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    24. 

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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.tut.lits"></a><a class="link" href="lits.html" title="Literal Types and constexpr Support">Literal Types and <code class="computeroutput"><span class="keyword">constexpr</span></code> Support</a>
    28. 

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

  29. <p>
    30. 

  30.         There are two kinds of <code class="computeroutput"><span class="keyword">constexpr</span></code>
    31. 

  31.         support in this library:
    32. 

  32.       </p>
    33. 

  33. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    34. 

  34. <li class="listitem">
    35. 

  35.             The more basic version requires only C++11 and allow the construction
    36. 

  36.             of some number types as literals.
    37. 

  37.           </li>
    38. 

  38. <li class="listitem">
    39. 

  39.             The more advanced support permits constexpr arithmetic and requires at
    40. 

  40.             least C++14 constexpr support, and for many operations C++2a support
    41. 

  41.           </li>
    42. 

  42. </ul></div>
    43. 

  43. <h5>
    44. 

  44. <a name="boost_multiprecision.tut.lits.h0"></a>
    45. 

  45.         <span class="phrase"><a name="boost_multiprecision.tut.lits.declaring_numeric_literals"></a></span><a class="link" href="lits.html#boost_multiprecision.tut.lits.declaring_numeric_literals">Declaring
    46. 

  46.         numeric literals</a>
    47. 

  47.       </h5>
    48. 

  48. <p>
    49. 

  49.         There are two backend types which are literals:
    50. 

  50.       </p>
    51. 

  51. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    52. 

  52. <li class="listitem">
    53. 

  53.             <a class="link" href="floats/float128.html" title="float128">float128</a>
    54. 

  54.             (which requires GCC), and
    55. 

  55.           </li>
    56. 

  56. <li class="listitem">
    57. 

  57.             Instantiations of <code class="computeroutput"><span class="identifier">cpp_int_backend</span></code>
    58. 

  58.             where the Allocator parameter is type <code class="computeroutput"><span class="keyword">void</span></code>.
    59. 

  59.             In addition, prior to C++14 the Checked parameter must be <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">unchecked</span></code>.
    60. 

  60.           </li>
    61. 

  61. </ul></div>
    62. 

  62. <p>
    63. 

  63.         For example:
    64. 

  64.       </p>
    65. 

  65. <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span>
    66. 

  66. 

  67. <span class="keyword">constexpr</span> <span class="identifier">float128</span>            <span class="identifier">f</span> <span class="special">=</span> <span class="number">0.1</span><span class="identifier">Q</span>   <span class="comment">// OK, float128's are always literals in C++11</span>
    68. 

  68. 

  69. <span class="keyword">constexpr</span> <span class="identifier">int128_t</span>            <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>     <span class="comment">// OK, fixed precision int128_t has no allocator.</span>
    70. 

  70. <span class="keyword">constexpr</span> <span class="identifier">uint1024_t</span>          <span class="identifier">j</span> <span class="special">=</span> <span class="number">0</span><span class="identifier">xFFFFFFFF00000000uLL</span><span class="special">;</span>  <span class="comment">// OK, fixed precision uint1024_t has no allocator.</span>
    71. 

  71. 

  72. <span class="keyword">constexpr</span> <span class="identifier">checked_uint128_t</span>   <span class="identifier">k</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="comment">// OK from C++14 and later, not supported for C++11.</span>
    73. 

  73. <span class="keyword">constexpr</span> <span class="identifier">checked_uint128_t</span>   <span class="identifier">k</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">;</span> <span class="comment">// Error, as this would normally lead to a runtime failure (exception).</span>
    74. 

  74. <span class="keyword">constexpr</span> <span class="identifier">cpp_int</span>             <span class="identifier">l</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>  <span class="comment">// Error, type is not a literal as it performs memory management.</span>
    75. 

  75. </pre>
    76. 

  76. <p>
    77. 

  77.         There is also support for user defined-literals with <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
    78. 

  78.         - these are limited to unchecked, fixed precision <code class="computeroutput"><span class="identifier">cpp_int</span></code>'s
    79. 

  79.         which are specified in hexadecimal notation. The suffixes supported are:
    80. 

  80.       </p>
    81. 

  81. <div class="informaltable"><table class="table">
    82. 

  82. <colgroup>
    83. 

  83. <col>
    84. 

  84. <col>
    85. 

  85. </colgroup>
    86. 

  86. <thead><tr>
    87. 

  87. <th>
    88. 

  88.                 <p>
    89. 

  89.                   Suffix
    90. 

  90.                 </p>
    91. 

  91.               </th>
    92. 

  92. <th>
    93. 

  93.                 <p>
    94. 

  94.                   Meaning
    95. 

  95.                 </p>
    96. 

  96.               </th>
    97. 

  97. </tr></thead>
    98. 

  98. <tbody>
    99. 

  99. <tr>
    100. 

  100. <td>
    101. 

  101.                 <p>
    102. 

  102.                   _cppi
    103. 

  103.                 </p>
    104. 

  104.               </td>
    105. 

  105. <td>
    106. 

  106.                 <p>
    107. 

  107.                   Specifies a value of type: <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">cpp_int_backend</span><span class="special">&lt;</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">signed_magnitude</span><span class="special">,</span><span class="identifier">unchecked</span><span class="special">,</span><span class="keyword">void</span><span class="special">&gt;</span> <span class="special">&gt;</span></code>,
    108. 

  108.                   where N is chosen to contain just enough digits to hold the number
    109. 

  109.                   specified.
    110. 

  110.                 </p>
    111. 

  111.               </td>
    112. 

  112. </tr>
    113. 

  113. <tr>
    114. 

  114. <td>
    115. 

  115.                 <p>
    116. 

  116.                   _cppui
    117. 

  117.                 </p>
    118. 

  118.               </td>
    119. 

  119. <td>
    120. 

  120.                 <p>
    121. 

  121.                   Specifies a value of type: <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">cpp_int_backend</span><span class="special">&lt;</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">unsigned_magnitude</span><span class="special">,</span><span class="identifier">unchecked</span><span class="special">,</span><span class="keyword">void</span><span class="special">&gt;</span> <span class="special">&gt;</span></code>,
    122. 

  122.                   where N is chosen to contain just enough digits to hold the number
    123. 

  123.                   specified.
    124. 

  124.                 </p>
    125. 

  125.               </td>
    126. 

  126. </tr>
    127. 

  127. <tr>
    128. 

  128. <td>
    129. 

  129.                 <p>
    130. 

  130.                   _cppi<span class="emphasis"><em>N</em></span>
    131. 

  131.                 </p>
    132. 

  132.               </td>
    133. 

  133. <td>
    134. 

  134.                 <p>
    135. 

  135.                   Specifies a value of type <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">cpp_int_backend</span><span class="special">&lt;</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">signed_magnitude</span><span class="special">,</span><span class="identifier">unchecked</span><span class="special">,</span><span class="keyword">void</span><span class="special">&gt;</span> <span class="special">&gt;</span></code>.
    136. 

  136.                 </p>
    137. 

  137.               </td>
    138. 

  138. </tr>
    139. 

  139. <tr>
    140. 

  140. <td>
    141. 

  141.                 <p>
    142. 

  142.                   _cppui<span class="emphasis"><em>N</em></span>
    143. 

  143.                 </p>
    144. 

  144.               </td>
    145. 

  145. <td>
    146. 

  146.                 <p>
    147. 

  147.                   Specifies a value of type <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">cpp_int_backend</span><span class="special">&lt;</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">signed_magnitude</span><span class="special">,</span><span class="identifier">unchecked</span><span class="special">,</span><span class="keyword">void</span><span class="special">&gt;</span> <span class="special">&gt;</span></code>.
    148. 

  148.                 </p>
    149. 

  149.               </td>
    150. 

  150. </tr>
    151. 

  151. </tbody>
    152. 

  152. </table></div>
    153. 

  153. <p>
    154. 

  154.         In each case, use of these suffixes with hexadecimal values produces a <code class="computeroutput"><span class="keyword">constexpr</span></code> result.
    155. 

  155.       </p>
    156. 

  156. <p>
    157. 

  157.         Examples:
    158. 

  158.       </p>
    159. 

  159. <pre class="programlisting"><span class="comment">//</span>
    160. 

  160. <span class="comment">// Any use of user defined literals requires that we import the literal-operators</span>
    161. 

  161. <span class="comment">// into current scope first:</span>
    162. 

  162. <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">literals</span><span class="special">;</span>
    163. 

  163. <span class="comment">//</span>
    164. 

  164. <span class="comment">// To keep things simple in the example, we'll make our types used visible to this scope as well:</span>
    165. 

  165. <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span>
    166. 

  166. <span class="comment">//</span>
    167. 

  167. <span class="comment">// The value zero as a number&lt;cpp_int_backend&lt;4,4,signed_magnitude,unchecked,void&gt; &gt;:</span>
    168. 

  168. <span class="keyword">constexpr</span> <span class="keyword">auto</span> <span class="identifier">a</span> <span class="special">=</span> <span class="number">0x0</span><span class="identifier">_cppi</span><span class="special">;</span>
    169. 

  169. <span class="comment">// The type of each constant has 4 bits per hexadecimal digit,</span>
    170. 

  170. <span class="comment">// so this is of type uint256_t (ie number&lt;cpp_int_backend&lt;256,256,unsigned_magnitude,unchecked,void&gt; &gt;):</span>
    171. 

  171. <span class="keyword">constexpr</span> <span class="keyword">auto</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">0</span><span class="identifier">xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF_cppui</span><span class="special">;</span>
    172. 

  172. <span class="comment">//</span>
    173. 

  173. <span class="comment">// Smaller values can be assigned to larger values:</span>
    174. 

  174. <span class="identifier">int256_t</span> <span class="identifier">c</span> <span class="special">=</span> <span class="number">0x1234</span><span class="identifier">_cppi</span><span class="special">;</span> <span class="comment">// OK</span>
    175. 

  175. <span class="comment">//</span>
    176. 

  176. <span class="comment">// However, this only works in constexpr contexts from C++14 onwards:</span>
    177. 

  177. <span class="keyword">constexpr</span> <span class="identifier">int256_t</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">0x1</span><span class="identifier">_cppi</span><span class="special">;</span> <span class="comment">// Compiler error in C++11, requires C++14</span>
    178. 

  178. <span class="comment">//</span>
    179. 

  179. <span class="comment">// Constants can be padded out with leading zeros to generate wider types:</span>
    180. 

  180. <span class="keyword">constexpr</span> <span class="identifier">uint256_t</span> <span class="identifier">e</span> <span class="special">=</span> <span class="number">0</span><span class="identifier">x0000000000000000000000000000000000000000000FFFFFFFFFFFFFFFFFFFFF_cppui</span><span class="special">;</span> <span class="comment">// OK</span>
    181. 

  181. <span class="comment">//</span>
    182. 

  182. <span class="comment">// However, specific width types are best produced with specific-width suffixes,</span>
    183. 

  183. <span class="comment">// ones supported by default are `_cpp[u]i128`, `_cpp[u]i256`, `_cpp[u]i512`, `_cpp[u]i1024`.</span>
    184. 

  184. <span class="comment">//</span>
    185. 

  185. <span class="keyword">constexpr</span> <span class="identifier">int128_t</span> <span class="identifier">f</span> <span class="special">=</span> <span class="number">0x1234</span><span class="identifier">_cppi128</span><span class="special">;</span> <span class="comment">// OK, always produces an int128_t as the result.</span>
    186. 

  186. <span class="keyword">constexpr</span> <span class="identifier">uint1024_t</span> <span class="identifier">g</span> <span class="special">=</span> <span class="number">0</span><span class="identifier">xaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbbccccccccccccccccccccc_cppui1024</span><span class="special">;</span>
    187. 

  187. <span class="comment">//</span>
    188. 

  188. <span class="comment">// If other specific width types are required, then there is a macro for generating the operators</span>
    189. 

  189. <span class="comment">// for these.  The macro can be used at namespace scope only:</span>
    190. 

  190. <span class="comment">//</span>
    191. 

  191. <span class="identifier">BOOST_MP_DEFINE_SIZED_CPP_INT_LITERAL</span><span class="special">(</span><span class="number">2048</span><span class="special">);</span>
    192. 

  192. <span class="comment">//</span>
    193. 

  193. <span class="comment">// Now we can create 2048-bit literals as well:</span>
    194. 

  194. <span class="keyword">constexpr</span> <span class="keyword">auto</span> <span class="identifier">h</span> <span class="special">=</span> <span class="number">0xff</span><span class="identifier">_cppi2048</span><span class="special">;</span> <span class="comment">// h is of type number&lt;cpp_int_backend&lt;2048,2048,signed_magnitude,unchecked,void&gt; &gt;</span>
    195. 

  195. <span class="comment">//</span>
    196. 

  196. <span class="comment">// Finally negative values are handled via the unary minus operator:</span>
    197. 

  197. <span class="comment">//</span>
    198. 

  198. <span class="keyword">constexpr</span> <span class="identifier">int1024_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="special">-</span><span class="number">0</span><span class="identifier">xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF_cppui1024</span><span class="special">;</span>
    199. 

  199. <span class="comment">//</span>
    200. 

  200. <span class="comment">// Which means this also works:</span>
    201. 

  201. <span class="keyword">constexpr</span> <span class="identifier">int1024_t</span> <span class="identifier">j</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">g</span><span class="special">;</span>   <span class="comment">// OK: unary minus operator is constexpr.</span>
    202. 

  202. </pre>
    203. 

  203. <h5>
    204. 

  204. <a name="boost_multiprecision.tut.lits.h1"></a>
    205. 

  205.         <span class="phrase"><a name="boost_multiprecision.tut.lits.constexpr_arithmetic"></a></span><a class="link" href="lits.html#boost_multiprecision.tut.lits.constexpr_arithmetic">constexpr
    206. 

  206.         arithmetic</a>
    207. 

  207.       </h5>
    208. 

  208. <p>
    209. 

  209.         The front end of the library is all <code class="computeroutput"><span class="keyword">constexpr</span></code>
    210. 

  210.         from C++14 and later. Currently there are only two back end types that are
    211. 

  211.         <code class="computeroutput"><span class="keyword">constexpr</span></code> aware: <a class="link" href="floats/float128.html" title="float128">float128</a>
    212. 

  212.         and <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>.
    213. 

  213.         More backends will follow at a later date.
    214. 

  214.       </p>
    215. 

  215. <p>
    216. 

  216.         Provided the compiler is GCC, type <a class="link" href="floats/float128.html" title="float128">float128</a>
    217. 

  217.         support <code class="computeroutput"><span class="keyword">constexpr</span></code> operations
    218. 

  218.         on all arithmetic operations from C++14, comparisons, <code class="computeroutput"><span class="identifier">abs</span></code>,
    219. 

  219.         <code class="computeroutput"><span class="identifier">fabs</span></code>, <code class="computeroutput"><span class="identifier">fpclassify</span></code>,
    220. 

  220.         <code class="computeroutput"><span class="identifier">isnan</span></code>, <code class="computeroutput"><span class="identifier">isinf</span></code>,
    221. 

  221.         <code class="computeroutput"><span class="identifier">isfinite</span></code> and <code class="computeroutput"><span class="identifier">isnormal</span></code> are also fully supported, but
    222. 

  222.         the transcendental functions are not.
    223. 

  223.       </p>
    224. 

  224. <p>
    225. 

  225.         The <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
    226. 

  226.         types support constexpr arithmetic, provided it is a fixed precision type
    227. 

  227.         with no allocator. It may also be a checked integer: in which case a compiler
    228. 

  228.         error will be generated on overflow or undefined behaviour. In addition the
    229. 

  229.         free functions <code class="computeroutput"><span class="identifier">abs</span></code>, <code class="computeroutput"><span class="identifier">swap</span></code>, <code class="computeroutput"><span class="identifier">multiply</span></code>,
    230. 

  230.         <code class="computeroutput"><span class="identifier">add</span></code>, <code class="computeroutput"><span class="identifier">subtract</span></code>,
    231. 

  231.         <code class="computeroutput"><span class="identifier">divide_qr</span></code>, <code class="computeroutput"><span class="identifier">integer_modulus</span></code>, <code class="computeroutput"><span class="identifier">powm</span></code>,
    232. 

  232.         <code class="computeroutput"><span class="identifier">lsb</span></code>, <code class="computeroutput"><span class="identifier">msb</span></code>,
    233. 

  233.         <code class="computeroutput"><span class="identifier">bit_test</span></code>, <code class="computeroutput"><span class="identifier">bit_set</span></code>,
    234. 

  234.         <code class="computeroutput"><span class="identifier">bit_unset</span></code>, <code class="computeroutput"><span class="identifier">bit_flip</span></code>, <code class="computeroutput"><span class="identifier">sqrt</span></code>,
    235. 

  235.         <code class="computeroutput"><span class="identifier">gcd</span></code>, <code class="computeroutput"><span class="identifier">lcm</span></code>
    236. 

  236.         are all supported. Use of <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
    237. 

  237.         in this way requires either a C++2a compiler (one which supports <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">is_constant_evaluated</span><span class="special">()</span></code> - currently only gcc-9 or clang-9 or later),
    238. 

  238.         or GCC-6 or later in C++14 mode. Compilers other than GCC and without <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">is_constant_evaluated</span><span class="special">()</span></code> will support a very limited set of operations:
    239. 

  239.         expect to hit roadblocks rather easily.
    240. 

  240.       </p>
    241. 

  241. <p>
    242. 

  242.         For example given:
    243. 

  243.       </p>
    244. 

  244. <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>
    245. 

  245. <span class="keyword">inline</span> <span class="keyword">constexpr</span> <span class="identifier">T</span> <span class="identifier">circumference</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">radius</span><span class="special">)</span>
    246. 

  246. <span class="special">{</span>
    247. 

  247.    <span class="keyword">return</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;()</span> <span class="special">*</span> <span class="identifier">radius</span><span class="special">;</span>
    248. 

  248. <span class="special">}</span>
    249. 

  249. 

  250. <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>
    251. 

  251. <span class="keyword">inline</span> <span class="keyword">constexpr</span> <span class="identifier">T</span> <span class="identifier">area</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">radius</span><span class="special">)</span>
    252. 

  252. <span class="special">{</span>
    253. 

  253.    <span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;()</span> <span class="special">*</span> <span class="identifier">radius</span> <span class="special">*</span> <span class="identifier">radius</span><span class="special">;</span>
    254. 

  254. <span class="special">}</span>
    255. 

  255. </pre>
    256. 

  256. <p>
    257. 

  257.         We can now calculate areas and circumferences using all constexpr arithmetic:
    258. 

  258.       </p>
    259. 

  259. <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">float128</span><span class="special">;</span>
    260. 

  260. 

  261. <span class="keyword">constexpr</span> <span class="identifier">float128</span> <span class="identifier">radius</span> <span class="special">=</span> <span class="number">2.25</span><span class="special">;</span>
    262. 

  262. <span class="keyword">constexpr</span> <span class="identifier">float128</span> <span class="identifier">c</span>      <span class="special">=</span> <span class="identifier">circumference</span><span class="special">(</span><span class="identifier">radius</span><span class="special">);</span>
    263. 

  263. <span class="keyword">constexpr</span> <span class="identifier">float128</span> <span class="identifier">a</span>      <span class="special">=</span> <span class="identifier">area</span><span class="special">(</span><span class="identifier">radius</span><span class="special">);</span>
    264. 

  264. 

  265. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Circumference = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">c</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    266. 

  266. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Area = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">a</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
    267. 

  267. </pre>
    268. 

  268. <p>
    269. 

  269.         Note that these make use of the numeric constants from the Math library,
    270. 

  270.         which also happen to be <code class="computeroutput"><span class="keyword">constexpr</span></code>.
    271. 

  271.       </p>
    272. 

  272. <p>
    273. 

  273.         For a more interesting example, in <a href="../../../../example/constexpr_float_arithmetic_examples.cpp" target="_top">constexpr_float_arithmetic_examples.cpp</a>
    274. 

  274.         we define a simple class for <code class="computeroutput"><span class="keyword">constexpr</span></code>
    275. 

  275.         polynomial arithmetic:
    276. 

  276.       </p>
    277. 

  277. <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">,</span> <span class="keyword">unsigned</span> <span class="identifier">Order</span><span class="special">&gt;</span>
    278. 

  278. <span class="keyword">struct</span> <span class="identifier">const_polynomial</span><span class="special">;</span>
    279. 

  279. </pre>
    280. 

  280. <p>
    281. 

  281.         Given this, we can use recurrence relations to calculate the coefficients
    282. 

  282.         for various orthogonal polynomials - in the example we use the Hermite polynomials,
    283. 

  283.         only the constructor does any work - it uses the recurrence relations to
    284. 

  284.         calculate the coefficient array:
    285. 

  285.       </p>
    286. 

  286. <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">,</span> <span class="keyword">unsigned</span> <span class="identifier">Order</span><span class="special">&gt;</span>
    287. 

  287. <span class="keyword">class</span> <span class="identifier">hermite_polynomial</span>
    288. 

  288. <span class="special">{</span>
    289. 

  289.    <span class="identifier">const_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="identifier">m_data</span><span class="special">;</span>
    290. 

  290. 

  291.  <span class="keyword">public</span><span class="special">:</span>
    292. 

  292.    <span class="keyword">constexpr</span> <span class="identifier">hermite_polynomial</span><span class="special">()</span> <span class="special">:</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">hermite_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">Order</span> <span class="special">-</span> <span class="number">1</span><span class="special">&gt;().</span><span class="identifier">data</span><span class="special">()</span> <span class="special">*</span> <span class="identifier">const_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</span><span class="special">&gt;{</span><span class="number">0</span><span class="special">,</span> <span class="number">2</span><span class="special">}</span> <span class="special">-</span> <span class="identifier">hermite_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">Order</span> <span class="special">-</span> <span class="number">1</span><span class="special">&gt;().</span><span class="identifier">data</span><span class="special">().</span><span class="identifier">derivative</span><span class="special">())</span>
    293. 

  293.    <span class="special">{</span>
    294. 

  294.    <span class="special">}</span>
    295. 

  295.    <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">const_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;&amp;</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span>
    296. 

  296.    <span class="special">{</span>
    297. 

  297.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
    298. 

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

  299.    <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">)</span><span class="keyword">const</span>
    300. 

  300.    <span class="special">{</span>
    301. 

  301.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span>
    302. 

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

  303.    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
    304. 

  304.    <span class="keyword">constexpr</span> <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">U</span> <span class="identifier">val</span><span class="special">)</span><span class="keyword">const</span>
    305. 

  305.    <span class="special">{</span>
    306. 

  306.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span>
    307. 

  307.    <span class="special">}</span>
    308. 

  308. <span class="special">};</span>
    309. 

  309. </pre>
    310. 

  310. <p>
    311. 

  311.         Now we just need to define H<sub>0</sub> and H<sub>1</sub> as termination conditions for the recurrence:
    312. 

  312.       </p>
    313. 

  313. <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>
    314. 

  314. <span class="keyword">class</span> <span class="identifier">hermite_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="number">0</span><span class="special">&gt;</span>
    315. 

  315. <span class="special">{</span>
    316. 

  316.    <span class="identifier">const_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="number">0</span><span class="special">&gt;</span> <span class="identifier">m_data</span><span class="special">;</span>
    317. 

  317. 

  318.  <span class="keyword">public</span><span class="special">:</span>
    319. 

  319.    <span class="keyword">constexpr</span> <span class="identifier">hermite_polynomial</span><span class="special">()</span> <span class="special">:</span> <span class="identifier">m_data</span><span class="special">{</span><span class="number">1</span><span class="special">}</span> <span class="special">{}</span>
    320. 

  320.    <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">const_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="number">0</span><span class="special">&gt;&amp;</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span>
    321. 

  321.    <span class="special">{</span>
    322. 

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

  323.    <span class="special">}</span>
    324. 

  324.    <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">)</span> <span class="keyword">const</span>
    325. 

  325.    <span class="special">{</span>
    326. 

  326.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span>
    327. 

  327.    <span class="special">}</span>
    328. 

  328.    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
    329. 

  329.    <span class="keyword">constexpr</span> <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">U</span> <span class="identifier">val</span><span class="special">)</span>
    330. 

  330.    <span class="special">{</span>
    331. 

  331.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span>
    332. 

  332.    <span class="special">}</span>
    333. 

  333. <span class="special">};</span>
    334. 

  334. 

  335. <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>
    336. 

  336. <span class="keyword">class</span> <span class="identifier">hermite_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</span><span class="special">&gt;</span>
    337. 

  337. <span class="special">{</span>
    338. 

  338.    <span class="identifier">const_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</span><span class="special">&gt;</span> <span class="identifier">m_data</span><span class="special">;</span>
    339. 

  339. 

  340.  <span class="keyword">public</span><span class="special">:</span>
    341. 

  341.    <span class="keyword">constexpr</span> <span class="identifier">hermite_polynomial</span><span class="special">()</span> <span class="special">:</span> <span class="identifier">m_data</span><span class="special">{</span><span class="number">0</span><span class="special">,</span> <span class="number">2</span><span class="special">}</span> <span class="special">{}</span>
    342. 

  342.    <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">const_polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</span><span class="special">&gt;&amp;</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span>
    343. 

  343.    <span class="special">{</span>
    344. 

  344.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
    345. 

  345.    <span class="special">}</span>
    346. 

  346.    <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">)</span> <span class="keyword">const</span>
    347. 

  347.    <span class="special">{</span>
    348. 

  348.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span>
    349. 

  349.    <span class="special">}</span>
    350. 

  350.    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
    351. 

  351.    <span class="keyword">constexpr</span> <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">U</span> <span class="identifier">val</span><span class="special">)</span>
    352. 

  352.    <span class="special">{</span>
    353. 

  353.       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span>
    354. 

  354.    <span class="special">}</span>
    355. 

  355. <span class="special">};</span>
    356. 

  356. </pre>
    357. 

  357. <p>
    358. 

  358.         We can now declare H<sub>9</sub> as a constexpr object, access the coefficients, and
    359. 

  359.         evaluate at an abscissa value, all using <code class="computeroutput"><span class="keyword">constexpr</span></code>
    360. 

  360.         arithmetic:
    361. 

  361.       </p>
    362. 

  362. <pre class="programlisting"><span class="keyword">constexpr</span> <span class="identifier">hermite_polynomial</span><span class="special">&lt;</span><span class="identifier">float128</span><span class="special">,</span> <span class="number">9</span><span class="special">&gt;</span> <span class="identifier">h9</span><span class="special">;</span>
    363. 

  363. <span class="comment">//</span>
    364. 

  364. <span class="comment">// Verify that the polynomial's coefficients match the known values:</span>
    365. 

  365. <span class="comment">//</span>
    366. 

  366. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">==</span> <span class="number">0</span><span class="special">);</span>
    367. 

  367. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">==</span> <span class="number">30240</span><span class="special">);</span>
    368. 

  368. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">==</span> <span class="number">0</span><span class="special">);</span>
    369. 

  369. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">3</span><span class="special">]</span> <span class="special">==</span> <span class="special">-</span><span class="number">80640</span><span class="special">);</span>
    370. 

  370. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">4</span><span class="special">]</span> <span class="special">==</span> <span class="number">0</span><span class="special">);</span>
    371. 

  371. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">5</span><span class="special">]</span> <span class="special">==</span> <span class="number">48384</span><span class="special">);</span>
    372. 

  372. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">6</span><span class="special">]</span> <span class="special">==</span> <span class="number">0</span><span class="special">);</span>
    373. 

  373. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">7</span><span class="special">]</span> <span class="special">==</span> <span class="special">-</span><span class="number">9216</span><span class="special">);</span>
    374. 

  374. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">8</span><span class="special">]</span> <span class="special">==</span> <span class="number">0</span><span class="special">);</span>
    375. 

  375. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">[</span><span class="number">9</span><span class="special">]</span> <span class="special">==</span> <span class="number">512</span><span class="special">);</span>
    376. 

  376. <span class="comment">//</span>
    377. 

  377. <span class="comment">// Define an abscissa value to evaluate at:</span>
    378. 

  378. <span class="comment">//</span>
    379. 

  379. <span class="keyword">constexpr</span> <span class="identifier">float128</span> <span class="identifier">abscissa</span><span class="special">(</span><span class="number">0.5</span><span class="special">);</span>
    380. 

  380. <span class="comment">//</span>
    381. 

  381. <span class="comment">// Evaluate H_9(0.5) using all constexpr arithmetic:</span>
    382. 

  382. <span class="comment">//</span>
    383. 

  383. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">h9</span><span class="special">(</span><span class="identifier">abscissa</span><span class="special">)</span> <span class="special">==</span> <span class="number">6481</span><span class="special">);</span>
    384. 

  384. </pre>
    385. 

  385. <p>
    386. 

  386.         Also since the coefficients to the Hermite polynomials are integers, we can
    387. 

  387.         also generate the Hermite coefficients using (fixed precision) cpp_int's:
    388. 

  388.         see <a href="../../../../test/constexpr_test_cpp_int_6.cpp" target="_top">constexpr_test_cpp_int_6.cpp</a>.
    389. 

  389.       </p>
    390. 

  390. <p>
    391. 

  391.         We can also generate factorials (and validate the result) like so:
    392. 

  392.       </p>
    393. 

  393. <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>
    394. 

  394. <span class="keyword">constexpr</span> <span class="identifier">T</span> <span class="identifier">factorial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">)</span>
    395. 

  395. <span class="special">{</span>
    396. 

  396.    <span class="keyword">return</span> <span class="identifier">a</span> <span class="special">?</span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">factorial</span><span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">:</span> <span class="number">1</span><span class="special">;</span>
    397. 

  397. <span class="special">}</span>
    398. 

  398. </pre>
    399. 

  399. <pre class="programlisting"><span class="keyword">constexpr</span> <span class="identifier">uint1024_t</span> <span class="identifier">f1</span> <span class="special">=</span> <span class="identifier">factorial</span><span class="special">(</span><span class="identifier">uint1024_t</span><span class="special">(</span><span class="number">31</span><span class="special">));</span>
    400. 

  400. <span class="keyword">static_assert</span><span class="special">(</span><span class="identifier">f1</span> <span class="special">==</span> <span class="number">0</span><span class="identifier">x1956ad0aae33a4560c5cd2c000000_cppi</span><span class="special">);</span>
    401. 

  401. </pre>
    402. 

  402. <p>
    403. 

  403.         Another example in <a href="../../../../test/constexpr_test_cpp_int_7.cpp" target="_top">constexpr_test_cpp_int_7.cpp</a>
    404. 

  404.         generates a fresh multiprecision random number each time the file is compiled.
    405. 

  405.       </p>
    406. 

  406. </div>
    407. 

  407. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    408. 

  408. <td align="left"></td>
    409. 

  409. <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2019 John Maddock
    410. 

  410.       and Christopher Kormanyos<p>
    411. 

  411.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    412. 

  412.         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>)
    413. 

  413.       </p>
    414. 

  414. </div></td>
    415. 

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    416. 

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