ceph/src/boost/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 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
  21 //
  22 // This is the only way we can avoid
  23 // warning: non-standard suffix on floating constant [-Wpedantic]
  24 // when building with -Wall -pedantic.  Neither __extension__
  25 // nor #pragma diagnostic ignored work :(
  26 //
  27 #pragma GCC system_header
  28 #endif
  29
  30 namespace boost{ namespace math{
  31
  32 namespace detail
  33 {
  34
  35 //
  36 // Asymptotic series for large z:
  37 //
  38 template <class T>
  39 struct erf_asympt_series_t
  40 {
  41    erf_asympt_series_t(T z) : xx(2 * -z * z), tk(1)
  42    {
  43       BOOST_MATH_STD_USING
  44       result = -exp(-z * z) / sqrt(boost::math::constants::pi<T>());
  45       result /= z;
  46    }
  47
  48    typedef T result_type;
  49
  50    T operator()()
  51    {
  52       BOOST_MATH_STD_USING
  53       T r = result;
  54       result *= tk / xx;
  55       tk += 2;
  56       if( fabs(r) < fabs(result))
  57          result = 0;
  58       return r;
  59    }
  60 private:
  61    T result;
  62    T xx;
  63    int tk;
  64 };
  65 //
  66 // How large z has to be in order to ensure that the series converges:
  67 //
  68 template <class T>
  69 inline float erf_asymptotic_limit_N(const T&)
  70 {
  71    return (std::numeric_limits<float>::max)();
  72 }
  73 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 24>&)
  74 {
  75    return 2.8F;
  76 }
  77 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 53>&)
  78 {
  79    return 4.3F;
  80 }
  81 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 64>&)
  82 {
  83    return 4.8F;
  84 }
  85 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 106>&)
  86 {
  87    return 6.5F;
  88 }
  89 inline float erf_asymptotic_limit_N(const boost::integral_constant<int, 113>&)
  90 {
  91    return 6.8F;
  92 }
  93
  94 template <class T, class Policy>
  95 inline T erf_asymptotic_limit()
  96 {
  97    typedef typename policies::precision<T, Policy>::type precision_type;
  98    typedef boost::integral_constant<int,
  99       precision_type::value <= 0 ? 0 :
 100       precision_type::value <= 24 ? 24 :
 101       precision_type::value <= 53 ? 53 :
 102       precision_type::value <= 64 ? 64 :
 103       precision_type::value <= 113 ? 113 : 0
 104    > tag_type;
 105    return erf_asymptotic_limit_N(tag_type());
 106 }
 107
 108 template <class T, class Policy, class Tag>
 109 T erf_imp(T z, bool invert, const Policy& pol, const Tag& t)
 110 {
 111    BOOST_MATH_STD_USING
 112
 113    BOOST_MATH_INSTRUMENT_CODE("Generic erf_imp called");
 114
 115    if(z < 0)
 116    {
 117       if(!invert)
 118          return -erf_imp(T(-z), invert, pol, t);
 119       else
 120          return 1 + erf_imp(T(-z), false, pol, t);
 121    }
 122
 123    T result;
 124
 125    if(!invert && (z > detail::erf_asymptotic_limit<T, Policy>()))
 126    {
 127       detail::erf_asympt_series_t<T> s(z);
 128       boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
 129       result = boost::math::tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, 1);
 130       policies::check_series_iterations<T>("boost::math::erf<%1%>(%1%, %1%)", max_iter, pol);
 131    }
 132    else
 133    {
 134       T x = z * z;
 135       if(x < 0.6)
 136       {
 137          // Compute P:
 138          result = z * exp(-x);
 139          result /= sqrt(boost::math::constants::pi<T>());
 140          if(result != 0)
 141             result *= 2 * detail::lower_gamma_series(T(0.5f), x, pol);
 142       }
 143       else if(x < 1.1f)
 144       {
 145          // Compute Q:
 146          invert = !invert;
 147          result = tgamma_small_upper_part(T(0.5f), x, pol);
 148          result /= sqrt(boost::math::constants::pi<T>());
 149       }
 150       else if(x > 1 / tools::epsilon<T>())
 151       {
 152          // http://functions.wolfram.com/06.27.06.0006.02
 153          invert = !invert;
 154          result = exp(-x) / (constants::root_pi<T>() * z);
 155       }
 156       else
 157       {
 158          // Compute Q:
 159          invert = !invert;
 160          result = z * exp(-x);
 161          result /= boost::math::constants::root_pi<T>();
 162          result *= upper_gamma_fraction(T(0.5f), x, policies::get_epsilon<T, Policy>());
 163       }
 164    }
 165    if(invert)
 166       result = 1 - result;
 167    return result;
 168 }
 169
 170 template <class T, class Policy>
 171 T erf_imp(T z, bool invert, const Policy& pol, const boost::integral_constant<int, 53>& t)
 172 {
 173    BOOST_MATH_STD_USING
 174
 175    BOOST_MATH_INSTRUMENT_CODE("53-bit precision erf_imp called");
 176
 177    if ((boost::math::isnan)(z))
 178       return policies::raise_denorm_error("boost::math::erf<%1%>(%1%)", "Expected a finite argument but got %1%", z, pol);
 179
 180    if(z < 0)
 181    {
 182       if(!invert)
 183          return -erf_imp(T(-z), invert, pol, t);
 184       else if(z < -0.5)
 185          return 2 - erf_imp(T(-z), invert, pol, t);
 186       else
 187          return 1 + erf_imp(T(-z), false, pol, t);
 188    }
 189
 190    T result;
 191
 192    //
 193    // Big bunch of selection statements now to pick
 194    // which implementation to use,
 195    // try to put most likely options first:
 196    //
 197    if(z < 0.5)
 198    {
 199       //
 200       // We're going to calculate erf:
 201       //
 202       if(z < 1e-10)
 203       {
 204          if(z == 0)
 205          {
 206             result = T(0);
 207          }
 208          else
 209          {
 210             static const T c = BOOST_MATH_BIG_CONSTANT(T, 53, 0.003379167095512573896158903121545171688);
 211             result = static_cast<T>(z * 1.125f + z * c);
 212          }
 213       }
 214       else
 215       {
 216          // Maximum Deviation Found:                     1.561e-17
 217          // Expected Error Term:                         1.561e-17
 218          // Maximum Relative Change in Control Points:   1.155e-04
 219          // Max Error found at double precision =        2.961182e-17
 220
 221          static const T Y = 1.044948577880859375f;
 222          static const T P[] = {
 223             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0834305892146531832907),
 224             BOOST_MATH_BIG_CONSTANT(T, 53, -0.338165134459360935041),
 225             BOOST_MATH_BIG_CONSTANT(T, 53, -0.0509990735146777432841),
 226             BOOST_MATH_BIG_CONSTANT(T, 53, -0.00772758345802133288487),
 227             BOOST_MATH_BIG_CONSTANT(T, 53, -0.000322780120964605683831),
 228          };
 229          static const T Q[] = {
 230             BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 231             BOOST_MATH_BIG_CONSTANT(T, 53, 0.455004033050794024546),
 232             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0875222600142252549554),
 233             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00858571925074406212772),
 234             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000370900071787748000569),
 235          };
 236          T zz = z * z;
 237          result = z * (Y + tools::evaluate_polynomial(P, zz) / tools::evaluate_polynomial(Q, zz));
 238       }
 239    }
 240    else if(invert ? (z < 28) : (z < 5.8f))
 241    {
 242       //
 243       // We'll be calculating erfc:
 244       //
 245       invert = !invert;
 246       if(z < 1.5f)
 247       {
 248          // Maximum Deviation Found:                     3.702e-17
 249          // Expected Error Term:                         3.702e-17
 250          // Maximum Relative Change in Control Points:   2.845e-04
 251          // Max Error found at double precision =        4.841816e-17
 252          static const T Y = 0.405935764312744140625f;
 253          static const T P[] = {
 254             BOOST_MATH_BIG_CONSTANT(T, 53, -0.098090592216281240205),
 255             BOOST_MATH_BIG_CONSTANT(T, 53, 0.178114665841120341155),
 256             BOOST_MATH_BIG_CONSTANT(T, 53, 0.191003695796775433986),
 257             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0888900368967884466578),
 258             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0195049001251218801359),
 259             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00180424538297014223957),
 260          };
 261          static const T Q[] = {
 262             BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 263             BOOST_MATH_BIG_CONSTANT(T, 53, 1.84759070983002217845),
 264             BOOST_MATH_BIG_CONSTANT(T, 53, 1.42628004845511324508),
 265             BOOST_MATH_BIG_CONSTANT(T, 53, 0.578052804889902404909),
 266             BOOST_MATH_BIG_CONSTANT(T, 53, 0.12385097467900864233),
 267             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0113385233577001411017),
 268             BOOST_MATH_BIG_CONSTANT(T, 53, 0.337511472483094676155e-5),
 269          };
 270          BOOST_MATH_INSTRUMENT_VARIABLE(Y);
 271          BOOST_MATH_INSTRUMENT_VARIABLE(P[0]);
 272          BOOST_MATH_INSTRUMENT_VARIABLE(Q[0]);
 273          BOOST_MATH_INSTRUMENT_VARIABLE(z);
 274          result = Y + tools::evaluate_polynomial(P, T(z - 0.5)) / tools::evaluate_polynomial(Q, T(z - 0.5));
 275          BOOST_MATH_INSTRUMENT_VARIABLE(result);
 276          result *= exp(-z * z) / z;
 277          BOOST_MATH_INSTRUMENT_VARIABLE(result);
 278       }
 279       else if(z < 2.5f)
 280       {
 281          // Max Error found at double precision =        6.599585e-18
 282          // Maximum Deviation Found:                     3.909e-18
 283          // Expected Error Term:                         3.909e-18
 284          // Maximum Relative Change in Control Points:   9.886e-05
 285          static const T Y = 0.50672817230224609375f;
 286          static const T P[] = {
 287             BOOST_MATH_BIG_CONSTANT(T, 53, -0.0243500476207698441272),
 288             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0386540375035707201728),
 289             BOOST_MATH_BIG_CONSTANT(T, 53, 0.04394818964209516296),
 290             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175679436311802092299),
 291             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00323962406290842133584),
 292             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000235839115596880717416),
 293          };
 294          static const T Q[] = {
 295             BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 296             BOOST_MATH_BIG_CONSTANT(T, 53, 1.53991494948552447182),
 297             BOOST_MATH_BIG_CONSTANT(T, 53, 0.982403709157920235114),
 298             BOOST_MATH_BIG_CONSTANT(T, 53, 0.325732924782444448493),
 299             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0563921837420478160373),
 300             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00410369723978904575884),
 301          };
 302          result = Y + tools::evaluate_polynomial(P, T(z - 1.5)) / tools::evaluate_polynomial(Q, T(z - 1.5));
 303          T hi, lo;
 304          int expon;
 305          hi = floor(ldexp(frexp(z, &expon), 26));
 306          hi = ldexp(hi, expon - 26);
 307          lo = z - hi;
 308          T sq = z * z;
 309          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 310          result *= exp(-sq) * exp(-err_sqr) / z;
 311       }
 312       else if(z < 4.5f)
 313       {
 314          // Maximum Deviation Found:                     1.512e-17
 315          // Expected Error Term:                         1.512e-17
 316          // Maximum Relative Change in Control Points:   2.222e-04
 317          // Max Error found at double precision =        2.062515e-17
 318          static const T Y = 0.5405750274658203125f;
 319          static const T P[] = {
 320             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00295276716530971662634),
 321             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0137384425896355332126),
 322             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00840807615555585383007),
 323             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00212825620914618649141),
 324             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000250269961544794627958),
 325             BOOST_MATH_BIG_CONSTANT(T, 53, 0.113212406648847561139e-4),
 326          };
 327          static const T Q[] = {
 328             BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 329             BOOST_MATH_BIG_CONSTANT(T, 53, 1.04217814166938418171),
 330             BOOST_MATH_BIG_CONSTANT(T, 53, 0.442597659481563127003),
 331             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0958492726301061423444),
 332             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0105982906484876531489),
 333             BOOST_MATH_BIG_CONSTANT(T, 53, 0.000479411269521714493907),
 334          };
 335          result = Y + tools::evaluate_polynomial(P, T(z - 3.5)) / tools::evaluate_polynomial(Q, T(z - 3.5));
 336          T hi, lo;
 337          int expon;
 338          hi = floor(ldexp(frexp(z, &expon), 26));
 339          hi = ldexp(hi, expon - 26);
 340          lo = z - hi;
 341          T sq = z * z;
 342          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 343          result *= exp(-sq) * exp(-err_sqr) / z;
 344       }
 345       else
 346       {
 347          // Max Error found at double precision =        2.997958e-17
 348          // Maximum Deviation Found:                     2.860e-17
 349          // Expected Error Term:                         2.859e-17
 350          // Maximum Relative Change in Control Points:   1.357e-05
 351          static const T Y = 0.5579090118408203125f;
 352          static const T P[] = {
 353             BOOST_MATH_BIG_CONSTANT(T, 53, 0.00628057170626964891937),
 354             BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175389834052493308818),
 355             BOOST_MATH_BIG_CONSTANT(T, 53, -0.212652252872804219852),
 356             BOOST_MATH_BIG_CONSTANT(T, 53, -0.687717681153649930619),
 357             BOOST_MATH_BIG_CONSTANT(T, 53, -2.5518551727311523996),
 358             BOOST_MATH_BIG_CONSTANT(T, 53, -3.22729451764143718517),
 359             BOOST_MATH_BIG_CONSTANT(T, 53, -2.8175401114513378771),
 360          };
 361          static const T Q[] = {
 362             BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 363             BOOST_MATH_BIG_CONSTANT(T, 53, 2.79257750980575282228),
 364             BOOST_MATH_BIG_CONSTANT(T, 53, 11.0567237927800161565),
 365             BOOST_MATH_BIG_CONSTANT(T, 53, 15.930646027911794143),
 366             BOOST_MATH_BIG_CONSTANT(T, 53, 22.9367376522880577224),
 367             BOOST_MATH_BIG_CONSTANT(T, 53, 13.5064170191802889145),
 368             BOOST_MATH_BIG_CONSTANT(T, 53, 5.48409182238641741584),
 369          };
 370          result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
 371          T hi, lo;
 372          int expon;
 373          hi = floor(ldexp(frexp(z, &expon), 26));
 374          hi = ldexp(hi, expon - 26);
 375          lo = z - hi;
 376          T sq = z * z;
 377          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 378          result *= exp(-sq) * exp(-err_sqr) / z;
 379       }
 380    }
 381    else
 382    {
 383       //
 384       // Any value of z larger than 28 will underflow to zero:
 385       //
 386       result = 0;
 387       invert = !invert;
 388    }
 389
 390    if(invert)
 391    {
 392       result = 1 - result;
 393    }
 394
 395    return result;
 396 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const boost::integral_constant<int, 53>& t)
 397
 398
 399 template <class T, class Policy>
 400 T erf_imp(T z, bool invert, const Policy& pol, const boost::integral_constant<int, 64>& t)
 401 {
 402    BOOST_MATH_STD_USING
 403
 404    BOOST_MATH_INSTRUMENT_CODE("64-bit precision erf_imp called");
 405
 406    if(z < 0)
 407    {
 408       if(!invert)
 409          return -erf_imp(T(-z), invert, pol, t);
 410       else if(z < -0.5)
 411          return 2 - erf_imp(T(-z), invert, pol, t);
 412       else
 413          return 1 + erf_imp(T(-z), false, pol, t);
 414    }
 415
 416    T result;
 417
 418    //
 419    // Big bunch of selection statements now to pick which
 420    // implementation to use, try to put most likely options
 421    // first:
 422    //
 423    if(z < 0.5)
 424    {
 425       //
 426       // We're going to calculate erf:
 427       //
 428       if(z == 0)
 429       {
 430          result = 0;
 431       }
 432       else if(z < 1e-10)
 433       {
 434          static const T c = BOOST_MATH_BIG_CONSTANT(T, 64, 0.003379167095512573896158903121545171688);
 435          result = z * 1.125 + z * c;
 436       }
 437       else
 438       {
 439          // Max Error found at long double precision =   1.623299e-20
 440          // Maximum Deviation Found:                     4.326e-22
 441          // Expected Error Term:                         -4.326e-22
 442          // Maximum Relative Change in Control Points:   1.474e-04
 443          static const T Y = 1.044948577880859375f;
 444          static const T P[] = {
 445             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0834305892146531988966),
 446             BOOST_MATH_BIG_CONSTANT(T, 64, -0.338097283075565413695),
 447             BOOST_MATH_BIG_CONSTANT(T, 64, -0.0509602734406067204596),
 448             BOOST_MATH_BIG_CONSTANT(T, 64, -0.00904906346158537794396),
 449             BOOST_MATH_BIG_CONSTANT(T, 64, -0.000489468651464798669181),
 450             BOOST_MATH_BIG_CONSTANT(T, 64, -0.200305626366151877759e-4),
 451          };
 452          static const T Q[] = {
 453             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 454             BOOST_MATH_BIG_CONSTANT(T, 64, 0.455817300515875172439),
 455             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0916537354356241792007),
 456             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0102722652675910031202),
 457             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000650511752687851548735),
 458             BOOST_MATH_BIG_CONSTANT(T, 64, 0.189532519105655496778e-4),
 459          };
 460          result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
 461       }
 462    }
 463    else if(invert ? (z < 110) : (z < 6.4f))
 464    {
 465       //
 466       // We'll be calculating erfc:
 467       //
 468       invert = !invert;
 469       if(z < 1.5)
 470       {
 471          // Max Error found at long double precision =   3.239590e-20
 472          // Maximum Deviation Found:                     2.241e-20
 473          // Expected Error Term:                         -2.241e-20
 474          // Maximum Relative Change in Control Points:   5.110e-03
 475          static const T Y = 0.405935764312744140625f;
 476          static const T P[] = {
 477             BOOST_MATH_BIG_CONSTANT(T, 64, -0.0980905922162812031672),
 478             BOOST_MATH_BIG_CONSTANT(T, 64, 0.159989089922969141329),
 479             BOOST_MATH_BIG_CONSTANT(T, 64, 0.222359821619935712378),
 480             BOOST_MATH_BIG_CONSTANT(T, 64, 0.127303921703577362312),
 481             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0384057530342762400273),
 482             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00628431160851156719325),
 483             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000441266654514391746428),
 484             BOOST_MATH_BIG_CONSTANT(T, 64, 0.266689068336295642561e-7),
 485          };
 486          static const T Q[] = {
 487             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 488             BOOST_MATH_BIG_CONSTANT(T, 64, 2.03237474985469469291),
 489             BOOST_MATH_BIG_CONSTANT(T, 64, 1.78355454954969405222),
 490             BOOST_MATH_BIG_CONSTANT(T, 64, 0.867940326293760578231),
 491             BOOST_MATH_BIG_CONSTANT(T, 64, 0.248025606990021698392),
 492             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0396649631833002269861),
 493             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00279220237309449026796),
 494          };
 495          result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
 496          T hi, lo;
 497          int expon;
 498          hi = floor(ldexp(frexp(z, &expon), 32));
 499          hi = ldexp(hi, expon - 32);
 500          lo = z - hi;
 501          T sq = z * z;
 502          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 503          result *= exp(-sq) * exp(-err_sqr) / z;
 504       }
 505       else if(z < 2.5)
 506       {
 507          // Max Error found at long double precision =   3.686211e-21
 508          // Maximum Deviation Found:                     1.495e-21
 509          // Expected Error Term:                         -1.494e-21
 510          // Maximum Relative Change in Control Points:   1.793e-04
 511          static const T Y = 0.50672817230224609375f;
 512          static const T P[] = {
 513             BOOST_MATH_BIG_CONSTANT(T, 64, -0.024350047620769840217),
 514             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0343522687935671451309),
 515             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0505420824305544949541),
 516             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0257479325917757388209),
 517             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00669349844190354356118),
 518             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00090807914416099524444),
 519             BOOST_MATH_BIG_CONSTANT(T, 64, 0.515917266698050027934e-4),
 520          };
 521          static const T Q[] = {
 522             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 523             BOOST_MATH_BIG_CONSTANT(T, 64, 1.71657861671930336344),
 524             BOOST_MATH_BIG_CONSTANT(T, 64, 1.26409634824280366218),
 525             BOOST_MATH_BIG_CONSTANT(T, 64, 0.512371437838969015941),
 526             BOOST_MATH_BIG_CONSTANT(T, 64, 0.120902623051120950935),
 527             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0158027197831887485261),
 528             BOOST_MATH_BIG_CONSTANT(T, 64, 0.000897871370778031611439),
 529          };
 530          result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
 531          T hi, lo;
 532          int expon;
 533          hi = floor(ldexp(frexp(z, &expon), 32));
 534          hi = ldexp(hi, expon - 32);
 535          lo = z - hi;
 536          T sq = z * z;
 537          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 538          result *= exp(-sq) * exp(-err_sqr) / z;
 539       }
 540       else if(z < 4.5)
 541       {
 542          // Maximum Deviation Found:                     1.107e-20
 543          // Expected Error Term:                         -1.106e-20
 544          // Maximum Relative Change in Control Points:   1.709e-04
 545          // Max Error found at long double precision =   1.446908e-20
 546          static const T Y  = 0.5405750274658203125f;
 547          static const T P[] = {
 548             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0029527671653097284033),
 549             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0141853245895495604051),
 550             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0104959584626432293901),
 551             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00343963795976100077626),
 552             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00059065441194877637899),
 553             BOOST_MATH_BIG_CONSTANT(T, 64, 0.523435380636174008685e-4),
 554             BOOST_MATH_BIG_CONSTANT(T, 64, 0.189896043050331257262e-5),
 555          };
 556          static const T Q[] = {
 557             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 558             BOOST_MATH_BIG_CONSTANT(T, 64, 1.19352160185285642574),
 559             BOOST_MATH_BIG_CONSTANT(T, 64, 0.603256964363454392857),
 560             BOOST_MATH_BIG_CONSTANT(T, 64, 0.165411142458540585835),
 561             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0259729870946203166468),
 562             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00221657568292893699158),
 563             BOOST_MATH_BIG_CONSTANT(T, 64, 0.804149464190309799804e-4),
 564          };
 565          result = Y + tools::evaluate_polynomial(P, T(z - 3.5f)) / tools::evaluate_polynomial(Q, T(z - 3.5f));
 566          T hi, lo;
 567          int expon;
 568          hi = floor(ldexp(frexp(z, &expon), 32));
 569          hi = ldexp(hi, expon - 32);
 570          lo = z - hi;
 571          T sq = z * z;
 572          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 573          result *= exp(-sq) * exp(-err_sqr) / z;
 574       }
 575       else
 576       {
 577          // Max Error found at long double precision =   7.961166e-21
 578          // Maximum Deviation Found:                     6.677e-21
 579          // Expected Error Term:                         6.676e-21
 580          // Maximum Relative Change in Control Points:   2.319e-05
 581          static const T Y = 0.55825519561767578125f;
 582          static const T P[] = {
 583             BOOST_MATH_BIG_CONSTANT(T, 64, 0.00593438793008050214106),
 584             BOOST_MATH_BIG_CONSTANT(T, 64, 0.0280666231009089713937),
 585             BOOST_MATH_BIG_CONSTANT(T, 64, -0.141597835204583050043),
 586             BOOST_MATH_BIG_CONSTANT(T, 64, -0.978088201154300548842),
 587             BOOST_MATH_BIG_CONSTANT(T, 64, -5.47351527796012049443),
 588             BOOST_MATH_BIG_CONSTANT(T, 64, -13.8677304660245326627),
 589             BOOST_MATH_BIG_CONSTANT(T, 64, -27.1274948720539821722),
 590             BOOST_MATH_BIG_CONSTANT(T, 64, -29.2545152747009461519),
 591             BOOST_MATH_BIG_CONSTANT(T, 64, -16.8865774499799676937),
 592          };
 593          static const T Q[] = {
 594             BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 595             BOOST_MATH_BIG_CONSTANT(T, 64, 4.72948911186645394541),
 596             BOOST_MATH_BIG_CONSTANT(T, 64, 23.6750543147695749212),
 597             BOOST_MATH_BIG_CONSTANT(T, 64, 60.0021517335693186785),
 598             BOOST_MATH_BIG_CONSTANT(T, 64, 131.766251645149522868),
 599             BOOST_MATH_BIG_CONSTANT(T, 64, 178.167924971283482513),
 600             BOOST_MATH_BIG_CONSTANT(T, 64, 182.499390505915222699),
 601             BOOST_MATH_BIG_CONSTANT(T, 64, 104.365251479578577989),
 602             BOOST_MATH_BIG_CONSTANT(T, 64, 30.8365511891224291717),
 603          };
 604          result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
 605          T hi, lo;
 606          int expon;
 607          hi = floor(ldexp(frexp(z, &expon), 32));
 608          hi = ldexp(hi, expon - 32);
 609          lo = z - hi;
 610          T sq = z * z;
 611          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 612          result *= exp(-sq) * exp(-err_sqr) / z;
 613       }
 614    }
 615    else
 616    {
 617       //
 618       // Any value of z larger than 110 will underflow to zero:
 619       //
 620       result = 0;
 621       invert = !invert;
 622    }
 623
 624    if(invert)
 625    {
 626       result = 1 - result;
 627    }
 628
 629    return result;
 630 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const boost::integral_constant<int, 64>& t)
 631
 632
 633 template <class T, class Policy>
 634 T erf_imp(T z, bool invert, const Policy& pol, const boost::integral_constant<int, 113>& t)
 635 {
 636    BOOST_MATH_STD_USING
 637
 638    BOOST_MATH_INSTRUMENT_CODE("113-bit precision erf_imp called");
 639
 640    if(z < 0)
 641    {
 642       if(!invert)
 643          return -erf_imp(T(-z), invert, pol, t);
 644       else if(z < -0.5)
 645          return 2 - erf_imp(T(-z), invert, pol, t);
 646       else
 647          return 1 + erf_imp(T(-z), false, pol, t);
 648    }
 649
 650    T result;
 651
 652    //
 653    // Big bunch of selection statements now to pick which
 654    // implementation to use, try to put most likely options
 655    // first:
 656    //
 657    if(z < 0.5)
 658    {
 659       //
 660       // We're going to calculate erf:
 661       //
 662       if(z == 0)
 663       {
 664          result = 0;
 665       }
 666       else if(z < 1e-20)
 667       {
 668          static const T c = BOOST_MATH_BIG_CONSTANT(T, 113, 0.003379167095512573896158903121545171688);
 669          result = z * 1.125 + z * c;
 670       }
 671       else
 672       {
 673          // Max Error found at long double precision =   2.342380e-35
 674          // Maximum Deviation Found:                     6.124e-36
 675          // Expected Error Term:                         -6.124e-36
 676          // Maximum Relative Change in Control Points:   3.492e-10
 677          static const T Y = 1.0841522216796875f;
 678          static const T P[] = {
 679             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0442269454158250738961589031215451778),
 680             BOOST_MATH_BIG_CONSTANT(T, 113, -0.35549265736002144875335323556961233),
 681             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0582179564566667896225454670863270393),
 682             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0112694696904802304229950538453123925),
 683             BOOST_MATH_BIG_CONSTANT(T, 113, -0.000805730648981801146251825329609079099),
 684             BOOST_MATH_BIG_CONSTANT(T, 113, -0.566304966591936566229702842075966273e-4),
 685             BOOST_MATH_BIG_CONSTANT(T, 113, -0.169655010425186987820201021510002265e-5),
 686             BOOST_MATH_BIG_CONSTANT(T, 113, -0.344448249920445916714548295433198544e-7),
 687          };
 688          static const T Q[] = {
 689             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 690             BOOST_MATH_BIG_CONSTANT(T, 113, 0.466542092785657604666906909196052522),
 691             BOOST_MATH_BIG_CONSTANT(T, 113, 0.100005087012526447295176964142107611),
 692             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0128341535890117646540050072234142603),
 693             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00107150448466867929159660677016658186),
 694             BOOST_MATH_BIG_CONSTANT(T, 113, 0.586168368028999183607733369248338474e-4),
 695             BOOST_MATH_BIG_CONSTANT(T, 113, 0.196230608502104324965623171516808796e-5),
 696             BOOST_MATH_BIG_CONSTANT(T, 113, 0.313388521582925207734229967907890146e-7),
 697          };
 698          result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
 699       }
 700    }
 701    else if(invert ? (z < 110) : (z < 8.65f))
 702    {
 703       //
 704       // We'll be calculating erfc:
 705       //
 706       invert = !invert;
 707       if(z < 1)
 708       {
 709          // Max Error found at long double precision =   3.246278e-35
 710          // Maximum Deviation Found:                     1.388e-35
 711          // Expected Error Term:                         1.387e-35
 712          // Maximum Relative Change in Control Points:   6.127e-05
 713          static const T Y = 0.371877193450927734375f;
 714          static const T P[] = {
 715             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0640320213544647969396032886581290455),
 716             BOOST_MATH_BIG_CONSTANT(T, 113, 0.200769874440155895637857443946706731),
 717             BOOST_MATH_BIG_CONSTANT(T, 113, 0.378447199873537170666487408805779826),
 718             BOOST_MATH_BIG_CONSTANT(T, 113, 0.30521399466465939450398642044975127),
 719             BOOST_MATH_BIG_CONSTANT(T, 113, 0.146890026406815277906781824723458196),
 720             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0464837937749539978247589252732769567),
 721             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00987895759019540115099100165904822903),
 722             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00137507575429025512038051025154301132),
 723             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0001144764551085935580772512359680516),
 724             BOOST_MATH_BIG_CONSTANT(T, 113, 0.436544865032836914773944382339900079e-5),
 725          };
 726          static const T Q[] = {
 727             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 728             BOOST_MATH_BIG_CONSTANT(T, 113, 2.47651182872457465043733800302427977),
 729             BOOST_MATH_BIG_CONSTANT(T, 113, 2.78706486002517996428836400245547955),
 730             BOOST_MATH_BIG_CONSTANT(T, 113, 1.87295924621659627926365005293130693),
 731             BOOST_MATH_BIG_CONSTANT(T, 113, 0.829375825174365625428280908787261065),
 732             BOOST_MATH_BIG_CONSTANT(T, 113, 0.251334771307848291593780143950311514),
 733             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0522110268876176186719436765734722473),
 734             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00718332151250963182233267040106902368),
 735             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000595279058621482041084986219276392459),
 736             BOOST_MATH_BIG_CONSTANT(T, 113, 0.226988669466501655990637599399326874e-4),
 737             BOOST_MATH_BIG_CONSTANT(T, 113, 0.270666232259029102353426738909226413e-10),
 738          };
 739          result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
 740          T hi, lo;
 741          int expon;
 742          hi = floor(ldexp(frexp(z, &expon), 56));
 743          hi = ldexp(hi, expon - 56);
 744          lo = z - hi;
 745          T sq = z * z;
 746          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 747          result *= exp(-sq) * exp(-err_sqr) / z;
 748       }
 749       else if(z < 1.5)
 750       {
 751          // Max Error found at long double precision =   2.215785e-35
 752          // Maximum Deviation Found:                     1.539e-35
 753          // Expected Error Term:                         1.538e-35
 754          // Maximum Relative Change in Control Points:   6.104e-05
 755          static const T Y = 0.45658016204833984375f;
 756          static const T P[] = {
 757             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0289965858925328393392496555094848345),
 758             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0868181194868601184627743162571779226),
 759             BOOST_MATH_BIG_CONSTANT(T, 113, 0.169373435121178901746317404936356745),
 760             BOOST_MATH_BIG_CONSTANT(T, 113, 0.13350446515949251201104889028133486),
 761             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0617447837290183627136837688446313313),
 762             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0185618495228251406703152962489700468),
 763             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00371949406491883508764162050169531013),
 764             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000485121708792921297742105775823900772),
 765             BOOST_MATH_BIG_CONSTANT(T, 113, 0.376494706741453489892108068231400061e-4),
 766             BOOST_MATH_BIG_CONSTANT(T, 113, 0.133166058052466262415271732172490045e-5),
 767          };
 768          static const T Q[] = {
 769             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 770             BOOST_MATH_BIG_CONSTANT(T, 113, 2.32970330146503867261275580968135126),
 771             BOOST_MATH_BIG_CONSTANT(T, 113, 2.46325715420422771961250513514928746),
 772             BOOST_MATH_BIG_CONSTANT(T, 113, 1.55307882560757679068505047390857842),
 773             BOOST_MATH_BIG_CONSTANT(T, 113, 0.644274289865972449441174485441409076),
 774             BOOST_MATH_BIG_CONSTANT(T, 113, 0.182609091063258208068606847453955649),
 775             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0354171651271241474946129665801606795),
 776             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00454060370165285246451879969534083997),
 777             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000349871943711566546821198612518656486),
 778             BOOST_MATH_BIG_CONSTANT(T, 113, 0.123749319840299552925421880481085392e-4),
 779          };
 780          result = Y + tools::evaluate_polynomial(P, T(z - 1.0f)) / tools::evaluate_polynomial(Q, T(z - 1.0f));
 781          T hi, lo;
 782          int expon;
 783          hi = floor(ldexp(frexp(z, &expon), 56));
 784          hi = ldexp(hi, expon - 56);
 785          lo = z - hi;
 786          T sq = z * z;
 787          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 788          result *= exp(-sq) * exp(-err_sqr) / z;
 789       }
 790       else if(z < 2.25)
 791       {
 792          // Maximum Deviation Found:                     1.418e-35
 793          // Expected Error Term:                         1.418e-35
 794          // Maximum Relative Change in Control Points:   1.316e-04
 795          // Max Error found at long double precision =   1.998462e-35
 796          static const T Y = 0.50250148773193359375f;
 797          static const T P[] = {
 798             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0201233630504573402185161184151016606),
 799             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0331864357574860196516686996302305002),
 800             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0716562720864787193337475444413405461),
 801             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0545835322082103985114927569724880658),
 802             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0236692635189696678976549720784989593),
 803             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00656970902163248872837262539337601845),
 804             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00120282643299089441390490459256235021),
 805             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000142123229065182650020762792081622986),
 806             BOOST_MATH_BIG_CONSTANT(T, 113, 0.991531438367015135346716277792989347e-5),
 807             BOOST_MATH_BIG_CONSTANT(T, 113, 0.312857043762117596999398067153076051e-6),
 808          };
 809          static const T Q[] = {
 810             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 811             BOOST_MATH_BIG_CONSTANT(T, 113, 2.13506082409097783827103424943508554),
 812             BOOST_MATH_BIG_CONSTANT(T, 113, 2.06399257267556230937723190496806215),
 813             BOOST_MATH_BIG_CONSTANT(T, 113, 1.18678481279932541314830499880691109),
 814             BOOST_MATH_BIG_CONSTANT(T, 113, 0.447733186643051752513538142316799562),
 815             BOOST_MATH_BIG_CONSTANT(T, 113, 0.11505680005657879437196953047542148),
 816             BOOST_MATH_BIG_CONSTANT(T, 113, 0.020163993632192726170219663831914034),
 817             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00232708971840141388847728782209730585),
 818             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000160733201627963528519726484608224112),
 819             BOOST_MATH_BIG_CONSTANT(T, 113, 0.507158721790721802724402992033269266e-5),
 820             BOOST_MATH_BIG_CONSTANT(T, 113, 0.18647774409821470950544212696270639e-12),
 821          };
 822          result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
 823          T hi, lo;
 824          int expon;
 825          hi = floor(ldexp(frexp(z, &expon), 56));
 826          hi = ldexp(hi, expon - 56);
 827          lo = z - hi;
 828          T sq = z * z;
 829          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 830          result *= exp(-sq) * exp(-err_sqr) / z;
 831       }
 832       else if (z < 3)
 833       {
 834          // Maximum Deviation Found:                     3.575e-36
 835          // Expected Error Term:                         3.575e-36
 836          // Maximum Relative Change in Control Points:   7.103e-05
 837          // Max Error found at long double precision =   5.794737e-36
 838          static const T Y = 0.52896785736083984375f;
 839          static const T P[] = {
 840             BOOST_MATH_BIG_CONSTANT(T, 113, -0.00902152521745813634562524098263360074),
 841             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0145207142776691539346923710537580927),
 842             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0301681239582193983824211995978678571),
 843             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0215548540823305814379020678660434461),
 844             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00864683476267958365678294164340749949),
 845             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00219693096885585491739823283511049902),
 846             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000364961639163319762492184502159894371),
 847             BOOST_MATH_BIG_CONSTANT(T, 113, 0.388174251026723752769264051548703059e-4),
 848             BOOST_MATH_BIG_CONSTANT(T, 113, 0.241918026931789436000532513553594321e-5),
 849             BOOST_MATH_BIG_CONSTANT(T, 113, 0.676586625472423508158937481943649258e-7),
 850          };
 851          static const T Q[] = {
 852             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 853             BOOST_MATH_BIG_CONSTANT(T, 113, 1.93669171363907292305550231764920001),
 854             BOOST_MATH_BIG_CONSTANT(T, 113, 1.69468476144051356810672506101377494),
 855             BOOST_MATH_BIG_CONSTANT(T, 113, 0.880023580986436640372794392579985511),
 856             BOOST_MATH_BIG_CONSTANT(T, 113, 0.299099106711315090710836273697708402),
 857             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0690593962363545715997445583603382337),
 858             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0108427016361318921960863149875360222),
 859             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00111747247208044534520499324234317695),
 860             BOOST_MATH_BIG_CONSTANT(T, 113, 0.686843205749767250666787987163701209e-4),
 861             BOOST_MATH_BIG_CONSTANT(T, 113, 0.192093541425429248675532015101904262e-5),
 862          };
 863          result = Y + tools::evaluate_polynomial(P, T(z - 2.25f)) / tools::evaluate_polynomial(Q, T(z - 2.25f));
 864          T hi, lo;
 865          int expon;
 866          hi = floor(ldexp(frexp(z, &expon), 56));
 867          hi = ldexp(hi, expon - 56);
 868          lo = z - hi;
 869          T sq = z * z;
 870          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 871          result *= exp(-sq) * exp(-err_sqr) / z;
 872       }
 873       else if(z < 3.5)
 874       {
 875          // Maximum Deviation Found:                     8.126e-37
 876          // Expected Error Term:                         -8.126e-37
 877          // Maximum Relative Change in Control Points:   1.363e-04
 878          // Max Error found at long double precision =   1.747062e-36
 879          static const T Y = 0.54037380218505859375f;
 880          static const T P[] = {
 881             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0033703486408887424921155540591370375),
 882             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0104948043110005245215286678898115811),
 883             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0148530118504000311502310457390417795),
 884             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00816693029245443090102738825536188916),
 885             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00249716579989140882491939681805594585),
 886             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0004655591010047353023978045800916647),
 887             BOOST_MATH_BIG_CONSTANT(T, 113, 0.531129557920045295895085236636025323e-4),
 888             BOOST_MATH_BIG_CONSTANT(T, 113, 0.343526765122727069515775194111741049e-5),
 889             BOOST_MATH_BIG_CONSTANT(T, 113, 0.971120407556888763695313774578711839e-7),
 890          };
 891          static const T Q[] = {
 892             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 893             BOOST_MATH_BIG_CONSTANT(T, 113, 1.59911256167540354915906501335919317),
 894             BOOST_MATH_BIG_CONSTANT(T, 113, 1.136006830764025173864831382946934),
 895             BOOST_MATH_BIG_CONSTANT(T, 113, 0.468565867990030871678574840738423023),
 896             BOOST_MATH_BIG_CONSTANT(T, 113, 0.122821824954470343413956476900662236),
 897             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0209670914950115943338996513330141633),
 898             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00227845718243186165620199012883547257),
 899             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000144243326443913171313947613547085553),
 900             BOOST_MATH_BIG_CONSTANT(T, 113, 0.407763415954267700941230249989140046e-5),
 901          };
 902          result = Y + tools::evaluate_polynomial(P, T(z - 3.0f)) / tools::evaluate_polynomial(Q, T(z - 3.0f));
 903          T hi, lo;
 904          int expon;
 905          hi = floor(ldexp(frexp(z, &expon), 56));
 906          hi = ldexp(hi, expon - 56);
 907          lo = z - hi;
 908          T sq = z * z;
 909          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 910          result *= exp(-sq) * exp(-err_sqr) / z;
 911       }
 912       else if(z < 5.5)
 913       {
 914          // Maximum Deviation Found:                     5.804e-36
 915          // Expected Error Term:                         -5.803e-36
 916          // Maximum Relative Change in Control Points:   2.475e-05
 917          // Max Error found at long double precision =   1.349545e-35
 918          static const T Y = 0.55000019073486328125f;
 919          static const T P[] = {
 920             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00118142849742309772151454518093813615),
 921             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0072201822885703318172366893469382745),
 922             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0078782276276860110721875733778481505),
 923             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00418229166204362376187593976656261146),
 924             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00134198400587769200074194304298642705),
 925             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000283210387078004063264777611497435572),
 926             BOOST_MATH_BIG_CONSTANT(T, 113, 0.405687064094911866569295610914844928e-4),
 927             BOOST_MATH_BIG_CONSTANT(T, 113, 0.39348283801568113807887364414008292e-5),
 928             BOOST_MATH_BIG_CONSTANT(T, 113, 0.248798540917787001526976889284624449e-6),
 929             BOOST_MATH_BIG_CONSTANT(T, 113, 0.929502490223452372919607105387474751e-8),
 930             BOOST_MATH_BIG_CONSTANT(T, 113, 0.156161469668275442569286723236274457e-9),
 931          };
 932          static const T Q[] = {
 933             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 934             BOOST_MATH_BIG_CONSTANT(T, 113, 1.52955245103668419479878456656709381),
 935             BOOST_MATH_BIG_CONSTANT(T, 113, 1.06263944820093830054635017117417064),
 936             BOOST_MATH_BIG_CONSTANT(T, 113, 0.441684612681607364321013134378316463),
 937             BOOST_MATH_BIG_CONSTANT(T, 113, 0.121665258426166960049773715928906382),
 938             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0232134512374747691424978642874321434),
 939             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00310778180686296328582860464875562636),
 940             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000288361770756174705123674838640161693),
 941             BOOST_MATH_BIG_CONSTANT(T, 113, 0.177529187194133944622193191942300132e-4),
 942             BOOST_MATH_BIG_CONSTANT(T, 113, 0.655068544833064069223029299070876623e-6),
 943             BOOST_MATH_BIG_CONSTANT(T, 113, 0.11005507545746069573608988651927452e-7),
 944          };
 945          result = Y + tools::evaluate_polynomial(P, T(z - 4.5f)) / tools::evaluate_polynomial(Q, T(z - 4.5f));
 946          T hi, lo;
 947          int expon;
 948          hi = floor(ldexp(frexp(z, &expon), 56));
 949          hi = ldexp(hi, expon - 56);
 950          lo = z - hi;
 951          T sq = z * z;
 952          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 953          result *= exp(-sq) * exp(-err_sqr) / z;
 954       }
 955       else if(z < 7.5)
 956       {
 957          // Maximum Deviation Found:                     1.007e-36
 958          // Expected Error Term:                         1.007e-36
 959          // Maximum Relative Change in Control Points:   1.027e-03
 960          // Max Error found at long double precision =   2.646420e-36
 961          static const T Y = 0.5574436187744140625f;
 962          static const T P[] = {
 963             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000293236907400849056269309713064107674),
 964             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00225110719535060642692275221961480162),
 965             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190984458121502831421717207849429799),
 966             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000747757733460111743833929141001680706),
 967             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000170663175280949889583158597373928096),
 968             BOOST_MATH_BIG_CONSTANT(T, 113, 0.246441188958013822253071608197514058e-4),
 969             BOOST_MATH_BIG_CONSTANT(T, 113, 0.229818000860544644974205957895688106e-5),
 970             BOOST_MATH_BIG_CONSTANT(T, 113, 0.134886977703388748488480980637704864e-6),
 971             BOOST_MATH_BIG_CONSTANT(T, 113, 0.454764611880548962757125070106650958e-8),
 972             BOOST_MATH_BIG_CONSTANT(T, 113, 0.673002744115866600294723141176820155e-10),
 973          };
 974          static const T Q[] = {
 975             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 976             BOOST_MATH_BIG_CONSTANT(T, 113, 1.12843690320861239631195353379313367),
 977             BOOST_MATH_BIG_CONSTANT(T, 113, 0.569900657061622955362493442186537259),
 978             BOOST_MATH_BIG_CONSTANT(T, 113, 0.169094404206844928112348730277514273),
 979             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0324887449084220415058158657252147063),
 980             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00419252877436825753042680842608219552),
 981             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00036344133176118603523976748563178578),
 982             BOOST_MATH_BIG_CONSTANT(T, 113, 0.204123895931375107397698245752850347e-4),
 983             BOOST_MATH_BIG_CONSTANT(T, 113, 0.674128352521481412232785122943508729e-6),
 984             BOOST_MATH_BIG_CONSTANT(T, 113, 0.997637501418963696542159244436245077e-8),
 985          };
 986          result = Y + tools::evaluate_polynomial(P, T(z - 6.5f)) / tools::evaluate_polynomial(Q, T(z - 6.5f));
 987          T hi, lo;
 988          int expon;
 989          hi = floor(ldexp(frexp(z, &expon), 56));
 990          hi = ldexp(hi, expon - 56);
 991          lo = z - hi;
 992          T sq = z * z;
 993          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
 994          result *= exp(-sq) * exp(-err_sqr) / z;
 995       }
 996       else if(z < 11.5)
 997       {
 998          // Maximum Deviation Found:                     8.380e-36
 999          // Expected Error Term:                         8.380e-36
1000          // Maximum Relative Change in Control Points:   2.632e-06
1001          // Max Error found at long double precision =   9.849522e-36
1002          static const T Y = 0.56083202362060546875f;
1003          static const T P[] = {
1004             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000282420728751494363613829834891390121),
1005             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00175387065018002823433704079355125161),
1006             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0021344978564889819420775336322920375),
1007             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00124151356560137532655039683963075661),
1008             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000423600733566948018555157026862139644),
1009             BOOST_MATH_BIG_CONSTANT(T, 113, 0.914030340865175237133613697319509698e-4),
1010             BOOST_MATH_BIG_CONSTANT(T, 113, 0.126999927156823363353809747017945494e-4),
1011             BOOST_MATH_BIG_CONSTANT(T, 113, 0.110610959842869849776179749369376402e-5),
1012             BOOST_MATH_BIG_CONSTANT(T, 113, 0.55075079477173482096725348704634529e-7),
1013             BOOST_MATH_BIG_CONSTANT(T, 113, 0.119735694018906705225870691331543806e-8),
1014          };
1015          static const T Q[] = {
1016             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1017             BOOST_MATH_BIG_CONSTANT(T, 113, 1.69889613396167354566098060039549882),
1018             BOOST_MATH_BIG_CONSTANT(T, 113, 1.28824647372749624464956031163282674),
1019             BOOST_MATH_BIG_CONSTANT(T, 113, 0.572297795434934493541628008224078717),
1020             BOOST_MATH_BIG_CONSTANT(T, 113, 0.164157697425571712377043857240773164),
1021             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0315311145224594430281219516531649562),
1022             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00405588922155632380812945849777127458),
1023             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000336929033691445666232029762868642417),
1024             BOOST_MATH_BIG_CONSTANT(T, 113, 0.164033049810404773469413526427932109e-4),
1025             BOOST_MATH_BIG_CONSTANT(T, 113, 0.356615210500531410114914617294694857e-6),
1026          };
1027          result = Y + tools::evaluate_polynomial(P, T(z / 2 - 4.75f)) / tools::evaluate_polynomial(Q, T(z / 2 - 4.75f));
1028          T hi, lo;
1029          int expon;
1030          hi = floor(ldexp(frexp(z, &expon), 56));
1031          hi = ldexp(hi, expon - 56);
1032          lo = z - hi;
1033          T sq = z * z;
1034          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
1035          result *= exp(-sq) * exp(-err_sqr) / z;
1036       }
1037       else
1038       {
1039          // Maximum Deviation Found:                     1.132e-35
1040          // Expected Error Term:                         -1.132e-35
1041          // Maximum Relative Change in Control Points:   4.674e-04
1042          // Max Error found at long double precision =   1.162590e-35
1043          static const T Y = 0.5632686614990234375f;
1044          static const T P[] = {
1045             BOOST_MATH_BIG_CONSTANT(T, 113, 0.000920922048732849448079451574171836943),
1046             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00321439044532288750501700028748922439),
1047             BOOST_MATH_BIG_CONSTANT(T, 113, -0.250455263029390118657884864261823431),
1048             BOOST_MATH_BIG_CONSTANT(T, 113, -0.906807635364090342031792404764598142),
1049             BOOST_MATH_BIG_CONSTANT(T, 113, -8.92233572835991735876688745989985565),
1050             BOOST_MATH_BIG_CONSTANT(T, 113, -21.7797433494422564811782116907878495),
1051             BOOST_MATH_BIG_CONSTANT(T, 113, -91.1451915251976354349734589601171659),
1052             BOOST_MATH_BIG_CONSTANT(T, 113, -144.1279109655993927069052125017673),
1053             BOOST_MATH_BIG_CONSTANT(T, 113, -313.845076581796338665519022313775589),
1054             BOOST_MATH_BIG_CONSTANT(T, 113, -273.11378811923343424081101235736475),
1055             BOOST_MATH_BIG_CONSTANT(T, 113, -271.651566205951067025696102600443452),
1056             BOOST_MATH_BIG_CONSTANT(T, 113, -60.0530577077238079968843307523245547),
1057          };
1058          static const T Q[] = {
1059             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1060             BOOST_MATH_BIG_CONSTANT(T, 113, 3.49040448075464744191022350947892036),
1061             BOOST_MATH_BIG_CONSTANT(T, 113, 34.3563592467165971295915749548313227),
1062             BOOST_MATH_BIG_CONSTANT(T, 113, 84.4993232033879023178285731843850461),
1063             BOOST_MATH_BIG_CONSTANT(T, 113, 376.005865281206894120659401340373818),
1064             BOOST_MATH_BIG_CONSTANT(T, 113, 629.95369438888946233003926191755125),
1065             BOOST_MATH_BIG_CONSTANT(T, 113, 1568.35771983533158591604513304269098),
1066             BOOST_MATH_BIG_CONSTANT(T, 113, 1646.02452040831961063640827116581021),
1067             BOOST_MATH_BIG_CONSTANT(T, 113, 2299.96860633240298708910425594484895),
1068             BOOST_MATH_BIG_CONSTANT(T, 113, 1222.73204392037452750381340219906374),
1069             BOOST_MATH_BIG_CONSTANT(T, 113, 799.359797306084372350264298361110448),
1070             BOOST_MATH_BIG_CONSTANT(T, 113, 72.7415265778588087243442792401576737),
1071          };
1072          result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
1073          T hi, lo;
1074          int expon;
1075          hi = floor(ldexp(frexp(z, &expon), 56));
1076          hi = ldexp(hi, expon - 56);
1077          lo = z - hi;
1078          T sq = z * z;
1079          T err_sqr = ((hi * hi - sq) + 2 * hi * lo) + lo * lo;
1080          result *= exp(-sq) * exp(-err_sqr) / z;
1081       }
1082    }
1083    else
1084    {
1085       //
1086       // Any value of z larger than 110 will underflow to zero:
1087       //
1088       result = 0;
1089       invert = !invert;
1090    }
1091
1092    if(invert)
1093    {
1094       result = 1 - result;
1095    }
1096
1097    return result;
1098 } // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const boost::integral_constant<int, 113>& t)
1099
1100 template <class T, class Policy, class tag>
1101 struct erf_initializer
1102 {
1103    struct init
1104    {
1105       init()
1106       {
1107          do_init(tag());
1108       }
1109       static void do_init(const boost::integral_constant<int, 0>&){}
1110       static void do_init(const boost::integral_constant<int, 53>&)
1111       {
1112          boost::math::erf(static_cast<T>(1e-12), Policy());
1113          boost::math::erf(static_cast<T>(0.25), Policy());
1114          boost::math::erf(static_cast<T>(1.25), Policy());
1115          boost::math::erf(static_cast<T>(2.25), Policy());
1116          boost::math::erf(static_cast<T>(4.25), Policy());
1117          boost::math::erf(static_cast<T>(5.25), Policy());
1118       }
1119       static void do_init(const boost::integral_constant<int, 64>&)
1120       {
1121          boost::math::erf(static_cast<T>(1e-12), Policy());
1122          boost::math::erf(static_cast<T>(0.25), Policy());
1123          boost::math::erf(static_cast<T>(1.25), Policy());
1124          boost::math::erf(static_cast<T>(2.25), Policy());
1125          boost::math::erf(static_cast<T>(4.25), Policy());
1126          boost::math::erf(static_cast<T>(5.25), Policy());
1127       }
1128       static void do_init(const boost::integral_constant<int, 113>&)
1129       {
1130          boost::math::erf(static_cast<T>(1e-22), Policy());
1131          boost::math::erf(static_cast<T>(0.25), Policy());
1132          boost::math::erf(static_cast<T>(1.25), Policy());
1133          boost::math::erf(static_cast<T>(2.125), Policy());
1134          boost::math::erf(static_cast<T>(2.75), Policy());
1135          boost::math::erf(static_cast<T>(3.25), Policy());
1136          boost::math::erf(static_cast<T>(5.25), Policy());
1137          boost::math::erf(static_cast<T>(7.25), Policy());
1138          boost::math::erf(static_cast<T>(11.25), Policy());
1139          boost::math::erf(static_cast<T>(12.5), Policy());
1140       }
1141       void force_instantiate()const{}
1142    };
1143    static const init initializer;
1144    static void force_instantiate()
1145    {
1146       initializer.force_instantiate();
1147    }
1148 };
1149
1150 template <class T, class Policy, class tag>
1151 const typename erf_initializer<T, Policy, tag>::init erf_initializer<T, Policy, tag>::initializer;
1152
1153 } // namespace detail
1154
1155 template <class T, class Policy>
1156 inline typename tools::promote_args<T>::type erf(T z, const Policy& /* pol */)
1157 {
1158    typedef typename tools::promote_args<T>::type result_type;
1159    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1160    typedef typename policies::precision<result_type, Policy>::type precision_type;
1161    typedef typename policies::normalise<
1162       Policy,
1163       policies::promote_float<false>,
1164       policies::promote_double<false>,
1165       policies::discrete_quantile<>,
1166       policies::assert_undefined<> >::type forwarding_policy;
1167
1168    BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
1169    BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
1170    BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
1171
1172    typedef boost::integral_constant<int,
1173       precision_type::value <= 0 ? 0 :
1174       precision_type::value <= 53 ? 53 :
1175       precision_type::value <= 64 ? 64 :
1176       precision_type::value <= 113 ? 113 : 0
1177    > tag_type;
1178
1179    BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
1180
1181    detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
1182
1183    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
1184       static_cast<value_type>(z),
1185       false,
1186       forwarding_policy(),
1187       tag_type()), "boost::math::erf<%1%>(%1%, %1%)");
1188 }
1189
1190 template <class T, class Policy>
1191 inline typename tools::promote_args<T>::type erfc(T z, const Policy& /* pol */)
1192 {
1193    typedef typename tools::promote_args<T>::type result_type;
1194    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1195    typedef typename policies::precision<result_type, Policy>::type precision_type;
1196    typedef typename policies::normalise<
1197       Policy,
1198       policies::promote_float<false>,
1199       policies::promote_double<false>,
1200       policies::discrete_quantile<>,
1201       policies::assert_undefined<> >::type forwarding_policy;
1202
1203    BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
1204    BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
1205    BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
1206
1207    typedef boost::integral_constant<int,
1208       precision_type::value <= 0 ? 0 :
1209       precision_type::value <= 53 ? 53 :
1210       precision_type::value <= 64 ? 64 :
1211       precision_type::value <= 113 ? 113 : 0
1212    > tag_type;
1213
1214    BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
1215
1216    detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
1217
1218    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
1219       static_cast<value_type>(z),
1220       true,
1221       forwarding_policy(),
1222       tag_type()), "boost::math::erfc<%1%>(%1%, %1%)");
1223 }
1224
1225 template <class T>
1226 inline typename tools::promote_args<T>::type erf(T z)
1227 {
1228    return boost::math::erf(z, policies::policy<>());
1229 }
1230
1231 template <class T>
1232 inline typename tools::promote_args<T>::type erfc(T z)
1233 {
1234    return boost::math::erfc(z, policies::policy<>());
1235 }
1236
1237 } // namespace math
1238 } // namespace boost
1239
1240 #include <boost/math/special_functions/detail/erf_inv.hpp>
1241
1242 #endif // BOOST_MATH_SPECIAL_ERF_HPP
1243
1244
1245
1246