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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.high_precision.use_multiprecision"></a><a class="link" href="use_multiprecision.html" title="Using Boost.Multiprecision">Using |
| Boost.Multiprecision</a> |
| </h3></div></div></div> |
| <p> |
| <span class="bold"><strong>All new projects are recommended to use <a href="http://www.boost.org/doc/libs/1_53_0_beta1/libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span> |
| </p> |
| <h6> |
| <a name="math_toolkit.high_precision.use_multiprecision.h0"></a> |
| <span class="phrase"><a name="math_toolkit.high_precision.use_multiprecision.using_boost_multiprecision_cpp_f"></a></span><a class="link" href="use_multiprecision.html#math_toolkit.high_precision.use_multiprecision.using_boost_multiprecision_cpp_f">Using |
| Boost.Multiprecision <code class="computeroutput"><span class="identifier">cpp_float</span></code> |
| for numerical calculations with high precision.</a> |
| </h6> |
| <p> |
| The Boost.Multiprecision library can be used for computations requiring precision |
| exceeding that of standard built-in types such as float, double and long |
| double. For extended-precision calculations, Boost.Multiprecision supplies |
| a template data type called cpp_dec_float. The number of decimal digits of |
| precision is fixed at compile-time via template parameter. |
| </p> |
| <p> |
| To use these floating-point types and constants, we need some includes: |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">constants</span><span class="special">/</span><span class="identifier">constants</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| |
| <span class="preprocessor">#include</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</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="comment">// using boost::multiprecision::cpp_dec_float</span> |
| |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> |
| </pre> |
| <p> |
| So now we can demonstrate with some trivial calculations: |
| </p> |
| <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> |
| <span class="special">{</span> |
| </pre> |
| <p> |
| Using <code class="computeroutput"><span class="keyword">typedef</span> <span class="identifier">cpp_dec_float_50</span></code> |
| hides the complexity of multiprecision to allow us to define variables with |
| 50 decimal digit precision just like built-in <code class="computeroutput"><span class="keyword">double</span></code>. |
| </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">cpp_dec_float_50</span><span class="special">;</span> |
| |
| <span class="identifier">cpp_dec_float_50</span> <span class="identifier">seventh</span> <span class="special">=</span> <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span> <span class="number">7</span><span class="special">;</span> |
| </pre> |
| <p> |
| By default, output would only show the standard 6 decimal digits, so set |
| precision to show all 50 significant digits. |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">seventh</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| which outputs: |
| </p> |
| <pre class="programlisting"><span class="number">0.14285714285714285714285714285714285714285714285714</span> |
| </pre> |
| <p> |
| We can also use constants, guaranteed to be initialized with the very last |
| bit of precision. |
| </p> |
| <pre class="programlisting"><span class="identifier">cpp_dec_float_50</span> <span class="identifier">circumference</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>()</span> <span class="special">*</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">seventh</span><span class="special">;</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">circumference</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| which outputs |
| </p> |
| <pre class="programlisting"><span class="number">0.89759790102565521098932668093700082405633411410717</span> |
| </pre> |
| <h6> |
| <a name="math_toolkit.high_precision.use_multiprecision.h1"></a> |
| <span class="phrase"><a name="math_toolkit.high_precision.use_multiprecision.using_boost_multiprecision_to_ge"></a></span><a class="link" href="use_multiprecision.html#math_toolkit.high_precision.use_multiprecision.using_boost_multiprecision_to_ge">Using |
| Boost.Multiprecision to generate a high-precision array of sin coefficents |
| for use with FFT.</a> |
| </h6> |
| <p> |
| The Boost.Multiprecision library can be used for computations requiring precision |
| exceeding that of standard built-in types such as <code class="computeroutput"><span class="keyword">float</span></code>, |
| <code class="computeroutput"><span class="keyword">double</span></code> and <code class="computeroutput"><span class="keyword">long</span> |
| <span class="keyword">double</span></code>. For extended-precision calculations, |
| Boost.Multiprecision supplies a template data type called <code class="computeroutput"><span class="identifier">cpp_dec_float</span></code>. |
| The number of decimal digits of precision is fixed at compile-time via template |
| parameter. |
| </p> |
| <p> |
| To use these floating-point types and constants, we need some includes: |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">constants</span><span class="special">/</span><span class="identifier">constants</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="comment">// using boost::math::constants::pi;</span> |
| |
| <span class="preprocessor">#include</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</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="comment">// using boost::multiprecision::cpp_dec_float</span> |
| |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">algorithm</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iterator</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">fstream</span><span class="special">></span> |
| </pre> |
| <p> |
| Define a text string which is a C++ comment with the program licence, copyright |
| etc. You could of course, tailor this to your needs, including your copyright |
| claim. There are versions of <code class="computeroutput"><span class="identifier">array</span></code> |
| provided by Boost.Array in <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code> |
| or the C++11 std::array, but since not all platforms provide C++11 support, |
| this program provides the Boost version as fallback. |
| </p> |
| <pre class="programlisting"><span class="keyword">static</span> <span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span> <span class="identifier">prolog</span> <span class="special">=</span> |
| <span class="special">{</span> |
| <span class="string">"// Use, modification and distribution are subject to the\n"</span> |
| <span class="string">"// Boost Software License, Version 1.0.\n"</span> |
| <span class="string">"// (See accompanying file LICENSE_1_0.txt\n"</span> |
| <span class="string">"// or copy at "</span><span class="string">"http://www.boost.org/LICENSE_1_0.txt)\n\n"</span> |
| |
| <span class="string">"// Copyright ???? 2013.\n\n"</span> |
| |
| <span class="string">"// Use boost/array if std::array (C++11 feature) is not available.\n"</span> |
| <span class="string">"#ifdef BOOST_NO_CXX11_HDR_ARRAY\n"</span> |
| <span class="string">"#include <boost/array/array.hpp>\n"</span> |
| <span class="string">"#else\n"</span> |
| <span class="string">"#include <array>\n"</span> |
| <span class="string">"#endif\n\n"</span> |
| <span class="special">};</span> |
| |
| |
| <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">cpp_dec_float_50</span><span class="special">;</span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">;</span> |
| <span class="comment">// VS 2010 (wrongly) requires these at file scope, not local scope in `main`.</span> |
| <span class="comment">// This program also requires `-std=c++11` option to compile using Clang and GCC.</span> |
| |
| <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> |
| <span class="special">{</span> |
| </pre> |
| <p> |
| One often needs to compute tables of numbers in mathematical software. |
| </p> |
| <p> |
| A fast Fourier transform (FFT), for example, may use a table of the values |
| of sin((π/2<sup>n</sup>) in its implementation details. In order to maximize the precision |
| in the FFT implementation, the precision of the tabulated trigonometric values |
| should exceed that of the built-in floating-point type used in the FFT. |
| </p> |
| <p> |
| The sample below computes a table of the values of sin(π/2<sup>n</sup>) in the range |
| 1 <= n <= 31. |
| </p> |
| <p> |
| This program makes use of, among other program elements, the data type <code class="computeroutput"><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></code> for a precision of 50 |
| decimal digits from Boost.Multiprecision, the value of constant π retrieved |
| from Boost.Math, guaranteed to be initialized with the very last bit of precision |
| for the type, here <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>, |
| and a C++11 lambda function combined with <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">for_each</span><span class="special">()</span></code>. |
| </p> |
| <p> |
| define the number of values in the array. |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">size</span> <span class="special">=</span> <span class="number">32U</span><span class="special">;</span> |
| <span class="identifier">cpp_dec_float_50</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>();</span> |
| <span class="identifier">cpp_dec_float_50</span> <span class="identifier">p2</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>();</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span> <span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">></span> <span class="identifier">sin_values</span> <span class="special">(</span><span class="identifier">size</span><span class="special">);</span> |
| <span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">1U</span><span class="special">;</span> |
| <span class="comment">// Generate the sine values.</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">for_each</span> |
| <span class="special">(</span> |
| <span class="identifier">sin_values</span><span class="special">.</span><span class="identifier">begin</span> <span class="special">(),</span> |
| <span class="identifier">sin_values</span><span class="special">.</span><span class="identifier">end</span> <span class="special">(),</span> |
| <span class="special">[&</span><span class="identifier">n</span><span class="special">](</span><span class="identifier">cpp_dec_float_50</span><span class="special">&</span> <span class="identifier">y</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>()</span> <span class="special">/</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">cpp_dec_float_50</span> <span class="special">(</span><span class="number">2</span><span class="special">),</span> <span class="identifier">n</span><span class="special">));</span> |
| <span class="special">++</span><span class="identifier">n</span><span class="special">;</span> |
| <span class="special">}</span> |
| <span class="special">);</span> |
| </pre> |
| <p> |
| Define the floating-point type for the generated file, either built-in <code class="computeroutput"><span class="keyword">double</span><span class="special">,</span> </code>float, |
| or <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>, |
| or a user defined type like <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>. |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">fp_type</span> <span class="special">=</span> <span class="string">"double"</span><span class="special">;</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Generating an `std::array` or `boost::array` for floating-point type: "</span> |
| <span class="special"><<</span> <span class="identifier">fp_type</span> <span class="special"><<</span> <span class="string">". "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| By default, output would only show the standard 6 decimal digits, so set |
| precision to show enough significant digits for the chosen floating-point |
| type. For <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code> |
| is 50. (50 decimal digits should be ample for most applications). |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">;</span> |
| |
| <span class="comment">// std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">precision</span> <span class="special"><<</span> <span class="string">" decimal digits precision. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| Of course, one could also choose less, for example, 36 would be sufficient |
| for the most precise current <code class="computeroutput"><span class="keyword">long</span> |
| <span class="keyword">double</span></code> implementations using 128-bit. |
| In general, it should be a couple of decimal digits more (guard digits) than |
| <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">RealType</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> for the target system floating-point |
| type. If the implementation does not provide <code class="computeroutput"><span class="identifier">max_digits10</span></code>, |
| the the Kahan formula <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">RealType</span><span class="special">>::</span><span class="identifier">digits</span> |
| <span class="special">*</span> <span class="number">3010</span><span class="special">/</span><span class="number">10000</span> <span class="special">+</span> |
| <span class="number">2</span></code> can be used instead. |
| </p> |
| <p> |
| The compiler will read these values as decimal digits strings and use the |
| nearest representation for the floating-point type. |
| </p> |
| <p> |
| Now output all the sine table, to a file of your chosen name. |
| </p> |
| <pre class="programlisting"> <span class="keyword">const</span> <span class="keyword">char</span> <span class="identifier">sines_name</span><span class="special">[]</span> <span class="special">=</span> <span class="string">"sines.hpp"</span><span class="special">;</span> <span class="comment">// In same directory as .exe</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">ofstream</span> <span class="identifier">fout</span><span class="special">(</span><span class="identifier">sines_name</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ios_base</span><span class="special">::</span><span class="identifier">out</span><span class="special">);</span> <span class="comment">// Creates if no file exists,</span> |
| <span class="comment">// & uses default overwrite/ ios::replace.</span> |
| <span class="keyword">if</span> <span class="special">(</span><span class="identifier">fout</span><span class="special">.</span><span class="identifier">is_open</span><span class="special">()</span> <span class="special">==</span> <span class="keyword">false</span><span class="special">)</span> |
| <span class="special">{</span> <span class="comment">// failed to open OK!</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Open file "</span> <span class="special"><<</span> <span class="identifier">sines_name</span> <span class="special"><<</span> <span class="string">" failed!"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="keyword">return</span> <span class="identifier">EXIT_FAILURE</span><span class="special">;</span> |
| <span class="special">}</span> |
| <span class="keyword">else</span> |
| <span class="special">{</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Open file "</span> <span class="special"><<</span> <span class="identifier">sines_name</span> <span class="special"><<</span> <span class="string">" for output OK."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">fout</span> <span class="special"><<</span> <span class="identifier">prolog</span> <span class="special"><<</span> <span class="string">"// Table of "</span> <span class="special"><<</span> <span class="identifier">sin_values</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special"><<</span> <span class="string">" values with "</span> |
| <span class="special"><<</span> <span class="identifier">precision</span> <span class="special"><<</span> <span class="string">" decimal digits precision,\n"</span> |
| <span class="string">"// generated by program fft_sines_table.cpp.\n"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">fout</span> <span class="special"><<</span> |
| <span class="string">"#ifdef BOOST_NO_CXX11_HDR_ARRAY"</span><span class="string">"\n"</span> |
| <span class="string">" static const boost::array<double, "</span> <span class="special"><<</span> <span class="identifier">size</span> <span class="special"><<</span> <span class="string">"> sines =\n"</span> |
| <span class="string">"#else"</span><span class="string">"\n"</span> |
| <span class="string">" static const std::array<double, "</span> <span class="special"><<</span> <span class="identifier">size</span> <span class="special"><<</span> <span class="string">"> sines =\n"</span> |
| <span class="string">"#endif"</span><span class="string">"\n"</span> |
| <span class="string">"{{\n"</span><span class="special">;</span> <span class="comment">// 2nd { needed for some GCC compiler versions.</span> |
| <span class="identifier">fout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">precision</span><span class="special">);</span> |
| |
| <span class="keyword">for</span> <span class="special">(</span><span class="keyword">unsigned</span> <span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0U</span><span class="special">;</span> <span class="special">;)</span> |
| <span class="special">{</span> |
| <span class="identifier">fout</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">sin_values</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span> |
| <span class="keyword">if</span> <span class="special">(</span><span class="identifier">i</span> <span class="special">==</span> <span class="identifier">sin_values</span><span class="special">.</span><span class="identifier">size</span><span class="special">()-</span><span class="number">1</span><span class="special">)</span> |
| <span class="special">{</span> <span class="comment">// next is last value.</span> |
| <span class="identifier">fout</span> <span class="special"><<</span> <span class="string">"\n}};\n"</span><span class="special">;</span> <span class="comment">// 2nd } needed for some earlier GCC compiler versions.</span> |
| <span class="keyword">break</span><span class="special">;</span> |
| <span class="special">}</span> |
| <span class="keyword">else</span> |
| <span class="special">{</span> |
| <span class="identifier">fout</span> <span class="special"><<</span> <span class="string">",\n"</span><span class="special">;</span> |
| <span class="identifier">i</span><span class="special">++;</span> |
| <span class="special">}</span> |
| <span class="special">}</span> |
| |
| <span class="identifier">fout</span><span class="special">.</span><span class="identifier">close</span><span class="special">();</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Close file "</span> <span class="special"><<</span> <span class="identifier">sines_name</span> <span class="special"><<</span> <span class="string">" for output OK."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="special">}</span> |
| </pre> |
| <p> |
| The output file generated can be seen at <a href="../../../../example/sines.hpp" target="_top">../../example/sines.hpp</a> |
| </p> |
| <p> |
| The table output is: |
| </p> |
| <pre class="programlisting"><span class="identifier">The</span> <span class="identifier">printed</span> <span class="identifier">table</span> <span class="identifier">is</span><span class="special">:</span> |
| |
| <span class="number">1</span> |
| <span class="number">0.70710678118654752440084436210484903928483593768847</span> |
| <span class="number">0.38268343236508977172845998403039886676134456248563</span> |
| <span class="number">0.19509032201612826784828486847702224092769161775195</span> |
| <span class="number">0.098017140329560601994195563888641845861136673167501</span> |
| <span class="number">0.049067674327418014254954976942682658314745363025753</span> |
| <span class="number">0.024541228522912288031734529459282925065466119239451</span> |
| <span class="number">0.012271538285719926079408261951003212140372319591769</span> |
| <span class="number">0.0061358846491544753596402345903725809170578863173913</span> |
| <span class="number">0.003067956762965976270145365490919842518944610213452</span> |
| <span class="number">0.0015339801862847656123036971502640790799548645752374</span> |
| <span class="number">0.00076699031874270452693856835794857664314091945206328</span> |
| <span class="number">0.00038349518757139558907246168118138126339502603496474</span> |
| <span class="number">0.00019174759731070330743990956198900093346887403385916</span> |
| <span class="number">9.5873799095977345870517210976476351187065612851145e-05</span> |
| <span class="number">4.7936899603066884549003990494658872746866687685767e-05</span> |
| <span class="number">2.3968449808418218729186577165021820094761474895673e-05</span> |
| <span class="number">1.1984224905069706421521561596988984804731977538387e-05</span> |
| <span class="number">5.9921124526424278428797118088908617299871778780951e-06</span> |
| <span class="number">2.9960562263346607504548128083570598118251878683408e-06</span> |
| <span class="number">1.4980281131690112288542788461553611206917585861527e-06</span> |
| <span class="number">7.4901405658471572113049856673065563715595930217207e-07</span> |
| <span class="number">3.7450702829238412390316917908463317739740476297248e-07</span> |
| <span class="number">1.8725351414619534486882457659356361712045272098287e-07</span> |
| <span class="number">9.3626757073098082799067286680885620193236507169473e-08</span> |
| <span class="number">4.681337853654909269511551813854009695950362701667e-08</span> |
| <span class="number">2.3406689268274552759505493419034844037886207223779e-08</span> |
| <span class="number">1.1703344634137277181246213503238103798093456639976e-08</span> |
| <span class="number">5.8516723170686386908097901008341396943900085051757e-09</span> |
| <span class="number">2.9258361585343193579282304690689559020175857150074e-09</span> |
| <span class="number">1.4629180792671596805295321618659637103742615227834e-09</span> |
| <span class="special">*/</span> |
| </pre> |
| <p> |
| The output can be copied as text and readily integrated into a given source |
| code. Alternatively, the output can be written to a text or even be used |
| within a self-written automatic code generator as this example. |
| </p> |
| <p> |
| A computer algebra system can be used to verify the results obtained from |
| Boost.Math and Boost.Multiprecision. For example, the <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram |
| Mathematica</a> computer algebra system can obtain a similar table with |
| the command: |
| </p> |
| <pre class="programlisting"><span class="identifier">Table</span><span class="special">[</span><span class="identifier">N</span><span class="special">[</span><span class="identifier">Sin</span><span class="special">[</span><span class="identifier">Pi</span> <span class="special">/</span> <span class="special">(</span><span class="number">2</span><span class="special">^</span><span class="identifier">n</span><span class="special">)],</span> <span class="number">50</span><span class="special">],</span> <span class="special">{</span><span class="identifier">n</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">31</span><span class="special">,</span> <span class="number">1</span><span class="special">}]</span> |
| </pre> |
| <p> |
| The <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> computational |
| knowledge engine can also be used to generate this table. The same command |
| can be pasted into the compute box. |
| </p> |
| </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 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| 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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