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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="boost_multiprecision.tut.floats.fp_eg.jel"></a><a class="link" href="jel.html" title="Defining a Special Function.">Defining
  28           a Special Function.</a>
  29 </h5></div></div></div>
  30 <p>
  31             In this example we'll show several implementations of the <a href="http://mathworld.wolfram.com/LambdaFunction.html" target="_top">Jahnke
  32             and Emden Lambda function</a>, each implementation a little more
  33             sophisticated than the last.
  34           </p>
  35 <p>
  36             The Jahnke-Emden Lambda function is defined by the equation:
  37           </p>
  38 <div class="blockquote"><blockquote class="blockquote"><p>
  39               <span class="emphasis"><em>JahnkeEmden(v, z) = &#915;(v+1) * J<sub>v</sub>(z) / (z / 2)<sup>v</sup></em></span>
  40             </p></blockquote></div>
  41 <p>
  42             If we were to implement this at double precision using Boost.Math's facilities
  43             for the Gamma and Bessel function calls it would look like this:
  44           </p>
  45 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">JEL1</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">z</span><span class="special">)</span>
  46 <span class="special">{</span>
  47    <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">tgamma</span><span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</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">cyl_bessel_j</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span><span class="special">(</span><span class="identifier">z</span> <span class="special">/</span> <span class="number">2</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
  48 <span class="special">}</span>
  49 </pre>
  50 <p>
  51             Calling this function as:
  52           </p>
  53 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span>
  54 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">JEL1</span><span class="special">(</span><span class="number">2.5</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</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>
  55 </pre>
  56 <p>
  57             Yields the output:
  58           </p>
  59 <pre class="programlisting">9.822663964796047e-001</pre>
  60 <p>
  61             Now let's implement the function again, but this time using the multiprecision
  62             type <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>
  63             as the argument type:
  64           </p>
  65 <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span>
  66    <span class="identifier">JEL2</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">z</span><span class="special">)</span>
  67 <span class="special">{</span>
  68    <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">tgamma</span><span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</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">cyl_bessel_j</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">pow</span><span class="special">(</span><span class="identifier">z</span> <span class="special">/</span> <span class="number">2</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
  69 <span class="special">}</span>
  70 </pre>
  71 <p>
  72             The implementation is almost the same as before, but with one key difference
  73             - we can no longer call <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span></code>,
  74             instead we must call the version inside the <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span></code>
  75             namespace. In point of fact, we could have omitted the namespace prefix
  76             on the call to <code class="computeroutput"><span class="identifier">pow</span></code> since
  77             the right overload would have been found via <a href="http://en.wikipedia.org/wiki/Argument-dependent_name_lookup" target="_top">argument
  78             dependent lookup</a> in any case.
  79           </p>
  80 <p>
  81             Note also that the first argument to <code class="computeroutput"><span class="identifier">pow</span></code>
  82             along with the argument to <code class="computeroutput"><span class="identifier">tgamma</span></code>
  83             in the above code are actually expression templates. The <code class="computeroutput"><span class="identifier">pow</span></code> and <code class="computeroutput"><span class="identifier">tgamma</span></code>
  84             functions will handle these arguments just fine.
  85           </p>
  86 <p>
  87             Here's an example of how the function may be called:
  88           </p>
  89 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span>
  90 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">JEL2</span><span class="special">(</span><span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">2.5</span><span class="special">),</span> <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">0.5</span><span class="special">))</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>
  91 </pre>
  92 <p>
  93             Which outputs:
  94           </p>
  95 <pre class="programlisting">9.82266396479604757017335009796882833995903762577173e-01</pre>
  96 <p>
  97             Now that we've seen some non-template examples, lets repeat the code
  98             again, but this time as a template that can be called either with a builtin
  99             type (<code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code> etc), or with a multiprecision
 100             type:
 101           </p>
 102 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Float</span><span class="special">&gt;</span>
 103 <span class="identifier">Float</span> <span class="identifier">JEL3</span><span class="special">(</span><span class="identifier">Float</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">Float</span> <span class="identifier">z</span><span class="special">)</span>
 104 <span class="special">{</span>
 105    <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span><span class="special">;</span>
 106    <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">tgamma</span><span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</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">cyl_bessel_j</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">z</span> <span class="special">/</span> <span class="number">2</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
 107 <span class="special">}</span>
 108 </pre>
 109 <p>
 110             Once again the code is almost the same as before, but the call to <code class="computeroutput"><span class="identifier">pow</span></code> has changed yet again. We need
 111             the call to resolve to either <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span></code>
 112             (when the argument is a builtin type), or to <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">pow</span></code>
 113             (when the argument is a multiprecision type). We do that by making the
 114             call unqualified so that versions of <code class="computeroutput"><span class="identifier">pow</span></code>
 115             defined in the same namespace as type <code class="computeroutput"><span class="identifier">Float</span></code>
 116             are found via argument dependent lookup, while the <code class="computeroutput"><span class="keyword">using</span>
 117             <span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span></code> directive makes the standard library
 118             versions visible for builtin floating point types.
 119           </p>
 120 <p>
 121             Let's call the function with both <code class="computeroutput"><span class="keyword">double</span></code>
 122             and multiprecision arguments:
 123           </p>
 124 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span>
 125 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">JEL3</span><span class="special">(</span><span class="number">2.5</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</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>
 126 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span>
 127 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">JEL3</span><span class="special">(</span><span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">2.5</span><span class="special">),</span> <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">0.5</span><span class="special">))</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>
 128 </pre>
 129 <p>
 130             Which outputs:
 131           </p>
 132 <pre class="programlisting">9.822663964796047e-001
 133 9.82266396479604757017335009796882833995903762577173e-01
 134 </pre>
 135 <p>
 136             Unfortunately there is a problem with this version: if we were to call
 137             it like this:
 138           </p>
 139 <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">v</span><span class="special">(</span><span class="number">2</span><span class="special">),</span> <span class="identifier">z</span><span class="special">(</span><span class="number">0.5</span><span class="special">);</span>
 140 <span class="identifier">JEL3</span><span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="number">0.5</span><span class="special">,</span> <span class="identifier">z</span><span class="special">);</span>
 141 </pre>
 142 <p>
 143             Then we would get a long and inscrutable error message from the compiler:
 144             the problem here is that the first argument to <code class="computeroutput"><span class="identifier">JEL3</span></code>
 145             is not a number type, but an expression template. We could obviously
 146             add a typecast to fix the issue:
 147           </p>
 148 <pre class="programlisting"><span class="identifier">JEL</span><span class="special">(</span><span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="number">0.5</span><span class="special">),</span> <span class="identifier">z</span><span class="special">);</span>
 149 </pre>
 150 <p>
 151             However, if we want the function JEL to be truly reusable, then a better
 152             solution might be preferred. To achieve this we can borrow some code
 153             from Boost.Math which calculates the return type of mixed-argument functions,
 154             here's how the new code looks now:
 155           </p>
 156 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Float1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Float2</span><span class="special">&gt;</span>
 157 <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special">&lt;</span><span class="identifier">Float1</span><span class="special">,</span> <span class="identifier">Float2</span><span class="special">&gt;::</span><span class="identifier">type</span>
 158    <span class="identifier">JEL4</span><span class="special">(</span><span class="identifier">Float1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">Float2</span> <span class="identifier">z</span><span class="special">)</span>
 159 <span class="special">{</span>
 160    <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span><span class="special">;</span>
 161    <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">tgamma</span><span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</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">cyl_bessel_j</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">z</span> <span class="special">/</span> <span class="number">2</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
 162 <span class="special">}</span>
 163 </pre>
 164 <p>
 165             As you can see the two arguments to the function are now separate template
 166             types, and the return type is computed using the <code class="computeroutput"><span class="identifier">promote_args</span></code>
 167             metafunction from Boost.Math.
 168           </p>
 169 <p>
 170             Now we can call:
 171           </p>
 172 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_100</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span>
 173 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">JEL4</span><span class="special">(</span><span class="identifier">cpp_dec_float_100</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">+</span> <span class="number">0.5</span><span class="special">,</span> <span class="identifier">cpp_dec_float_100</span><span class="special">(</span><span class="number">0.5</span><span class="special">))</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>
 174 </pre>
 175 <p>
 176             And get 100 digits of output:
 177           </p>
 178 <pre class="programlisting">9.8226639647960475701733500979688283399590376257717309069410413822165082248153638454147004236848917775e-01</pre>
 179 <p>
 180             As a bonus, we can now call the function not just with expression templates,
 181             but with other mixed types as well: for example <code class="computeroutput"><span class="keyword">float</span></code>
 182             and <code class="computeroutput"><span class="keyword">double</span></code> or <code class="computeroutput"><span class="keyword">int</span></code> and <code class="computeroutput"><span class="keyword">double</span></code>,
 183             and the correct return type will be computed in each case.
 184           </p>
 185 <p>
 186             Note that while in this case we didn't have to change the body of the
 187             function, in the general case any function like this which creates local
 188             variables internally would have to use <code class="computeroutput"><span class="identifier">promote_args</span></code>
 189             to work out what type those variables should be, for example:
 190           </p>
 191 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Float1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Float2</span><span class="special">&gt;</span>
 192 <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special">&lt;</span><span class="identifier">Float1</span><span class="special">,</span> <span class="identifier">Float2</span><span class="special">&gt;::</span><span class="identifier">type</span>
 193    <span class="identifier">JEL5</span><span class="special">(</span><span class="identifier">Float1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">Float2</span> <span class="identifier">z</span><span class="special">)</span>
 194 <span class="special">{</span>
 195    <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span><span class="special">;</span>
 196    <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special">&lt;</span><span class="identifier">Float1</span><span class="special">,</span> <span class="identifier">Float2</span><span class="special">&gt;::</span><span class="identifier">type</span> <span class="identifier">variable_type</span><span class="special">;</span>
 197    <span class="identifier">variable_type</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">z</span> <span class="special">/</span> <span class="number">2</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
 198    <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">tgamma</span><span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</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">cyl_bessel_j</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">t</span><span class="special">;</span>
 199 <span class="special">}</span>
 200 </pre>
 201 </div>
 202 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 203 <td align="left"></td>
 204 <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2019 John Maddock
 205       and Christopher Kormanyos<p>
 206         Distributed under the Boost Software License, Version 1.0. (See accompanying
 207         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>)
 208       </p>
 209 </div></td>
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