1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Literal Types and constexpr Support</title> 5<link rel="stylesheet" href="../../multiprecision.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Chapter 1. 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"> 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"><</span><span class="identifier">cpp_int_backend</span><span class="special"><</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">></span> <span class="special">></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"><</span><span class="identifier">cpp_int_backend</span><span class="special"><</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">></span> <span class="special">></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"><</span><span class="identifier">cpp_int_backend</span><span class="special"><</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">></span> <span class="special">></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"><</span><span class="identifier">cpp_int_backend</span><span class="special"><</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">></span> <span class="special">></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">// Any use of user defined literals requires that we import the literal-operators into current scope first:</span> 160<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> 161<span class="comment">//</span> 162<span class="comment">// To keep things simple in the example, we'll make our types used visible to this scope as well:</span> 163<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> 164<span class="comment">//</span> 165<span class="comment">// The value zero as a number<cpp_int_backend<4,4,signed_magnitude,unchecked,void> >:</span> 166<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> 167<span class="comment">// The type of each constant has 4 bits per hexadecimal digit,</span> 168<span class="comment">// so this is of type uint256_t (ie number<cpp_int_backend<256,256,unsigned_magnitude,unchecked,void> >):</span> 169<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> 170<span class="comment">//</span> 171<span class="comment">// Smaller values can be assigned to larger values:</span> 172<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> 173<span class="comment">//</span> 174<span class="comment">// However, this only works in constexpr contexts from C++14 onwards:</span> 175<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> 176<span class="comment">//</span> 177<span class="comment">// Constants can be padded out with leading zeros to generate wider types:</span> 178<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> 179<span class="comment">//</span> 180<span class="comment">// However, specific-width types are best produced with specific-width suffixes,</span> 181<span class="comment">// ones supported by default are `_cpp[u]i128`, `_cpp[u]i256`, `_cpp[u]i512`, `_cpp[u]i1024`.</span> 182<span class="comment">//</span> 183<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> 184<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> <span class="comment">// OK,</span> 185<span class="comment">// always produces an uint1024_t as the result.</span> 186<span class="comment">//</span> 187<span class="comment">// If other specific-width types are required, then there is a macro for generating the operators for these.</span> 188<span class="comment">// The macro can be used at namespace scope only:</span> 189<span class="comment">//</span> 190<span class="identifier">BOOST_MP_DEFINE_SIZED_CPP_INT_LITERAL</span><span class="special">(</span><span class="number">2048</span><span class="special">);</span> 191<span class="comment">//</span> 192<span class="comment">// Now we can create 2048-bit literals as well:</span> 193<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<cpp_int_backend<2048,2048,signed_magnitude,unchecked,void> ></span> 194<span class="comment">//</span> 195<span class="comment">// Finally, negative values are handled via the unary minus operator:</span> 196<span class="comment">//</span> 197<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> 198<span class="comment">//</span> 199<span class="comment">// Which means this also works:</span> 200<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> 201</pre> 202<h5> 203<a name="boost_multiprecision.tut.lits.h1"></a> 204 <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 205 arithmetic</a> 206 </h5> 207<p> 208 The front end of the library is all <code class="computeroutput"><span class="keyword">constexpr</span></code> 209 from C++14 and later. Currently there are only two backend types that are 210 <code class="computeroutput"><span class="keyword">constexpr</span></code> aware: __float128 211 and <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>. 212 More backends will follow at a later date. 213 </p> 214<p> 215 Provided the compiler is GCC, type <a class="link" href="floats/float128.html" title="float128">float128</a> 216 support <code class="computeroutput"><span class="keyword">constexpr</span></code> operations 217 on all arithmetic operations from C++14, comparisons, <code class="computeroutput"><span class="identifier">abs</span></code>, 218 <code class="computeroutput"><span class="identifier">fabs</span></code>, <code class="computeroutput"><span class="identifier">fpclassify</span></code>, 219 <code class="computeroutput"><span class="identifier">isnan</span></code>, <code class="computeroutput"><span class="identifier">isinf</span></code>, 220 <code class="computeroutput"><span class="identifier">isfinite</span></code> and <code class="computeroutput"><span class="identifier">isnormal</span></code> are also fully supported, but 221 the transcendental functions are not. 222 </p> 223<p> 224 The <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a> 225 types support constexpr arithmetic, provided it is a fixed precision type 226 with no allocator. It may also be a checked integer: in which case a compiler 227 error will be generated on overflow or undefined behaviour. In addition the 228 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>, 229 <code class="computeroutput"><span class="identifier">add</span></code>, <code class="computeroutput"><span class="identifier">subtract</span></code>, 230 <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>, 231 <code class="computeroutput"><span class="identifier">lsb</span></code>, <code class="computeroutput"><span class="identifier">msb</span></code>, 232 <code class="computeroutput"><span class="identifier">bit_test</span></code>, <code class="computeroutput"><span class="identifier">bit_set</span></code>, 233 <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>, 234 <code class="computeroutput"><span class="identifier">gcd</span></code>, <code class="computeroutput"><span class="identifier">lcm</span></code> 235 are all supported. Use of <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a> 236 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), 237 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: 238 expect to hit roadblocks rather easily. 239 </p> 240<p> 241 See <a href="https://en.cppreference.com/w/cpp/compiler_support" target="_top">compiler 242 support</a> for <a href="https://en.cppreference.com/w/cpp/types/is_constant_evaluated" target="_top"><code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">is_constant_evaluated</span></code></a>; 243 </p> 244<p> 245 For example given: 246 </p> 247<pre class="programlisting"><span class="preprocessor">#include</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">constants</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// For constant pi with full precision of type T.</span> 248<span class="comment">// using boost::math::constants::pi;</span> 249 250<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 251<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> 252<span class="special">{</span> 253 <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"><</span><span class="identifier">T</span><span class="special">>()</span> <span class="special">*</span> <span class="identifier">radius</span><span class="special">;</span> 254<span class="special">}</span> 255 256<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 257<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> 258<span class="special">{</span> 259 <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"><</span><span class="identifier">T</span><span class="special">>()</span> <span class="special">*</span> <span class="identifier">radius</span> <span class="special">*</span> <span class="identifier">radius</span><span class="special">;</span> 260<span class="special">}</span> 261</pre> 262<p> 263 We can now calculate areas and circumferences, using all compile-time <code class="computeroutput"><span class="keyword">constexpr</span></code> arithmetic: 264 </p> 265<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> 266 267<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> 268<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> 269<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> 270 271<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Circumference = "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 272<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Area = "</span> <span class="special"><<</span> <span class="identifier">a</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 273</pre> 274<p> 275 Note that these make use of the numeric constants from the <a href="https://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/constants.html" target="_top">Boost.Math 276 constants</a> library, which also happen to be <code class="computeroutput"><span class="keyword">constexpr</span></code>. 277 These usually have the full precision of the floating-point type, here 128-bit, 278 about 36 decimal digits. 279 </p> 280<h6> 281<a name="boost_multiprecision.tut.lits.h2"></a> 282 <span class="phrase"><a name="boost_multiprecision.tut.lits.hermite_poly_coeffics"></a></span><a class="link" href="lits.html#boost_multiprecision.tut.lits.hermite_poly_coeffics">Calculating 283 Hermite Polynomial coefficients at compile time</a> 284 </h6> 285<p> 286 For a more interesting example, in <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/doc/html/../../example/constexpr_float_arithmetic_examples.cpp" target="_top">constexpr_float_arithmetic_examples.cpp</a> 287 we define a simple class for <code class="computeroutput"><span class="keyword">constexpr</span></code> 288 polynomial arithmetic: 289 </p> 290<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> 291<span class="keyword">struct</span> <span class="identifier">const_polynomial</span><span class="special">;</span> 292</pre> 293<p> 294 Given this, we can use recurrence relations to calculate the coefficients 295 for various orthogonal polynomials - in the example we use the Hermite polynomials. 296 Only the constructor does any work - it uses the recurrence relations to 297 calculate the coefficient array: 298 </p> 299<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> 300<span class="keyword">class</span> <span class="identifier">hermite_polynomial</span> 301<span class="special">{</span> 302 <span class="identifier">const_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">></span> <span class="identifier">m_data</span><span class="special">;</span> 303 304 <span class="keyword">public</span><span class="special">:</span> 305 <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"><</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">>().</span><span class="identifier">data</span><span class="special">()</span> <span class="special">*</span> <span class="identifier">const_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</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> <span class="identifier">hermite_polynomial</span><span class="special"><</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">>().</span><span class="identifier">data</span><span class="special">().</span><span class="identifier">derivative</span><span class="special">())</span> 306 <span class="special">{</span> 307 <span class="special">}</span> 308 <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">const_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span> 309 <span class="special">{</span> 310 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span> 311 <span class="special">}</span> 312 <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</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> 313 <span class="special">{</span> 314 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span> 315 <span class="special">}</span> 316 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> 317 <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> 318 <span class="special">{</span> 319 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span> 320 <span class="special">}</span> 321<span class="special">};</span> 322</pre> 323<p> 324 Now we just need to define <span class="emphasis"><em>H<sub>0</sub></em></span> and <span class="emphasis"><em>H<sub>1</sub></em></span> 325 as termination conditions for the recurrence: 326 </p> 327<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 328<span class="keyword">class</span> <span class="identifier">hermite_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">0</span><span class="special">></span> 329<span class="special">{</span> 330 <span class="identifier">const_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">0</span><span class="special">></span> <span class="identifier">m_data</span><span class="special">;</span> 331 332 <span class="keyword">public</span><span class="special">:</span> 333 <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> 334 <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">const_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">0</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span> 335 <span class="special">{</span> 336 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span> 337 <span class="special">}</span> 338 <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</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> 339 <span class="special">{</span> 340 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span> 341 <span class="special">}</span> 342 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> 343 <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> 344 <span class="special">{</span> 345 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span> 346 <span class="special">}</span> 347<span class="special">};</span> 348 349<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 350<span class="keyword">class</span> <span class="identifier">hermite_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</span><span class="special">></span> 351<span class="special">{</span> 352 <span class="identifier">const_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</span><span class="special">></span> <span class="identifier">m_data</span><span class="special">;</span> 353 354 <span class="keyword">public</span><span class="special">:</span> 355 <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> 356 <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">const_polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">1</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span> 357 <span class="special">{</span> 358 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span> 359 <span class="special">}</span> 360 <span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</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> 361 <span class="special">{</span> 362 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span> 363 <span class="special">}</span> 364 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> 365 <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> 366 <span class="special">{</span> 367 <span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span> 368 <span class="special">}</span> 369<span class="special">};</span> 370</pre> 371<p> 372 We can now declare <span class="emphasis"><em>H<sub>9</sub></em></span> as a <code class="computeroutput"><span class="keyword">constexpr</span></code> 373 object, access the coefficients, and evaluate at an abscissa value, all at 374 compile-time using <code class="computeroutput"><span class="keyword">constexpr</span></code> 375 arithmetic: 376 </p> 377<pre class="programlisting"><span class="keyword">constexpr</span> <span class="identifier">hermite_polynomial</span><span class="special"><</span><span class="identifier">float128</span><span class="special">,</span> <span class="number">9</span><span class="special">></span> <span class="identifier">h9</span><span class="special">;</span> 378<span class="comment">//</span> 379<span class="comment">// Verify that the polynomial's coefficients match the known values:</span> 380<span class="comment">//</span> 381<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> 382<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> 383<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> 384<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> 385<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> 386<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> 387<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> 388<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> 389<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> 390<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> 391<span class="comment">//</span> 392<span class="comment">// Define an abscissa value to evaluate at:</span> 393<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> 394<span class="comment">//</span> 395<span class="comment">// Evaluate H_9(0.5) using all constexpr arithmetic, and check that it has the expected result:</span> 396<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> 397</pre> 398<p> 399 See <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/doc/html/../../example/constexpr_float_arithmetic_examples.cpp" target="_top">constexpr_float_arithmetic_examples.cpp</a> 400 for working code. 401 </p> 402<p> 403 Also since the coefficients to the Hermite polynomials are integers, we can 404 also generate the Hermite coefficients using (fixed precision) <code class="computeroutput"><span class="identifier">cpp_int</span></code>s: see <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/doc/html/../../test/constexpr_test_cpp_int_6.cpp" target="_top">constexpr_test_cpp_int_6.cpp</a>. 405 </p> 406<h6> 407<a name="boost_multiprecision.tut.lits.h3"></a> 408 <span class="phrase"><a name="boost_multiprecision.tut.lits.factorial_constexpr"></a></span><a class="link" href="lits.html#boost_multiprecision.tut.lits.factorial_constexpr"><code class="computeroutput"><span class="keyword">constexpr</span></code> Factorials</a> 409 </h6> 410<p> 411 We can also generate integer factorials in <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/doc/html/../../test/constexpr_test_cpp_int_5.cpp" target="_top">constexpr_test_cpp_int_5.cpp</a> 412 like so: 413 </p> 414<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 415<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">&</span> <span class="identifier">a</span><span class="special">)</span> 416<span class="special">{</span> 417 <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> 418<span class="special">}</span> 419</pre> 420<p> 421 and validate the result: 422 </p> 423<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> <span class="comment">// Factorial 31!</span> 424<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> <span class="comment">// Expected result as an Boost.Multiprecision integer literal. </span> 425</pre> 426<h6> 427<a name="boost_multiprecision.tut.lits.h4"></a> 428 <span class="phrase"><a name="boost_multiprecision.tut.lits.random_constexpr"></a></span><a class="link" href="lits.html#boost_multiprecision.tut.lits.random_constexpr">Random 429 <code class="computeroutput"><span class="keyword">constexpr</span></code> values</a> 430 </h6> 431<p> 432 Another example in <a href="http://www.boost.org/doc/libs/release/libs/multiprecision/doc/html/../../test/constexpr_test_cpp_int_7.cpp" target="_top">constexpr_test_cpp_int_7.cpp</a> 433 generates a fresh multiprecision random number each time the file is compiled. 434 It includes an C++ template implementation of the <a href="https://en.wikipedia.org/wiki/KISS_(algorithm)" target="_top">KISS 435 random number algorithm by George Marsaglia</a> for <code class="computeroutput"><span class="identifier">cpp_int</span></code> 436 integers. 437 </p> 438<pre class="programlisting"><span class="keyword">constexpr</span> <span class="identifier">uint1024_t</span> <span class="identifier">rand</span> <span class="special">=</span> <span class="identifier">nth_random_value</span><span class="special"><</span><span class="identifier">uint1024_t</span><span class="special">>(</span><span class="number">1000</span><span class="special">);</span> 439<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">rand</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 440</pre> 441<p> 442 See also the <a class="link" href="random.html" title="Generating Random Numbers">random number 443 generation</a> section. 444 </p> 445</div> 446<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 447<td align="left"></td> 448<td align="right"><div class="copyright-footer">Copyright © 2002-2020 John 449 Maddock and Christopher Kormanyos<p> 450 Distributed under the Boost Software License, Version 1.0. (See accompanying 451 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>) 452 </p> 453</div></td> 454</tr></table> 455<hr> 456<div class="spirit-nav"> 457<a accesskey="p" href="primetest.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../tut.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="import_export.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 458</div> 459</body> 460</html> 461
