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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.high_precision.use_multiprecision"></a><a class="link" href="use_multiprecision.html" title="Using Boost.Multiprecision">Using
  28       Boost.Multiprecision</a>
  29 </h3></div></div></div>
  30 <p>
  31         <span class="bold"><strong>All new projects are recommended to use <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
  32       </p>
  33 <h6>
  34 <a name="math_toolkit.high_precision.use_multiprecision.h0"></a>
  35         <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
  36         Boost.Multiprecision <code class="computeroutput"><span class="identifier">cpp_float</span></code>
  37         for numerical calculations with high precision.</a>
  38       </h6>
  39 <p>
  40         The Boost.Multiprecision library can be used for computations requiring precision
  41         exceeding that of standard built-in types such as float, double and long
  42         double. For extended-precision calculations, Boost.Multiprecision supplies
  43         a template data type called cpp_dec_float. The number of decimal digits of
  44         precision is fixed at compile-time via template parameter.
  45       </p>
  46 <p>
  47         To use these floating-point types and constants, we need some includes:
  48       </p>
  49 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
  50
  51 <span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
  52 <span class="comment">// using boost::multiprecision::cpp_dec_float</span>
  53
  54 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
  55 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
  56 </pre>
  57 <p>
  58         So now we can demonstrate with some trivial calculations:
  59       </p>
  60 <p>
  61         Using <code class="computeroutput"><span class="keyword">typedef</span> <span class="identifier">cpp_dec_float_50</span></code>
  62         hides the complexity of multiprecision, allows us to define variables with
  63         50 decimal digit precision just like built-in <code class="computeroutput"><span class="keyword">double</span></code>.
  64       </p>
  65 <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>
  66
  67 <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>  <span class="comment">// 1 / 7</span>
  68 </pre>
  69 <p>
  70         By default, output would only show the standard 6 decimal digits, so set
  71         precision to show all 50 significant digits, including any trailing zeros.
  72       </p>
  73 <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">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">);</span>
  74 <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">showpoint</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Append any trailing zeros.</span>
  75 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">seventh</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>
  76 </pre>
  77 <p>
  78         which outputs:
  79       </p>
  80 <pre class="programlisting"><span class="number">0.14285714285714285714285714285714285714285714285714</span>
  81 </pre>
  82 <p>
  83         We can also use Boost.Math <a class="link" href="../constants.html" title="The Mathematical Constants">constants</a>
  84         like &#960;, guaranteed to be initialized with the very last bit of precision for
  85         the floating-point type.
  86       </p>
  87 <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="string">"pi = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;()</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>
  88 <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">&lt;</span><span class="identifier">cpp_dec_float_50</span><span class="special">&gt;()</span> <span class="special">*</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">seventh</span><span class="special">;</span>
  89 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"c =  "</span><span class="special">&lt;&lt;</span> <span class="identifier">circumference</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>
  90 </pre>
  91 <p>
  92         which outputs
  93       </p>
  94 <pre class="programlisting"><span class="identifier">pi</span> <span class="special">=</span> <span class="number">3.1415926535897932384626433832795028841971693993751</span>
  95
  96 <span class="identifier">c</span> <span class="special">=</span>  <span class="number">0.89759790102565521098932668093700082405633411410717</span>
  97 </pre>
  98 <p>
  99         The full source of this example is at <a href="../../../../example/big_seventh.cpp" target="_top">big_seventh.cpp</a>
 100       </p>
 101 <h6>
 102 <a name="math_toolkit.high_precision.use_multiprecision.h1"></a>
 103         <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
 104         Boost.Multiprecision to generate a high-precision array of sine coefficents
 105         for use with FFT.</a>
 106       </h6>
 107 <p>
 108         The Boost.Multiprecision library can be used for computations requiring precision
 109         exceeding that of standard built-in types such as <code class="computeroutput"><span class="keyword">float</span></code>,
 110         <code class="computeroutput"><span class="keyword">double</span></code> and <code class="computeroutput"><span class="keyword">long</span>
 111         <span class="keyword">double</span></code>. For extended-precision calculations,
 112         Boost.Multiprecision supplies a template data type called <code class="computeroutput"><span class="identifier">cpp_bin_float</span></code>.
 113         The number of decimal digits of precision is fixed at compile-time via a
 114         template parameter.
 115       </p>
 116 <p>
 117         One often needs to compute tables of numbers in mathematical software. To
 118         avoid the <a href="https://en.wikipedia.org/wiki/Rounding#Table-maker's_dilemma" target="_top">Table-maker's
 119         dilemma</a> it is necessary to use a higher precision type to compute
 120         the table values so that they have the nearest representable bit-pattern
 121         for the type, say <code class="computeroutput"><span class="keyword">double</span></code>, of
 122         the table value.
 123       </p>
 124 <p>
 125         This example is a program <code class="computeroutput"><span class="identifier">fft_since_table</span><span class="special">.</span><span class="identifier">cpp</span></code> that
 126         writes a header file <code class="computeroutput"><span class="identifier">sines</span><span class="special">.</span><span class="identifier">hpp</span></code> containing
 127         an array of sine coefficients for use with a Fast Fourier Transform (FFT),
 128         that can be included by the FFT program.
 129       </p>
 130 <p>
 131         To use Boost.Multiprecision's high-precision floating-point types and constants,
 132         we need some includes:
 133       </p>
 134 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
 135 <span class="comment">// using boost::math::constants::pi;</span>
 136
 137 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_bin_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for</span>
 138 <span class="comment">// using boost::multiprecision::cpp_bin_float and</span>
 139 <span class="comment">// using boost::multiprecision::cpp_bin_float_50;</span>
 140 <span class="comment">// using boost::multiprecision::cpp_bin_float_quad;</span>
 141
 142 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">array</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// or &lt;array&gt; for std::array</span>
 143
 144 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 145 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
 146 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
 147 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">algorithm</span><span class="special">&gt;</span>
 148 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
 149 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iterator</span><span class="special">&gt;</span>
 150 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">fstream</span><span class="special">&gt;</span>
 151 </pre>
 152 <p>
 153         First, this example defines a prolog text string which is a C++ comment with
 154         the program licence, copyright etc. (You would of course, tailor this to
 155         your needs, including <span class="bold"><strong>your</strong></span> copyright claim).
 156         This will appear at the top of the written header file <code class="computeroutput"><span class="identifier">sines</span><span class="special">.</span><span class="identifier">hpp</span></code>.
 157       </p>
 158 <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_bin_float_50</span><span class="special">;</span>
 159 <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>
 160 </pre>
 161 <p>
 162         A fast Fourier transform (FFT), for example, may use a table of the values
 163         of sin((&#960;/2<sup>n</sup>) in its implementation details. In order to maximize the precision
 164         in the FFT implementation, the precision of the tabulated trigonometric values
 165         should exceed that of the built-in floating-point type used in the FFT.
 166       </p>
 167 <p>
 168         The sample below computes a table of the values of sin(&#960;/2<sup>n</sup>) in the range
 169         1 &lt;= n &lt;= 31.
 170       </p>
 171 <p>
 172         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_bin_float_50</span></code> for a precision of 50
 173         decimal digits from Boost.Multiprecision, the value of constant &#960; retrieved
 174         from Boost.Math, guaranteed to be initialized with the very last bit of precision
 175         for the type, here <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>,
 176         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>.
 177       </p>
 178 <p>
 179         define the number of values (32) in the array of sines.
 180       </p>
 181 <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>
 182 <span class="comment">//cpp_bin_float_50 p = pi&lt;cpp_bin_float_50&gt;();</span>
 183 <span class="identifier">cpp_bin_float_50</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">&lt;</span><span class="identifier">cpp_bin_float_50</span><span class="special">&gt;();</span>
 184
 185 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span> <span class="special">&lt;</span><span class="identifier">cpp_bin_float_50</span><span class="special">&gt;</span> <span class="identifier">sin_values</span> <span class="special">(</span><span class="identifier">size</span><span class="special">);</span>
 186 <span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">1U</span><span class="special">;</span>
 187 <span class="comment">// Generate the sine values.</span>
 188 <span class="identifier">std</span><span class="special">::</span><span class="identifier">for_each</span>
 189 <span class="special">(</span>
 190   <span class="identifier">sin_values</span><span class="special">.</span><span class="identifier">begin</span> <span class="special">(),</span>
 191   <span class="identifier">sin_values</span><span class="special">.</span><span class="identifier">end</span> <span class="special">(),</span>
 192   <span class="special">[&amp;</span><span class="identifier">n</span><span class="special">](</span><span class="identifier">cpp_bin_float_50</span><span class="special">&amp;</span> <span class="identifier">y</span><span class="special">)</span>
 193   <span class="special">{</span>
 194     <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">&lt;</span><span class="identifier">cpp_bin_float_50</span><span class="special">&gt;()</span> <span class="special">/</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">cpp_bin_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>
 195     <span class="special">++</span><span class="identifier">n</span><span class="special">;</span>
 196   <span class="special">}</span>
 197 <span class="special">);</span>
 198 </pre>
 199 <p>
 200         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,
 201         or <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>,
 202         or a user defined type like <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>.
 203       </p>
 204 <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>
 205
 206 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Generating an `std::array` or `boost::array` for floating-point type: "</span>
 207   <span class="special">&lt;&lt;</span> <span class="identifier">fp_type</span> <span class="special">&lt;&lt;</span> <span class="string">". "</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>
 208 </pre>
 209 <p>
 210         By default, output would only show the standard 6 decimal digits, so set
 211         precision to show enough significant digits for the chosen floating-point
 212         type. For <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>
 213         is 50. (50 decimal digits should be ample for most applications).
 214       </p>
 215 <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">&lt;</span><span class="identifier">cpp_bin_float_50</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">;</span>
 216
 217 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Sines table precision is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">precision</span> <span class="special">&lt;&lt;</span> <span class="string">" decimal digits. "</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>
 218 </pre>
 219 <p>
 220         Of course, one could also choose a lower precision for the table values,
 221         for example,
 222       </p>
 223 <p>
 224         <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">precision</span>
 225         <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_bin_float_quad</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span><span class="special">;</span></code>
 226       </p>
 227 <p>
 228         128-bit 'quad' precision of 36 decimal digits would be sufficient for the
 229         most precise current <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
 230         implementations using 128-bit. In general, it should be a couple of decimal
 231         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">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span></code>
 232         for the target system floating-point type. (If the implementation does not
 233         provide <code class="computeroutput"><span class="identifier">max_digits10</span></code>, the
 234         the Kahan formula <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">digits</span> <span class="special">*</span> <span class="number">3010</span><span class="special">/</span><span class="number">10000</span>
 235         <span class="special">+</span> <span class="number">2</span></code>
 236         can be used instead).
 237       </p>
 238 <p>
 239         The compiler will read these values as decimal digits strings and use the
 240         nearest representation for the floating-point type.
 241       </p>
 242 <p>
 243         Now output all the sine table, to a file of your chosen name.
 244       </p>
 245 <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">// Assuming in same directory as .exe</span>
 246
 247 <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>
 248 <span class="comment">// &amp; uses default overwrite/ ios::replace.</span>
 249 <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>
 250 <span class="special">{</span>  <span class="comment">// failed to open OK!</span>
 251   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Open file "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sines_name</span> <span class="special">&lt;&lt;</span> <span class="string">" failed!"</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>
 252   <span class="keyword">return</span> <span class="identifier">EXIT_FAILURE</span><span class="special">;</span>
 253 <span class="special">}</span>
 254 <span class="keyword">else</span>
 255 <span class="special">{</span> <span class="comment">// Write prolog etc as a C++ comment.</span>
 256   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Open file "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sines_name</span> <span class="special">&lt;&lt;</span> <span class="string">" for output OK."</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>
 257   <span class="identifier">fout</span> <span class="special">&lt;&lt;</span> <span class="identifier">prolog</span>
 258   <span class="special">&lt;&lt;</span> <span class="string">"// Table of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sin_values</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">" values with "</span>
 259     <span class="special">&lt;&lt;</span> <span class="identifier">precision</span> <span class="special">&lt;&lt;</span> <span class="string">" decimal digits precision,\n"</span>
 260     <span class="string">"// generated by program fft_sines_table.cpp.\n"</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>
 261
 262 <span class="identifier">fout</span> <span class="special">&lt;&lt;</span> <span class="string">"#include &lt;array&gt; // std::array"</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>
 263
 264 <span class="comment">// Write the table of sines as a C++ array.</span>
 265   <span class="identifier">fout</span> <span class="special">&lt;&lt;</span>  <span class="string">"\nstatic const std::array&lt;double, "</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span> <span class="special">&lt;&lt;</span> <span class="string">"&gt; sines =\n"</span>
 266   <span class="string">"{{\n"</span><span class="special">;</span> <span class="comment">// 2nd { needed for some old GCC compiler versions.</span>
 267   <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>
 268
 269   <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>
 270   <span class="special">{</span>
 271     <span class="identifier">fout</span> <span class="special">&lt;&lt;</span> <span class="string">"  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sin_values</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
 272     <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>
 273     <span class="special">{</span> <span class="comment">// next is last value.</span>
 274       <span class="identifier">fout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n}};  // array sines\n"</span><span class="special">;</span> <span class="comment">// 2nd } needed for some old GCC compiler versions.</span>
 275       <span class="keyword">break</span><span class="special">;</span>
 276     <span class="special">}</span>
 277     <span class="keyword">else</span>
 278     <span class="special">{</span>
 279       <span class="identifier">fout</span> <span class="special">&lt;&lt;</span> <span class="string">",\n"</span><span class="special">;</span>
 280       <span class="identifier">i</span><span class="special">++;</span>
 281     <span class="special">}</span>
 282   <span class="special">}</span> <span class="comment">// for</span>
 283
 284   <span class="identifier">fout</span><span class="special">.</span><span class="identifier">close</span><span class="special">();</span>
 285   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Closed file "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sines_name</span> <span class="special">&lt;&lt;</span> <span class="string">" for output."</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>
 286 <span class="special">}</span>
 287 </pre>
 288 <p>
 289         The output file generated can be seen at <a href="../../../../example/sines.hpp" target="_top">../../example/sines.hpp</a>
 290       </p>
 291 <p>
 292         The table output is:
 293       </p>
 294 <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>
 295
 296   <span class="number">1</span>
 297   <span class="number">0.70710678118654752440084436210484903928483593768847</span>
 298   <span class="number">0.38268343236508977172845998403039886676134456248563</span>
 299   <span class="number">0.19509032201612826784828486847702224092769161775195</span>
 300   <span class="number">0.098017140329560601994195563888641845861136673167501</span>
 301   <span class="number">0.049067674327418014254954976942682658314745363025753</span>
 302   <span class="number">0.024541228522912288031734529459282925065466119239451</span>
 303   <span class="number">0.012271538285719926079408261951003212140372319591769</span>
 304   <span class="number">0.0061358846491544753596402345903725809170578863173913</span>
 305   <span class="number">0.003067956762965976270145365490919842518944610213452</span>
 306   <span class="number">0.0015339801862847656123036971502640790799548645752374</span>
 307   <span class="number">0.00076699031874270452693856835794857664314091945206328</span>
 308   <span class="number">0.00038349518757139558907246168118138126339502603496474</span>
 309   <span class="number">0.00019174759731070330743990956198900093346887403385916</span>
 310   <span class="number">9.5873799095977345870517210976476351187065612851145e-05</span>
 311   <span class="number">4.7936899603066884549003990494658872746866687685767e-05</span>
 312   <span class="number">2.3968449808418218729186577165021820094761474895673e-05</span>
 313   <span class="number">1.1984224905069706421521561596988984804731977538387e-05</span>
 314   <span class="number">5.9921124526424278428797118088908617299871778780951e-06</span>
 315   <span class="number">2.9960562263346607504548128083570598118251878683408e-06</span>
 316   <span class="number">1.4980281131690112288542788461553611206917585861527e-06</span>
 317   <span class="number">7.4901405658471572113049856673065563715595930217207e-07</span>
 318   <span class="number">3.7450702829238412390316917908463317739740476297248e-07</span>
 319   <span class="number">1.8725351414619534486882457659356361712045272098287e-07</span>
 320   <span class="number">9.3626757073098082799067286680885620193236507169473e-08</span>
 321   <span class="number">4.681337853654909269511551813854009695950362701667e-08</span>
 322   <span class="number">2.3406689268274552759505493419034844037886207223779e-08</span>
 323   <span class="number">1.1703344634137277181246213503238103798093456639976e-08</span>
 324   <span class="number">5.8516723170686386908097901008341396943900085051757e-09</span>
 325   <span class="number">2.9258361585343193579282304690689559020175857150074e-09</span>
 326   <span class="number">1.4629180792671596805295321618659637103742615227834e-09</span>
 327 <span class="special">*/</span>
 328 </pre>
 329 <p>
 330         The output can be copied as text and readily integrated into a given source
 331         code. Alternatively, the output can be written to a text or even be used
 332         within a self-written automatic code generator as this example.
 333       </p>
 334 <p>
 335         A computer algebra system can be used to verify the results obtained from
 336         Boost.Math and Boost.Multiprecision. For example, the <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram
 337         Mathematica</a> computer algebra system can obtain a similar table with
 338         the command:
 339       </p>
 340 <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>
 341 </pre>
 342 <p>
 343         The <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> computational
 344         knowledge engine can also be used to generate this table. The same command
 345         can be pasted into the compute box.
 346       </p>
 347 <p>
 348         The full source of this example is at <a href="../../../../example/fft_sines_table.cpp" target="_top">fft_sines_table.cpp</a>
 349       </p>
 350 </div>
 351 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 352 <td align="left"></td>
 353 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 354       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 355       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 356       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 357       Daryle Walker and Xiaogang Zhang<p>
 358         Distributed under the Boost Software License, Version 1.0. (See accompanying
 359         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>)
 360       </p>
 361 </div></td>
 362 </tr></table>
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