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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_inv"></a><a class="link" href="error_inv.html" title="Error Function Inverses">Error Function Inverses</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_erf.error_inv.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.synopsis"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</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;20.&#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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</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;20.&#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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#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;20.&#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;20.&#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_inv.h1"></a>
  63         <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.description"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.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_inv</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;20.&#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_inv</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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  70 </pre>
  71 <p>
  72         Returns the <a href="http://functions.wolfram.com/GammaBetaErf/InverseErf/" target="_top">inverse
  73         error function</a> of z, that is a value x such that:
  74       </p>
  75 <div class="blockquote"><blockquote class="blockquote"><p>
  76           <span class="serif_italic"><span class="emphasis"><em>p = erf(x);</em></span></span>
  77         </p></blockquote></div>
  78 <div class="blockquote"><blockquote class="blockquote"><p>
  79           <span class="inlinemediaobject"><img src="../../../graphs/erf_inv.svg" align="middle"></span>
  80
  81         </p></blockquote></div>
  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_inv</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;20.&#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_inv</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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  87 </pre>
  88 <p>
  89         Returns the inverse of the complement of the error function of z, that is
  90         a value x such that:
  91       </p>
  92 <div class="blockquote"><blockquote class="blockquote"><p>
  93           <span class="serif_italic"><span class="emphasis"><em>p = erfc(x);</em></span></span>
  94         </p></blockquote></div>
  95 <div class="blockquote"><blockquote class="blockquote"><p>
  96           <span class="inlinemediaobject"><img src="../../../graphs/erfc_inv.svg" align="middle"></span>
  97
  98         </p></blockquote></div>
  99 <h5>
 100 <a name="math_toolkit.sf_erf.error_inv.h2"></a>
 101         <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.accuracy"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.accuracy">Accuracy</a>
 102       </h5>
 103 <p>
 104         For types up to and including 80-bit long doubles the approximations used
 105         are accurate to less than ~ 2 epsilon. For higher precision types these functions
 106         have the same accuracy as the <a class="link" href="error_function.html" title="Error Function erf and complement erfc">forward
 107         error functions</a>.
 108       </p>
 109 <div class="table">
 110 <a name="math_toolkit.sf_erf.error_inv.table_erf_inv"></a><p class="title"><b>Table&#160;8.30.&#160;Error rates for erf_inv</b></p>
 111 <div class="table-contents"><table class="table" summary="Error rates for erf_inv">
 112 <colgroup>
 113 <col>
 114 <col>
 115 <col>
 116 <col>
 117 <col>
 118 </colgroup>
 119 <thead><tr>
 120 <th>
 121               </th>
 122 <th>
 123                 <p>
 124                   GNU C++ version 7.1.0<br> linux<br> double
 125                 </p>
 126               </th>
 127 <th>
 128                 <p>
 129                   GNU C++ version 7.1.0<br> linux<br> long double
 130                 </p>
 131               </th>
 132 <th>
 133                 <p>
 134                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 135                 </p>
 136               </th>
 137 <th>
 138                 <p>
 139                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 140                 </p>
 141               </th>
 142 </tr></thead>
 143 <tbody><tr>
 144 <td>
 145                 <p>
 146                   Inverse Erf Function
 147                 </p>
 148               </td>
 149 <td>
 150                 <p>
 151                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 152                 </p>
 153               </td>
 154 <td>
 155                 <p>
 156                   <span class="blue">Max = 0.996&#949; (Mean = 0.389&#949;)</span>
 157                 </p>
 158               </td>
 159 <td>
 160                 <p>
 161                   <span class="blue">Max = 1.08&#949; (Mean = 0.395&#949;)</span>
 162                 </p>
 163               </td>
 164 <td>
 165                 <p>
 166                   <span class="blue">Max = 1.09&#949; (Mean = 0.502&#949;)</span>
 167                 </p>
 168               </td>
 169 </tr></tbody>
 170 </table></div>
 171 </div>
 172 <br class="table-break"><div class="table">
 173 <a name="math_toolkit.sf_erf.error_inv.table_erfc_inv"></a><p class="title"><b>Table&#160;8.31.&#160;Error rates for erfc_inv</b></p>
 174 <div class="table-contents"><table class="table" summary="Error rates for erfc_inv">
 175 <colgroup>
 176 <col>
 177 <col>
 178 <col>
 179 <col>
 180 <col>
 181 </colgroup>
 182 <thead><tr>
 183 <th>
 184               </th>
 185 <th>
 186                 <p>
 187                   GNU C++ version 7.1.0<br> linux<br> double
 188                 </p>
 189               </th>
 190 <th>
 191                 <p>
 192                   GNU C++ version 7.1.0<br> linux<br> long double
 193                 </p>
 194               </th>
 195 <th>
 196                 <p>
 197                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 198                 </p>
 199               </th>
 200 <th>
 201                 <p>
 202                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 203                 </p>
 204               </th>
 205 </tr></thead>
 206 <tbody>
 207 <tr>
 208 <td>
 209                 <p>
 210                   Inverse Erfc Function
 211                 </p>
 212               </td>
 213 <td>
 214                 <p>
 215                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 216                 </p>
 217               </td>
 218 <td>
 219                 <p>
 220                   <span class="blue">Max = 0.996&#949; (Mean = 0.397&#949;)</span>
 221                 </p>
 222               </td>
 223 <td>
 224                 <p>
 225                   <span class="blue">Max = 1.08&#949; (Mean = 0.403&#949;)</span>
 226                 </p>
 227               </td>
 228 <td>
 229                 <p>
 230                   <span class="blue">Max = 1&#949; (Mean = 0.491&#949;)</span>
 231                 </p>
 232               </td>
 233 </tr>
 234 <tr>
 235 <td>
 236                 <p>
 237                   Inverse Erfc Function: extreme values
 238                 </p>
 239               </td>
 240 <td>
 241               </td>
 242 <td>
 243                 <p>
 244                   <span class="blue">Max = 1.62&#949; (Mean = 0.383&#949;)</span>
 245                 </p>
 246               </td>
 247 <td>
 248                 <p>
 249                   <span class="blue">Max = 1.62&#949; (Mean = 0.383&#949;)</span>
 250                 </p>
 251               </td>
 252 <td>
 253               </td>
 254 </tr>
 255 </tbody>
 256 </table></div>
 257 </div>
 258 <br class="table-break"><p>
 259         The following error plot are based on an exhaustive search of the functions
 260         domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
 261         precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
 262         <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
 263       </p>
 264 <div class="blockquote"><blockquote class="blockquote"><p>
 265           <span class="inlinemediaobject"><img src="../../../graphs/erfc__double.svg" align="middle"></span>
 266
 267         </p></blockquote></div>
 268 <div class="blockquote"><blockquote class="blockquote"><p>
 269           <span class="inlinemediaobject"><img src="../../../graphs/erfc__80_bit_long_double.svg" align="middle"></span>
 270
 271         </p></blockquote></div>
 272 <div class="blockquote"><blockquote class="blockquote"><p>
 273           <span class="inlinemediaobject"><img src="../../../graphs/erfc____float128.svg" align="middle"></span>
 274
 275         </p></blockquote></div>
 276 <h5>
 277 <a name="math_toolkit.sf_erf.error_inv.h3"></a>
 278         <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.testing"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.testing">Testing</a>
 279       </h5>
 280 <p>
 281         There are two sets of tests:
 282       </p>
 283 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 284 <li class="listitem">
 285             Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>x</em></span>
 286             to <span class="emphasis"><em>p</em></span> and back again. These tests have quite generous
 287             tolerances: in general both the error functions and their inverses change
 288             so rapidly in some places that round tripping to more than a couple of
 289             significant digits isn't possible. This is especially true when <span class="emphasis"><em>p</em></span>
 290             is very near one: in this case there isn't enough "information content"
 291             in the input to the inverse function to get back where you started.
 292           </li>
 293 <li class="listitem">
 294             Accuracy checks using high-precision test values. These measure the accuracy
 295             of the result, given <span class="emphasis"><em>exact</em></span> input values.
 296           </li>
 297 </ul></div>
 298 <h5>
 299 <a name="math_toolkit.sf_erf.error_inv.h4"></a>
 300         <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.implementation"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.implementation">Implementation</a>
 301       </h5>
 302 <p>
 303         These functions use a rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 304         by JM</a> to calculate an initial approximation to the result that is
 305         accurate to ~10<sup>-19</sup>, then only if that has insufficient accuracy compared
 306         to the epsilon for T, do we clean up the result using <a href="http://en.wikipedia.org/wiki/Simple_rational_approximation" target="_top">Halley
 307         iteration</a>.
 308       </p>
 309 <p>
 310         Constructing rational approximations to the erf/erfc functions is actually
 311         surprisingly hard, especially at high precision. For this reason no attempt
 312         has been made to achieve 10<sup>-34 </sup> accuracy suitable for use with 128-bit reals.
 313       </p>
 314 <p>
 315         In the following discussion, <span class="emphasis"><em>p</em></span> is the value passed to
 316         erf_inv, and <span class="emphasis"><em>q</em></span> is the value passed to erfc_inv, so that
 317         <span class="emphasis"><em>p = 1 - q</em></span> and <span class="emphasis"><em>q = 1 - p</em></span> and in
 318         both cases we want to solve for the same result <span class="emphasis"><em>x</em></span>.
 319       </p>
 320 <p>
 321         For <span class="emphasis"><em>p &lt; 0.5</em></span> the inverse erf function is reasonably
 322         smooth and the approximation:
 323       </p>
 324 <div class="blockquote"><blockquote class="blockquote"><p>
 325           <span class="serif_italic"><span class="emphasis"><em>x = p(p + 10)(Y + R(p))</em></span></span>
 326         </p></blockquote></div>
 327 <p>
 328         Gives a good result for a constant Y, and R(p) optimised for low absolute
 329         error compared to |Y|.
 330       </p>
 331 <p>
 332         For q &lt; 0.5 things get trickier, over the interval <span class="emphasis"><em>0.5 &gt;
 333         q &gt; 0.25</em></span> the following approximation works well:
 334       </p>
 335 <div class="blockquote"><blockquote class="blockquote"><p>
 336           <span class="serif_italic"><span class="emphasis"><em>x = sqrt(-2log(q)) / (Y + R(q))</em></span></span>
 337         </p></blockquote></div>
 338 <p>
 339         While for q &lt; 0.25, let
 340       </p>
 341 <div class="blockquote"><blockquote class="blockquote"><p>
 342           <span class="serif_italic"><span class="emphasis"><em>z = sqrt(-log(q))</em></span></span>
 343         </p></blockquote></div>
 344 <p>
 345         Then the result is given by:
 346       </p>
 347 <div class="blockquote"><blockquote class="blockquote"><p>
 348           <span class="serif_italic"><span class="emphasis"><em>x = z(Y + R(z - B))</em></span></span>
 349         </p></blockquote></div>
 350 <p>
 351         As before Y is a constant and the rational function R is optimised for low
 352         absolute error compared to |Y|. B is also a constant: it is the smallest
 353         value of <span class="emphasis"><em>z</em></span> for which each approximation is valid. There
 354         are several approximations of this form each of which reaches a little further
 355         into the tail of the erfc function (at <code class="computeroutput"><span class="keyword">long</span>
 356         <span class="keyword">double</span></code> precision the extended exponent
 357         range compared to <code class="computeroutput"><span class="keyword">double</span></code> means
 358         that the tail goes on for a very long way indeed).
 359       </p>
 360 </div>
 361 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 362 <td align="left"></td>
 363 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 364       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 365       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 366       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 367       Daryle Walker and Xiaogang Zhang<p>
 368         Distributed under the Boost Software License, Version 1.0. (See accompanying
 369         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>)
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