1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Error Function erf and complement erfc</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../sf_erf.html" title="Error Functions"> 9<link rel="prev" href="../sf_erf.html" title="Error Functions"> 10<link rel="next" href="error_inv.html" title="Error Function Inverses"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../sf_erf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_erf.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="error_inv.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 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 Function erf and complement erfc">Error Function erf 28 and complement erfc</a> 29</h3></div></div></div> 30<h5> 31<a name="math_toolkit.sf_erf.error_function.h0"></a> 32 <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> 33 </h5> 34<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">erf</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 35</pre> 36<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> 37 38<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 39<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> 40 41<span class="keyword">template</span> <span class="special"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 42<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 43 44<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 45<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> 46 47<span class="keyword">template</span> <span class="special"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 48<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 49 50<span class="special">}}</span> <span class="comment">// namespaces</span> 51</pre> 52<p> 53 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 54 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. 55 </p> 56<p> 57 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 58 be used to control the behaviour of the function: how it handles errors, 59 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 60 documentation for more details</a>. 61 </p> 62<h5> 63<a name="math_toolkit.sf_erf.error_function.h1"></a> 64 <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> 65 </h5> 66<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 67<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> 68 69<span class="keyword">template</span> <span class="special"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 70<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 71</pre> 72<p> 73 Returns the <a href="http://en.wikipedia.org/wiki/Error_function" target="_top">error 74 function</a> <a href="http://functions.wolfram.com/GammaBetaErf/Erf/" target="_top">erf</a> 75 of z: 76 </p> 77<div class="blockquote"><blockquote class="blockquote"><p> 78 <span class="inlinemediaobject"><img src="../../../equations/erf1.svg"></span> 79 80 </p></blockquote></div> 81<div class="blockquote"><blockquote class="blockquote"><p> 82 <span class="inlinemediaobject"><img src="../../../graphs/erf.svg" align="middle"></span> 83 84 </p></blockquote></div> 85<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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> 87 88<span class="keyword">template</span> <span class="special"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 89<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 90</pre> 91<p> 92 Returns the complement of the <a href="http://functions.wolfram.com/GammaBetaErf/Erfc/" target="_top">error 93 function</a> of z: 94 </p> 95<div class="blockquote"><blockquote class="blockquote"><p> 96 <span class="inlinemediaobject"><img src="../../../equations/erf2.svg"></span> 97 98 </p></blockquote></div> 99<div class="blockquote"><blockquote class="blockquote"><p> 100 <span class="inlinemediaobject"><img src="../../../graphs/erfc.svg" align="middle"></span> 101 102 </p></blockquote></div> 103<h5> 104<a name="math_toolkit.sf_erf.error_function.h2"></a> 105 <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> 106 </h5> 107<p> 108 The following table shows the peak errors (in units of epsilon) found on 109 various platforms with various floating-point types, along with comparisons 110 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 111 C Library</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> 112 libraries. Unless otherwise specified any floating-point type that is narrower 113 than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 114 zero error</a>. 115 </p> 116<div class="table"> 117<a name="math_toolkit.sf_erf.error_function.table_erf"></a><p class="title"><b>Table 8.28. Error rates for erf</b></p> 118<div class="table-contents"><table class="table" summary="Error rates for erf"> 119<colgroup> 120<col> 121<col> 122<col> 123<col> 124<col> 125</colgroup> 126<thead><tr> 127<th> 128 </th> 129<th> 130 <p> 131 GNU C++ version 7.1.0<br> linux<br> long double 132 </p> 133 </th> 134<th> 135 <p> 136 GNU C++ version 7.1.0<br> linux<br> double 137 </p> 138 </th> 139<th> 140 <p> 141 Sun compiler version 0x5150<br> Sun Solaris<br> long double 142 </p> 143 </th> 144<th> 145 <p> 146 Microsoft Visual C++ version 14.1<br> Win32<br> double 147 </p> 148 </th> 149</tr></thead> 150<tbody> 151<tr> 152<td> 153 <p> 154 Erf Function: Small Values 155 </p> 156 </td> 157<td> 158 <p> 159 <span class="blue">Max = 0.925ε (Mean = 0.193ε)</span><br> <br> 160 (<span class="emphasis"><em><cmath>:</em></span> Max = 0.944ε (Mean = 0.191ε))<br> 161 (<span class="emphasis"><em><math.h>:</em></span> Max = 0.944ε (Mean = 0.191ε)) 162 </p> 163 </td> 164<td> 165 <p> 166 <span class="blue">Max = 0.841ε (Mean = 0.0687ε)</span><br> 167 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.06ε (Mean = 0.319ε)) 168 </p> 169 </td> 170<td> 171 <p> 172 <span class="blue">Max = 0.925ε (Mean = 0.193ε)</span><br> <br> 173 (<span class="emphasis"><em><math.h>:</em></span> Max = 0.944ε (Mean = 0.194ε)) 174 </p> 175 </td> 176<td> 177 <p> 178 <span class="blue">Max = 0.996ε (Mean = 0.182ε)</span><br> <br> 179 (<span class="emphasis"><em><math.h>:</em></span> Max = 1.57ε (Mean = 0.317ε)) 180 </p> 181 </td> 182</tr> 183<tr> 184<td> 185 <p> 186 Erf Function: Medium Values 187 </p> 188 </td> 189<td> 190 <p> 191 <span class="blue">Max = 1.5ε (Mean = 0.193ε)</span><br> <br> 192 (<span class="emphasis"><em><cmath>:</em></span> Max = 0.921ε (Mean = 0.0723ε))<br> 193 (<span class="emphasis"><em><math.h>:</em></span> Max = 0.921ε (Mean = 0.0723ε)) 194 </p> 195 </td> 196<td> 197 <p> 198 <span class="blue">Max = 1ε (Mean = 0.119ε)</span><br> <br> 199 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.31ε (Mean = 0.368ε)) 200 </p> 201 </td> 202<td> 203 <p> 204 <span class="blue">Max = 1.5ε (Mean = 0.197ε)</span><br> <br> 205 (<span class="emphasis"><em><math.h>:</em></span> Max = 0.921ε (Mean = 0.071ε)) 206 </p> 207 </td> 208<td> 209 <p> 210 <span class="blue">Max = 1ε (Mean = 0.171ε)</span><br> <br> 211 (<span class="emphasis"><em><math.h>:</em></span> Max = 1.19ε (Mean = 0.244ε)) 212 </p> 213 </td> 214</tr> 215<tr> 216<td> 217 <p> 218 Erf Function: Large Values 219 </p> 220 </td> 221<td> 222 <p> 223 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><cmath>:</em></span> 224 Max = 0ε (Mean = 0ε))<br> (<span class="emphasis"><em><math.h>:</em></span> 225 Max = 0ε (Mean = 0ε)) 226 </p> 227 </td> 228<td> 229 <p> 230 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 231 2.1:</em></span> Max = 0ε (Mean = 0ε)) 232 </p> 233 </td> 234<td> 235 <p> 236 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span> 237 Max = 0ε (Mean = 0ε)) 238 </p> 239 </td> 240<td> 241 <p> 242 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span> 243 Max = 0ε (Mean = 0ε)) 244 </p> 245 </td> 246</tr> 247</tbody> 248</table></div> 249</div> 250<br class="table-break"><div class="table"> 251<a name="math_toolkit.sf_erf.error_function.table_erfc"></a><p class="title"><b>Table 8.29. Error rates for erfc</b></p> 252<div class="table-contents"><table class="table" summary="Error rates for erfc"> 253<colgroup> 254<col> 255<col> 256<col> 257<col> 258<col> 259</colgroup> 260<thead><tr> 261<th> 262 </th> 263<th> 264 <p> 265 GNU C++ version 7.1.0<br> linux<br> long double 266 </p> 267 </th> 268<th> 269 <p> 270 GNU C++ version 7.1.0<br> linux<br> double 271 </p> 272 </th> 273<th> 274 <p> 275 Sun compiler version 0x5150<br> Sun Solaris<br> long double 276 </p> 277 </th> 278<th> 279 <p> 280 Microsoft Visual C++ version 14.1<br> Win32<br> double 281 </p> 282 </th> 283</tr></thead> 284<tbody> 285<tr> 286<td> 287 <p> 288 Erf Function: Small Values 289 </p> 290 </td> 291<td> 292 <p> 293 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><cmath>:</em></span> 294 Max = 0ε (Mean = 0ε))<br> (<span class="emphasis"><em><math.h>:</em></span> 295 Max = 0ε (Mean = 0ε)) 296 </p> 297 </td> 298<td> 299 <p> 300 <span class="blue">Max = 0.658ε (Mean = 0.0537ε)</span><br> 301 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.01ε (Mean = 0.485ε)) 302 </p> 303 </td> 304<td> 305 <p> 306 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span> 307 Max = 0ε (Mean = 0ε)) 308 </p> 309 </td> 310<td> 311 <p> 312 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span> 313 Max = 0ε (Mean = 0ε)) 314 </p> 315 </td> 316</tr> 317<tr> 318<td> 319 <p> 320 Erf Function: Medium Values 321 </p> 322 </td> 323<td> 324 <p> 325 <span class="blue">Max = 1.76ε (Mean = 0.365ε)</span><br> <br> 326 (<span class="emphasis"><em><cmath>:</em></span> Max = 1.35ε (Mean = 0.307ε))<br> 327 (<span class="emphasis"><em><math.h>:</em></span> Max = 1.35ε (Mean = 0.307ε)) 328 </p> 329 </td> 330<td> 331 <p> 332 <span class="blue">Max = 0.983ε (Mean = 0.213ε)</span><br> <br> 333 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.64ε (Mean = 0.662ε)) 334 </p> 335 </td> 336<td> 337 <p> 338 <span class="blue">Max = 1.76ε (Mean = 0.38ε)</span><br> <br> 339 (<span class="emphasis"><em><math.h>:</em></span> Max = 2.81ε (Mean = 0.739ε)) 340 </p> 341 </td> 342<td> 343 <p> 344 <span class="blue">Max = 1.65ε (Mean = 0.373ε)</span><br> <br> 345 (<span class="emphasis"><em><math.h>:</em></span> Max = 2.36ε (Mean = 0.539ε)) 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 = 1.57ε (Mean = 0.542ε)</span><br> <br> 358 (<span class="emphasis"><em><cmath>:</em></span> Max = 1.26ε (Mean = 0.441ε))<br> 359 (<span class="emphasis"><em><math.h>:</em></span> Max = 1.26ε (Mean = 0.441ε)) 360 </p> 361 </td> 362<td> 363 <p> 364 <span class="blue">Max = 0.868ε (Mean = 0.147ε)</span><br> <br> 365 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.9ε (Mean = 0.472ε)) 366 </p> 367 </td> 368<td> 369 <p> 370 <span class="blue">Max = 1.57ε (Mean = 0.564ε)</span><br> <br> 371 (<span class="emphasis"><em><math.h>:</em></span> Max = 4.91ε (Mean = 1.54ε)) 372 </p> 373 </td> 374<td> 375 <p> 376 <span class="blue">Max = 1.14ε (Mean = 0.248ε)</span><br> <br> 377 (<span class="emphasis"><em><math.h>:</em></span> Max = 1.84ε (Mean = 0.331ε)) 378 </p> 379 </td> 380</tr> 381</tbody> 382</table></div> 383</div> 384<br class="table-break"><p> 385 The following error plot are based on an exhaustive search of the functions 386 domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code> 387 precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span> 388 <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>. 389 </p> 390<div class="blockquote"><blockquote class="blockquote"><p> 391 <span class="inlinemediaobject"><img src="../../../graphs/erf__double.svg" align="middle"></span> 392 393 </p></blockquote></div> 394<div class="blockquote"><blockquote class="blockquote"><p> 395 <span class="inlinemediaobject"><img src="../../../graphs/erf__80_bit_long_double.svg" align="middle"></span> 396 397 </p></blockquote></div> 398<div class="blockquote"><blockquote class="blockquote"><p> 399 <span class="inlinemediaobject"><img src="../../../graphs/erf____float128.svg" align="middle"></span> 400 401 </p></blockquote></div> 402<h5> 403<a name="math_toolkit.sf_erf.error_function.h3"></a> 404 <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> 405 </h5> 406<p> 407 The tests for these functions come in two parts: basic sanity checks use 408 spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf" target="_top">Mathworld's 409 online evaluator</a>, while accuracy checks use high-precision test values 410 calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> 411 and this implementation. Note that the generic and type-specific versions 412 of these functions use differing implementations internally, so this gives 413 us reasonably independent test data. Using our test data to test other "known 414 good" implementations also provides an additional sanity check. 415 </p> 416<h5> 417<a name="math_toolkit.sf_erf.error_function.h4"></a> 418 <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> 419 </h5> 420<p> 421 All versions of these functions first use the usual reflection formulas to 422 make their arguments positive: 423 </p> 424<div class="blockquote"><blockquote class="blockquote"><p> 425 <span class="serif_italic"><span class="emphasis"><em>erf(-z) = 1 - erf(z);</em></span> 426 </span> 427 </p></blockquote></div> 428<div class="blockquote"><blockquote class="blockquote"><p> 429 <span class="serif_italic"><span class="emphasis"><em>erfc(-z) = 2 - erfc(z);</em></span> 430 // preferred when -z < -0.5</span> 431 </p></blockquote></div> 432<div class="blockquote"><blockquote class="blockquote"><p> 433 <span class="serif_italic"><span class="emphasis"><em>erfc(-z) = 1 + erf(z);</em></span> 434 // preferred when -0.5 <= -z < 0</span> 435 </p></blockquote></div> 436<p> 437 The generic versions of these functions are implemented in terms of the incomplete 438 gamma function. 439 </p> 440<p> 441 When the significand (mantissa) size is recognised (currently for 53, 64 442 and 113-bit reals, plus single-precision 24-bit handled via promotion to 443 double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised 444 by JM</a> are used. 445 </p> 446<p> 447 For <code class="computeroutput"><span class="identifier">z</span> <span class="special"><=</span> 448 <span class="number">0.5</span></code> then a rational approximation to 449 erf is used, based on the observation that erf is an odd function and therefore 450 erf is calculated using: 451 </p> 452<div class="blockquote"><blockquote class="blockquote"><p> 453 <span class="serif_italic"><span class="emphasis"><em>erf(z) = z * (C + R(z*z));</em></span></span> 454 </p></blockquote></div> 455<p> 456 where the rational approximation <span class="emphasis"><em>R(z*z)</em></span> is optimised 457 for absolute error: as long as its absolute error is small enough compared 458 to the constant C, then any round-off error incurred during the computation 459 of R(z*z) will effectively disappear from the result. As a result the error 460 for erf and erfc in this region is very low: the last bit is incorrect in 461 only a very small number of cases. 462 </p> 463<p> 464 For <code class="computeroutput"><span class="identifier">z</span> <span class="special">></span> 465 <span class="number">0.5</span></code> we observe that over a small interval 466 [<span class="emphasis"><em>a, b)</em></span> then: 467 </p> 468<div class="blockquote"><blockquote class="blockquote"><p> 469 <span class="serif_italic"><span class="emphasis"><em>erfc(z) * exp(z*z) * z ~ c</em></span></span> 470 </p></blockquote></div> 471<p> 472 for some constant c. 473 </p> 474<p> 475 Therefore for <code class="computeroutput"><span class="identifier">z</span> <span class="special">></span> 476 <span class="number">0.5</span></code> we calculate <code class="computeroutput"><span class="identifier">erfc</span></code> 477 using: 478 </p> 479<div class="blockquote"><blockquote class="blockquote"><p> 480 <span class="serif_italic"><span class="emphasis"><em>erfc(z) = exp(-z*z) * (C + R(z - 481 B)) / z;</em></span></span> 482 </p></blockquote></div> 483<p> 484 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> 485 <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> 486 <span class="identifier">z</span></code> taken at the endpoints of the 487 range. Once again, as long as the absolute error in R(z - B) is small compared 488 to <code class="computeroutput"><span class="identifier">c</span></code> then <code class="computeroutput"><span class="identifier">c</span> 489 <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span> <span class="special">-</span> 490 <span class="identifier">B</span><span class="special">)</span></code> 491 will be correctly rounded, and the error in the result will depend only on 492 the accuracy of the exp function. In practice, in all but a very small number 493 of cases, the error is confined to the last bit of the result. The constant 494 <code class="computeroutput"><span class="identifier">B</span></code> is chosen so that the left 495 hand end of the range of the rational approximation is 0. 496 </p> 497<p> 498 For large <code class="computeroutput"><span class="identifier">z</span></code> over a range 499 [a, +∞] the above approximation is modified to: 500 </p> 501<div class="blockquote"><blockquote class="blockquote"><p> 502 <span class="serif_italic"><span class="emphasis"><em>erfc(z) = exp(-z*z) * (C + R(1 / 503 z)) / z;</em></span></span> 504 </p></blockquote></div> 505</div> 506<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 507<td align="left"></td> 508<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 509 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 510 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 511 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 512 Daryle Walker and Xiaogang Zhang<p> 513 Distributed under the Boost Software License, Version 1.0. (See accompanying 514 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>) 515 </p> 516</div></td> 517</tr></table> 518<hr> 519<div class="spirit-nav"> 520<a accesskey="p" href="../sf_erf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_erf.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="error_inv.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 521</div> 522</body> 523</html> 524
