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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Implementation</title>
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<table cellpadding="2" width="100%"><tr>
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<a accesskey="p" href="../special_tut.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../special_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="special_tut_test.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.special_tut.special_tut_impl"></a><a class="link" href="special_tut_impl.html" title="Implementation">Implementation</a>
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</h3></div></div></div>
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<p>
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In this section, we'll provide a "recipe" for adding a new special
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function to this library to make life easier for future authors wishing to
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contribute. We'll assume the function returns a single floating-point result,
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and takes two floating-point arguments. For the sake of exposition we'll
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give the function the name <em class="replaceable"><code>my_special</code></em>.
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</p>
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<p>
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Normally, the implementation of such a function is split into two layers
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- a public user layer, and an internal implementation layer that does the
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actual work. The implementation layer is declared inside a <code class="computeroutput"><span class="identifier">detail</span></code> namespace and has a simple signature:
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</p>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">detail</span> <span class="special">{</span>
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<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">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<span class="identifier">T</span> <span class="identifier">my_special_imp</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> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span><span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="comment">/* Implementation goes here */</span>
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<span class="special">}</span>
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<span class="special">}}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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We'll come back to what can go inside the implementation later, but first
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lets look at the user layer. This consists of two overloads of the function,
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with and without a <a class="link" href="../../policy.html" title="Chapter 18. Policies: Controlling Precision, Error Handling etc">Policy</a> argument:
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</p>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<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">class</span> <span class="identifier">U</span><span class="special">></span>
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<span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</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> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
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<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">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</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> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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Note how each argument has a different template type - this allows for mixed
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type arguments - the return type is computed from a traits class and is the
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"common type" of all the arguments after any integer arguments
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have been promoted to type <code class="computeroutput"><span class="keyword">double</span></code>.
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</p>
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<p>
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The implementation of the non-policy overload is trivial:
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</p>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<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">class</span> <span class="identifier">U</span><span class="special">></span>
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<span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</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> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="comment">// Simply forward with a default policy:</span>
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<span class="keyword">return</span> <span class="identifier">my_special</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><>();</span>
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<span class="special">}</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The implementation of the other overload is somewhat more complex, as there's
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some meta-programming to do, but from a runtime perspective is still a one-line
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forwarding function. Here it is with comments explaining what each line does:
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</p>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<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">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</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> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="comment">//</span>
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<span class="comment">// We've found some standard library functions to misbehave if any FPU exception flags</span>
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<span class="comment">// are set prior to their call, this code will clear those flags, then reset them</span>
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<span class="comment">// on exit:</span>
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<span class="comment">//</span>
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<span class="identifier">BOOST_FPU_EXCEPTION_GUARD</span>
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<span class="comment">//</span>
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<span class="comment">// The type of the result - the common type of T and U after</span>
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<span class="comment">// any integer types have been promoted to double:</span>
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<span class="comment">//</span>
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<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">result_type</span><span class="special">;</span>
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<span class="comment">//</span>
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<span class="comment">// The type used for the calculation. This may be a wider type than</span>
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<span class="comment">// the result in order to ensure full precision:</span>
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<span class="comment">//</span>
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<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">evaluation</span><span class="special"><</span><span class="identifier">result_type</span><span class="special">,</span> <span class="identifier">Policy</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="comment">//</span>
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<span class="comment">// The type of the policy to forward to the actual implementation.</span>
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<span class="comment">// We disable promotion of float and double as that's [possibly]</span>
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<span class="comment">// happened already in the line above. Also reset to the default</span>
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<span class="comment">// any policies we don't use (reduces code bloat if we're called</span>
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<span class="comment">// multiple times with differing policies we don't actually use).</span>
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<span class="comment">// Also normalise the type, again to reduce code bloat in case we're</span>
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<span class="comment">// called multiple times with functionally identical policies that happen</span>
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<span class="comment">// to be different types.</span>
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<span class="comment">//</span>
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<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">normalise</span><span class="special"><</span>
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<span class="identifier">Policy</span><span class="special">,</span>
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<span class="identifier">policies</span><span class="special">::</span><span class="identifier">promote_float</span><span class="special"><</span><span class="keyword">false</span><span class="special">>,</span>
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<span class="identifier">policies</span><span class="special">::</span><span class="identifier">promote_double</span><span class="special"><</span><span class="keyword">false</span><span class="special">>,</span>
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<span class="identifier">policies</span><span class="special">::</span><span class="identifier">discrete_quantile</span><span class="special"><>,</span>
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<span class="identifier">policies</span><span class="special">::</span><span class="identifier">assert_undefined</span><span class="special"><></span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">forwarding_policy</span><span class="special">;</span>
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<span class="comment">//</span>
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<span class="comment">// Whew. Now we can make the actual call to the implementation.</span>
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<span class="comment">// Arguments are explicitly cast to the evaluation type, and the result</span>
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<span class="comment">// passed through checked_narrowing_cast which handles things like overflow</span>
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<span class="comment">// according to the policy passed:</span>
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<span class="comment">//</span>
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<span class="keyword">return</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">checked_narrowing_cast</span><span class="special"><</span><span class="identifier">result_type</span><span class="special">,</span> <span class="identifier">forwarding_policy</span><span class="special">>(</span>
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<span class="identifier">detail</span><span class="special">::</span><span class="identifier">my_special_imp</span><span class="special">(</span>
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<span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">value_type</span><span class="special">>(</span><span class="identifier">a</span><span class="special">),</span>
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<span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">value_type</span><span class="special">>(</span><span class="identifier">x</span><span class="special">),</span>
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<span class="identifier">forwarding_policy</span><span class="special">()),</span>
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<span class="string">"boost::math::my_special<%1%>(%1%, %1%)"</span><span class="special">);</span>
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<span class="special">}</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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We're now almost there, we just need to flesh out the details of the implementation
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layer:
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</p>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">detail</span> <span class="special">{</span>
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<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">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<span class="identifier">T</span> <span class="identifier">my_special_imp</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> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span><span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="comment">/* Implementation goes here */</span>
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<span class="special">}</span>
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<span class="special">}}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The following guidelines indicate what (other than basic arithmetic) can
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go in the implementation:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Error conditions (for example bad arguments) should be handled by calling
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one of the <a class="link" href="../error_handling.html#math_toolkit.error_handling.finding_more_information">policy
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based error handlers</a>.
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</li>
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<li class="listitem">
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Calls to standard library functions should be made unqualified (this
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allows argument dependent lookup to find standard library functions for
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user-defined floating point types such as those from <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>).
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In addition, the macro <code class="computeroutput"><span class="identifier">BOOST_MATH_STD_USING</span></code>
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should appear at the start of the function (note no semi-colon afterwards!)
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so that all the math functions in <code class="computeroutput"><span class="keyword">namespace</span>
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<span class="identifier">std</span></code> are visible in the current
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scope.
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</li>
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<li class="listitem">
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Calls to other special functions should be made as fully qualified calls,
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and include the policy parameter as the last argument, for example <code class="computeroutput"><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">a</span><span class="special">,</span> <span class="identifier">pol</span><span class="special">)</span></code>.
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</li>
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<li class="listitem">
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Where possible, evaluation of series, continued fractions, polynomials,
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or root finding should use one of the <a class="link" href="../internals_overview.html" title="Overview">boiler-plate
|
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functions</a>. In any case, after any iterative method, you should
|
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verify that the number of iterations did not exceed the maximum specified
|
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in the <a class="link" href="../../policy.html" title="Chapter 18. Policies: Controlling Precision, Error Handling etc">Policy</a> type, and if it did terminate
|
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as a result of exceeding the maximum, then the appropriate error handler
|
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should be called (see existing code for examples).
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</li>
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<li class="listitem">
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Numeric constants such as π etc should be obtained via a call to the <a class="link" href="../constants.html" title="The Mathematical Constants">appropriate function</a>, for example:
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<code class="computeroutput"><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></code>.
|
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</li>
|
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<li class="listitem">
|
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Where tables of coefficients are used (for example for rational approximations),
|
|
care should be taken to ensure these are initialized at program startup
|
|
to ensure thread safety when using user-defined number types. See for
|
|
example the use of <code class="computeroutput"><span class="identifier">erf_initializer</span></code>
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in <a href="../../../../include/boost/math/special_functions/erf.hpp" target="_top">erf.hpp</a>.
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</li>
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</ul></div>
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<p>
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Here are some other useful internal functions:
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</p>
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<div class="informaltable"><table class="table">
|
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<colgroup>
|
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<col>
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<col>
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</colgroup>
|
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<thead><tr>
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<th>
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<p>
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function
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</p>
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</th>
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<th>
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<p>
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Meaning
|
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</p>
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</th>
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</tr></thead>
|
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<tbody>
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<tr>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">policies</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span>
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<span class="identifier">Policy</span><span class="special">>()</span></code>
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</p>
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</td>
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<td>
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<p>
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Returns number of binary digits in T (possible overridden by the
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policy).
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">policies</span><span class="special">::</span><span class="identifier">get_max_series_iterations</span><span class="special"><</span><span class="identifier">Policy</span><span class="special">>()</span></code>
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</p>
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</td>
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<td>
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<p>
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Maximum number of iterations for series evaluation.
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</p>
|
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</td>
|
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</tr>
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<tr>
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<td>
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<p>
|
|
<code class="computeroutput"><span class="identifier">policies</span><span class="special">::</span><span class="identifier">get_max_root_iterations</span><span class="special"><</span><span class="identifier">Policy</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Maximum number of iterations for root finding.
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">polices</span><span class="special">::</span><span class="identifier">get_epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span>
|
|
<span class="identifier">Policy</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Epsilon for type T, possibly overridden by the Policy.
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Returns the number of binary digits in T.
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">max_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Equivalent to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">min_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Equivalent to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span></code>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_max_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Equivalent to the natural logarithm of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_min_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Equivalent to the natural logarithm of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span></code>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Equivalent to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>.
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Equivalent to the square root of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>.
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">forth_root_epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Equivalent to the forth root of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>.
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014, 2017 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam
|
|
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
|
|
and Xiaogang Zhang<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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
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