ceph/src/boost/libs/multiprecision/doc/html/boost_multiprecision/tut/interval/mpfi.html

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  27 <a name="boost_multiprecision.tut.interval.mpfi"></a><a class="link" href="mpfi.html" title="mpfi_float">mpfi_float</a>
  28 </h4></div></div></div>
  29 <p>
  30           <code class="computeroutput"><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">mpfi</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>
  31         </p>
  32 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">multiprecision</span><span class="special">{</span>
  33
  34 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">mpfi_float_backend</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="number">50</span><span class="special">&gt;</span> <span class="special">&gt;</span>    <span class="identifier">mpfi_float_50</span><span class="special">;</span>
  38 <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="number">100</span><span class="special">&gt;</span> <span class="special">&gt;</span>   <span class="identifier">mpfifloat_100</span><span class="special">;</span>
  39 <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="number">500</span><span class="special">&gt;</span> <span class="special">&gt;</span>   <span class="identifier">mpfifloat_500</span><span class="special">;</span>
  40 <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="number">1000</span><span class="special">&gt;</span> <span class="special">&gt;</span>  <span class="identifier">mpfi_float_1000</span><span class="special">;</span>
  41 <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="number">0</span><span class="special">&gt;</span> <span class="special">&gt;</span>     <span class="identifier">mpfi_float</span><span class="special">;</span>
  42
  43 <span class="special">}}</span> <span class="comment">// namespaces</span>
  44 </pre>
  45 <p>
  46           The <code class="computeroutput"><span class="identifier">mpfi_float_backend</span></code>
  47           type is used in conjunction with <code class="computeroutput"><span class="identifier">number</span></code>:
  48           It acts as a thin wrapper around the <a href="http://perso.ens-lyon.fr/nathalie.revol/software.html" target="_top">MPFI</a>
  49           <code class="computeroutput"><span class="identifier">mpfi_t</span></code> to provide an real-number
  50           type that is a drop-in replacement for the native C++ floating-point types,
  51           but with much greater precision and implementing interval arithmetic.
  52         </p>
  53 <p>
  54           Type <code class="computeroutput"><span class="identifier">mpfi_float_backend</span></code>
  55           can be used at fixed precision by specifying a non-zero <code class="computeroutput"><span class="identifier">Digits10</span></code>
  56           template parameter, or at variable precision by setting the template argument
  57           to zero. The typedefs mpfi_float_50, mpfi_float_100, mpfi_float_500, mpfi_float_1000
  58           provide arithmetic types at 50, 100, 500 and 1000 decimal digits precision
  59           respectively. The typedef mpfi_float provides a variable precision type
  60           whose precision can be controlled via the <code class="computeroutput"><span class="identifier">number</span></code>s
  61           member functions.
  62         </p>
  63 <div class="note"><table border="0" summary="Note">
  64 <tr>
  65 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  66 <th align="left">Note</th>
  67 </tr>
  68 <tr><td align="left" valign="top"><p>
  69             This type only provides <code class="computeroutput"><span class="identifier">numeric_limits</span></code>
  70             support when the precision is fixed at compile time.
  71           </p></td></tr>
  72 </table></div>
  73 <p>
  74           As well as the usual conversions from arithmetic and string types, instances
  75           of <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">N</span><span class="special">&gt;</span> <span class="special">&gt;</span></code> are copy constructible and assignable
  76           from:
  77         </p>
  78 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  79 <li class="listitem">
  80               The <a href="http://perso.ens-lyon.fr/nathalie.revol/software.html" target="_top">MPFI</a>
  81               native type <code class="computeroutput"><span class="identifier">mpfi_t</span></code>.
  82             </li>
  83 <li class="listitem">
  84               The <code class="computeroutput"><span class="identifier">number</span></code> wrappers
  85               around <a href="http://perso.ens-lyon.fr/nathalie.revol/software.html" target="_top">MPFI</a>
  86               or <a href="http://www.mpfr.org" target="_top">MPFR</a>: <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">M</span><span class="special">&gt;</span> <span class="special">&gt;</span></code>
  87               and <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfr_float</span><span class="special">&lt;</span><span class="identifier">M</span><span class="special">&gt;</span> <span class="special">&gt;</span></code>.
  88             </li>
  89 <li class="listitem">
  90               There is a two argument constructor taking two <code class="computeroutput"><span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfr_float</span><span class="special">&lt;</span><span class="identifier">M</span><span class="special">&gt;</span> <span class="special">&gt;</span></code>
  91               arguments specifying the interval.
  92             </li>
  93 </ul></div>
  94 <p>
  95           It's also possible to access the underlying <code class="computeroutput"><span class="identifier">mpfi_t</span></code>
  96           via the data() member function of <code class="computeroutput"><span class="identifier">mpfi_float_backend</span></code>.
  97         </p>
  98 <p>
  99           Things you should know when using this type:
 100         </p>
 101 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 102 <li class="listitem">
 103               A default constructed <code class="computeroutput"><span class="identifier">mpfi_float_backend</span></code>
 104               is set to zero (<span class="emphasis"><em>Note that this is <span class="bold"><strong>not</strong></span>
 105               the default <a href="http://perso.ens-lyon.fr/nathalie.revol/software.html" target="_top">MPFI</a>
 106               behavior</em></span>).
 107             </li>
 108 <li class="listitem">
 109               No changes are made to <a href="http://gmplib.org" target="_top">GMP</a> or
 110               <a href="http://www.mpfr.org" target="_top">MPFR</a> global settings, so this
 111               type can coexist with existing <a href="http://www.mpfr.org" target="_top">MPFR</a>
 112               or <a href="http://gmplib.org" target="_top">GMP</a> code.
 113             </li>
 114 <li class="listitem">
 115               The code can equally use <a href="http://mpir.org/" target="_top">MPIR</a>
 116               in place of <a href="http://gmplib.org" target="_top">GMP</a> - indeed that
 117               is the preferred option on Win32.
 118             </li>
 119 <li class="listitem">
 120               This backend supports rvalue-references and is move-aware, making instantiations
 121               of <code class="computeroutput"><span class="identifier">number</span></code> on this backend
 122               move aware.
 123             </li>
 124 <li class="listitem">
 125               Conversion from a string results in a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">runtime_error</span></code>
 126               being thrown if the string can not be interpreted as a valid floating-point
 127               number.
 128             </li>
 129 <li class="listitem">
 130               Division by zero results in an infinity.
 131             </li>
 132 </ul></div>
 133 <p>
 134           There are some additional non member functions for working on intervals:
 135         </p>
 136 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 137 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfr_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="identifier">lower</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">);</span>
 138 </pre>
 139 <p>
 140           Returns the lower end of the interval.
 141         </p>
 142 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 143 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfr_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="identifier">upper</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">);</span>
 144 </pre>
 145 <p>
 146           Returns the upper end of the interval.
 147         </p>
 148 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 149 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfr_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="identifier">median</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">);</span>
 150 </pre>
 151 <p>
 152           Returns the mid point of the interval.
 153         </p>
 154 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 155 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfr_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="identifier">width</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">val</span><span class="special">);</span>
 156 </pre>
 157 <p>
 158           Returns the absolute width of the interval.
 159         </p>
 160 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 161 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="identifier">intersect</span><span class="special">(</span>
 162   <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 163   <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span>  <span class="identifier">b</span><span class="special">);</span>
 164 </pre>
 165 <p>
 166           Returns the interval which is the intersection of the <span class="emphasis"><em>a</em></span>
 167           and <span class="emphasis"><em>b</em></span>. Returns an unspecified empty interval if there
 168           is no such intersection.
 169         </p>
 170 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 171 <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span> <span class="identifier">hull</span><span class="special">(</span>
 172   <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 173   <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span>  <span class="identifier">b</span><span class="special">);</span>
 174 </pre>
 175 <p>
 176           Returns the interval which is the union of <span class="emphasis"><em>a</em></span> and
 177           <span class="emphasis"><em>b</em></span>.
 178         </p>
 179 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 180 <span class="keyword">bool</span> <span class="identifier">overlap</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 181              <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span>  <span class="identifier">b</span><span class="special">);</span>
 182 </pre>
 183 <p>
 184           Returns <code class="computeroutput"><span class="keyword">true</span></code> only if the intervals
 185           <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> overlap.
 186         </p>
 187 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates1</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates2</span><span class="special">&gt;</span>
 188 <span class="keyword">bool</span> <span class="identifier">in</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfr_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates1</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 189         <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates2</span><span class="special">&gt;&amp;</span>  <span class="identifier">b</span><span class="special">);</span>
 190 </pre>
 191 <p>
 192           Returns <code class="computeroutput"><span class="keyword">true</span></code> only if point
 193           <span class="emphasis"><em>a</em></span> is contained within the interval <span class="emphasis"><em>b</em></span>.
 194         </p>
 195 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 196 <span class="keyword">bool</span> <span class="identifier">zero_in</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">);</span>
 197 </pre>
 198 <p>
 199           Returns <code class="computeroutput"><span class="keyword">true</span></code> only if the interval
 200           <span class="emphasis"><em>a</em></span> contains the value zero.
 201         </p>
 202 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 203 <span class="keyword">bool</span> <span class="identifier">subset</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 204             <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 205 </pre>
 206 <p>
 207           Returns <code class="computeroutput"><span class="keyword">true</span></code> only if <span class="emphasis"><em>a</em></span>
 208           is a subset of <span class="emphasis"><em>b</em></span>.
 209         </p>
 210 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 211 <span class="keyword">bool</span> <span class="identifier">proper_subset</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span>
 212                    <span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 213 </pre>
 214 <p>
 215           Returns <code class="computeroutput"><span class="keyword">true</span></code> only if <span class="emphasis"><em>a</em></span>
 216           is a proper subset of <span class="emphasis"><em>b</em></span>.
 217         </p>
 218 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 219 <span class="keyword">bool</span> <span class="identifier">empty</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">);</span>
 220 </pre>
 221 <p>
 222           Returns <code class="computeroutput"><span class="keyword">true</span></code> only if <span class="emphasis"><em>a</em></span>
 223           is an empty interval, equivalent to <code class="computeroutput"><span class="identifier">upper</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span>
 224           <span class="special">&lt;</span> <span class="identifier">lower</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span></code>.
 225         </p>
 226 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">unsigned</span> <span class="identifier">Digits10</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;</span>
 227 <span class="keyword">bool</span> <span class="identifier">singleton</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special">&lt;</span><span class="identifier">mpfi_float_backend</span><span class="special">&lt;</span><span class="identifier">Digits10</span><span class="special">&gt;,</span> <span class="identifier">ExpressionTemplates</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">);</span>
 228 </pre>
 229 <p>
 230           Returns <code class="computeroutput"><span class="keyword">true</span></code> if <code class="computeroutput"><span class="identifier">lower</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span> <span class="special">==</span>
 231           <span class="identifier">upper</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span></code>.
 232         </p>
 233 <h6>
 234 <a name="boost_multiprecision.tut.interval.mpfi.h0"></a>
 235           <span class="phrase"><a name="boost_multiprecision.tut.interval.mpfi.mpfi_example"></a></span><a class="link" href="mpfi.html#boost_multiprecision.tut.interval.mpfi.mpfi_example"> MPFI
 236           example:</a>
 237         </h6>
 238 <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">multiprecision</span><span class="special">/</span><span class="identifier">mpfi</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 239 <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">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 240 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 241
 242 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 243 <span class="special">{</span>
 244    <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span>
 245
 246    <span class="comment">// Operations at variable precision and no numeric_limits support:</span>
 247    <span class="identifier">mpfi_float</span> <span class="identifier">a</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>
 248    <span class="identifier">mpfi_float</span><span class="special">::</span><span class="identifier">default_precision</span><span class="special">(</span><span class="number">1000</span><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="identifier">mpfi_float</span><span class="special">::</span><span class="identifier">default_precision</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="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</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> <span class="comment">// print root-2</span>
 251
 252    <span class="comment">// Operations at fixed precision and full numeric_limits support:</span>
 253    <span class="identifier">mpfi_float_100</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>
 254    <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">numeric_limits</span><span class="special">&lt;</span><span class="identifier">mpfi_float_100</span><span class="special">&gt;::</span><span class="identifier">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>
 255    <span class="comment">// We can use any C++ std lib function:</span>
 256    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">b</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> <span class="comment">// print log(2)</span>
 257
 258    <span class="comment">// Access the underlying data:</span>
 259    <span class="identifier">mpfi_t</span> <span class="identifier">r</span><span class="special">;</span>
 260    <span class="identifier">mpfi_init</span><span class="special">(</span><span class="identifier">r</span><span class="special">);</span>
 261    <span class="identifier">mpfi_set</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">b</span><span class="special">.</span><span class="identifier">backend</span><span class="special">().</span><span class="identifier">data</span><span class="special">());</span>
 262
 263    <span class="comment">// Construct some explicit intervals and perform set operations:</span>
 264    <span class="identifier">mpfi_float_50</span> <span class="identifier">i1</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">2</span><span class="special">),</span> <span class="identifier">i2</span><span class="special">(</span><span class="number">1.5</span><span class="special">,</span> <span class="number">2.5</span><span class="special">);</span>
 265    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">intersect</span><span class="special">(</span><span class="identifier">i1</span><span class="special">,</span> <span class="identifier">i2</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>
 266    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">hull</span><span class="special">(</span><span class="identifier">i1</span><span class="special">,</span> <span class="identifier">i2</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>
 267    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">overlap</span><span class="special">(</span><span class="identifier">i1</span><span class="special">,</span> <span class="identifier">i2</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>
 268    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">subset</span><span class="special">(</span><span class="identifier">i1</span><span class="special">,</span> <span class="identifier">i2</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>
 269    <span class="identifier">mpfi_clear</span><span class="special">(</span><span class="identifier">r</span><span class="special">);</span>
 270    <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
 271 <span class="special">}</span>
 272 </pre>
 273 </div>
 274 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 275 <td align="left"></td>
 276 <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2013 John Maddock and Christopher Kormanyos<p>
 277         Distributed under the Boost Software License, Version 1.0. (See accompanying
 278         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>)
 279       </p>
 280 </div></td>
 281 </tr></table>
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