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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_erf.error_function"></a><a class="link" href="error_function.html" title="Error Functions">Error Functions</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_erf.error_function.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_erf.error_function.synopsis"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.synopsis">Synopsis</a>
  32       </h5>
  33 <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">special_functions</span><span class="special">/</span><span class="identifier">erf</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  38 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  44 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  47 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  48
  49 <span class="special">}}</span> <span class="comment">// namespaces</span>
  50 </pre>
  51 <p>
  52         The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  53         type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
  54       </p>
  55 <p>
  56         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  57         be used to control the behaviour of the function: how it handles errors,
  58         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  59         documentation for more details</a>.
  60       </p>
  61 <h5>
  62 <a name="math_toolkit.sf_erf.error_function.h1"></a>
  63         <span class="phrase"><a name="math_toolkit.sf_erf.error_function.description"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.description">Description</a>
  64       </h5>
  65 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  66 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  67
  68 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  69 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  70 </pre>
  71 <p>
  72         Returns the <a href="http://en.wikipedia.org/wiki/Error_function" target="_top">error
  73         function</a> <a href="http://functions.wolfram.com/GammaBetaErf/Erf/" target="_top">erf</a>
  74         of z:
  75       </p>
  76 <p>
  77         <span class="inlinemediaobject"><img src="../../../equations/erf1.svg"></span>
  78       </p>
  79 <p>
  80         <span class="inlinemediaobject"><img src="../../../graphs/erf.svg" align="middle"></span>
  81       </p>
  82 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  83 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  84
  85 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  86 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  87 </pre>
  88 <p>
  89         Returns the complement of the <a href="http://functions.wolfram.com/GammaBetaErf/Erfc/" target="_top">error
  90         function</a> of z:
  91       </p>
  92 <p>
  93         <span class="inlinemediaobject"><img src="../../../equations/erf2.svg"></span>
  94       </p>
  95 <p>
  96         <span class="inlinemediaobject"><img src="../../../graphs/erfc.svg" align="middle"></span>
  97       </p>
  98 <h5>
  99 <a name="math_toolkit.sf_erf.error_function.h2"></a>
 100         <span class="phrase"><a name="math_toolkit.sf_erf.error_function.accuracy"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.accuracy">Accuracy</a>
 101       </h5>
 102 <p>
 103         The following table shows the peak errors (in units of epsilon) found on
 104         various platforms with various floating point types, along with comparisons
 105         to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>, <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>, <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
 106         C Library</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
 107         libraries. Unless otherwise specified any floating point type that is narrower
 108         than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 109         zero error</a>.
 110       </p>
 111 <div class="table">
 112 <a name="math_toolkit.sf_erf.error_function.table_erf"></a><p class="title"><b>Table&#160;6.28.&#160;Error rates for erf</b></p>
 113 <div class="table-contents"><table class="table" summary="Error rates for erf">
 114 <colgroup>
 115 <col>
 116 <col>
 117 <col>
 118 <col>
 119 <col>
 120 </colgroup>
 121 <thead><tr>
 122 <th>
 123               </th>
 124 <th>
 125                 <p>
 126                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 127                 </p>
 128               </th>
 129 <th>
 130                 <p>
 131                   GNU C++ version 5.1.0<br> linux<br> long double
 132                 </p>
 133               </th>
 134 <th>
 135                 <p>
 136                   GNU C++ version 5.1.0<br> linux<br> double
 137                 </p>
 138               </th>
 139 <th>
 140                 <p>
 141                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 142                 </p>
 143               </th>
 144 </tr></thead>
 145 <tbody>
 146 <tr>
 147 <td>
 148                 <p>
 149                   Erf Function: Small Values
 150                 </p>
 151               </td>
 152 <td>
 153                 <p>
 154                   <span class="blue">Max = 0.996&#949; (Mean = 0.182&#949;)</span><br> <br>
 155                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.57&#949; (Mean = 0.317&#949;))
 156                 </p>
 157               </td>
 158 <td>
 159                 <p>
 160                   <span class="blue">Max = 0.925&#949; (Mean = 0.193&#949;)</span><br> <br>
 161                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.944&#949; (Mean = 0.191&#949;))<br>
 162                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.944&#949; (Mean = 0.191&#949;))
 163                 </p>
 164               </td>
 165 <td>
 166                 <p>
 167                   <span class="blue">Max = 0.841&#949; (Mean = 0.0687&#949;)</span><br>
 168                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.06&#949; (Mean = 0.319&#949;))<br>
 169                   (<span class="emphasis"><em>Cephes:</em></span> Max = 1.13&#949; (Mean = 0.442&#949;))
 170                 </p>
 171               </td>
 172 <td>
 173                 <p>
 174                   <span class="blue">Max = 0.925&#949; (Mean = 0.193&#949;)</span><br> <br>
 175                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.944&#949; (Mean = 0.194&#949;))
 176                 </p>
 177               </td>
 178 </tr>
 179 <tr>
 180 <td>
 181                 <p>
 182                   Erf Function: Medium Values
 183                 </p>
 184               </td>
 185 <td>
 186                 <p>
 187                   <span class="blue">Max = 1&#949; (Mean = 0.169&#949;)</span><br> <br>
 188                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.19&#949; (Mean = 0.244&#949;))
 189                 </p>
 190               </td>
 191 <td>
 192                 <p>
 193                   <span class="blue">Max = 1.5&#949; (Mean = 0.193&#949;)</span><br> <br>
 194                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.921&#949; (Mean = 0.0723&#949;))<br>
 195                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.921&#949; (Mean = 0.0723&#949;))
 196                 </p>
 197               </td>
 198 <td>
 199                 <p>
 200                   <span class="blue">Max = 1&#949; (Mean = 0.119&#949;)</span><br> <br>
 201                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.31&#949; (Mean = 0.368&#949;))<br>
 202                   (<span class="emphasis"><em>Cephes:</em></span> Max = 1.34&#949; (Mean = 0.279&#949;))
 203                 </p>
 204               </td>
 205 <td>
 206                 <p>
 207                   <span class="blue">Max = 1.5&#949; (Mean = 0.202&#949;)</span><br> <br>
 208                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.921&#949; (Mean = 0.071&#949;))
 209                 </p>
 210               </td>
 211 </tr>
 212 <tr>
 213 <td>
 214                 <p>
 215                   Erf Function: Large Values
 216                 </p>
 217               </td>
 218 <td>
 219                 <p>
 220                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
 221                   Max = 0&#949; (Mean = 0&#949;))
 222                 </p>
 223               </td>
 224 <td>
 225                 <p>
 226                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span>
 227                   Max = 0&#949; (Mean = 0&#949;))<br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
 228                   Max = 0&#949; (Mean = 0&#949;))
 229                 </p>
 230               </td>
 231 <td>
 232                 <p>
 233                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 234                   1.16:</em></span> Max = 0&#949; (Mean = 0&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 235                   Max = 0&#949; (Mean = 0&#949;))
 236                 </p>
 237               </td>
 238 <td>
 239                 <p>
 240                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
 241                   Max = 0&#949; (Mean = 0&#949;))
 242                 </p>
 243               </td>
 244 </tr>
 245 </tbody>
 246 </table></div>
 247 </div>
 248 <br class="table-break"><div class="table">
 249 <a name="math_toolkit.sf_erf.error_function.table_erfc"></a><p class="title"><b>Table&#160;6.29.&#160;Error rates for erfc</b></p>
 250 <div class="table-contents"><table class="table" summary="Error rates for erfc">
 251 <colgroup>
 252 <col>
 253 <col>
 254 <col>
 255 <col>
 256 <col>
 257 </colgroup>
 258 <thead><tr>
 259 <th>
 260               </th>
 261 <th>
 262                 <p>
 263                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 264                 </p>
 265               </th>
 266 <th>
 267                 <p>
 268                   GNU C++ version 5.1.0<br> linux<br> long double
 269                 </p>
 270               </th>
 271 <th>
 272                 <p>
 273                   GNU C++ version 5.1.0<br> linux<br> double
 274                 </p>
 275               </th>
 276 <th>
 277                 <p>
 278                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 279                 </p>
 280               </th>
 281 </tr></thead>
 282 <tbody>
 283 <tr>
 284 <td>
 285                 <p>
 286                   Erf Function: Small Values
 287                 </p>
 288               </td>
 289 <td>
 290                 <p>
 291                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
 292                   Max = 0&#949; (Mean = 0&#949;))
 293                 </p>
 294               </td>
 295 <td>
 296                 <p>
 297                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span>
 298                   Max = 0&#949; (Mean = 0&#949;))<br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
 299                   Max = 0&#949; (Mean = 0&#949;))
 300                 </p>
 301               </td>
 302 <td>
 303                 <p>
 304                   <span class="blue">Max = 0.658&#949; (Mean = 0.0537&#949;)</span><br>
 305                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.01&#949; (Mean = 0.485&#949;))<br>
 306                   (<span class="emphasis"><em>Cephes:</em></span> Max = 0.786&#949; (Mean = 0.0642&#949;))
 307                 </p>
 308               </td>
 309 <td>
 310                 <p>
 311                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
 312                   Max = 0&#949; (Mean = 0&#949;))
 313                 </p>
 314               </td>
 315 </tr>
 316 <tr>
 317 <td>
 318                 <p>
 319                   Erf Function: Medium Values
 320                 </p>
 321               </td>
 322 <td>
 323                 <p>
 324                   <span class="blue">Max = 1.65&#949; (Mean = 0.373&#949;)</span><br> <br>
 325                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.36&#949; (Mean = 0.539&#949;))
 326                 </p>
 327               </td>
 328 <td>
 329                 <p>
 330                   <span class="blue">Max = 1.76&#949; (Mean = 0.365&#949;)</span><br> <br>
 331                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.35&#949; (Mean = 0.307&#949;))<br>
 332                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.35&#949; (Mean = 0.307&#949;))
 333                 </p>
 334               </td>
 335 <td>
 336                 <p>
 337                   <span class="blue">Max = 0.983&#949; (Mean = 0.213&#949;)</span><br> <br>
 338                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.64&#949; (Mean = 0.662&#949;))<br>
 339                   (<span class="emphasis"><em>Cephes:</em></span> Max = 3.59&#949; (Mean = 0.779&#949;))
 340                 </p>
 341               </td>
 342 <td>
 343                 <p>
 344                   <span class="blue">Max = 1.76&#949; (Mean = 0.383&#949;)</span><br> <br>
 345                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.81&#949; (Mean = 0.739&#949;))
 346                 </p>
 347               </td>
 348 </tr>
 349 <tr>
 350 <td>
 351                 <p>
 352                   Erf Function: Large Values
 353                 </p>
 354               </td>
 355 <td>
 356                 <p>
 357                   <span class="blue">Max = 1.14&#949; (Mean = 0.248&#949;)</span><br> <br>
 358                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.84&#949; (Mean = 0.331&#949;))
 359                 </p>
 360               </td>
 361 <td>
 362                 <p>
 363                   <span class="blue">Max = 1.57&#949; (Mean = 0.542&#949;)</span><br> <br>
 364                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.26&#949; (Mean = 0.441&#949;))<br>
 365                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.26&#949; (Mean = 0.441&#949;))
 366                 </p>
 367               </td>
 368 <td>
 369                 <p>
 370                   <span class="blue">Max = 0.868&#949; (Mean = 0.147&#949;)</span><br> <br>
 371                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 3.9&#949; (Mean = 0.472&#949;))<br>
 372                   (<span class="emphasis"><em>Cephes:</em></span> Max = 2.74&#949; (Mean = 0.413&#949;))
 373                 </p>
 374               </td>
 375 <td>
 376                 <p>
 377                   <span class="blue">Max = 1.57&#949; (Mean = 0.564&#949;)</span><br> <br>
 378                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 4.91&#949; (Mean = 1.54&#949;))
 379                 </p>
 380               </td>
 381 </tr>
 382 </tbody>
 383 </table></div>
 384 </div>
 385 <br class="table-break"><h5>
 386 <a name="math_toolkit.sf_erf.error_function.h3"></a>
 387         <span class="phrase"><a name="math_toolkit.sf_erf.error_function.testing"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.testing">Testing</a>
 388       </h5>
 389 <p>
 390         The tests for these functions come in two parts: basic sanity checks use
 391         spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf" target="_top">Mathworld's
 392         online evaluator</a>, while accuracy checks use high-precision test values
 393         calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
 394         and this implementation. Note that the generic and type-specific versions
 395         of these functions use differing implementations internally, so this gives
 396         us reasonably independent test data. Using our test data to test other "known
 397         good" implementations also provides an additional sanity check.
 398       </p>
 399 <h5>
 400 <a name="math_toolkit.sf_erf.error_function.h4"></a>
 401         <span class="phrase"><a name="math_toolkit.sf_erf.error_function.implementation"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.implementation">Implementation</a>
 402       </h5>
 403 <p>
 404         All versions of these functions first use the usual reflection formulas to
 405         make their arguments positive:
 406       </p>
 407 <pre class="programlisting"><span class="identifier">erf</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
 408
 409 <span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">2</span> <span class="special">-</span> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>  <span class="comment">// preferred when -z &lt; -0.5</span>
 410
 411 <span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">1</span> <span class="special">+</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>   <span class="comment">// preferred when -0.5 &lt;= -z &lt; 0</span>
 412 </pre>
 413 <p>
 414         The generic versions of these functions are implemented in terms of the incomplete
 415         gamma function.
 416       </p>
 417 <p>
 418         When the significand (mantissa) size is recognised (currently for 53, 64
 419         and 113-bit reals, plus single-precision 24-bit handled via promotion to
 420         double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 421         by JM</a> are used.
 422       </p>
 423 <p>
 424         For <code class="computeroutput"><span class="identifier">z</span> <span class="special">&lt;=</span>
 425         <span class="number">0.5</span></code> then a rational approximation to
 426         erf is used, based on the observation that erf is an odd function and therefore
 427         erf is calculated using:
 428       </p>
 429 <pre class="programlisting"><span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">z</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">C</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">));</span>
 430 </pre>
 431 <p>
 432         where the rational approximation R(z*z) is optimised for absolute error:
 433         as long as its absolute error is small enough compared to the constant C,
 434         then any round-off error incurred during the computation of R(z*z) will effectively
 435         disappear from the result. As a result the error for erf and erfc in this
 436         region is very low: the last bit is incorrect in only a very small number
 437         of cases.
 438       </p>
 439 <p>
 440         For <code class="computeroutput"><span class="identifier">z</span> <span class="special">&gt;</span>
 441         <span class="number">0.5</span></code> we observe that over a small interval
 442         [a, b) then:
 443       </p>
 444 <pre class="programlisting"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">z</span> <span class="special">~</span> <span class="identifier">c</span>
 445 </pre>
 446 <p>
 447         for some constant c.
 448       </p>
 449 <p>
 450         Therefore for <code class="computeroutput"><span class="identifier">z</span> <span class="special">&gt;</span>
 451         <span class="number">0.5</span></code> we calculate erfc using:
 452       </p>
 453 <pre class="programlisting"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">C</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span> <span class="special">-</span> <span class="identifier">B</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">;</span>
 454 </pre>
 455 <p>
 456         Again R(z - B) is optimised for absolute error, and the constant <code class="computeroutput"><span class="identifier">C</span></code> is the average of <code class="computeroutput"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span>
 457         <span class="special">*</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span>
 458         <span class="identifier">z</span></code> taken at the endpoints of the
 459         range. Once again, as long as the absolute error in R(z - B) is small compared
 460         to <code class="computeroutput"><span class="identifier">c</span></code> then <code class="computeroutput"><span class="identifier">c</span>
 461         <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span> <span class="special">-</span>
 462         <span class="identifier">B</span><span class="special">)</span></code>
 463         will be correctly rounded, and the error in the result will depend only on
 464         the accuracy of the exp function. In practice, in all but a very small number
 465         of cases, the error is confined to the last bit of the result. The constant
 466         <code class="computeroutput"><span class="identifier">B</span></code> is chosen so that the left
 467         hand end of the range of the rational approximation is 0.
 468       </p>
 469 <p>
 470         For large <code class="computeroutput"><span class="identifier">z</span></code> over a range
 471         [a, +&#8734;] the above approximation is modified to:
 472       </p>
 473 <pre class="programlisting"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">C</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="number">1</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">;</span>
 474 </pre>
 475 </div>
 476 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 477 <td align="left"></td>
 478 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 479       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 480       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 481       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 482         Distributed under the Boost Software License, Version 1.0. (See accompanying
 483         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>)
 484       </p>
 485 </div></td>
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