. 

  1340.         <span class="identifier">a</span></code>.
    1341. 

  1341.       </p>
    1342. 

  1342. <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>
    1343. 

  1343. <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>
    1344. 

  1344. </pre>
    1345. 

  1345. <p>
    1346. 

  1346.         Returns the largest integer <code class="computeroutput"><span class="identifier">x</span></code>
    1347. 

  1347.         such that <code class="computeroutput"><span class="identifier">x</span> <span class="special">*</span>
    1348. 

  1348.         <span class="identifier">x</span> <span class="special">&lt;</span>
    1349. 

  1349.         <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>
    1350. 

  1350.         <span class="special">=</span> <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">x</span> <span class="special">*</span>
    1351. 

  1351.         <span class="identifier">x</span></code>.
    1352. 

  1352.       </p>
    1353. 

  1353. <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>
    1354. 

  1354. <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>
    1355. 

  1355.                <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>
    1356. 

  1356. </pre>
    1357. 

  1357. <p>
    1358. 

  1358.         Divides x by y and returns both the quotient and remainder. After the call
    1359. 

  1359.         <code class="computeroutput"><span class="identifier">q</span> <span class="special">=</span>
    1360. 

  1360.         <span class="identifier">x</span> <span class="special">/</span> <span class="identifier">y</span></code> and <code class="computeroutput"><span class="identifier">r</span>
    1361. 

  1361.         <span class="special">=</span> <span class="identifier">x</span> <span class="special">%</span> <span class="identifier">y</span></code>.
    1362. 

  1362.       </p>
    1363. 

  1363. <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>
    1364. 

  1364. <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>
    1365. 

  1365. </pre>
    1366. 

  1366. <p>
    1367. 

  1367.         Returns the absolute value of <code class="computeroutput"><span class="identifier">x</span>
    1368. 

  1368.         <span class="special">%</span> <span class="identifier">val</span></code>.
    1369. 

  1369.       </p>
    1370. 

  1370. <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>
    1371. 

  1371. </pre>
    1372. 

  1372. <p>
    1373. 

  1373.         Returns the (zero-based) index of the least significant bit that is set to
    1374. 

  1374.         1.
    1375. 

  1375.       </p>
    1376. 

  1376. <p>
    1377. 

  1377.         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.
    1378. 

  1378.       </p>
    1379. 

  1379. <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>
    1380. 

  1380. </pre>
    1381. 

  1381. <p>
    1382. 

  1382.         Returns the (zero-based) index of the most significant bit.
    1383. 

  1383.       </p>
    1384. 

  1384. <p>
    1385. 

  1385.         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.
    1386. 

  1386.       </p>
    1387. 

  1387. <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>
    1388. 

  1388. <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>
    1389. 

  1389. </pre>
    1390. 

  1390. <p>
    1391. 

  1391.         Returns <code class="computeroutput"><span class="keyword">true</span></code> if the bit at
    1392. 

  1392.         <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span> is set.
    1393. 

  1393.       </p>
    1394. 

  1394. <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>
    1395. 

  1395. <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>
    1396. 

  1396. </pre>
    1397. 

  1397. <p>
    1398. 

  1398.         Sets the bit at <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span>, and
    1399. 

  1399.         returns <span class="emphasis"><em>val</em></span>.
    1400. 

  1400.       </p>
    1401. 

  1401. <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>
    1402. 

  1402. <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>
    1403. 

  1403. </pre>
    1404. 

  1404. <p>
    1405. 

  1405.         Unsets the bit at <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span>,
    1406. 

  1406.         and returns <span class="emphasis"><em>val</em></span>.
    1407. 

  1407.       </p>
    1408. 

  1408. <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>
    1409. 

  1409. <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>
    1410. 

  1410. </pre>
    1411. 

  1411. <p>
    1412. 

  1412.         Flips the bit at <span class="emphasis"><em>index</em></span> in <span class="emphasis"><em>val</em></span>,
    1413. 

  1413.         and returns <span class="emphasis"><em>val</em></span>.
    1414. 

  1414.       </p>
    1415. 

  1415. <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>
    1416. 

  1416. <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>
    1417. 

  1417. <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>
    1418. 

  1418. </pre>
    1419. 

  1419. <p>
    1420. 

  1420.         Tests to see if the number <span class="emphasis"><em>n</em></span> is probably prime - the
    1421. 

  1421.         test excludes the vast majority of composite numbers by excluding small prime
    1422. 

  1422.         factors and performing a single Fermat test. Then performs <span class="emphasis"><em>trials</em></span>
    1423. 

  1423.         Miller-Rabin tests. Returns <code class="computeroutput"><span class="keyword">false</span></code>
    1424. 

  1424.         if <span class="emphasis"><em>n</em></span> is definitely composite, or <code class="computeroutput"><span class="keyword">true</span></code>
    1425. 

  1425.         if <span class="emphasis"><em>n</em></span> is probably prime with the probability of it being
    1426. 

  1426.         composite less than 0.25^trials. Fixed precision types are promoted internally
    1427. 

  1427.         to ensure accuracy.
    1428. 

  1428.       </p>
    1429. 

  1429. <h5>
    1430. 

  1430. <a name="boost_multiprecision.ref.number.h9"></a>
    1431. 

  1431.         <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
    1432. 

  1432.         Number Functions</a>
    1433. 

  1433.       </h5>
    1434. 

  1434. <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>
    1435. 

  1435. <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>
    1436. 

  1436. </pre>
    1437. 

  1437. <p>
    1438. 

  1438.         These functions return the numerator and denominator of a rational number
    1439. 

  1439.         respectively.
    1440. 

  1440.       </p>
    1441. 

  1441. <h5>
    1442. 

  1442. <a name="boost_multiprecision.ref.number.h10"></a>
    1443. 

  1443.         <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
    1444. 

  1444.         Interoperability Support</a>
    1445. 

  1445.       </h5>
    1446. 

  1446. <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>
    1447. 

  1447. 

  1448. <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>
    1449. 

  1449. <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>
    1450. 

  1450. <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>
    1451. 

  1451. <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>
    1452. 

  1452. <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>
    1453. 

  1453. 

  1454. <span class="special">}}</span> <span class="comment">// namespaces</span>
    1455. 

  1455. </pre>
    1456. 

  1456. <p>
    1457. 

  1457.         These floating-point classification functions behave exactly as their Boost.Math
    1458. 

  1458.         equivalents.
    1459. 

  1459.       </p>
    1460. 

  1460. <p>
    1461. 

  1461.         Other Boost.Math functions and templates may also be specialized or overloaded
    1462. 

  1462.         to ensure interoperability.
    1463. 

  1463.       </p>
    1464. 

  1464. <h5>
    1465. 

  1465. <a name="boost_multiprecision.ref.number.h11"></a>
    1466. 

  1466.         <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
    1467. 

  1467.         support</a>
    1468. 

  1468.       </h5>
    1469. 

  1469. <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">{</span>
    1470. 

  1470. 

  1471. <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>
    1472. 

  1472. <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>
    1473. 

  1473. <span class="special">{</span>
    1474. 

  1474.    <span class="comment">/* Usual members here */</span>
    1475. 

  1475. <span class="special">};</span>
    1476. 

  1476. 

  1477. <span class="special">}</span>
    1478. 

  1478. </pre>
    1479. 

  1479. <p>
    1480. 

  1480.         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
    1481. 

  1481.         of <code class="computeroutput"><span class="identifier">number</span></code> whose precision
    1482. 

  1482.         is known at compile time, plus those types whose precision is unlimited (though
    1483. 

  1483.         it is much less useful in those cases). It is not specialized for types whose
    1484. 

  1484.         precision can vary at compile time (such as <code class="computeroutput"><span class="identifier">mpf_float</span></code>).
    1485. 

  1485.       </p>
    1486. 

  1486. </div>
    1487. 

  1487. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    1488. 

  1488. <td align="left"></td>
    1489. 

  1489. <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2019 John Maddock
    1490. 

  1490.       and Christopher Kormanyos<p>
    1491. 

  1491.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    1492. 

  1492.         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>)
    1493. 

  1493.       </p>
    1494. 

  1494. </div></td>
    1495. 

  1495. </tr></table>
    1496. 

  1496. <hr>
    1497. 

  1497. <div class="spirit-nav">
    1498. 

  1498. <a accesskey="p" href="../ref.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ref.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cpp_int_ref.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
    1499. 

  1499. </div>
    1500. 

  1500. </body>
    1501. 

  1501. </html>
    1502.