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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="boost_multiprecision.tut.floats.fp_eg.variable_precision"></a><a class="link" href="variable_precision.html" title="Variable Precision Newton Evaluation">Variable
  28           Precision Newton Evaluation</a>
  29 </h5></div></div></div>
  30 <p>
  31             This example illustrates the use of variable-precision arithmetic with
  32             the <code class="computeroutput"><span class="identifier">mpfr_float</span></code> number
  33             type. We'll calculate the median of the beta distribution to an absurdly
  34             high precision and compare the accuracy and times taken for various methods.
  35             That is, we want to calculate the value of <code class="computeroutput"><span class="identifier">x</span></code>
  36             for which <span class="emphasis"><em>I<sub>x</sub>(a, b) = 0.5</em></span>.
  37           </p>
  38 <p>
  39             Ultimately we'll use Newtons method and set the precision of mpfr_float
  40             to have just enough digits at each iteration.
  41           </p>
  42 <p>
  43             The full source of the this program is in <a href="../../../../../../example/mpfr_precision.cpp" target="_top">../../example/mpfr_precision.cpp</a>
  44           </p>
  45 <p>
  46             We'll skip over the #includes and using declations, and go straight to
  47             some support code, first off a simple stopwatch for performance measurement:
  48           </p>
  49 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">clock_type</span><span class="special">&gt;</span>
  50 <span class="keyword">struct</span> <span class="identifier">stopwatch</span> <span class="special">{</span> <span class="comment">/*details \*/</span> <span class="special">};</span>
  51 </pre>
  52 <p>
  53             We'll use <code class="computeroutput"><span class="identifier">stopwatch</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">chono</span><span class="special">::</span><span class="identifier">high_resolution_clock</span><span class="special">&gt;</span></code>
  54             as our performance measuring device.
  55           </p>
  56 <p>
  57             We also have a small utility class for controlling the current precision
  58             of mpfr_float:
  59           </p>
  60 <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">scoped_precision</span>
  61 <span class="special">{</span>
  62    <span class="keyword">unsigned</span> <span class="identifier">p</span><span class="special">;</span>
  63    <span class="identifier">scoped_precision</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">new_p</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">p</span><span class="special">(</span><span class="identifier">mpfr_float</span><span class="special">::</span><span class="identifier">default_precision</span><span class="special">())</span>
  64    <span class="special">{</span>
  65       <span class="identifier">mpfr_float</span><span class="special">::</span><span class="identifier">default_precision</span><span class="special">(</span><span class="identifier">new_p</span><span class="special">);</span>
  66    <span class="special">}</span>
  67    <span class="special">~</span><span class="identifier">scoped_precision</span><span class="special">()</span>
  68    <span class="special">{</span>
  69       <span class="identifier">mpfr_float</span><span class="special">::</span><span class="identifier">default_precision</span><span class="special">(</span><span class="identifier">p</span><span class="special">);</span>
  70    <span class="special">}</span>
  71 <span class="special">};</span>
  72 </pre>
  73 <p>
  74             We'll begin with a reference method that simply calls the Boost.Math
  75             function <code class="computeroutput"><span class="identifier">ibeta_inv</span></code> and
  76             uses the full working precision of the arguments throughout. Our reference
  77             function takes 3 arguments:
  78           </p>
  79 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  80 <li class="listitem">
  81                 The 2 parameters <code class="computeroutput"><span class="identifier">a</span></code>
  82                 and <code class="computeroutput"><span class="identifier">b</span></code> of the beta
  83                 distribution, and
  84               </li>
  85 <li class="listitem">
  86                 The number of decimal digits precision to achieve in the result.
  87               </li>
  88 </ul></div>
  89 <p>
  90             We begin by setting the default working precision to that requested,
  91             and then, since we don't know where our arguments <code class="computeroutput"><span class="identifier">a</span></code>
  92             and <code class="computeroutput"><span class="identifier">b</span></code> have been or what
  93             precision they have, we make a copy of them - note that since copying
  94             also copies the precision as well as the value, we have to set the precision
  95             expicitly with a second argument to the copy. Then we can simply return
  96             the result of <code class="computeroutput"><span class="identifier">ibeta_inv</span></code>:
  97           </p>
  98 <pre class="programlisting"><span class="identifier">mpfr_float</span> <span class="identifier">beta_distribution_median_method_1</span><span class="special">(</span><span class="identifier">mpfr_float</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">a_</span><span class="special">,</span> <span class="identifier">mpfr_float</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">b_</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">)</span>
  99 <span class="special">{</span>
 100    <span class="identifier">scoped_precision</span> <span class="identifier">sp</span><span class="special">(</span><span class="identifier">digits10</span><span class="special">);</span>
 101    <span class="identifier">mpfr_float</span> <span class="identifier">half</span><span class="special">(</span><span class="number">0.5</span><span class="special">),</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">a_</span><span class="special">,</span> <span class="identifier">digits10</span><span class="special">),</span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">b_</span><span class="special">,</span> <span class="identifier">digits10</span><span class="special">);</span>
 102    <span class="keyword">return</span> <span class="identifier">ibeta_inv</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">half</span><span class="special">);</span>
 103 <span class="special">}</span>
 104 </pre>
 105 <p>
 106             You be wondering why we needed to change the precision of our variables
 107             <code class="computeroutput"><span class="identifier">a</span></code> and <code class="computeroutput"><span class="identifier">b</span></code>
 108             as well as setting the default - there are in fact two ways in which
 109             this can go wrong if we don't do that:
 110           </p>
 111 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 112 <li class="listitem">
 113                 The variables have too much precision - this will cause all arithmetic
 114                 operations involving those types to be promoted to the higher precision
 115                 wasting precious calculation time.
 116               </li>
 117 <li class="listitem">
 118                 The variables have too little precision - this will cause expressions
 119                 involving only those variables to be calculated at the lower precision
 120                 - for example if we calculate <code class="computeroutput"><span class="identifier">exp</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span></code> internally, this will be evaluated
 121                 at the precision of <code class="computeroutput"><span class="identifier">a</span></code>,
 122                 and not the current default.
 123               </li>
 124 </ul></div>
 125 <p>
 126             Since our reference method carries out all calculations at the full precision
 127             requested, an obvious refinement would be to calculate a first approximation
 128             to <code class="computeroutput"><span class="keyword">double</span></code> precision and
 129             then to use Newton steps to refine it further.
 130           </p>
 131 <p>
 132             Our function begins the same as before: set the new default precision
 133             and then make copies of our arguments at the correct precision. We then
 134             call <code class="computeroutput"><span class="identifier">ibeta_inv</span></code> with all
 135             double precision arguments, promote the result to an <code class="computeroutput"><span class="identifier">mpfr_float</span></code>
 136             and perform Newton steps to obtain the result. Note that our termination
 137             condition is somewhat cude: we simply assume that we have approximately
 138             14 digits correct from the double-precision approximation and that the
 139             precision doubles with each step. We also cheat, and use an internal
 140             Boost.Math function that calculates <span class="emphasis"><em>I<sub>x</sub>(a, b)</em></span> and
 141             it's derivative in one go:
 142           </p>
 143 <pre class="programlisting"><span class="identifier">mpfr_float</span> <span class="identifier">beta_distribution_median_method_2</span><span class="special">(</span><span class="identifier">mpfr_float</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">a_</span><span class="special">,</span> <span class="identifier">mpfr_float</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">b_</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">)</span>
 144 <span class="special">{</span>
 145    <span class="identifier">scoped_precision</span> <span class="identifier">sp</span><span class="special">(</span><span class="identifier">digits10</span><span class="special">);</span>
 146    <span class="identifier">mpfr_float</span> <span class="identifier">half</span><span class="special">(</span><span class="number">0.5</span><span class="special">),</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">a_</span><span class="special">,</span> <span class="identifier">digits10</span><span class="special">),</span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">b_</span><span class="special">,</span> <span class="identifier">digits10</span><span class="special">);</span>
 147    <span class="identifier">mpfr_float</span> <span class="identifier">guess</span> <span class="special">=</span> <span class="identifier">ibeta_inv</span><span class="special">((</span><span class="keyword">double</span><span class="special">)</span><span class="identifier">a</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span><span class="identifier">b</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
 148    <span class="keyword">unsigned</span> <span class="identifier">current_digits</span> <span class="special">=</span> <span class="number">14</span><span class="special">;</span>
 149    <span class="identifier">mpfr_float</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">f1</span><span class="special">;</span>
 150    <span class="keyword">while</span> <span class="special">(</span><span class="identifier">current_digits</span> <span class="special">&lt;</span> <span class="identifier">digits10</span><span class="special">)</span>
 151    <span class="special">{</span>
 152       <span class="identifier">f</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">detail</span><span class="special">::</span><span class="identifier">ibeta_imp</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">guess</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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;&gt;(),</span> <span class="keyword">false</span><span class="special">,</span> <span class="keyword">true</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">f1</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">half</span><span class="special">;</span>
 153       <span class="identifier">guess</span> <span class="special">-=</span> <span class="identifier">f</span> <span class="special">/</span> <span class="identifier">f1</span><span class="special">;</span>
 154       <span class="identifier">current_digits</span> <span class="special">*=</span> <span class="number">2</span><span class="special">;</span>
 155    <span class="special">}</span>
 156    <span class="keyword">return</span> <span class="identifier">guess</span><span class="special">;</span>
 157 <span class="special">}</span>
 158 </pre>
 159 <p>
 160             Before we refine the method further, it might be wise to take stock and
 161             see how method's 1 and 2 compare. We'll ask them both for 1500 digit
 162             precision, and compare against the value produced by <code class="computeroutput"><span class="identifier">ibeta_inv</span></code>
 163             at 1700 digits. Here's what the results look like:
 164           </p>
 165 <pre class="programlisting">Method 1 time = 0.611647
 166 Relative error: 2.99991e-1501
 167 Method 2 time = 0.646746
 168 Relative error: 7.55843e-1501
 169 </pre>
 170 <p>
 171             Clearly they are both equally accurate, but Method 1 is actually faster
 172             and our plan for improved performance hasn't actually worked. It turns
 173             out that we're not actually comparing like with like, because <code class="computeroutput"><span class="identifier">ibeta_inv</span></code> uses Halley iteration internally
 174             which churns out more digits of precision rather more rapidly than Newton
 175             iteration. So the time we save by refining an initial <code class="computeroutput"><span class="keyword">double</span></code>
 176             approximation, then loose it again by taking more iterations to get to
 177             the result.
 178           </p>
 179 <p>
 180             Time for a more refined approach. It follows the same form as Method
 181             2, but now we set the working precision within the Newton iteration loop,
 182             to just enough digits to cover the expected precision at each step. That
 183             means we also create new copies of our arguments at the correct precision
 184             within the loop, and likewise change the precision of the current <code class="computeroutput"><span class="identifier">guess</span></code> each time through:
 185           </p>
 186 <pre class="programlisting"><span class="identifier">mpfr_float</span> <span class="identifier">beta_distribution_median_method_3</span><span class="special">(</span><span class="identifier">mpfr_float</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">a_</span><span class="special">,</span> <span class="identifier">mpfr_float</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">b_</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">)</span>
 187 <span class="special">{</span>
 188    <span class="identifier">mpfr_float</span> <span class="identifier">guess</span> <span class="special">=</span> <span class="identifier">ibeta_inv</span><span class="special">((</span><span class="keyword">double</span><span class="special">)</span><span class="identifier">a_</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span><span class="identifier">b_</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
 189    <span class="keyword">unsigned</span> <span class="identifier">current_digits</span> <span class="special">=</span> <span class="number">14</span><span class="special">;</span>
 190    <span class="identifier">mpfr_float</span> <span class="identifier">f</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">current_digits</span><span class="special">),</span> <span class="identifier">f1</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">current_digits</span><span class="special">),</span> <span class="identifier">delta</span><span class="special">(</span><span class="number">1</span><span class="special">);</span>
 191    <span class="keyword">while</span> <span class="special">(</span><span class="identifier">current_digits</span> <span class="special">&lt;</span> <span class="identifier">digits10</span><span class="special">)</span>
 192    <span class="special">{</span>
 193       <span class="identifier">current_digits</span> <span class="special">*=</span> <span class="number">2</span><span class="special">;</span>
 194       <span class="identifier">scoped_precision</span> <span class="identifier">sp</span><span class="special">((</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">min</span><span class="special">)(</span><span class="identifier">current_digits</span><span class="special">,</span> <span class="identifier">digits10</span><span class="special">));</span>
 195       <span class="identifier">mpfr_float</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">a_</span><span class="special">,</span> <span class="identifier">mpfr_float</span><span class="special">::</span><span class="identifier">default_precision</span><span class="special">()),</span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">b_</span><span class="special">,</span> <span class="identifier">mpfr_float</span><span class="special">::</span><span class="identifier">default_precision</span><span class="special">());</span>
 196       <span class="identifier">guess</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">mpfr_float</span><span class="special">::</span><span class="identifier">default_precision</span><span class="special">());</span>
 197       <span class="identifier">f</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">detail</span><span class="special">::</span><span class="identifier">ibeta_imp</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">guess</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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">&lt;&gt;(),</span> <span class="keyword">false</span><span class="special">,</span> <span class="keyword">true</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">f1</span><span class="special">)</span> <span class="special">-</span> <span class="number">0.5f</span><span class="special">;</span>
 198       <span class="identifier">guess</span> <span class="special">-=</span> <span class="identifier">f</span> <span class="special">/</span> <span class="identifier">f1</span><span class="special">;</span>
 199    <span class="special">}</span>
 200    <span class="keyword">return</span> <span class="identifier">guess</span><span class="special">;</span>
 201 <span class="special">}</span>
 202 </pre>
 203 <p>
 204             The new performance results look much more promising:
 205           </p>
 206 <pre class="programlisting">Method 1 time = 0.591244
 207 Relative error: 2.99991e-1501
 208 Method 2 time = 0.622679
 209 Relative error: 7.55843e-1501
 210 Method 3 time = 0.143393
 211 Relative error: 4.03898e-1501
 212 </pre>
 213 <p>
 214             This time we're 4x faster than <code class="computeroutput"><span class="identifier">ibeta_inv</span></code>,
 215             and no doubt that could be improved a little more by carefully optimising
 216             the number of iterations and the method (Halley vs Newton) taken.
 217           </p>
 218 <p>
 219             Finally, here's the driver code for the above methods:
 220           </p>
 221 <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 222 <span class="special">{</span>
 223    <span class="keyword">try</span> <span class="special">{</span>
 224       <span class="identifier">mpfr_float</span> <span class="identifier">a</span><span class="special">(</span><span class="number">10</span><span class="special">),</span> <span class="identifier">b</span><span class="special">(</span><span class="number">20</span><span class="special">);</span>
 225
 226       <span class="identifier">mpfr_float</span> <span class="identifier">true_value</span> <span class="special">=</span> <span class="identifier">beta_distribution_median_method_1</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="number">1700</span><span class="special">);</span>
 227
 228       <span class="identifier">stopwatch</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">chrono</span><span class="special">::</span><span class="identifier">high_resolution_clock</span><span class="special">&gt;</span> <span class="identifier">my_stopwatch</span><span class="special">;</span>
 229
 230       <span class="identifier">mpfr_float</span> <span class="identifier">v1</span> <span class="special">=</span> <span class="identifier">beta_distribution_median_method_1</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="number">1500</span><span class="special">);</span>
 231       <span class="keyword">float</span> <span class="identifier">hp_time</span> <span class="special">=</span> <span class="identifier">my_stopwatch</span><span class="special">.</span><span class="identifier">elapsed</span><span class="special">();</span>
 232       <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Method 1 time = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">hp_time</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>
 233       <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Relative error: "</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">relative_difference</span><span class="special">(</span><span class="identifier">v1</span><span class="special">,</span> <span class="identifier">true_value</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 234
 235       <span class="identifier">my_stopwatch</span><span class="special">.</span><span class="identifier">reset</span><span class="special">();</span>
 236       <span class="identifier">mpfr_float</span> <span class="identifier">v2</span> <span class="special">=</span> <span class="identifier">beta_distribution_median_method_2</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="number">1500</span><span class="special">);</span>
 237       <span class="identifier">hp_time</span> <span class="special">=</span> <span class="identifier">my_stopwatch</span><span class="special">.</span><span class="identifier">elapsed</span><span class="special">();</span>
 238       <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Method 2 time = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">hp_time</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>
 239       <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Relative error: "</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">relative_difference</span><span class="special">(</span><span class="identifier">v2</span><span class="special">,</span> <span class="identifier">true_value</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 240
 241       <span class="identifier">my_stopwatch</span><span class="special">.</span><span class="identifier">reset</span><span class="special">();</span>
 242       <span class="identifier">mpfr_float</span> <span class="identifier">v3</span> <span class="special">=</span> <span class="identifier">beta_distribution_median_method_3</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="number">1500</span><span class="special">);</span>
 243       <span class="identifier">hp_time</span> <span class="special">=</span> <span class="identifier">my_stopwatch</span><span class="special">.</span><span class="identifier">elapsed</span><span class="special">();</span>
 244       <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Method 3 time = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">hp_time</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>
 245       <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Relative error: "</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">relative_difference</span><span class="special">(</span><span class="identifier">v3</span><span class="special">,</span> <span class="identifier">true_value</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 246    <span class="special">}</span>
 247    <span class="keyword">catch</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">e</span><span class="special">)</span>
 248    <span class="special">{</span>
 249       <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Found exception with message: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 250    <span class="special">}</span>
 251    <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
 252 <span class="special">}</span>
 253 </pre>
 254 </div>
 255 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 256 <td align="left"></td>
 257 <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2019 John Maddock
 258       and Christopher Kormanyos<p>
 259         Distributed under the Boost Software License, Version 1.0. (See accompanying
 260         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>)
 261       </p>
 262 </div></td>
 263 </tr></table>
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