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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  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 </h3></div></div></div>
  29 <p>
  30         There are two kinds of <code class="computeroutput"><span class="keyword">constexpr</span></code>
  31         support in this library:
  32       </p>
  33 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  34 <li class="listitem">
  35             The more basic version requires only C++11 and allow the construction
  36             of some number types as literals.
  37           </li>
  38 <li class="listitem">
  39             The more advanced support permits constexpr arithmetic and requires at
  40             least C++14 constexpr support, and for many operations C++2a support
  41           </li>
  42 </ul></div>
  43 <h5>
  44 <a name="boost_multiprecision.tut.lits.h0"></a>
  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         numeric literals</a>
  47       </h5>
  48 <p>
  49         There are two backend types which are literals:
  50       </p>
  51 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  52 <li class="listitem">
  53             <a class="link" href="floats/float128.html" title="float128">float128</a>
  54             (which requires GCC), and
  55           </li>
  56 <li class="listitem">
  57             Instantiations of <code class="computeroutput"><span class="identifier">cpp_int_backend</span></code>
  58             where the Allocator parameter is type <code class="computeroutput"><span class="keyword">void</span></code>.
  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           </li>
  61 </ul></div>
  62 <p>
  63         For example:
  64       </p>
  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
  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
  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 <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
  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 <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 <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 </pre>
  76 <p>
  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         - these are limited to unchecked, fixed precision <code class="computeroutput"><span class="identifier">cpp_int</span></code>'s
  79         which are specified in hexadecimal notation. The suffixes supported are:
  80       </p>
  81 <div class="informaltable"><table class="table">
  82 <colgroup>
  83 <col>
  84 <col>
  85 </colgroup>
  86 <thead><tr>
  87 <th>
  88                 <p>
  89                   Suffix
  90                 </p>
  91               </th>
  92 <th>
  93                 <p>
  94                   Meaning
  95                 </p>
  96               </th>
  97 </tr></thead>
  98 <tbody>
  99 <tr>
 100 <td>
 101                 <p>
 102                   _cppi
 103                 </p>
 104               </td>
 105 <td>
 106                 <p>
 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                   where N is chosen to contain just enough digits to hold the number
 109                   specified.
 110                 </p>
 111               </td>
 112 </tr>
 113 <tr>
 114 <td>
 115                 <p>
 116                   _cppui
 117                 </p>
 118               </td>
 119 <td>
 120                 <p>
 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                   where N is chosen to contain just enough digits to hold the number
 123                   specified.
 124                 </p>
 125               </td>
 126 </tr>
 127 <tr>
 128 <td>
 129                 <p>
 130                   _cppi<span class="emphasis"><em>N</em></span>
 131                 </p>
 132               </td>
 133 <td>
 134                 <p>
 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                 </p>
 137               </td>
 138 </tr>
 139 <tr>
 140 <td>
 141                 <p>
 142                   _cppui<span class="emphasis"><em>N</em></span>
 143                 </p>
 144               </td>
 145 <td>
 146                 <p>
 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                 </p>
 149               </td>
 150 </tr>
 151 </tbody>
 152 </table></div>
 153 <p>
 154         In each case, use of these suffixes with hexadecimal values produces a <code class="computeroutput"><span class="keyword">constexpr</span></code> result.
 155       </p>
 156 <p>
 157         Examples:
 158       </p>
 159 <pre class="programlisting"><span class="comment">//</span>
 160 <span class="comment">// Any use of user defined literals requires that we import the literal-operators</span>
 161 <span class="comment">// into current scope first:</span>
 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 <span class="comment">//</span>
 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 <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 <span class="comment">//</span>
 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 <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 <span class="comment">// The type of each constant has 4 bits per hexadecimal digit,</span>
 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 <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 <span class="comment">//</span>
 173 <span class="comment">// Smaller values can be assigned to larger values:</span>
 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 <span class="comment">//</span>
 176 <span class="comment">// However, this only works in constexpr contexts from C++14 onwards:</span>
 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 <span class="comment">//</span>
 179 <span class="comment">// Constants can be padded out with leading zeros to generate wider types:</span>
 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 <span class="comment">//</span>
 182 <span class="comment">// However, specific width types are best produced with specific-width suffixes,</span>
 183 <span class="comment">// ones supported by default are `_cpp[u]i128`, `_cpp[u]i256`, `_cpp[u]i512`, `_cpp[u]i1024`.</span>
 184 <span class="comment">//</span>
 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 <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 <span class="comment">//</span>
 188 <span class="comment">// If other specific width types are required, then there is a macro for generating the operators</span>
 189 <span class="comment">// for these.  The macro can be used at namespace scope only:</span>
 190 <span class="comment">//</span>
 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 <span class="comment">//</span>
 193 <span class="comment">// Now we can create 2048-bit literals as well:</span>
 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 <span class="comment">//</span>
 196 <span class="comment">// Finally negative values are handled via the unary minus operator:</span>
 197 <span class="comment">//</span>
 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 <span class="comment">//</span>
 200 <span class="comment">// Which means this also works:</span>
 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 </pre>
 203 <h5>
 204 <a name="boost_multiprecision.tut.lits.h1"></a>
 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         arithmetic</a>
 207       </h5>
 208 <p>
 209         The front end of the library is all <code class="computeroutput"><span class="keyword">constexpr</span></code>
 210         from C++14 and later. Currently there are only two back end types that are
 211         <code class="computeroutput"><span class="keyword">constexpr</span></code> aware: <a class="link" href="floats/float128.html" title="float128">float128</a>
 212         and <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>.
 213         More backends will follow at a later date.
 214       </p>
 215 <p>
 216         Provided the compiler is GCC, type <a class="link" href="floats/float128.html" title="float128">float128</a>
 217         support <code class="computeroutput"><span class="keyword">constexpr</span></code> operations
 218         on all arithmetic operations from C++14, comparisons, <code class="computeroutput"><span class="identifier">abs</span></code>,
 219         <code class="computeroutput"><span class="identifier">fabs</span></code>, <code class="computeroutput"><span class="identifier">fpclassify</span></code>,
 220         <code class="computeroutput"><span class="identifier">isnan</span></code>, <code class="computeroutput"><span class="identifier">isinf</span></code>,
 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         the transcendental functions are not.
 223       </p>
 224 <p>
 225         The <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
 226         types support constexpr arithmetic, provided it is a fixed precision type
 227         with no allocator. It may also be a checked integer: in which case a compiler
 228         error will be generated on overflow or undefined behaviour. In addition the
 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         <code class="computeroutput"><span class="identifier">add</span></code>, <code class="computeroutput"><span class="identifier">subtract</span></code>,
 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         <code class="computeroutput"><span class="identifier">lsb</span></code>, <code class="computeroutput"><span class="identifier">msb</span></code>,
 233         <code class="computeroutput"><span class="identifier">bit_test</span></code>, <code class="computeroutput"><span class="identifier">bit_set</span></code>,
 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         <code class="computeroutput"><span class="identifier">gcd</span></code>, <code class="computeroutput"><span class="identifier">lcm</span></code>
 236         are all supported. Use of <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
 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         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         expect to hit roadblocks rather easily.
 240       </p>
 241 <p>
 242         For example given:
 243       </p>
 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 <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 <span class="special">{</span>
 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 <span class="special">}</span>
 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 <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 <span class="special">{</span>
 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 <span class="special">}</span>
 255 </pre>
 256 <p>
 257         We can now calculate areas and circumferences using all constexpr arithmetic:
 258       </p>
 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
 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 <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 <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
 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 <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 </pre>
 268 <p>
 269         Note that these make use of the numeric constants from the Math library,
 270         which also happen to be <code class="computeroutput"><span class="keyword">constexpr</span></code>.
 271       </p>
 272 <p>
 273         For a more interesting example, in <a href="../../../../example/constexpr_float_arithmetic_examples.cpp" target="_top">constexpr_float_arithmetic_examples.cpp</a>
 274         we define a simple class for <code class="computeroutput"><span class="keyword">constexpr</span></code>
 275         polynomial arithmetic:
 276       </p>
 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 <span class="keyword">struct</span> <span class="identifier">const_polynomial</span><span class="special">;</span>
 279 </pre>
 280 <p>
 281         Given this, we can use recurrence relations to calculate the coefficients
 282         for various orthogonal polynomials - in the example we use the Hermite polynomials,
 283         only the constructor does any work - it uses the recurrence relations to
 284         calculate the coefficient array:
 285       </p>
 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 <span class="keyword">class</span> <span class="identifier">hermite_polynomial</span>
 288 <span class="special">{</span>
 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
 291  <span class="keyword">public</span><span class="special">:</span>
 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    <span class="special">{</span>
 294    <span class="special">}</span>
 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    <span class="special">{</span>
 297       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
 298    <span class="special">}</span>
 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    <span class="special">{</span>
 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    <span class="special">}</span>
 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    <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    <span class="special">{</span>
 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    <span class="special">}</span>
 308 <span class="special">};</span>
 309 </pre>
 310 <p>
 311         Now we just need to define H<sub>0</sub> and H<sub>1</sub> as termination conditions for the recurrence:
 312       </p>
 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 <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 <span class="special">{</span>
 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
 318  <span class="keyword">public</span><span class="special">:</span>
 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    <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    <span class="special">{</span>
 322       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
 323    <span class="special">}</span>
 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    <span class="special">{</span>
 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    <span class="special">}</span>
 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    <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    <span class="special">{</span>
 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    <span class="special">}</span>
 333 <span class="special">};</span>
 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 <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 <span class="special">{</span>
 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
 340  <span class="keyword">public</span><span class="special">:</span>
 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    <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    <span class="special">{</span>
 344       <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
 345    <span class="special">}</span>
 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    <span class="special">{</span>
 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    <span class="special">}</span>
 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    <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    <span class="special">{</span>
 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    <span class="special">}</span>
 355 <span class="special">};</span>
 356 </pre>
 357 <p>
 358         We can now declare H<sub>9</sub> as a constexpr object, access the coefficients, and
 359         evaluate at an abscissa value, all using <code class="computeroutput"><span class="keyword">constexpr</span></code>
 360         arithmetic:
 361       </p>
 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 <span class="comment">//</span>
 364 <span class="comment">// Verify that the polynomial's coefficients match the known values:</span>
 365 <span class="comment">//</span>
 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 <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 <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 <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 <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 <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 <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 <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 <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 <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 <span class="comment">//</span>
 377 <span class="comment">// Define an abscissa value to evaluate at:</span>
 378 <span class="comment">//</span>
 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 <span class="comment">//</span>
 381 <span class="comment">// Evaluate H_9(0.5) using all constexpr arithmetic:</span>
 382 <span class="comment">//</span>
 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 </pre>
 385 <p>
 386         Also since the coefficients to the Hermite polynomials are integers, we can
 387         also generate the Hermite coefficients using (fixed precision) cpp_int's:
 388         see <a href="../../../../test/constexpr_test_cpp_int_6.cpp" target="_top">constexpr_test_cpp_int_6.cpp</a>.
 389       </p>
 390 <p>
 391         We can also generate factorials (and validate the result) like so:
 392       </p>
 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 <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 <span class="special">{</span>
 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 <span class="special">}</span>
 398 </pre>
 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 <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 </pre>
 402 <p>
 403         Another example in <a href="../../../../test/constexpr_test_cpp_int_7.cpp" target="_top">constexpr_test_cpp_int_7.cpp</a>
 404         generates a fresh multiprecision random number each time the file is compiled.
 405       </p>
 406 </div>
 407 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 408 <td align="left"></td>
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