boost/math/special_functions/erf.hpp

   1 //  (C) Copyright John Maddock 2006.
   2 //  Use, modification and distribution are subject to the
   3 //  Boost Software License, Version 1.0. (See accompanying file
   4 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   5
   6 #ifndef BOOST_MATH_SPECIAL_ERF_HPP
   7 #define BOOST_MATH_SPECIAL_ERF_HPP
   8
   9 #ifdef _MSC_VER
  10 #pragma once
  11 #endif
  12
  13 #include <boost/math/special_functions/math_fwd.hpp>
  14 #include <boost/math/tools/config.hpp>
  15 #include <boost/math/special_functions/gamma.hpp>
  16 #include <boost/math/tools/roots.hpp>
  17 #include <boost/math/policies/error_handling.hpp>
  18 #include <boost/math/tools/big_constant.hpp>
  19
  20 namespace boost{ namespace math{
  21
  22 namespace detail
  23 {
  24
  25 //
  26 // Asymptotic series for large z:
  27 //
  28 template <class T>
  29 struct erf_asympt_series_t
  30 {
  31    erf_asympt_series_t(T z) : xx(2 * -z * z), tk(1)
  32    {
  33       BOOST_MATH_STD_USING
  34       result = -exp(-z * z) / sqrt(boost::math::constants::pi<T>());
  35       result /= z;
  36    }
  37
  38    typedef T result_type;
  39
  40    T operator()()
  41    {
  42       BOOST_MATH_STD_USING
  43       T r = result;
  44       result *= tk / xx;
  45       tk += 2;
  46       if( fabs(r) < fabs(result))
  47          result = 0;
  48       return r;
  49    }
  50 private:
  51    T result;
  52    T xx;
  53    int tk;
  54 };
  55 //
  56 // How large z has to be in order to ensure that the series converges:
  57 //
  58 template <class T>
  59 inline float erf_asymptotic_limit_N(const T&)
  60 {
  61    return (std::numeric_limits<float>::max)();
  62 }
  63 inline float erf_asymptotic_limit_N(const mpl::int_<24>&)
  64 {
  65    return 2.8F;
  66 }
  67 inline float erf_asymptotic_limit_N(const mpl::int_<53>&)
  68 {
  69    return 4.3F;
  70 }
  71 inline float erf_asymptotic_limit_N(const mpl::int_<64>&)
  72 {
  73    return 4.8F;
  74 }
  75 inline float erf_asymptotic_limit_N(const mpl::int_<106>&)
  76 {
  77    return 6.5F;
  78 }
  79 inline float erf_asymptotic_limit_N(const mpl::int_<113>&)
  80 {
  81    return 6.8F;
  82 }
  83
  84 template <class T, class Policy>
  85 inline T erf_asymptotic_limit()
  86 {
  87    typedef typename policies::precision<T, Policy>::type precision_type;
  88    typedef typename mpl::if_<
  89       mpl::less_equal<precision_type, mpl::int_<24> >,
  90       typename mpl::if_<
  91          mpl::less_equal<precision_type, mpl::int_<0> >,
  92          mpl::int_<0>,
  93          mpl::int_<24>
  94       >::type,
  95       typename mpl::if_<
  96          mpl::less_equal<precision_type, mpl::int_<53> >,
  97          mpl::int_<53>,
  98          typename mpl::if_<
  99             mpl::less_equal<precision_type, mpl::int_<64> >,
 100             mpl::int_<64>,
 101             typename mpl::if_<
 102                mpl::less_equal<precision_type, mpl::int_<106> >,
 103                mpl::int_<106>,
 104                typename mpl::if_<
 105                   mpl::less_equal<precision_type, mpl::int_<113> >,
 106                   mpl::int_<113>,
 107                   mpl::int_<0>
 108                >::type
 109             >::type
 110          >::type
 111       >::type
 112    >::type tag_type;
 113    return erf_asymptotic_limit_N(tag_type());
 114 }
 115
 116 template <class T, class Policy, class Tag>
 117 T erf_imp(T z, bool invert, const Policy& pol, const Tag& t)
 118 {
 119    BOOST_MATH_STD_USING
 120
 121    BOOST_MATH_INSTRUMENT_CODE("Generic erf_imp called");
 122
 123    if(z < 0)
 124    {
 125       if(!invert)
 126          return -erf_imp(T(-z), invert, pol, t);
 127       else
 128          return 1 + erf_imp(T(-z), false, pol, t);
 129    }
 130
 131    T result;
 132
 133    if(!invert && (z > detail::erf_asymptotic_limit<T, Policy>()))
 134    {
 135       detail::erf_asympt_series_t<T> s(z);
 136       boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
 137       result = boost::math::tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, 1);
 138       policies::check_series_iterations<T>("boost::math::erf<%1%>(%1%, %1%)", max_iter, pol);
 139    }
 140    else
 141    {
 142       T x = z * z;
 143       if(x < 0.6)
 144       {
 145          // Compute P:
 146          result = z * exp(-x);
 147          result /= sqrt(boost::math::constants::pi<T>());
 148          if(result != 0)
 149             result *= 2 * detail::lower_gamma_series(T(0.5f), x, pol);
 150       }
 151       else if(x < 1.1f)
 152       {
 153          // Compute Q:
 154          invert = !invert;
 155          result = tgamma_small_upper_part(T(0.5f), x, pol);
 156          result /= sqrt(boost::math::constants::pi<T>());
 157       }
 158       else
 159       {
 160          // Compute Q:
 161          invert = !invert;
 162          result = z * exp(-x);
 163          result /= sqrt(boost::math::constants::pi<T>());
 164          result *= upper_gamma_fraction(T(0.5f), x, policies::get_epsilon<T, Policy>());
 165       }
 166    }
 167    if(invert)
 168       result = 1 - result;
 169    return result;
 170 }
 171
 172 template <class T, class Policy>
 173 T erf_imp(T z, bool invert, const Policy& pol, const mpl::int_<53>& t)
 174 {
 175    BOOST_MATH_STD_USING
 176
 177    BOOST_MATH_INSTRUMENT_CODE("53-bit precision erf_imp called");
 178
 179    if(z < 0)
 180    {
 181       if(!invert)
 182          return -erf_imp(T(-z), invert, pol, t);
 183       else if(z < -0.5)
 184          return 2 - erf_imp(T(-z), invert, pol, t);
 185       else
 186          return 1 + erf_imp(T(-z), false, pol, t);
 187    }
 188
 189    T result;
 190
 191    //
 192    // Big bunch of selection statements now to pick
 193    // which implementation to use,
 194    // try to put most likely options first:
 195    //
 196    if(z < 0.5)
 197    {
 198       //
 199       // We're going to calculate erf:
 200       //
 201       if(z < 1e-10)
 202       {
 203          if(z == 0)
 204          {
 205             result = T(0);
 206          }
 207          else
 208          {
 209             static const T c = BOOST_MATH_BIG_CONSTANT(T, 53, 0.003379167095512573896158903121545171688);
 210             result = static_cast<T>(z * 1.125f + z * c);
 211          }
 212       }
 213       else
 214       {
 215          // Maximum Deviation Found:                     1.561e-17
 216          // Expected Error Term:                         1.561e-17
 217          // Maximum Relative Change in Control Points:   1.155e-04
 218          // Max Error found at double precision =        2.961182e-17
 219
 220          static const T Y = 1.044948577880859375f;
 221          static const T P[] = {
 222             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0834305892146531832907),
 223             BOOST_MATH_BIG_CONSTANT(T, 53, -0.338165134459360935041),
 224             BOOST_MATH_BIG_CONSTANT(T, 53, -0.0509990735146777432841),
 225             BOOST_MATH_BIG_CONSTANT(T, 53, -0.00772758345802133288487),
 226             BOOST_MATH_BIG_CONSTANT(T, 53, -0.000322780120964605683831),
 227          };
 228          static const T Q[] = {
 229             BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 230             BOOST_MATH_BIG_CONSTANT(T, 53, 0.455004033050794024546),
 231             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0875222600142252549554),
 232             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00858571925074406212772),
 233             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000370900071787748000569),
 234          };
 235          T zz = z * z;
 236          result = z * (Y + tools::evaluate_polynomial(P, zz) / tools::evaluate_polynomial(Q, zz));
 237       }
 238    }
 239    else if(invert ? (z < 28) : (z < 5.8f))
 240    {
 241       //
 242       // We'll be calculating erfc:
 243       //
 244       invert = !invert;
 245       if(z < 1.5f)
 246       {
 247          // Maximum Deviation Found:                     3.702e-17
 248          // Expected Error Term:                         3.702e-17
 249          // Maximum Relative Change in Control Points:   2.845e-04
 250          // Max Error found at double precision =        4.841816e-17
 251          static const T Y = 0.405935764312744140625f;
 252          static const T P[] = {
 253             BOOST_MATH_BIG_CONSTANT(T, 53, -0.098090592216281240205),
 254             BOOST_MATH_BIG_CONSTANT(T, 53, 0.178114665841120341155),
 255             BOOST_MATH_BIG_CONSTANT(T, 53, 0.191003695796775433986),
 256             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0888900368967884466578),
 257             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0195049001251218801359),
 258             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00180424538297014223957),
 259          };
 260          static const T Q[] = {
 261             BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 262             BOOST_MATH_BIG_CONSTANT(T, 53, 1.84759070983002217845),
 263             BOOST_MATH_BIG_CONSTANT(T, 53, 1.42628004845511324508),
 264             BOOST_MATH_BIG_CONSTANT(T, 53, 0.578052804889902404909),
 265             BOOST_MATH_BIG_CONSTANT(T, 53, 0.12385097467900864233),
 266             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0113385233577001411017),
 267             BOOST_MATH_BIG_CONSTANT(T, 53, 0.337511472483094676155e-5),
 268          };
 269          BOOST_MATH_INSTRUMENT_VARIABLE(Y);
 270          BOOST_MATH_INSTRUMENT_VARIABLE(P[0]);
 271          BOOST_MATH_INSTRUMENT_VARIABLE(Q[0]);
 272          BOOST_MATH_INSTRUMENT_VARIABLE(z);
 273          result = Y + tools::evaluate_polynomial(P, T(z - 0.5)) / tools::evaluate_polynomial(Q, T(z - 0.5));
 274          BOOST_MATH_INSTRUMENT_VARIABLE(result);
 275          result *= exp(-z * z) / z;
 276          BOOST_MATH_INSTRUMENT_VARIABLE(result);
 277       }
 278       else if(z < 2.5f)
 279       {
 280          // Max Error found at double precision =        6.599585e-18
 281          // Maximum Deviation Found:                     3.909e-18
 282          // Expected Error Term:                         3.909e-18
 283          // Maximum Relative Change in Control Points:   9.886e-05
 284          static const T Y = 0.50672817230224609375f;
 285          static const T P[] = {
 286             BOOST_MATH_BIG_CONSTANT(T, 53, -0.0243500476207698441272),
 287             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0386540375035707201728),
 288             BOOST_MATH_BIG_CONSTANT(T, 53, 0.04394818964209516296),
 289             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175679436311802092299),
 290             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00323962406290842133584),
 291             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000235839115596880717416),
 292          };
 293          static const T Q[] = {
 294             BOOST_MATH_BIG_CONSTANT(T, 53, 1),
 295             BOOST_MATH_BIG_CONSTANT(T, 53, 1.53991494948552447182),
 296             BOOST_MATH_BIG_CONSTANT(T, 53, 0.982403709157920235114),
 297             BOOST_MATH_BIG_CONSTANT(T, 53, 0.325732924782444448493),
 298             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0563921837420478160373),
 299             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00410369723978904575884),
 300          };
 301          result = Y + tools::evaluate_polynomial(P, T(z - 1.5)) / tools::evaluate_polynomial(Q, T(z - 1.5));
 302          result *= exp(-z * z) / z;
 303       }
 304       else if(z < 4.5f)
 305       {
 306          // Maximum Deviation Found:                     1.512e-17
 307          // Expected Error Term:                         1.512e-17
 308          // Maximum Relative Change in Control Points:   2.222e-04
 309          // Max Error found at double precision =        2.062515e-17
 310          static const T Y = 0.5405750274658203125f;
 311          static const T P[] = {
 312             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00295276716530971662634),
 313             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0137384425896355332126),
 314             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00840807615555585383007),
 315             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00212825620914618649141),
 316             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000250269961544794627958),
 317             BOOST_MATH_BIG_CONSTANT(T, 53, 0.113212406648847561139e-4),
 318          };
 319          static const T Q[] = {
 320             BOOST_MATH_BIG_CONSTANT(T, 53, 1),
 321             BOOST_MATH_BIG_CONSTANT(T, 53, 1.04217814166938418171),
 322             BOOST_MATH_BIG_CONSTANT(T, 53, 0.442597659481563127003),
 323             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0958492726301061423444),
 324             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0105982906484876531489),
 325             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000479411269521714493907),
 326          };
 327          result = Y + tools::evaluate_polynomial(P, T(z - 3.5)) / tools::evaluate_polynomial(Q, T(z - 3.5));
 328          result *= exp(-z * z) / z;
 329       }
 330       else
 331       {
 332          // Max Error found at double precision =        2.997958e-17
 333          // Maximum Deviation Found:                     2.860e-17
 334          // Expected Error Term:                         2.859e-17
 335          // Maximum Relative Change in Control Points:   1.357e-05
 336          static const T Y = 0.5579090118408203125f;
 337          static const T P[] = {
 338             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00628057170626964891937),
 339             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175389834052493308818),
 340             BOOST_MATH_BIG_CONSTANT(T, 53, -0.212652252872804219852),
 341             BOOST_MATH_BIG_CONSTANT(T, 53, -0.687717681153649930619),
 342             BOOST_MATH_BIG_CONSTANT(T, 53, -2.5518551727311523996),
 343             BOOST_MATH_BIG_CONSTANT(T, 53, -3.22729451764143718517),
 344             BOOST_MATH_BIG_CONSTANT(T, 53, -2.8175401114513378771),
 345          };
 346          static const T Q[] = {
 347             BOOST_MATH_BIG_CONSTANT(T, 53, 1),
 348             BOOST_MATH_BIG_CONSTANT(T, 53, 2.79257750980575282228),
 349             BOOST_MATH_BIG_CONSTANT(T, 53, 11.0567237927800161565),
 350             BOOST_MATH_BIG_CONSTANT(T, 53, 15.930646027911794143),
 351             BOOST_MATH_BIG_CONSTANT(T, 53, 22.9367376522880577224),
 352             BOOST_MATH_BIG_CONSTANT(T, 53, 13.5064170191802889145),
 353             BOOST_MATH_BIG_CONSTANT(T, 53, 5.48409182238641741584),
 354          };
 355          result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
 356          result *= exp(-z * z) / z;
 357       }
 358    }
 359    else
 360    {
 361       //
 362       // Any value of z larger than 28 will underflow to zero:
 363       //
 364       result = 0;
 365       invert = !invert;
 366    }
 367
 368    if(invert)
 369    {
 370       result = 1 - result;
 371    }
 372
 373    return result;
 374 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const mpl::int_<53>& t)
 375
 376
 377 template <class T, class Policy>
 378 T erf_imp(T z, bool invert, const Policy& pol, const mpl::int_<64>& t)
 379 {
 380    BOOST_MATH_STD_USING
 381
 382    BOOST_MATH_INSTRUMENT_CODE("64-bit precision erf_imp called");
 383
 384    if(z < 0)
 385    {
 386       if(!invert)
 387          return -erf_imp(T(-z), invert, pol, t);
 388       else if(z < -0.5)
 389          return 2 - erf_imp(T(-z), invert, pol, t);
 390       else
 391          return 1 + erf_imp(T(-z), false, pol, t);
 392    }
 393
 394    T result;
 395
 396    //
 397    // Big bunch of selection statements now to pick which
 398    // implementation to use, try to put most likely options
 399    // first:
 400    //
 401    if(z < 0.5)
 402    {
 403       //
 404       // We're going to calculate erf:
 405       //
 406       if(z == 0)
 407       {
 408          result = 0;
 409       }
 410       else if(z < 1e-10)
 411       {
 412          static const T c = BOOST_MATH_BIG_CONSTANT(T, 64, 0.003379167095512573896158903121545171688);
 413          result = z * 1.125 + z * c;
 414       }
 415       else
 416       {
 417          // Max Error found at long double precision =   1.623299e-20
 418          // Maximum Deviation Found:                     4.326e-22
 419          // Expected Error Term:                         -4.326e-22
 420          // Maximum Relative Change in Control Points:   1.474e-04
 421          static const T Y = 1.044948577880859375f;
 422          static const T P[] = {
 423             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0834305892146531988966),
 424             BOOST_MATH_BIG_CONSTANT(T, 64, -0.338097283075565413695),
 425             BOOST_MATH_BIG_CONSTANT(T, 64, -0.0509602734406067204596),
 426             BOOST_MATH_BIG_CONSTANT(T, 64, -0.00904906346158537794396),
 427             BOOST_MATH_BIG_CONSTANT(T, 64, -0.000489468651464798669181),
 428             BOOST_MATH_BIG_CONSTANT(T, 64, -0.200305626366151877759e-4),
 429          };
 430          static const T Q[] = {
 431             BOOST_MATH_BIG_CONSTANT(T, 64, 1),
 432             BOOST_MATH_BIG_CONSTANT(T, 64, 0.455817300515875172439),
 433             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0916537354356241792007),
 434             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0102722652675910031202),
 435             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000650511752687851548735),
 436             BOOST_MATH_BIG_CONSTANT(T, 64, 0.189532519105655496778e-4),
 437          };
 438          result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
 439       }
 440    }
 441    else if(invert ? (z < 110) : (z < 6.4f))
 442    {
 443       //
 444       // We'll be calculating erfc:
 445       //
 446       invert = !invert;
 447       if(z < 1.5)
 448       {
 449          // Max Error found at long double precision =   3.239590e-20
 450          // Maximum Deviation Found:                     2.241e-20
 451          // Expected Error Term:                         -2.241e-20
 452          // Maximum Relative Change in Control Points:   5.110e-03
 453          static const T Y = 0.405935764312744140625f;
 454          static const T P[] = {
 455             BOOST_MATH_BIG_CONSTANT(T, 64, -0.0980905922162812031672),
 456             BOOST_MATH_BIG_CONSTANT(T, 64, 0.159989089922969141329),
 457             BOOST_MATH_BIG_CONSTANT(T, 64, 0.222359821619935712378),
 458             BOOST_MATH_BIG_CONSTANT(T, 64, 0.127303921703577362312),
 459             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0384057530342762400273),
 460             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00628431160851156719325),
 461             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000441266654514391746428),
 462             BOOST_MATH_BIG_CONSTANT(T, 64, 0.266689068336295642561e-7),
 463          };
 464          static const T Q[] = {
 465             BOOST_MATH_BIG_CONSTANT(T, 64, 1),
 466             BOOST_MATH_BIG_CONSTANT(T, 64, 2.03237474985469469291),
 467             BOOST_MATH_BIG_CONSTANT(T, 64, 1.78355454954969405222),
 468             BOOST_MATH_BIG_CONSTANT(T, 64, 0.867940326293760578231),
 469             BOOST_MATH_BIG_CONSTANT(T, 64, 0.248025606990021698392),
 470             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0396649631833002269861),
 471             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00279220237309449026796),
 472          };
 473          result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
 474          result *= exp(-z * z) / z;
 475       }
 476       else if(z < 2.5)
 477       {
 478          // Max Error found at long double precision =   3.686211e-21
 479          // Maximum Deviation Found:                     1.495e-21
 480          // Expected Error Term:                         -1.494e-21
 481          // Maximum Relative Change in Control Points:   1.793e-04
 482          static const T Y = 0.50672817230224609375f;
 483          static const T P[] = {
 484             BOOST_MATH_BIG_CONSTANT(T, 64, -0.024350047620769840217),
 485             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0343522687935671451309),
 486             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0505420824305544949541),
 487             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0257479325917757388209),
 488             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00669349844190354356118),
 489             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00090807914416099524444),
 490             BOOST_MATH_BIG_CONSTANT(T, 64, 0.515917266698050027934e-4),
 491          };
 492          static const T Q[] = {
 493             BOOST_MATH_BIG_CONSTANT(T, 64, 1),
 494             BOOST_MATH_BIG_CONSTANT(T, 64, 1.71657861671930336344),
 495             BOOST_MATH_BIG_CONSTANT(T, 64, 1.26409634824280366218),
 496             BOOST_MATH_BIG_CONSTANT(T, 64, 0.512371437838969015941),
 497             BOOST_MATH_BIG_CONSTANT(T, 64, 0.120902623051120950935),
 498             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0158027197831887485261),
 499             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000897871370778031611439),
 500          };
 501          result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
 502          result *= exp(-z * z) / z;
 503       }
 504       else if(z < 4.5)
 505       {
 506          // Maximum Deviation Found:                     1.107e-20
 507          // Expected Error Term:                         -1.106e-20
 508          // Maximum Relative Change in Control Points:   1.709e-04
 509          // Max Error found at long double precision =   1.446908e-20
 510          static const T Y  = 0.5405750274658203125f;
 511          static const T P[] = {
 512             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0029527671653097284033),
 513             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0141853245895495604051),
 514             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0104959584626432293901),
 515             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00343963795976100077626),
 516             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00059065441194877637899),
 517             BOOST_MATH_BIG_CONSTANT(T, 64, 0.523435380636174008685e-4),
 518             BOOST_MATH_BIG_CONSTANT(T, 64, 0.189896043050331257262e-5),
 519          };
 520          static const T Q[] = {
 521             BOOST_MATH_BIG_CONSTANT(T, 64, 1),
 522             BOOST_MATH_BIG_CONSTANT(T, 64, 1.19352160185285642574),
 523             BOOST_MATH_BIG_CONSTANT(T, 64, 0.603256964363454392857),
 524             BOOST_MATH_BIG_CONSTANT(T, 64, 0.165411142458540585835),
 525             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0259729870946203166468),
 526             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00221657568292893699158),
 527             BOOST_MATH_BIG_CONSTANT(T, 64, 0.804149464190309799804e-4),
 528          };
 529          result = Y + tools::evaluate_polynomial(P, T(z - 3.5f)) / tools::evaluate_polynomial(Q, T(z - 3.5f));
 530          result *= exp(-z * z) / z;
 531       }
 532       else
 533       {
 534          // Max Error found at long double precision =   7.961166e-21
 535          // Maximum Deviation Found:                     6.677e-21
 536          // Expected Error Term:                         6.676e-21
 537          // Maximum Relative Change in Control Points:   2.319e-05
 538          static const T Y = 0.55825519561767578125f;
 539          static const T P[] = {
 540             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00593438793008050214106),
 541             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0280666231009089713937),
 542             BOOST_MATH_BIG_CONSTANT(T, 64, -0.141597835204583050043),
 543             BOOST_MATH_BIG_CONSTANT(T, 64, -0.978088201154300548842),
 544             BOOST_MATH_BIG_CONSTANT(T, 64, -5.47351527796012049443),
 545             BOOST_MATH_BIG_CONSTANT(T, 64, -13.8677304660245326627),
 546             BOOST_MATH_BIG_CONSTANT(T, 64, -27.1274948720539821722),
 547             BOOST_MATH_BIG_CONSTANT(T, 64, -29.2545152747009461519),
 548             BOOST_MATH_BIG_CONSTANT(T, 64, -16.8865774499799676937),
 549          };
 550          static const T Q[] = {
 551             BOOST_MATH_BIG_CONSTANT(T, 64, 1),
 552             BOOST_MATH_BIG_CONSTANT(T, 64, 4.72948911186645394541),
 553             BOOST_MATH_BIG_CONSTANT(T, 64, 23.6750543147695749212),
 554             BOOST_MATH_BIG_CONSTANT(T, 64, 60.0021517335693186785),
 555             BOOST_MATH_BIG_CONSTANT(T, 64, 131.766251645149522868),
 556             BOOST_MATH_BIG_CONSTANT(T, 64, 178.167924971283482513),
 557             BOOST_MATH_BIG_CONSTANT(T, 64, 182.499390505915222699),
 558             BOOST_MATH_BIG_CONSTANT(T, 64, 104.365251479578577989),
 559             BOOST_MATH_BIG_CONSTANT(T, 64, 30.8365511891224291717),
 560          };
 561          result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
 562          result *= exp(-z * z) / z;
 563       }
 564    }
 565    else
 566    {
 567       //
 568       // Any value of z larger than 110 will underflow to zero:
 569       //
 570       result = 0;
 571       invert = !invert;
 572    }
 573
 574    if(invert)
 575    {
 576       result = 1 - result;
 577    }
 578
 579    return result;
 580 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const mpl::int_<64>& t)
 581
 582
 583 template <class T, class Policy>
 584 T erf_imp(T z, bool invert, const Policy& pol, const mpl::int_<113>& t)
 585 {
 586    BOOST_MATH_STD_USING
 587
 588    BOOST_MATH_INSTRUMENT_CODE("113-bit precision erf_imp called");
 589
 590    if(z < 0)
 591    {
 592       if(!invert)
 593          return -erf_imp(T(-z), invert, pol, t);
 594       else if(z < -0.5)
 595          return 2 - erf_imp(T(-z), invert, pol, t);
 596       else
 597          return 1 + erf_imp(T(-z), false, pol, t);
 598    }
 599
 600    T result;
 601
 602    //
 603    // Big bunch of selection statements now to pick which
 604    // implementation to use, try to put most likely options
 605    // first:
 606    //
 607    if(z < 0.5)
 608    {
 609       //
 610       // We're going to calculate erf:
 611       //
 612       if(z == 0)
 613       {
 614          result = 0;
 615       }
 616       else if(z < 1e-20)
 617       {
 618          static const T c = BOOST_MATH_BIG_CONSTANT(T, 113, 0.003379167095512573896158903121545171688);
 619          result = z * 1.125 + z * c;
 620       }
 621       else
 622       {
 623          // Max Error found at long double precision =   2.342380e-35
 624          // Maximum Deviation Found:                     6.124e-36
 625          // Expected Error Term:                         -6.124e-36
 626          // Maximum Relative Change in Control Points:   3.492e-10
 627          static const T Y = 1.0841522216796875f;
 628          static const T P[] = {
 629             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0442269454158250738961589031215451778),
 630             BOOST_MATH_BIG_CONSTANT(T, 113, -0.35549265736002144875335323556961233),
 631             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0582179564566667896225454670863270393),
 632             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0112694696904802304229950538453123925),
 633             BOOST_MATH_BIG_CONSTANT(T, 113, -0.000805730648981801146251825329609079099),
 634             BOOST_MATH_BIG_CONSTANT(T, 113, -0.566304966591936566229702842075966273e-4),
 635             BOOST_MATH_BIG_CONSTANT(T, 113, -0.169655010425186987820201021510002265e-5),
 636             BOOST_MATH_BIG_CONSTANT(T, 113, -0.344448249920445916714548295433198544e-7),
 637          };
 638          static const T Q[] = {
 639             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 640             BOOST_MATH_BIG_CONSTANT(T, 113, 0.466542092785657604666906909196052522),
 641             BOOST_MATH_BIG_CONSTANT(T, 113, 0.100005087012526447295176964142107611),
 642             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0128341535890117646540050072234142603),
 643             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00107150448466867929159660677016658186),
 644             BOOST_MATH_BIG_CONSTANT(T, 113, 0.586168368028999183607733369248338474e-4),
 645             BOOST_MATH_BIG_CONSTANT(T, 113, 0.196230608502104324965623171516808796e-5),
 646             BOOST_MATH_BIG_CONSTANT(T, 113, 0.313388521582925207734229967907890146e-7),
 647          };
 648          result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
 649       }
 650    }
 651    else if(invert ? (z < 110) : (z < 8.65f))
 652    {
 653       //
 654       // We'll be calculating erfc:
 655       //
 656       invert = !invert;
 657       if(z < 1)
 658       {
 659          // Max Error found at long double precision =   3.246278e-35
 660          // Maximum Deviation Found:                     1.388e-35
 661          // Expected Error Term:                         1.387e-35
 662          // Maximum Relative Change in Control Points:   6.127e-05
 663          static const T Y = 0.371877193450927734375f;
 664          static const T P[] = {
 665             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0640320213544647969396032886581290455),
 666             BOOST_MATH_BIG_CONSTANT(T, 113, 0.200769874440155895637857443946706731),
 667             BOOST_MATH_BIG_CONSTANT(T, 113, 0.378447199873537170666487408805779826),
 668             BOOST_MATH_BIG_CONSTANT(T, 113, 0.30521399466465939450398642044975127),
 669             BOOST_MATH_BIG_CONSTANT(T, 113, 0.146890026406815277906781824723458196),
 670             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0464837937749539978247589252732769567),
 671             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00987895759019540115099100165904822903),
 672             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00137507575429025512038051025154301132),
 673             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0001144764551085935580772512359680516),
 674             BOOST_MATH_BIG_CONSTANT(T, 113, 0.436544865032836914773944382339900079e-5),
 675          };
 676          static const T Q[] = {
 677             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 678             BOOST_MATH_BIG_CONSTANT(T, 113, 2.47651182872457465043733800302427977),
 679             BOOST_MATH_BIG_CONSTANT(T, 113, 2.78706486002517996428836400245547955),
 680             BOOST_MATH_BIG_CONSTANT(T, 113, 1.87295924621659627926365005293130693),
 681             BOOST_MATH_BIG_CONSTANT(T, 113, 0.829375825174365625428280908787261065),
 682             BOOST_MATH_BIG_CONSTANT(T, 113, 0.251334771307848291593780143950311514),
 683             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0522110268876176186719436765734722473),
 684             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00718332151250963182233267040106902368),
 685             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000595279058621482041084986219276392459),
 686             BOOST_MATH_BIG_CONSTANT(T, 113, 0.226988669466501655990637599399326874e-4),
 687             BOOST_MATH_BIG_CONSTANT(T, 113, 0.270666232259029102353426738909226413e-10),
 688          };
 689          result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
 690          result *= exp(-z * z) / z;
 691       }
 692       else if(z < 1.5)
 693       {
 694          // Max Error found at long double precision =   2.215785e-35
 695          // Maximum Deviation Found:                     1.539e-35
 696          // Expected Error Term:                         1.538e-35
 697          // Maximum Relative Change in Control Points:   6.104e-05
 698          static const T Y = 0.45658016204833984375f;
 699          static const T P[] = {
 700             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0289965858925328393392496555094848345),
 701             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0868181194868601184627743162571779226),
 702             BOOST_MATH_BIG_CONSTANT(T, 113, 0.169373435121178901746317404936356745),
 703             BOOST_MATH_BIG_CONSTANT(T, 113, 0.13350446515949251201104889028133486),
 704             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0617447837290183627136837688446313313),
 705             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0185618495228251406703152962489700468),
 706             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00371949406491883508764162050169531013),
 707             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000485121708792921297742105775823900772),
 708             BOOST_MATH_BIG_CONSTANT(T, 113, 0.376494706741453489892108068231400061e-4),
 709             BOOST_MATH_BIG_CONSTANT(T, 113, 0.133166058052466262415271732172490045e-5),
 710          };
 711          static const T Q[] = {
 712             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 713             BOOST_MATH_BIG_CONSTANT(T, 113, 2.32970330146503867261275580968135126),
 714             BOOST_MATH_BIG_CONSTANT(T, 113, 2.46325715420422771961250513514928746),
 715             BOOST_MATH_BIG_CONSTANT(T, 113, 1.55307882560757679068505047390857842),
 716             BOOST_MATH_BIG_CONSTANT(T, 113, 0.644274289865972449441174485441409076),
 717             BOOST_MATH_BIG_CONSTANT(T, 113, 0.182609091063258208068606847453955649),
 718             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0354171651271241474946129665801606795),
 719             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00454060370165285246451879969534083997),
 720             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000349871943711566546821198612518656486),
 721             BOOST_MATH_BIG_CONSTANT(T, 113, 0.123749319840299552925421880481085392e-4),
 722          };
 723          result = Y + tools::evaluate_polynomial(P, T(z - 1.0f)) / tools::evaluate_polynomial(Q, T(z - 1.0f));
 724          result *= exp(-z * z) / z;
 725       }
 726       else if(z < 2.25)
 727       {
 728          // Maximum Deviation Found:                     1.418e-35
 729          // Expected Error Term:                         1.418e-35
 730          // Maximum Relative Change in Control Points:   1.316e-04
 731          // Max Error found at long double precision =   1.998462e-35
 732          static const T Y = 0.50250148773193359375f;
 733          static const T P[] = {
 734             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0201233630504573402185161184151016606),
 735             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0331864357574860196516686996302305002),
 736             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0716562720864787193337475444413405461),
 737             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0545835322082103985114927569724880658),
 738             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0236692635189696678976549720784989593),
 739             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00656970902163248872837262539337601845),
 740             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00120282643299089441390490459256235021),
 741             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000142123229065182650020762792081622986),
 742             BOOST_MATH_BIG_CONSTANT(T, 113, 0.991531438367015135346716277792989347e-5),
 743             BOOST_MATH_BIG_CONSTANT(T, 113, 0.312857043762117596999398067153076051e-6),
 744          };
 745          static const T Q[] = {
 746             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 747             BOOST_MATH_BIG_CONSTANT(T, 113, 2.13506082409097783827103424943508554),
 748             BOOST_MATH_BIG_CONSTANT(T, 113, 2.06399257267556230937723190496806215),
 749             BOOST_MATH_BIG_CONSTANT(T, 113, 1.18678481279932541314830499880691109),
 750             BOOST_MATH_BIG_CONSTANT(T, 113, 0.447733186643051752513538142316799562),
 751             BOOST_MATH_BIG_CONSTANT(T, 113, 0.11505680005657879437196953047542148),
 752             BOOST_MATH_BIG_CONSTANT(T, 113, 0.020163993632192726170219663831914034),
 753             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00232708971840141388847728782209730585),
 754             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000160733201627963528519726484608224112),
 755             BOOST_MATH_BIG_CONSTANT(T, 113, 0.507158721790721802724402992033269266e-5),
 756             BOOST_MATH_BIG_CONSTANT(T, 113, 0.18647774409821470950544212696270639e-12),
 757          };
 758          result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
 759          result *= exp(-z * z) / z;
 760       }
 761       else if (z < 3)
 762       {
 763          // Maximum Deviation Found:                     3.575e-36
 764          // Expected Error Term:                         3.575e-36
 765          // Maximum Relative Change in Control Points:   7.103e-05
 766          // Max Error found at long double precision =   5.794737e-36
 767          static const T Y = 0.52896785736083984375f;
 768          static const T P[] = {
 769             BOOST_MATH_BIG_CONSTANT(T, 113, -0.00902152521745813634562524098263360074),
 770             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0145207142776691539346923710537580927),
 771             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0301681239582193983824211995978678571),
 772             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0215548540823305814379020678660434461),
 773             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00864683476267958365678294164340749949),
 774             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00219693096885585491739823283511049902),
 775             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000364961639163319762492184502159894371),
 776             BOOST_MATH_BIG_CONSTANT(T, 113, 0.388174251026723752769264051548703059e-4),
 777             BOOST_MATH_BIG_CONSTANT(T, 113, 0.241918026931789436000532513553594321e-5),
 778             BOOST_MATH_BIG_CONSTANT(T, 113, 0.676586625472423508158937481943649258e-7),
 779          };
 780          static const T Q[] = {
 781             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 782             BOOST_MATH_BIG_CONSTANT(T, 113, 1.93669171363907292305550231764920001),
 783             BOOST_MATH_BIG_CONSTANT(T, 113, 1.69468476144051356810672506101377494),
 784             BOOST_MATH_BIG_CONSTANT(T, 113, 0.880023580986436640372794392579985511),
 785             BOOST_MATH_BIG_CONSTANT(T, 113, 0.299099106711315090710836273697708402),
 786             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0690593962363545715997445583603382337),
 787             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0108427016361318921960863149875360222),
 788             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00111747247208044534520499324234317695),
 789             BOOST_MATH_BIG_CONSTANT(T, 113, 0.686843205749767250666787987163701209e-4),
 790             BOOST_MATH_BIG_CONSTANT(T, 113, 0.192093541425429248675532015101904262e-5),
 791          };
 792          result = Y + tools::evaluate_polynomial(P, T(z - 2.25f)) / tools::evaluate_polynomial(Q, T(z - 2.25f));
 793          result *= exp(-z * z) / z;
 794       }
 795       else if(z < 3.5)
 796       {
 797          // Maximum Deviation Found:                     8.126e-37
 798          // Expected Error Term:                         -8.126e-37
 799          // Maximum Relative Change in Control Points:   1.363e-04
 800          // Max Error found at long double precision =   1.747062e-36
 801          static const T Y = 0.54037380218505859375f;
 802          static const T P[] = {
 803             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0033703486408887424921155540591370375),
 804             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0104948043110005245215286678898115811),
 805             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0148530118504000311502310457390417795),
 806             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00816693029245443090102738825536188916),
 807             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00249716579989140882491939681805594585),
 808             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0004655591010047353023978045800916647),
 809             BOOST_MATH_BIG_CONSTANT(T, 113, 0.531129557920045295895085236636025323e-4),
 810             BOOST_MATH_BIG_CONSTANT(T, 113, 0.343526765122727069515775194111741049e-5),
 811             BOOST_MATH_BIG_CONSTANT(T, 113, 0.971120407556888763695313774578711839e-7),
 812          };
 813          static const T Q[] = {
 814             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 815             BOOST_MATH_BIG_CONSTANT(T, 113, 1.59911256167540354915906501335919317),
 816             BOOST_MATH_BIG_CONSTANT(T, 113, 1.136006830764025173864831382946934),
 817             BOOST_MATH_BIG_CONSTANT(T, 113, 0.468565867990030871678574840738423023),
 818             BOOST_MATH_BIG_CONSTANT(T, 113, 0.122821824954470343413956476900662236),
 819             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0209670914950115943338996513330141633),
 820             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00227845718243186165620199012883547257),
 821             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000144243326443913171313947613547085553),
 822             BOOST_MATH_BIG_CONSTANT(T, 113, 0.407763415954267700941230249989140046e-5),
 823          };
 824          result = Y + tools::evaluate_polynomial(P, T(z - 3.0f)) / tools::evaluate_polynomial(Q, T(z - 3.0f));
 825          result *= exp(-z * z) / z;
 826       }
 827       else if(z < 5.5)
 828       {
 829          // Maximum Deviation Found:                     5.804e-36
 830          // Expected Error Term:                         -5.803e-36
 831          // Maximum Relative Change in Control Points:   2.475e-05
 832          // Max Error found at long double precision =   1.349545e-35
 833          static const T Y = 0.55000019073486328125f;
 834          static const T P[] = {
 835             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00118142849742309772151454518093813615),
 836             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0072201822885703318172366893469382745),
 837             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0078782276276860110721875733778481505),
 838             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00418229166204362376187593976656261146),
 839             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00134198400587769200074194304298642705),
 840             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000283210387078004063264777611497435572),
 841             BOOST_MATH_BIG_CONSTANT(T, 113, 0.405687064094911866569295610914844928e-4),
 842             BOOST_MATH_BIG_CONSTANT(T, 113, 0.39348283801568113807887364414008292e-5),
 843             BOOST_MATH_BIG_CONSTANT(T, 113, 0.248798540917787001526976889284624449e-6),
 844             BOOST_MATH_BIG_CONSTANT(T, 113, 0.929502490223452372919607105387474751e-8),
 845             BOOST_MATH_BIG_CONSTANT(T, 113, 0.156161469668275442569286723236274457e-9),
 846          };
 847          static const T Q[] = {
 848             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 849             BOOST_MATH_BIG_CONSTANT(T, 113, 1.52955245103668419479878456656709381),
 850             BOOST_MATH_BIG_CONSTANT(T, 113, 1.06263944820093830054635017117417064),
 851             BOOST_MATH_BIG_CONSTANT(T, 113, 0.441684612681607364321013134378316463),
 852             BOOST_MATH_BIG_CONSTANT(T, 113, 0.121665258426166960049773715928906382),
 853             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0232134512374747691424978642874321434),
 854             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00310778180686296328582860464875562636),
 855             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000288361770756174705123674838640161693),
 856             BOOST_MATH_BIG_CONSTANT(T, 113, 0.177529187194133944622193191942300132e-4),
 857             BOOST_MATH_BIG_CONSTANT(T, 113, 0.655068544833064069223029299070876623e-6),
 858             BOOST_MATH_BIG_CONSTANT(T, 113, 0.11005507545746069573608988651927452e-7),
 859          };
 860          result = Y + tools::evaluate_polynomial(P, T(z - 4.5f)) / tools::evaluate_polynomial(Q, T(z - 4.5f));
 861          result *= exp(-z * z) / z;
 862       }
 863       else if(z < 7.5)
 864       {
 865          // Maximum Deviation Found:                     1.007e-36
 866          // Expected Error Term:                         1.007e-36
 867          // Maximum Relative Change in Control Points:   1.027e-03
 868          // Max Error found at long double precision =   2.646420e-36
 869          static const T Y = 0.5574436187744140625f;
 870          static const T P[] = {
 871             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000293236907400849056269309713064107674),
 872             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00225110719535060642692275221961480162),
 873             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190984458121502831421717207849429799),
 874             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000747757733460111743833929141001680706),
 875             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000170663175280949889583158597373928096),
 876             BOOST_MATH_BIG_CONSTANT(T, 113, 0.246441188958013822253071608197514058e-4),
 877             BOOST_MATH_BIG_CONSTANT(T, 113, 0.229818000860544644974205957895688106e-5),
 878             BOOST_MATH_BIG_CONSTANT(T, 113, 0.134886977703388748488480980637704864e-6),
 879             BOOST_MATH_BIG_CONSTANT(T, 113, 0.454764611880548962757125070106650958e-8),
 880             BOOST_MATH_BIG_CONSTANT(T, 113, 0.673002744115866600294723141176820155e-10),
 881          };
 882          static const T Q[] = {
 883             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 884             BOOST_MATH_BIG_CONSTANT(T, 113, 1.12843690320861239631195353379313367),
 885             BOOST_MATH_BIG_CONSTANT(T, 113, 0.569900657061622955362493442186537259),
 886             BOOST_MATH_BIG_CONSTANT(T, 113, 0.169094404206844928112348730277514273),
 887             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0324887449084220415058158657252147063),
 888             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00419252877436825753042680842608219552),
 889             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00036344133176118603523976748563178578),
 890             BOOST_MATH_BIG_CONSTANT(T, 113, 0.204123895931375107397698245752850347e-4),
 891             BOOST_MATH_BIG_CONSTANT(T, 113, 0.674128352521481412232785122943508729e-6),
 892             BOOST_MATH_BIG_CONSTANT(T, 113, 0.997637501418963696542159244436245077e-8),
 893          };
 894          result = Y + tools::evaluate_polynomial(P, T(z - 6.5f)) / tools::evaluate_polynomial(Q, T(z - 6.5f));
 895          result *= exp(-z * z) / z;
 896       }
 897       else if(z < 11.5)
 898       {
 899          // Maximum Deviation Found:                     8.380e-36
 900          // Expected Error Term:                         8.380e-36
 901          // Maximum Relative Change in Control Points:   2.632e-06
 902          // Max Error found at long double precision =   9.849522e-36
 903          static const T Y = 0.56083202362060546875f;
 904          static const T P[] = {
 905             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000282420728751494363613829834891390121),
 906             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00175387065018002823433704079355125161),
 907             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0021344978564889819420775336322920375),
 908             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00124151356560137532655039683963075661),
 909             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000423600733566948018555157026862139644),
 910             BOOST_MATH_BIG_CONSTANT(T, 113, 0.914030340865175237133613697319509698e-4),
 911             BOOST_MATH_BIG_CONSTANT(T, 113, 0.126999927156823363353809747017945494e-4),
 912             BOOST_MATH_BIG_CONSTANT(T, 113, 0.110610959842869849776179749369376402e-5),
 913             BOOST_MATH_BIG_CONSTANT(T, 113, 0.55075079477173482096725348704634529e-7),
 914             BOOST_MATH_BIG_CONSTANT(T, 113, 0.119735694018906705225870691331543806e-8),
 915          };
 916          static const T Q[] = {
 917             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 918             BOOST_MATH_BIG_CONSTANT(T, 113, 1.69889613396167354566098060039549882),
 919             BOOST_MATH_BIG_CONSTANT(T, 113, 1.28824647372749624464956031163282674),
 920             BOOST_MATH_BIG_CONSTANT(T, 113, 0.572297795434934493541628008224078717),
 921             BOOST_MATH_BIG_CONSTANT(T, 113, 0.164157697425571712377043857240773164),
 922             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0315311145224594430281219516531649562),
 923             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00405588922155632380812945849777127458),
 924             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000336929033691445666232029762868642417),
 925             BOOST_MATH_BIG_CONSTANT(T, 113, 0.164033049810404773469413526427932109e-4),
 926             BOOST_MATH_BIG_CONSTANT(T, 113, 0.356615210500531410114914617294694857e-6),
 927          };
 928          result = Y + tools::evaluate_polynomial(P, T(z / 2 - 4.75f)) / tools::evaluate_polynomial(Q, T(z / 2 - 4.75f));
 929          result *= exp(-z * z) / z;
 930       }
 931       else
 932       {
 933          // Maximum Deviation Found:                     1.132e-35
 934          // Expected Error Term:                         -1.132e-35
 935          // Maximum Relative Change in Control Points:   4.674e-04
 936          // Max Error found at long double precision =   1.162590e-35
 937          static const T Y = 0.5632686614990234375f;
 938          static const T P[] = {
 939             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000920922048732849448079451574171836943),
 940             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00321439044532288750501700028748922439),
 941             BOOST_MATH_BIG_CONSTANT(T, 113, -0.250455263029390118657884864261823431),
 942             BOOST_MATH_BIG_CONSTANT(T, 113, -0.906807635364090342031792404764598142),
 943             BOOST_MATH_BIG_CONSTANT(T, 113, -8.92233572835991735876688745989985565),
 944             BOOST_MATH_BIG_CONSTANT(T, 113, -21.7797433494422564811782116907878495),
 945             BOOST_MATH_BIG_CONSTANT(T, 113, -91.1451915251976354349734589601171659),
 946             BOOST_MATH_BIG_CONSTANT(T, 113, -144.1279109655993927069052125017673),
 947             BOOST_MATH_BIG_CONSTANT(T, 113, -313.845076581796338665519022313775589),
 948             BOOST_MATH_BIG_CONSTANT(T, 113, -273.11378811923343424081101235736475),
 949             BOOST_MATH_BIG_CONSTANT(T, 113, -271.651566205951067025696102600443452),
 950             BOOST_MATH_BIG_CONSTANT(T, 113, -60.0530577077238079968843307523245547),
 951          };
 952          static const T Q[] = {
 953             BOOST_MATH_BIG_CONSTANT(T, 113, 1),
 954             BOOST_MATH_BIG_CONSTANT(T, 113, 3.49040448075464744191022350947892036),
 955             BOOST_MATH_BIG_CONSTANT(T, 113, 34.3563592467165971295915749548313227),
 956             BOOST_MATH_BIG_CONSTANT(T, 113, 84.4993232033879023178285731843850461),
 957             BOOST_MATH_BIG_CONSTANT(T, 113, 376.005865281206894120659401340373818),
 958             BOOST_MATH_BIG_CONSTANT(T, 113, 629.95369438888946233003926191755125),
 959             BOOST_MATH_BIG_CONSTANT(T, 113, 1568.35771983533158591604513304269098),
 960             BOOST_MATH_BIG_CONSTANT(T, 113, 1646.02452040831961063640827116581021),
 961             BOOST_MATH_BIG_CONSTANT(T, 113, 2299.96860633240298708910425594484895),
 962             BOOST_MATH_BIG_CONSTANT(T, 113, 1222.73204392037452750381340219906374),
 963             BOOST_MATH_BIG_CONSTANT(T, 113, 799.359797306084372350264298361110448),
 964             BOOST_MATH_BIG_CONSTANT(T, 113, 72.7415265778588087243442792401576737),
 965          };
 966          result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
 967          result *= exp(-z * z) / z;
 968       }
 969    }
 970    else
 971    {
 972       //
 973       // Any value of z larger than 110 will underflow to zero:
 974       //
 975       result = 0;
 976       invert = !invert;
 977    }
 978
 979    if(invert)
 980    {
 981       result = 1 - result;
 982    }
 983
 984    return result;
 985 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const mpl::int_<113>& t)
 986
 987 template <class T, class Policy, class tag>
 988 struct erf_initializer
 989 {
 990    struct init
 991    {
 992       init()
 993       {
 994          do_init(tag());
 995       }
 996       static void do_init(const mpl::int_<0>&){}
 997       static void do_init(const mpl::int_<53>&)
 998       {
 999          boost::math::erf(static_cast<T>(1e-12), Policy());
1000          boost::math::erf(static_cast<T>(0.25), Policy());
1001          boost::math::erf(static_cast<T>(1.25), Policy());
1002          boost::math::erf(static_cast<T>(2.25), Policy());
1003          boost::math::erf(static_cast<T>(4.25), Policy());
1004          boost::math::erf(static_cast<T>(5.25), Policy());
1005       }
1006       static void do_init(const mpl::int_<64>&)
1007       {
1008          boost::math::erf(static_cast<T>(1e-12), Policy());
1009          boost::math::erf(static_cast<T>(0.25), Policy());
1010          boost::math::erf(static_cast<T>(1.25), Policy());
1011          boost::math::erf(static_cast<T>(2.25), Policy());
1012          boost::math::erf(static_cast<T>(4.25), Policy());
1013          boost::math::erf(static_cast<T>(5.25), Policy());
1014       }
1015       static void do_init(const mpl::int_<113>&)
1016       {
1017          boost::math::erf(static_cast<T>(1e-22), Policy());
1018          boost::math::erf(static_cast<T>(0.25), Policy());
1019          boost::math::erf(static_cast<T>(1.25), Policy());
1020          boost::math::erf(static_cast<T>(2.125), Policy());
1021          boost::math::erf(static_cast<T>(2.75), Policy());
1022          boost::math::erf(static_cast<T>(3.25), Policy());
1023          boost::math::erf(static_cast<T>(5.25), Policy());
1024          boost::math::erf(static_cast<T>(7.25), Policy());
1025          boost::math::erf(static_cast<T>(11.25), Policy());
1026          boost::math::erf(static_cast<T>(12.5), Policy());
1027       }
1028       void force_instantiate()const{}
1029    };
1030    static const init initializer;
1031    static void force_instantiate()
1032    {
1033       initializer.force_instantiate();
1034    }
1035 };
1036
1037 template <class T, class Policy, class tag>
1038 const typename erf_initializer<T, Policy, tag>::init erf_initializer<T, Policy, tag>::initializer;
1039
1040 } // namespace detail
1041
1042 template <class T, class Policy>
1043 inline typename tools::promote_args<T>::type erf(T z, const Policy& /* pol */)
1044 {
1045    typedef typename tools::promote_args<T>::type result_type;
1046    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1047    typedef typename policies::precision<result_type, Policy>::type precision_type;
1048    typedef typename policies::normalise<
1049       Policy,
1050       policies::promote_float<false>,
1051       policies::promote_double<false>,
1052       policies::discrete_quantile<>,
1053       policies::assert_undefined<> >::type forwarding_policy;
1054
1055    BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
1056    BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
1057    BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
1058
1059    typedef typename mpl::if_<
1060       mpl::less_equal<precision_type, mpl::int_<0> >,
1061       mpl::int_<0>,
1062       typename mpl::if_<
1063          mpl::less_equal<precision_type, mpl::int_<53> >,
1064          mpl::int_<53>,  // double
1065          typename mpl::if_<
1066             mpl::less_equal<precision_type, mpl::int_<64> >,
1067             mpl::int_<64>, // 80-bit long double
1068             typename mpl::if_<
1069                mpl::less_equal<precision_type, mpl::int_<113> >,
1070                mpl::int_<113>, // 128-bit long double
1071                mpl::int_<0> // too many bits, use generic version.
1072             >::type
1073          >::type
1074       >::type
1075    >::type tag_type;
1076
1077    BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
1078
1079    detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
1080
1081    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
1082       static_cast<value_type>(z),
1083       false,
1084       forwarding_policy(),
1085       tag_type()), "boost::math::erf<%1%>(%1%, %1%)");
1086 }
1087
1088 template <class T, class Policy>
1089 inline typename tools::promote_args<T>::type erfc(T z, const Policy& /* pol */)
1090 {
1091    typedef typename tools::promote_args<T>::type result_type;
1092    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1093    typedef typename policies::precision<result_type, Policy>::type precision_type;
1094    typedef typename policies::normalise<
1095       Policy,
1096       policies::promote_float<false>,
1097       policies::promote_double<false>,
1098       policies::discrete_quantile<>,
1099       policies::assert_undefined<> >::type forwarding_policy;
1100
1101    BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
1102    BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
1103    BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
1104
1105    typedef typename mpl::if_<
1106       mpl::less_equal<precision_type, mpl::int_<0> >,
1107       mpl::int_<0>,
1108       typename mpl::if_<
1109          mpl::less_equal<precision_type, mpl::int_<53> >,
1110          mpl::int_<53>,  // double
1111          typename mpl::if_<
1112             mpl::less_equal<precision_type, mpl::int_<64> >,
1113             mpl::int_<64>, // 80-bit long double
1114             typename mpl::if_<
1115                mpl::less_equal<precision_type, mpl::int_<113> >,
1116                mpl::int_<113>, // 128-bit long double
1117                mpl::int_<0> // too many bits, use generic version.
1118             >::type
1119          >::type
1120       >::type
1121    >::type tag_type;
1122
1123    BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
1124
1125    detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
1126
1127    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
1128       static_cast<value_type>(z),
1129       true,
1130       forwarding_policy(),
1131       tag_type()), "boost::math::erfc<%1%>(%1%, %1%)");
1132 }
1133
1134 template <class T>
1135 inline typename tools::promote_args<T>::type erf(T z)
1136 {
1137    return boost::math::erf(z, policies::policy<>());
1138 }
1139
1140 template <class T>
1141 inline typename tools::promote_args<T>::type erfc(T z)
1142 {
1143    return boost::math::erfc(z, policies::policy<>());
1144 }
1145
1146 } // namespace math
1147 } // namespace boost
1148
1149 #include <boost/math/special_functions/detail/erf_inv.hpp>
1150
1151 #endif // BOOST_MATH_SPECIAL_ERF_HPP
1152
1153
1154
1155