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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<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>
</h3></div></div></div>
<p>
There are two kinds of <code class="computeroutput"><span class="keyword">constexpr</span></code>
support in this library:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
The more basic version requires only C++11 and allow the construction
of some number types as literals.
</li>
<li class="listitem">
The more advanced support permits constexpr arithmetic and requires at
least C++14 constexpr support, and for many operations C++2a support
</li>
</ul></div>
<h5>
<a name="boost_multiprecision.tut.lits.h0"></a>
<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
numeric literals</a>
</h5>
<p>
There are two backend types which are literals:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<a class="link" href="floats/float128.html" title="float128">float128</a>
(which requires GCC), and
</li>
<li class="listitem">
Instantiations of <code class="computeroutput"><span class="identifier">cpp_int_backend</span></code>
where the Allocator parameter is type <code class="computeroutput"><span class="keyword">void</span></code>.
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>.
</li>
</ul></div>
<p>
For example:
</p>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
</pre>
<p>
There is also support for user defined-literals with <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
- these are limited to unchecked, fixed precision <code class="computeroutput"><span class="identifier">cpp_int</span></code>'s
which are specified in hexadecimal notation. The suffixes supported are:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Suffix
</p>
</th>
<th>
<p>
Meaning
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
_cppi
</p>
</td>
<td>
<p>
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>,
where N is chosen to contain just enough digits to hold the number
specified.
</p>
</td>
</tr>
<tr>
<td>
<p>
_cppui
</p>
</td>
<td>
<p>
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>,
where N is chosen to contain just enough digits to hold the number
specified.
</p>
</td>
</tr>
<tr>
<td>
<p>
_cppi<span class="emphasis"><em>N</em></span>
</p>
</td>
<td>
<p>
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>.
</p>
</td>
</tr>
<tr>
<td>
<p>
_cppui<span class="emphasis"><em>N</em></span>
</p>
</td>
<td>
<p>
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>.
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
In each case, use of these suffixes with hexadecimal values produces a <code class="computeroutput"><span class="keyword">constexpr</span></code> result.
</p>
<p>
Examples:
</p>
<pre class="programlisting"><span class="comment">//</span>
<span class="comment">// Any use of user defined literals requires that we import the literal-operators</span>
<span class="comment">// into current scope first:</span>
<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>
<span class="comment">//</span>
<span class="comment">// To keep things simple in the example, we'll make our types used visible to this scope as well:</span>
<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="comment">//</span>
<span class="comment">// The value zero as a number&lt;cpp_int_backend&lt;4,4,signed_magnitude,unchecked,void&gt; &gt;:</span>
<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>
<span class="comment">// The type of each constant has 4 bits per hexadecimal digit,</span>
<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>
<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>
<span class="comment">//</span>
<span class="comment">// Smaller values can be assigned to larger values:</span>
<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>
<span class="comment">//</span>
<span class="comment">// However, this only works in constexpr contexts from C++14 onwards:</span>
<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>
<span class="comment">//</span>
<span class="comment">// Constants can be padded out with leading zeros to generate wider types:</span>
<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>
<span class="comment">//</span>
<span class="comment">// However, specific width types are best produced with specific-width suffixes,</span>
<span class="comment">// ones supported by default are `_cpp[u]i128`, `_cpp[u]i256`, `_cpp[u]i512`, `_cpp[u]i1024`.</span>
<span class="comment">//</span>
<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>
<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">//</span>
<span class="comment">// If other specific width types are required, then there is a macro for generating the operators</span>
<span class="comment">// for these. The macro can be used at namespace scope only:</span>
<span class="comment">//</span>
<span class="identifier">BOOST_MP_DEFINE_SIZED_CPP_INT_LITERAL</span><span class="special">(</span><span class="number">2048</span><span class="special">);</span>
<span class="comment">//</span>
<span class="comment">// Now we can create 2048-bit literals as well:</span>
<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>
<span class="comment">//</span>
<span class="comment">// Finally negative values are handled via the unary minus operator:</span>
<span class="comment">//</span>
<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>
<span class="comment">//</span>
<span class="comment">// Which means this also works:</span>
<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>
</pre>
<h5>
<a name="boost_multiprecision.tut.lits.h1"></a>
<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
arithmetic</a>
</h5>
<p>
The front end of the library is all <code class="computeroutput"><span class="keyword">constexpr</span></code>
from C++14 and later. Currently there are only two back end types that are
<code class="computeroutput"><span class="keyword">constexpr</span></code> aware: <a class="link" href="floats/float128.html" title="float128">float128</a>
and <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>.
More backends will follow at a later date.
</p>
<p>
Provided the compiler is GCC, type <a class="link" href="floats/float128.html" title="float128">float128</a>
support <code class="computeroutput"><span class="keyword">constexpr</span></code> operations
on all arithmetic operations from C++14, comparisons, <code class="computeroutput"><span class="identifier">abs</span></code>,
<code class="computeroutput"><span class="identifier">fabs</span></code>, <code class="computeroutput"><span class="identifier">fpclassify</span></code>,
<code class="computeroutput"><span class="identifier">isnan</span></code>, <code class="computeroutput"><span class="identifier">isinf</span></code>,
<code class="computeroutput"><span class="identifier">isfinite</span></code> and <code class="computeroutput"><span class="identifier">isnormal</span></code> are also fully supported, but
the transcendental functions are not.
</p>
<p>
The <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
types support constexpr arithmetic, provided it is a fixed precision type
with no allocator. It may also be a checked integer: in which case a compiler
error will be generated on overflow or undefined behaviour. In addition the
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>,
<code class="computeroutput"><span class="identifier">add</span></code>, <code class="computeroutput"><span class="identifier">subtract</span></code>,
<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>,
<code class="computeroutput"><span class="identifier">lsb</span></code>, <code class="computeroutput"><span class="identifier">msb</span></code>,
<code class="computeroutput"><span class="identifier">bit_test</span></code>, <code class="computeroutput"><span class="identifier">bit_set</span></code>,
<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>,
<code class="computeroutput"><span class="identifier">gcd</span></code>, <code class="computeroutput"><span class="identifier">lcm</span></code>
are all supported. Use of <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>
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),
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:
expect to hit roadblocks rather easily.
</p>
<p>
For example given:
</p>
<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>
<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>
<span class="special">{</span>
<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>
<span class="special">}</span>
<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>
<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>
<span class="special">{</span>
<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>
<span class="special">}</span>
</pre>
<p>
We can now calculate areas and circumferences using all constexpr arithmetic:
</p>
<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>
<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>
<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>
<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>
<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>
<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>
</pre>
<p>
Note that these make use of the numeric constants from the Math library,
which also happen to be <code class="computeroutput"><span class="keyword">constexpr</span></code>.
</p>
<p>
For a more interesting example, in <a href="../../../../example/constexpr_float_arithmetic_examples.cpp" target="_top">constexpr_float_arithmetic_examples.cpp</a>
we define a simple class for <code class="computeroutput"><span class="keyword">constexpr</span></code>
polynomial arithmetic:
</p>
<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>
<span class="keyword">struct</span> <span class="identifier">const_polynomial</span><span class="special">;</span>
</pre>
<p>
Given this, we can use recurrence relations to calculate the coefficients
for various orthogonal polynomials - in the example we use the Hermite polynomials,
only the constructor does any work - it uses the recurrence relations to
calculate the coefficient array:
</p>
<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>
<span class="keyword">class</span> <span class="identifier">hermite_polynomial</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="identifier">Order</span><span class="special">&gt;</span> <span class="identifier">m_data</span><span class="special">;</span>
<span class="keyword">public</span><span class="special">:</span>
<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>
<span class="special">{</span>
<span class="special">}</span>
<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>
<span class="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
<span class="special">}</span>
<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>
<span class="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span>
<span class="special">}</span>
<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>
<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>
<span class="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span>
<span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
Now we just need to define H<sub>0</sub> and H<sub>1</sub> as termination conditions for the recurrence:
</p>
<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>
<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>
<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">0</span><span class="special">&gt;</span> <span class="identifier">m_data</span><span class="special">;</span>
<span class="keyword">public</span><span class="special">:</span>
<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>
<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>
<span class="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
<span class="special">}</span>
<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>
<span class="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span>
<span class="special">}</span>
<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>
<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="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span>
<span class="special">}</span>
<span class="special">};</span>
<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>
<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>
<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="identifier">m_data</span><span class="special">;</span>
<span class="keyword">public</span><span class="special">:</span>
<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>
<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>
<span class="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">;</span>
<span class="special">}</span>
<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>
<span class="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">[</span><span class="identifier">N</span><span class="special">];</span>
<span class="special">}</span>
<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>
<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="special">{</span>
<span class="keyword">return</span> <span class="identifier">m_data</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span>
<span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
We can now declare H<sub>9</sub> as a constexpr object, access the coefficients, and
evaluate at an abscissa value, all using <code class="computeroutput"><span class="keyword">constexpr</span></code>
arithmetic:
</p>
<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>
<span class="comment">//</span>
<span class="comment">// Verify that the polynomial's coefficients match the known values:</span>
<span class="comment">//</span>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<span class="comment">//</span>
<span class="comment">// Define an abscissa value to evaluate at:</span>
<span class="comment">//</span>
<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>
<span class="comment">//</span>
<span class="comment">// Evaluate H_9(0.5) using all constexpr arithmetic:</span>
<span class="comment">//</span>
<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>
</pre>
<p>
Also since the coefficients to the Hermite polynomials are integers, we can
also generate the Hermite coefficients using (fixed precision) cpp_int's:
see <a href="../../../../test/constexpr_test_cpp_int_6.cpp" target="_top">constexpr_test_cpp_int_6.cpp</a>.
</p>
<p>
We can also generate factorials (and validate the result) like so:
</p>
<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>
<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>
<span class="special">{</span>
<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>
<span class="special">}</span>
</pre>
<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="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>
</pre>
<p>
Another example in <a href="../../../../test/constexpr_test_cpp_int_7.cpp" target="_top">constexpr_test_cpp_int_7.cpp</a>
generates a fresh multiprecision random number each time the file is compiled.
</p>
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and Christopher Kormanyos<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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>)
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