ceph/src/boost/boost/math/special_functions/expint.hpp

   1 //  Copyright John Maddock 2007.
   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_EXPINT_HPP
   7 #define BOOST_MATH_EXPINT_HPP
   8
   9 #ifdef _MSC_VER
  10 #pragma once
  11 #pragma warning(push)
  12 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
  13 #endif
  14
  15 #include <boost/math/tools/precision.hpp>
  16 #include <boost/math/tools/promotion.hpp>
  17 #include <boost/math/tools/fraction.hpp>
  18 #include <boost/math/tools/series.hpp>
  19 #include <boost/math/policies/error_handling.hpp>
  20 #include <boost/math/special_functions/math_fwd.hpp>
  21 #include <boost/math/special_functions/digamma.hpp>
  22 #include <boost/math/special_functions/log1p.hpp>
  23 #include <boost/math/special_functions/pow.hpp>
  24
  25 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
  26 //
  27 // This is the only way we can avoid
  28 // warning: non-standard suffix on floating constant [-Wpedantic]
  29 // when building with -Wall -pedantic.  Neither __extension__
  30 // nor #pragma diagnostic ignored work :(
  31 //
  32 #pragma GCC system_header
  33 #endif
  34
  35 namespace boost{ namespace math{
  36
  37 template <class T, class Policy>
  38 inline typename tools::promote_args<T>::type
  39    expint(unsigned n, T z, const Policy& /*pol*/);
  40
  41 namespace detail{
  42
  43 template <class T>
  44 inline T expint_1_rational(const T& z, const boost::integral_constant<int, 0>&)
  45 {
  46    // this function is never actually called
  47    BOOST_ASSERT(0);
  48    return z;
  49 }
  50
  51 template <class T>
  52 T expint_1_rational(const T& z, const boost::integral_constant<int, 53>&)
  53 {
  54    BOOST_MATH_STD_USING
  55    T result;
  56    if(z <= 1)
  57    {
  58       // Maximum Deviation Found:                     2.006e-18
  59       // Expected Error Term:                         2.006e-18
  60       // Max error found at double precision:         2.760e-17
  61       static const T Y = 0.66373538970947265625F;
  62       static const T P[6] = {
  63          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0865197248079397976498),
  64          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0320913665303559189999),
  65          BOOST_MATH_BIG_CONSTANT(T, 53, -0.245088216639761496153),
  66          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0368031736257943745142),
  67          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00399167106081113256961),
  68          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000111507792921197858394)
  69       };
  70       static const T Q[6] = {
  71          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
  72          BOOST_MATH_BIG_CONSTANT(T, 53, 0.37091387659397013215),
  73          BOOST_MATH_BIG_CONSTANT(T, 53, 0.056770677104207528384),
  74          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00427347600017103698101),
  75          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000131049900798434683324),
  76          BOOST_MATH_BIG_CONSTANT(T, 53, -0.528611029520217142048e-6)
  77       };
  78       result = tools::evaluate_polynomial(P, z)
  79          / tools::evaluate_polynomial(Q, z);
  80       result += z - log(z) - Y;
  81    }
  82    else if(z < -boost::math::tools::log_min_value<T>())
  83    {
  84       // Maximum Deviation Found (interpolated):      1.444e-17
  85       // Max error found at double precision:         3.119e-17
  86       static const T P[11] = {
  87          BOOST_MATH_BIG_CONSTANT(T, 53, -0.121013190657725568138e-18),
  88          BOOST_MATH_BIG_CONSTANT(T, 53, -0.999999999999998811143),
  89          BOOST_MATH_BIG_CONSTANT(T, 53, -43.3058660811817946037),
  90          BOOST_MATH_BIG_CONSTANT(T, 53, -724.581482791462469795),
  91          BOOST_MATH_BIG_CONSTANT(T, 53, -6046.8250112711035463),
  92          BOOST_MATH_BIG_CONSTANT(T, 53, -27182.6254466733970467),
  93          BOOST_MATH_BIG_CONSTANT(T, 53, -66598.2652345418633509),
  94          BOOST_MATH_BIG_CONSTANT(T, 53, -86273.1567711649528784),
  95          BOOST_MATH_BIG_CONSTANT(T, 53, -54844.4587226402067411),
  96          BOOST_MATH_BIG_CONSTANT(T, 53, -14751.4895786128450662),
  97          BOOST_MATH_BIG_CONSTANT(T, 53, -1185.45720315201027667)
  98       };
  99       static const T Q[12] = {
 100          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 101          BOOST_MATH_BIG_CONSTANT(T, 53, 45.3058660811801465927),
 102          BOOST_MATH_BIG_CONSTANT(T, 53, 809.193214954550328455),
 103          BOOST_MATH_BIG_CONSTANT(T, 53, 7417.37624454689546708),
 104          BOOST_MATH_BIG_CONSTANT(T, 53, 38129.5594484818471461),
 105          BOOST_MATH_BIG_CONSTANT(T, 53, 113057.05869159631492),
 106          BOOST_MATH_BIG_CONSTANT(T, 53, 192104.047790227984431),
 107          BOOST_MATH_BIG_CONSTANT(T, 53, 180329.498380501819718),
 108          BOOST_MATH_BIG_CONSTANT(T, 53, 86722.3403467334749201),
 109          BOOST_MATH_BIG_CONSTANT(T, 53, 18455.4124737722049515),
 110          BOOST_MATH_BIG_CONSTANT(T, 53, 1229.20784182403048905),
 111          BOOST_MATH_BIG_CONSTANT(T, 53, -0.776491285282330997549)
 112       };
 113       T recip = 1 / z;
 114       result = 1 + tools::evaluate_polynomial(P, recip)
 115          / tools::evaluate_polynomial(Q, recip);
 116       result *= exp(-z) * recip;
 117    }
 118    else
 119    {
 120       result = 0;
 121    }
 122    return result;
 123 }
 124
 125 template <class T>
 126 T expint_1_rational(const T& z, const boost::integral_constant<int, 64>&)
 127 {
 128    BOOST_MATH_STD_USING
 129    T result;
 130    if(z <= 1)
 131    {
 132       // Maximum Deviation Found:                     3.807e-20
 133       // Expected Error Term:                         3.807e-20
 134       // Max error found at long double precision:    6.249e-20
 135
 136       static const T Y = 0.66373538970947265625F;
 137       static const T P[6] = {
 138          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0865197248079397956816),
 139          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0275114007037026844633),
 140          BOOST_MATH_BIG_CONSTANT(T, 64, -0.246594388074877139824),
 141          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0237624819878732642231),
 142          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00259113319641673986276),
 143          BOOST_MATH_BIG_CONSTANT(T, 64, 0.30853660894346057053e-4)
 144       };
 145       static const T Q[7] = {
 146          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 147          BOOST_MATH_BIG_CONSTANT(T, 64, 0.317978365797784100273),
 148          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0393622602554758722511),
 149          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00204062029115966323229),
 150          BOOST_MATH_BIG_CONSTANT(T, 64, 0.732512107100088047854e-5),
 151          BOOST_MATH_BIG_CONSTANT(T, 64, -0.202872781770207871975e-5),
 152          BOOST_MATH_BIG_CONSTANT(T, 64, 0.52779248094603709945e-7)
 153       };
 154       result = tools::evaluate_polynomial(P, z)
 155          / tools::evaluate_polynomial(Q, z);
 156       result += z - log(z) - Y;
 157    }
 158    else if(z < -boost::math::tools::log_min_value<T>())
 159    {
 160       // Maximum Deviation Found (interpolated):     2.220e-20
 161       // Max error found at long double precision:   1.346e-19
 162       static const T P[14] = {
 163          BOOST_MATH_BIG_CONSTANT(T, 64, -0.534401189080684443046e-23),
 164          BOOST_MATH_BIG_CONSTANT(T, 64, -0.999999999999999999905),
 165          BOOST_MATH_BIG_CONSTANT(T, 64, -62.1517806091379402505),
 166          BOOST_MATH_BIG_CONSTANT(T, 64, -1568.45688271895145277),
 167          BOOST_MATH_BIG_CONSTANT(T, 64, -21015.3431990874009619),
 168          BOOST_MATH_BIG_CONSTANT(T, 64, -164333.011755931661949),
 169          BOOST_MATH_BIG_CONSTANT(T, 64, -777917.270775426696103),
 170          BOOST_MATH_BIG_CONSTANT(T, 64, -2244188.56195255112937),
 171          BOOST_MATH_BIG_CONSTANT(T, 64, -3888702.98145335643429),
 172          BOOST_MATH_BIG_CONSTANT(T, 64, -3909822.65621952648353),
 173          BOOST_MATH_BIG_CONSTANT(T, 64, -2149033.9538897398457),
 174          BOOST_MATH_BIG_CONSTANT(T, 64, -584705.537139793925189),
 175          BOOST_MATH_BIG_CONSTANT(T, 64, -65815.2605361889477244),
 176          BOOST_MATH_BIG_CONSTANT(T, 64, -2038.82870680427258038)
 177       };
 178       static const T Q[14] = {
 179          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 180          BOOST_MATH_BIG_CONSTANT(T, 64, 64.1517806091379399478),
 181          BOOST_MATH_BIG_CONSTANT(T, 64, 1690.76044393722763785),
 182          BOOST_MATH_BIG_CONSTANT(T, 64, 24035.9534033068949426),
 183          BOOST_MATH_BIG_CONSTANT(T, 64, 203679.998633572361706),
 184          BOOST_MATH_BIG_CONSTANT(T, 64, 1074661.58459976978285),
 185          BOOST_MATH_BIG_CONSTANT(T, 64, 3586552.65020899358773),
 186          BOOST_MATH_BIG_CONSTANT(T, 64, 7552186.84989547621411),
 187          BOOST_MATH_BIG_CONSTANT(T, 64, 9853333.79353054111434),
 188          BOOST_MATH_BIG_CONSTANT(T, 64, 7689642.74550683631258),
 189          BOOST_MATH_BIG_CONSTANT(T, 64, 3385553.35146759180739),
 190          BOOST_MATH_BIG_CONSTANT(T, 64, 763218.072732396428725),
 191          BOOST_MATH_BIG_CONSTANT(T, 64, 73930.2995984054930821),
 192          BOOST_MATH_BIG_CONSTANT(T, 64, 2063.86994219629165937)
 193       };
 194       T recip = 1 / z;
 195       result = 1 + tools::evaluate_polynomial(P, recip)
 196          / tools::evaluate_polynomial(Q, recip);
 197       result *= exp(-z) * recip;
 198    }
 199    else
 200    {
 201       result = 0;
 202    }
 203    return result;
 204 }
 205
 206 template <class T>
 207 T expint_1_rational(const T& z, const boost::integral_constant<int, 113>&)
 208 {
 209    BOOST_MATH_STD_USING
 210    T result;
 211    if(z <= 1)
 212    {
 213       // Maximum Deviation Found:                     2.477e-35
 214       // Expected Error Term:                         2.477e-35
 215       // Max error found at long double precision:    6.810e-35
 216
 217       static const T Y = 0.66373538970947265625F;
 218       static const T P[10] = {
 219          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0865197248079397956434879099175975937),
 220          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0369066175910795772830865304506087759),
 221          BOOST_MATH_BIG_CONSTANT(T, 113, -0.24272036838415474665971599314725545),
 222          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0502166331248948515282379137550178307),
 223          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00768384138547489410285101483730424919),
 224          BOOST_MATH_BIG_CONSTANT(T, 113, -0.000612574337702109683505224915484717162),
 225          BOOST_MATH_BIG_CONSTANT(T, 113, -0.380207107950635046971492617061708534e-4),
 226          BOOST_MATH_BIG_CONSTANT(T, 113, -0.136528159460768830763009294683628406e-5),
 227          BOOST_MATH_BIG_CONSTANT(T, 113, -0.346839106212658259681029388908658618e-7),
 228          BOOST_MATH_BIG_CONSTANT(T, 113, -0.340500302777838063940402160594523429e-9)
 229       };
 230       static const T Q[10] = {
 231          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 232          BOOST_MATH_BIG_CONSTANT(T, 113, 0.426568827778942588160423015589537302),
 233          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0841384046470893490592450881447510148),
 234          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0100557215850668029618957359471132995),
 235          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000799334870474627021737357294799839363),
 236          BOOST_MATH_BIG_CONSTANT(T, 113, 0.434452090903862735242423068552687688e-4),
 237          BOOST_MATH_BIG_CONSTANT(T, 113, 0.15829674748799079874182885081231252e-5),
 238          BOOST_MATH_BIG_CONSTANT(T, 113, 0.354406206738023762100882270033082198e-7),
 239          BOOST_MATH_BIG_CONSTANT(T, 113, 0.369373328141051577845488477377890236e-9),
 240          BOOST_MATH_BIG_CONSTANT(T, 113, -0.274149801370933606409282434677600112e-12)
 241       };
 242       result = tools::evaluate_polynomial(P, z)
 243          / tools::evaluate_polynomial(Q, z);
 244       result += z - log(z) - Y;
 245    }
 246    else if(z <= 4)
 247    {
 248       // Max error in interpolated form:             5.614e-35
 249       // Max error found at long double precision:   7.979e-35
 250
 251       static const T Y = 0.70190334320068359375F;
 252
 253       static const T P[16] = {
 254          BOOST_MATH_BIG_CONSTANT(T, 113, 0.298096656795020369955077350585959794),
 255          BOOST_MATH_BIG_CONSTANT(T, 113, 12.9314045995266142913135497455971247),
 256          BOOST_MATH_BIG_CONSTANT(T, 113, 226.144334921582637462526628217345501),
 257          BOOST_MATH_BIG_CONSTANT(T, 113, 2070.83670924261732722117682067381405),
 258          BOOST_MATH_BIG_CONSTANT(T, 113, 10715.1115684330959908244769731347186),
 259          BOOST_MATH_BIG_CONSTANT(T, 113, 30728.7876355542048019664777316053311),
 260          BOOST_MATH_BIG_CONSTANT(T, 113, 38520.6078609349855436936232610875297),
 261          BOOST_MATH_BIG_CONSTANT(T, 113, -27606.0780981527583168728339620565165),
 262          BOOST_MATH_BIG_CONSTANT(T, 113, -169026.485055785605958655247592604835),
 263          BOOST_MATH_BIG_CONSTANT(T, 113, -254361.919204983608659069868035092282),
 264          BOOST_MATH_BIG_CONSTANT(T, 113, -195765.706874132267953259272028679935),
 265          BOOST_MATH_BIG_CONSTANT(T, 113, -83352.6826013533205474990119962408675),
 266          BOOST_MATH_BIG_CONSTANT(T, 113, -19251.6828496869586415162597993050194),
 267          BOOST_MATH_BIG_CONSTANT(T, 113, -2226.64251774578542836725386936102339),
 268          BOOST_MATH_BIG_CONSTANT(T, 113, -109.009437301400845902228611986479816),
 269          BOOST_MATH_BIG_CONSTANT(T, 113, -1.51492042209561411434644938098833499)
 270       };
 271       static const T Q[16] = {
 272          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 273          BOOST_MATH_BIG_CONSTANT(T, 113, 46.734521442032505570517810766704587),
 274          BOOST_MATH_BIG_CONSTANT(T, 113, 908.694714348462269000247450058595655),
 275          BOOST_MATH_BIG_CONSTANT(T, 113, 9701.76053033673927362784882748513195),
 276          BOOST_MATH_BIG_CONSTANT(T, 113, 63254.2815292641314236625196594947774),
 277          BOOST_MATH_BIG_CONSTANT(T, 113, 265115.641285880437335106541757711092),
 278          BOOST_MATH_BIG_CONSTANT(T, 113, 732707.841188071900498536533086567735),
 279          BOOST_MATH_BIG_CONSTANT(T, 113, 1348514.02492635723327306628712057794),
 280          BOOST_MATH_BIG_CONSTANT(T, 113, 1649986.81455283047769673308781585991),
 281          BOOST_MATH_BIG_CONSTANT(T, 113, 1326000.828522976970116271208812099),
 282          BOOST_MATH_BIG_CONSTANT(T, 113, 683643.09490612171772350481773951341),
 283          BOOST_MATH_BIG_CONSTANT(T, 113, 217640.505137263607952365685653352229),
 284          BOOST_MATH_BIG_CONSTANT(T, 113, 40288.3467237411710881822569476155485),
 285          BOOST_MATH_BIG_CONSTANT(T, 113, 3932.89353979531632559232883283175754),
 286          BOOST_MATH_BIG_CONSTANT(T, 113, 169.845369689596739824177412096477219),
 287          BOOST_MATH_BIG_CONSTANT(T, 113, 2.17607292280092201170768401876895354)
 288       };
 289       T recip = 1 / z;
 290       result = Y + tools::evaluate_polynomial(P, recip)
 291          / tools::evaluate_polynomial(Q, recip);
 292       result *= exp(-z) * recip;
 293    }
 294    else if(z < -boost::math::tools::log_min_value<T>())
 295    {
 296       // Max error in interpolated form:             4.413e-35
 297       // Max error found at long double precision:   8.928e-35
 298
 299       static const T P[19] = {
 300          BOOST_MATH_BIG_CONSTANT(T, 113, -0.559148411832951463689610809550083986e-40),
 301          BOOST_MATH_BIG_CONSTANT(T, 113, -0.999999999999999999999999999999999997),
 302          BOOST_MATH_BIG_CONSTANT(T, 113, -166.542326331163836642960118190147367),
 303          BOOST_MATH_BIG_CONSTANT(T, 113, -12204.639128796330005065904675153652),
 304          BOOST_MATH_BIG_CONSTANT(T, 113, -520807.069767086071806275022036146855),
 305          BOOST_MATH_BIG_CONSTANT(T, 113, -14435981.5242137970691490903863125326),
 306          BOOST_MATH_BIG_CONSTANT(T, 113, -274574945.737064301247496460758654196),
 307          BOOST_MATH_BIG_CONSTANT(T, 113, -3691611582.99810039356254671781473079),
 308          BOOST_MATH_BIG_CONSTANT(T, 113, -35622515944.8255047299363690814678763),
 309          BOOST_MATH_BIG_CONSTANT(T, 113, -248040014774.502043161750715548451142),
 310          BOOST_MATH_BIG_CONSTANT(T, 113, -1243190389769.53458416330946622607913),
 311          BOOST_MATH_BIG_CONSTANT(T, 113, -4441730126135.54739052731990368425339),
 312          BOOST_MATH_BIG_CONSTANT(T, 113, -11117043181899.7388524310281751971366),
 313          BOOST_MATH_BIG_CONSTANT(T, 113, -18976497615396.9717776601813519498961),
 314          BOOST_MATH_BIG_CONSTANT(T, 113, -21237496819711.1011661104761906067131),
 315          BOOST_MATH_BIG_CONSTANT(T, 113, -14695899122092.5161620333466757812848),
 316          BOOST_MATH_BIG_CONSTANT(T, 113, -5737221535080.30569711574295785864903),
 317          BOOST_MATH_BIG_CONSTANT(T, 113, -1077042281708.42654526404581272546244),
 318          BOOST_MATH_BIG_CONSTANT(T, 113, -68028222642.1941480871395695677675137)
 319       };
 320       static const T Q[20] = {
 321          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
 322          BOOST_MATH_BIG_CONSTANT(T, 113, 168.542326331163836642960118190147311),
 323          BOOST_MATH_BIG_CONSTANT(T, 113, 12535.7237814586576783518249115343619),
 324          BOOST_MATH_BIG_CONSTANT(T, 113, 544891.263372016404143120911148640627),
 325          BOOST_MATH_BIG_CONSTANT(T, 113, 15454474.7241010258634446523045237762),
 326          BOOST_MATH_BIG_CONSTANT(T, 113, 302495899.896629522673410325891717381),
 327          BOOST_MATH_BIG_CONSTANT(T, 113, 4215565948.38886507646911672693270307),
 328          BOOST_MATH_BIG_CONSTANT(T, 113, 42552409471.7951815668506556705733344),
 329          BOOST_MATH_BIG_CONSTANT(T, 113, 313592377066.753173979584098301610186),
 330          BOOST_MATH_BIG_CONSTANT(T, 113, 1688763640223.4541980740597514904542),
 331          BOOST_MATH_BIG_CONSTANT(T, 113, 6610992294901.59589748057620192145704),
 332          BOOST_MATH_BIG_CONSTANT(T, 113, 18601637235659.6059890851321772682606),
 333          BOOST_MATH_BIG_CONSTANT(T, 113, 36944278231087.2571020964163402941583),
 334          BOOST_MATH_BIG_CONSTANT(T, 113, 50425858518481.7497071917028793820058),
 335          BOOST_MATH_BIG_CONSTANT(T, 113, 45508060902865.0899967797848815980644),
 336          BOOST_MATH_BIG_CONSTANT(T, 113, 25649955002765.3817331501988304758142),
 337          BOOST_MATH_BIG_CONSTANT(T, 113, 8259575619094.6518520988612711292331),
 338          BOOST_MATH_BIG_CONSTANT(T, 113, 1299981487496.12607474362723586264515),
 339          BOOST_MATH_BIG_CONSTANT(T, 113, 70242279152.8241187845178443118302693),
 340          BOOST_MATH_BIG_CONSTANT(T, 113, -37633302.9409263839042721539363416685)
 341       };
 342       T recip = 1 / z;
 343       result = 1 + tools::evaluate_polynomial(P, recip)
 344          / tools::evaluate_polynomial(Q, recip);
 345       result *= exp(-z) * recip;
 346    }
 347    else
 348    {
 349       result = 0;
 350    }
 351    return result;
 352 }
 353
 354 template <class T>
 355 struct expint_fraction
 356 {
 357    typedef std::pair<T,T> result_type;
 358    expint_fraction(unsigned n_, T z_) : b(n_ + z_), i(-1), n(n_){}
 359    std::pair<T,T> operator()()
 360    {
 361       std::pair<T,T> result = std::make_pair(-static_cast<T>((i+1) * (n+i)), b);
 362       b += 2;
 363       ++i;
 364       return result;
 365    }
 366 private:
 367    T b;
 368    int i;
 369    unsigned n;
 370 };
 371
 372 template <class T, class Policy>
 373 inline T expint_as_fraction(unsigned n, T z, const Policy& pol)
 374 {
 375    BOOST_MATH_STD_USING
 376    BOOST_MATH_INSTRUMENT_VARIABLE(z)
 377    boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
 378    expint_fraction<T> f(n, z);
 379    T result = tools::continued_fraction_b(
 380       f,
 381       boost::math::policies::get_epsilon<T, Policy>(),
 382       max_iter);
 383    policies::check_series_iterations<T>("boost::math::expint_continued_fraction<%1%>(unsigned,%1%)", max_iter, pol);
 384    BOOST_MATH_INSTRUMENT_VARIABLE(result)
 385    BOOST_MATH_INSTRUMENT_VARIABLE(max_iter)
 386    result = exp(-z) / result;
 387    BOOST_MATH_INSTRUMENT_VARIABLE(result)
 388    return result;
 389 }
 390
 391 template <class T>
 392 struct expint_series
 393 {
 394    typedef T result_type;
 395    expint_series(unsigned k_, T z_, T x_k_, T denom_, T fact_)
 396       : k(k_), z(z_), x_k(x_k_), denom(denom_), fact(fact_){}
 397    T operator()()
 398    {
 399       x_k *= -z;
 400       denom += 1;
 401       fact *= ++k;
 402       return x_k / (denom * fact);
 403    }
 404 private:
 405    unsigned k;
 406    T z;
 407    T x_k;
 408    T denom;
 409    T fact;
 410 };
 411
 412 template <class T, class Policy>
 413 inline T expint_as_series(unsigned n, T z, const Policy& pol)
 414 {
 415    BOOST_MATH_STD_USING
 416    boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
 417
 418    BOOST_MATH_INSTRUMENT_VARIABLE(z)
 419
 420    T result = 0;
 421    T x_k = -1;
 422    T denom = T(1) - n;
 423    T fact = 1;
 424    unsigned k = 0;
 425    for(; k < n - 1;)
 426    {
 427       result += x_k / (denom * fact);
 428       denom += 1;
 429       x_k *= -z;
 430       fact *= ++k;
 431    }
 432    BOOST_MATH_INSTRUMENT_VARIABLE(result)
 433    result += pow(-z, static_cast<T>(n - 1))
 434       * (boost::math::digamma(static_cast<T>(n)) - log(z)) / fact;
 435    BOOST_MATH_INSTRUMENT_VARIABLE(result)
 436
 437    expint_series<T> s(k, z, x_k, denom, fact);
 438    result = tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, result);
 439    policies::check_series_iterations<T>("boost::math::expint_series<%1%>(unsigned,%1%)", max_iter, pol);
 440    BOOST_MATH_INSTRUMENT_VARIABLE(result)
 441    BOOST_MATH_INSTRUMENT_VARIABLE(max_iter)
 442    return result;
 443 }
 444
 445 template <class T, class Policy, class Tag>
 446 T expint_imp(unsigned n, T z, const Policy& pol, const Tag& tag)
 447 {
 448    BOOST_MATH_STD_USING
 449    static const char* function = "boost::math::expint<%1%>(unsigned, %1%)";
 450    if(z < 0)
 451       return policies::raise_domain_error<T>(function, "Function requires z >= 0 but got %1%.", z, pol);
 452    if(z == 0)
 453       return n == 1 ? policies::raise_overflow_error<T>(function, 0, pol) : T(1 / (static_cast<T>(n - 1)));
 454
 455    T result;
 456
 457    bool f;
 458    if(n < 3)
 459    {
 460       f = z < 0.5;
 461    }
 462    else
 463    {
 464       f = z < (static_cast<T>(n - 2) / static_cast<T>(n - 1));
 465    }
 466 #ifdef BOOST_MSVC
 467 #  pragma warning(push)
 468 #  pragma warning(disable:4127) // conditional expression is constant
 469 #endif
 470    if(n == 0)
 471       result = exp(-z) / z;
 472    else if((n == 1) && (Tag::value))
 473    {
 474       result = expint_1_rational(z, tag);
 475    }
 476    else if(f)
 477       result = expint_as_series(n, z, pol);
 478    else
 479       result = expint_as_fraction(n, z, pol);
 480 #ifdef BOOST_MSVC
 481 #  pragma warning(pop)
 482 #endif
 483
 484    return result;
 485 }
 486
 487 template <class T>
 488 struct expint_i_series
 489 {
 490    typedef T result_type;
 491    expint_i_series(T z_) : k(0), z_k(1), z(z_){}
 492    T operator()()
 493    {
 494       z_k *= z / ++k;
 495       return z_k / k;
 496    }
 497 private:
 498    unsigned k;
 499    T z_k;
 500    T z;
 501 };
 502
 503 template <class T, class Policy>
 504 T expint_i_as_series(T z, const Policy& pol)
 505 {
 506    BOOST_MATH_STD_USING
 507    T result = log(z); // (log(z) - log(1 / z)) / 2;
 508    result += constants::euler<T>();
 509    expint_i_series<T> s(z);
 510    boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
 511    result = tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, result);
 512    policies::check_series_iterations<T>("boost::math::expint_i_series<%1%>(%1%)", max_iter, pol);
 513    return result;
 514 }
 515
 516 template <class T, class Policy, class Tag>
 517 T expint_i_imp(T z, const Policy& pol, const Tag& tag)
 518 {
 519    static const char* function = "boost::math::expint<%1%>(%1%)";
 520    if(z < 0)
 521       return -expint_imp(1, T(-z), pol, tag);
 522    if(z == 0)
 523       return -policies::raise_overflow_error<T>(function, 0, pol);
 524    return expint_i_as_series(z, pol);
 525 }
 526
 527 template <class T, class Policy>
 528 T expint_i_imp(T z, const Policy& pol, const boost::integral_constant<int, 53>& tag)
 529 {
 530    BOOST_MATH_STD_USING
 531    static const char* function = "boost::math::expint<%1%>(%1%)";
 532    if(z < 0)
 533       return -expint_imp(1, T(-z), pol, tag);
 534    if(z == 0)
 535       return -policies::raise_overflow_error<T>(function, 0, pol);
 536
 537    T result;
 538
 539    if(z <= 6)
 540    {
 541       // Maximum Deviation Found:                     2.852e-18
 542       // Expected Error Term:                         2.852e-18
 543       // Max Error found at double precision =        Poly: 2.636335e-16   Cheb: 4.187027e-16
 544       static const T P[10] = {
 545          BOOST_MATH_BIG_CONSTANT(T, 53, 2.98677224343598593013),
 546          BOOST_MATH_BIG_CONSTANT(T, 53, 0.356343618769377415068),
 547          BOOST_MATH_BIG_CONSTANT(T, 53, 0.780836076283730801839),
 548          BOOST_MATH_BIG_CONSTANT(T, 53, 0.114670926327032002811),
 549          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0499434773576515260534),
 550          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00726224593341228159561),
 551          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00115478237227804306827),
 552          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000116419523609765200999),
 553          BOOST_MATH_BIG_CONSTANT(T, 53, 0.798296365679269702435e-5),
 554          BOOST_MATH_BIG_CONSTANT(T, 53, 0.2777056254402008721e-6)
 555       };
 556       static const T Q[8] = {
 557          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 558          BOOST_MATH_BIG_CONSTANT(T, 53, -1.17090412365413911947),
 559          BOOST_MATH_BIG_CONSTANT(T, 53, 0.62215109846016746276),
 560          BOOST_MATH_BIG_CONSTANT(T, 53, -0.195114782069495403315),
 561          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0391523431392967238166),
 562          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00504800158663705747345),
 563          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000389034007436065401822),
 564          BOOST_MATH_BIG_CONSTANT(T, 53, -0.138972589601781706598e-4)
 565       };
 566
 567       static const T c1 = BOOST_MATH_BIG_CONSTANT(T, 53, 1677624236387711.0);
 568       static const T c2 = BOOST_MATH_BIG_CONSTANT(T, 53, 4503599627370496.0);
 569       static const T r1 = static_cast<T>(c1 / c2);
 570       static const T r2 = BOOST_MATH_BIG_CONSTANT(T, 53, 0.131401834143860282009280387409357165515556574352422001206362e-16);
 571       static const T r = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 53, 0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392));
 572       T t = (z / 3) - 1;
 573       result = tools::evaluate_polynomial(P, t)
 574          / tools::evaluate_polynomial(Q, t);
 575       t = (z - r1) - r2;
 576       result *= t;
 577       if(fabs(t) < 0.1)
 578       {
 579          result += boost::math::log1p(t / r, pol);
 580       }
 581       else
 582       {
 583          result += log(z / r);
 584       }
 585    }
 586    else if (z <= 10)
 587    {
 588       // Maximum Deviation Found:                     6.546e-17
 589       // Expected Error Term:                         6.546e-17
 590       // Max Error found at double precision =        Poly: 6.890169e-17   Cheb: 6.772128e-17
 591       static const T Y = 1.158985137939453125F;
 592       static const T P[8] = {
 593          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00139324086199402804173),
 594          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0349921221823888744966),
 595          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0264095520754134848538),
 596          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00761224003005476438412),
 597          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00247496209592143627977),
 598          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000374885917942100256775),
 599          BOOST_MATH_BIG_CONSTANT(T, 53, -0.554086272024881826253e-4),
 600          BOOST_MATH_BIG_CONSTANT(T, 53, -0.396487648924804510056e-5)
 601       };
 602       static const T Q[8] = {
 603          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 604          BOOST_MATH_BIG_CONSTANT(T, 53, 0.744625566823272107711),
 605          BOOST_MATH_BIG_CONSTANT(T, 53, 0.329061095011767059236),
 606          BOOST_MATH_BIG_CONSTANT(T, 53, 0.100128624977313872323),
 607          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0223851099128506347278),
 608          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00365334190742316650106),
 609          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000402453408512476836472),
 610          BOOST_MATH_BIG_CONSTANT(T, 53, 0.263649630720255691787e-4)
 611       };
 612       T t = z / 2 - 4;
 613       result = Y + tools::evaluate_polynomial(P, t)
 614          / tools::evaluate_polynomial(Q, t);
 615       result *= exp(z) / z;
 616       result += z;
 617    }
 618    else if(z <= 20)
 619    {
 620       // Maximum Deviation Found:                     1.843e-17
 621       // Expected Error Term:                         -1.842e-17
 622       // Max Error found at double precision =        Poly: 4.375868e-17   Cheb: 5.860967e-17
 623
 624       static const T Y = 1.0869731903076171875F;
 625       static const T P[9] = {
 626          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00893891094356945667451),
 627          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0484607730127134045806),
 628          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0652810444222236895772),
 629          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0478447572647309671455),
 630          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0226059218923777094596),
 631          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00720603636917482065907),
 632          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00155941947035972031334),
 633          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000209750022660200888349),
 634          BOOST_MATH_BIG_CONSTANT(T, 53, -0.138652200349182596186e-4)
 635       };
 636       static const T Q[9] = {
 637          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 638          BOOST_MATH_BIG_CONSTANT(T, 53, 1.97017214039061194971),
 639          BOOST_MATH_BIG_CONSTANT(T, 53, 1.86232465043073157508),
 640          BOOST_MATH_BIG_CONSTANT(T, 53, 1.09601437090337519977),
 641          BOOST_MATH_BIG_CONSTANT(T, 53, 0.438873285773088870812),
 642          BOOST_MATH_BIG_CONSTANT(T, 53, 0.122537731979686102756),
 643          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0233458478275769288159),
 644          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00278170769163303669021),
 645          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000159150281166108755531)
 646       };
 647       T t = z / 5 - 3;
 648       result = Y + tools::evaluate_polynomial(P, t)
 649          / tools::evaluate_polynomial(Q, t);
 650       result *= exp(z) / z;
 651       result += z;
 652    }
 653    else if(z <= 40)
 654    {
 655       // Maximum Deviation Found:                     5.102e-18
 656       // Expected Error Term:                         5.101e-18
 657       // Max Error found at double precision =        Poly: 1.441088e-16   Cheb: 1.864792e-16
 658
 659
 660       static const T Y = 1.03937530517578125F;
 661       static const T P[9] = {
 662          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00356165148914447597995),
 663          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0229930320357982333406),
 664          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0449814350482277917716),
 665          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0453759383048193402336),
 666          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0272050837209380717069),
 667          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00994403059883350813295),
 668          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00207592267812291726961),
 669          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000192178045857733706044),
 670          BOOST_MATH_BIG_CONSTANT(T, 53, -0.113161784705911400295e-9)
 671       };
 672       static const T Q[9] = {
 673          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 674          BOOST_MATH_BIG_CONSTANT(T, 53, 2.84354408840148561131),
 675          BOOST_MATH_BIG_CONSTANT(T, 53, 3.6599610090072393012),
 676          BOOST_MATH_BIG_CONSTANT(T, 53, 2.75088464344293083595),
 677          BOOST_MATH_BIG_CONSTANT(T, 53, 1.2985244073998398643),
 678          BOOST_MATH_BIG_CONSTANT(T, 53, 0.383213198510794507409),
 679          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0651165455496281337831),
 680          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00488071077519227853585)
 681       };
 682       T t = z / 10 - 3;
 683       result = Y + tools::evaluate_polynomial(P, t)
 684          / tools::evaluate_polynomial(Q, t);
 685       result *= exp(z) / z;
 686       result += z;
 687    }
 688    else
 689    {
 690       // Max Error found at double precision =        3.381886e-17
 691       static const T exp40 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 53, 2.35385266837019985407899910749034804508871617254555467236651e17));
 692       static const T Y= 1.013065338134765625F;
 693       static const T P[6] = {
 694          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0130653381347656243849),
 695          BOOST_MATH_BIG_CONSTANT(T, 53, 0.19029710559486576682),
 696          BOOST_MATH_BIG_CONSTANT(T, 53, 94.7365094537197236011),
 697          BOOST_MATH_BIG_CONSTANT(T, 53, -2516.35323679844256203),
 698          BOOST_MATH_BIG_CONSTANT(T, 53, 18932.0850014925993025),
 699          BOOST_MATH_BIG_CONSTANT(T, 53, -38703.1431362056714134)
 700       };
 701       static const T Q[7] = {
 702          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
 703          BOOST_MATH_BIG_CONSTANT(T, 53, 61.9733592849439884145),
 704          BOOST_MATH_BIG_CONSTANT(T, 53, -2354.56211323420194283),
 705          BOOST_MATH_BIG_CONSTANT(T, 53, 22329.1459489893079041),
 706          BOOST_MATH_BIG_CONSTANT(T, 53, -70126.245140396567133),
 707          BOOST_MATH_BIG_CONSTANT(T, 53, 54738.2833147775537106),
 708          BOOST_MATH_BIG_CONSTANT(T, 53, 8297.16296356518409347)
 709       };
 710       T t = 1 / z;
 711       result = Y + tools::evaluate_polynomial(P, t)
 712          / tools::evaluate_polynomial(Q, t);
 713       if(z < 41)
 714          result *= exp(z) / z;
 715       else
 716       {
 717          // Avoid premature overflow if we can:
 718          t = z - 40;
 719          if(t > tools::log_max_value<T>())
 720          {
 721             result = policies::raise_overflow_error<T>(function, 0, pol);
 722          }
 723          else
 724          {
 725             result *= exp(z - 40) / z;
 726             if(result > tools::max_value<T>() / exp40)
 727             {
 728                result = policies::raise_overflow_error<T>(function, 0, pol);
 729             }
 730             else
 731             {
 732                result *= exp40;
 733             }
 734          }
 735       }
 736       result += z;
 737    }
 738    return result;
 739 }
 740
 741 template <class T, class Policy>
 742 T expint_i_imp(T z, const Policy& pol, const boost::integral_constant<int, 64>& tag)
 743 {
 744    BOOST_MATH_STD_USING
 745    static const char* function = "boost::math::expint<%1%>(%1%)";
 746    if(z < 0)
 747       return -expint_imp(1, T(-z), pol, tag);
 748    if(z == 0)
 749       return -policies::raise_overflow_error<T>(function, 0, pol);
 750
 751    T result;
 752
 753    if(z <= 6)
 754    {
 755       // Maximum Deviation Found:                     3.883e-21
 756       // Expected Error Term:                         3.883e-21
 757       // Max Error found at long double precision =   Poly: 3.344801e-19   Cheb: 4.989937e-19
 758
 759       static const T P[11] = {
 760          BOOST_MATH_BIG_CONSTANT(T, 64, 2.98677224343598593764),
 761          BOOST_MATH_BIG_CONSTANT(T, 64, 0.25891613550886736592),
 762          BOOST_MATH_BIG_CONSTANT(T, 64, 0.789323584998672832285),
 763          BOOST_MATH_BIG_CONSTANT(T, 64, 0.092432587824602399339),
 764          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0514236978728625906656),
 765          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00658477469745132977921),
 766          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00124914538197086254233),
 767          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000131429679565472408551),
 768          BOOST_MATH_BIG_CONSTANT(T, 64, 0.11293331317982763165e-4),
 769          BOOST_MATH_BIG_CONSTANT(T, 64, 0.629499283139417444244e-6),
 770          BOOST_MATH_BIG_CONSTANT(T, 64, 0.177833045143692498221e-7)
 771       };
 772       static const T Q[9] = {
 773          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 774          BOOST_MATH_BIG_CONSTANT(T, 64, -1.20352377969742325748),
 775          BOOST_MATH_BIG_CONSTANT(T, 64, 0.66707904942606479811),
 776          BOOST_MATH_BIG_CONSTANT(T, 64, -0.223014531629140771914),
 777          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0493340022262908008636),
 778          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00741934273050807310677),
 779          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00074353567782087939294),
 780          BOOST_MATH_BIG_CONSTANT(T, 64, -0.455861727069603367656e-4),
 781          BOOST_MATH_BIG_CONSTANT(T, 64, 0.131515429329812837701e-5)
 782       };
 783
 784       static const T c1 = BOOST_MATH_BIG_CONSTANT(T, 64, 1677624236387711.0);
 785       static const T c2 = BOOST_MATH_BIG_CONSTANT(T, 64, 4503599627370496.0);
 786       static const T r1 = c1 / c2;
 787       static const T r2 = BOOST_MATH_BIG_CONSTANT(T, 64, 0.131401834143860282009280387409357165515556574352422001206362e-16);
 788       static const T r = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392));
 789       T t = (z / 3) - 1;
 790       result = tools::evaluate_polynomial(P, t)
 791          / tools::evaluate_polynomial(Q, t);
 792       t = (z - r1) - r2;
 793       result *= t;
 794       if(fabs(t) < 0.1)
 795       {
 796          result += boost::math::log1p(t / r, pol);
 797       }
 798       else
 799       {
 800          result += log(z / r);
 801       }
 802    }
 803    else if (z <= 10)
 804    {
 805       // Maximum Deviation Found:                     2.622e-21
 806       // Expected Error Term:                         -2.622e-21
 807       // Max Error found at long double precision =   Poly: 1.208328e-20   Cheb: 1.073723e-20
 808
 809       static const T Y = 1.158985137939453125F;
 810       static const T P[9] = {
 811          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00139324086199409049399),
 812          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0345238388952337563247),
 813          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0382065278072592940767),
 814          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0156117003070560727392),
 815          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00383276012430495387102),
 816          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000697070540945496497992),
 817          BOOST_MATH_BIG_CONSTANT(T, 64, -0.877310384591205930343e-4),
 818          BOOST_MATH_BIG_CONSTANT(T, 64, -0.623067256376494930067e-5),
 819          BOOST_MATH_BIG_CONSTANT(T, 64, -0.377246883283337141444e-6)
 820       };
 821       static const T Q[10] = {
 822          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 823          BOOST_MATH_BIG_CONSTANT(T, 64, 1.08073635708902053767),
 824          BOOST_MATH_BIG_CONSTANT(T, 64, 0.553681133533942532909),
 825          BOOST_MATH_BIG_CONSTANT(T, 64, 0.176763647137553797451),
 826          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0387891748253869928121),
 827          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0060603004848394727017),
 828          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000670519492939992806051),
 829          BOOST_MATH_BIG_CONSTANT(T, 64, 0.4947357050100855646e-4),
 830          BOOST_MATH_BIG_CONSTANT(T, 64, 0.204339282037446434827e-5),
 831          BOOST_MATH_BIG_CONSTANT(T, 64, 0.146951181174930425744e-7)
 832       };
 833       T t = z / 2 - 4;
 834       result = Y + tools::evaluate_polynomial(P, t)
 835          / tools::evaluate_polynomial(Q, t);
 836       result *= exp(z) / z;
 837       result += z;
 838    }
 839    else if(z <= 20)
 840    {
 841       // Maximum Deviation Found:                     3.220e-20
 842       // Expected Error Term:                         3.220e-20
 843       // Max Error found at long double precision =   Poly: 7.696841e-20   Cheb: 6.205163e-20
 844
 845
 846       static const T Y = 1.0869731903076171875F;
 847       static const T P[10] = {
 848          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00893891094356946995368),
 849          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0487562980088748775943),
 850          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0670568657950041926085),
 851          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0509577352851442932713),
 852          BOOST_MATH_BIG_CONSTANT(T, 64, -0.02551800927409034206),
 853          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00892913759760086687083),
 854          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00224469630207344379888),
 855          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000392477245911296982776),
 856          BOOST_MATH_BIG_CONSTANT(T, 64, -0.44424044184395578775e-4),
 857          BOOST_MATH_BIG_CONSTANT(T, 64, -0.252788029251437017959e-5)
 858       };
 859       static const T Q[10] = {
 860          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 861          BOOST_MATH_BIG_CONSTANT(T, 64, 2.00323265503572414261),
 862          BOOST_MATH_BIG_CONSTANT(T, 64, 1.94688958187256383178),
 863          BOOST_MATH_BIG_CONSTANT(T, 64, 1.19733638134417472296),
 864          BOOST_MATH_BIG_CONSTANT(T, 64, 0.513137726038353385661),
 865          BOOST_MATH_BIG_CONSTANT(T, 64, 0.159135395578007264547),
 866          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0358233587351620919881),
 867          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0056716655597009417875),
 868          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000577048986213535829925),
 869          BOOST_MATH_BIG_CONSTANT(T, 64, 0.290976943033493216793e-4)
 870       };
 871       T t = z / 5 - 3;
 872       result = Y + tools::evaluate_polynomial(P, t)
 873          / tools::evaluate_polynomial(Q, t);
 874       result *= exp(z) / z;
 875       result += z;
 876    }
 877    else if(z <= 40)
 878    {
 879       // Maximum Deviation Found:                     2.940e-21
 880       // Expected Error Term:                         -2.938e-21
 881       // Max Error found at long double precision =   Poly: 3.419893e-19   Cheb: 3.359874e-19
 882
 883       static const T Y = 1.03937530517578125F;
 884       static const T P[12] = {
 885          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00356165148914447278177),
 886          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0240235006148610849678),
 887          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0516699967278057976119),
 888          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0586603078706856245674),
 889          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0409960120868776180825),
 890          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0185485073689590665153),
 891          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00537842101034123222417),
 892          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000920988084778273760609),
 893          BOOST_MATH_BIG_CONSTANT(T, 64, -0.716742618812210980263e-4),
 894          BOOST_MATH_BIG_CONSTANT(T, 64, -0.504623302166487346677e-9),
 895          BOOST_MATH_BIG_CONSTANT(T, 64, 0.712662196671896837736e-10),
 896          BOOST_MATH_BIG_CONSTANT(T, 64, -0.533769629702262072175e-11)
 897       };
 898       static const T Q[9] = {
 899          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 900          BOOST_MATH_BIG_CONSTANT(T, 64, 3.13286733695729715455),
 901          BOOST_MATH_BIG_CONSTANT(T, 64, 4.49281223045653491929),
 902          BOOST_MATH_BIG_CONSTANT(T, 64, 3.84900294427622911374),
 903          BOOST_MATH_BIG_CONSTANT(T, 64, 2.15205199043580378211),
 904          BOOST_MATH_BIG_CONSTANT(T, 64, 0.802912186540269232424),
 905          BOOST_MATH_BIG_CONSTANT(T, 64, 0.194793170017818925388),
 906          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0280128013584653182994),
 907          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00182034930799902922549)
 908       };
 909       T t = z / 10 - 3;
 910       result = Y + tools::evaluate_polynomial(P, t)
 911          / tools::evaluate_polynomial(Q, t);
 912       BOOST_MATH_INSTRUMENT_VARIABLE(result)
 913       result *= exp(z) / z;
 914       BOOST_MATH_INSTRUMENT_VARIABLE(result)
 915       result += z;
 916       BOOST_MATH_INSTRUMENT_VARIABLE(result)
 917    }
 918    else
 919    {
 920       // Maximum Deviation Found:                     3.536e-20
 921       // Max Error found at long double precision =   Poly: 1.310671e-19   Cheb: 8.630943e-11
 922
 923       static const T exp40 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.35385266837019985407899910749034804508871617254555467236651e17));
 924       static const T Y= 1.013065338134765625F;
 925       static const T P[9] = {
 926          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0130653381347656250004),
 927          BOOST_MATH_BIG_CONSTANT(T, 64, 0.644487780349757303739),
 928          BOOST_MATH_BIG_CONSTANT(T, 64, 143.995670348227433964),
 929          BOOST_MATH_BIG_CONSTANT(T, 64, -13918.9322758014173709),
 930          BOOST_MATH_BIG_CONSTANT(T, 64, 476260.975133624194484),
 931          BOOST_MATH_BIG_CONSTANT(T, 64, -7437102.15135982802122),
 932          BOOST_MATH_BIG_CONSTANT(T, 64, 53732298.8764767916542),
 933          BOOST_MATH_BIG_CONSTANT(T, 64, -160695051.957997452509),
 934          BOOST_MATH_BIG_CONSTANT(T, 64, 137839271.592778020028)
 935       };
 936       static const T Q[9] = {
 937          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
 938          BOOST_MATH_BIG_CONSTANT(T, 64, 27.2103343964943718802),
 939          BOOST_MATH_BIG_CONSTANT(T, 64, -8785.48528692879413676),
 940          BOOST_MATH_BIG_CONSTANT(T, 64, 397530.290000322626766),
 941          BOOST_MATH_BIG_CONSTANT(T, 64, -7356441.34957799368252),
 942          BOOST_MATH_BIG_CONSTANT(T, 64, 63050914.5343400957524),
 943          BOOST_MATH_BIG_CONSTANT(T, 64, -246143779.638307701369),
 944          BOOST_MATH_BIG_CONSTANT(T, 64, 384647824.678554961174),
 945          BOOST_MATH_BIG_CONSTANT(T, 64, -166288297.874583961493)
 946       };
 947       T t = 1 / z;
 948       result = Y + tools::evaluate_polynomial(P, t)
 949          / tools::evaluate_polynomial(Q, t);
 950       if(z < 41)
 951          result *= exp(z) / z;
 952       else
 953       {
 954          // Avoid premature overflow if we can:
 955          t = z - 40;
 956          if(t > tools::log_max_value<T>())
 957          {
 958             result = policies::raise_overflow_error<T>(function, 0, pol);
 959          }
 960          else
 961          {
 962             result *= exp(z - 40) / z;
 963             if(result > tools::max_value<T>() / exp40)
 964             {
 965                result = policies::raise_overflow_error<T>(function, 0, pol);
 966             }
 967             else
 968             {
 969                result *= exp40;
 970             }
 971          }
 972       }
 973       result += z;
 974    }
 975    return result;
 976 }
 977
 978 template <class T, class Policy>
 979 void expint_i_imp_113a(T& result, const T& z, const Policy& pol)
 980 {
 981    BOOST_MATH_STD_USING
 982    // Maximum Deviation Found:                     1.230e-36
 983    // Expected Error Term:                         -1.230e-36
 984    // Max Error found at long double precision =   Poly: 4.355299e-34   Cheb: 7.512581e-34
 985
 986
 987    static const T P[15] = {
 988       BOOST_MATH_BIG_CONSTANT(T, 113, 2.98677224343598593765287235997328555),
 989       BOOST_MATH_BIG_CONSTANT(T, 113, -0.333256034674702967028780537349334037),
 990       BOOST_MATH_BIG_CONSTANT(T, 113, 0.851831522798101228384971644036708463),
 991       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0657854833494646206186773614110374948),
 992       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0630065662557284456000060708977935073),
 993       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00311759191425309373327784154659649232),
 994       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00176213568201493949664478471656026771),
 995       BOOST_MATH_BIG_CONSTANT(T, 113, -0.491548660404172089488535218163952295e-4),
 996       BOOST_MATH_BIG_CONSTANT(T, 113, 0.207764227621061706075562107748176592e-4),
 997       BOOST_MATH_BIG_CONSTANT(T, 113, -0.225445398156913584846374273379402765e-6),
 998       BOOST_MATH_BIG_CONSTANT(T, 113, 0.996939977231410319761273881672601592e-7),
 999       BOOST_MATH_BIG_CONSTANT(T, 113, 0.212546902052178643330520878928100847e-9),
1000       BOOST_MATH_BIG_CONSTANT(T, 113, 0.154646053060262871360159325115980023e-9),
1001       BOOST_MATH_BIG_CONSTANT(T, 113, 0.143971277122049197323415503594302307e-11),
1002       BOOST_MATH_BIG_CONSTANT(T, 113, 0.306243138978114692252817805327426657e-13)
1003    };
1004    static const T Q[15] = {
1005       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1006       BOOST_MATH_BIG_CONSTANT(T, 113, -1.40178870313943798705491944989231793),
1007       BOOST_MATH_BIG_CONSTANT(T, 113, 0.943810968269701047641218856758605284),
1008       BOOST_MATH_BIG_CONSTANT(T, 113, -0.405026631534345064600850391026113165),
1009       BOOST_MATH_BIG_CONSTANT(T, 113, 0.123924153524614086482627660399122762),
1010       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0286364505373369439591132549624317707),
1011       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00516148845910606985396596845494015963),
1012       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000738330799456364820380739850924783649),
1013       BOOST_MATH_BIG_CONSTANT(T, 113, 0.843737760991856114061953265870882637e-4),
1014       BOOST_MATH_BIG_CONSTANT(T, 113, -0.767957673431982543213661388914587589e-5),
1015       BOOST_MATH_BIG_CONSTANT(T, 113, 0.549136847313854595809952100614840031e-6),
1016       BOOST_MATH_BIG_CONSTANT(T, 113, -0.299801381513743676764008325949325404e-7),
1017       BOOST_MATH_BIG_CONSTANT(T, 113, 0.118419479055346106118129130945423483e-8),
1018       BOOST_MATH_BIG_CONSTANT(T, 113, -0.30372295663095470359211949045344607e-10),
1019       BOOST_MATH_BIG_CONSTANT(T, 113, 0.382742953753485333207877784720070523e-12)
1020    };
1021
1022    static const T c1 = BOOST_MATH_BIG_CONSTANT(T, 113, 1677624236387711.0);
1023    static const T c2 = BOOST_MATH_BIG_CONSTANT(T, 113, 4503599627370496.0);
1024    static const T c3 = BOOST_MATH_BIG_CONSTANT(T, 113, 266514582277687.0);
1025    static const T c4 = BOOST_MATH_BIG_CONSTANT(T, 113, 4503599627370496.0);
1026    static const T c5 = BOOST_MATH_BIG_CONSTANT(T, 113, 4503599627370496.0);
1027    static const T r1 = c1 / c2;
1028    static const T r2 = c3 / c4 / c5;
1029    static const T r3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.283806480836357377069325311780969887585024578164571984232357e-31));
1030    static const T r = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392));
1031    T t = (z / 3) - 1;
1032    result = tools::evaluate_polynomial(P, t)
1033       / tools::evaluate_polynomial(Q, t);
1034    t = ((z - r1) - r2) - r3;
1035    result *= t;
1036    if(fabs(t) < 0.1)
1037    {
1038       result += boost::math::log1p(t / r, pol);
1039    }
1040    else
1041    {
1042       result += log(z / r);
1043    }
1044 }
1045
1046 template <class T>
1047 void expint_i_113b(T& result, const T& z)
1048 {
1049    BOOST_MATH_STD_USING
1050    // Maximum Deviation Found:                     7.779e-36
1051    // Expected Error Term:                         -7.779e-36
1052    // Max Error found at long double precision =   Poly: 2.576723e-35   Cheb: 1.236001e-34
1053
1054    static const T Y = 1.158985137939453125F;
1055    static const T P[15] = {
1056       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139324086199409049282472239613554817),
1057       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0338173111691991289178779840307998955),
1058       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0555972290794371306259684845277620556),
1059       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0378677976003456171563136909186202177),
1060       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0152221583517528358782902783914356667),
1061       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00428283334203873035104248217403126905),
1062       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000922782631491644846511553601323435286),
1063       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000155513428088853161562660696055496696),
1064       BOOST_MATH_BIG_CONSTANT(T, 113, -0.205756580255359882813545261519317096e-4),
1065       BOOST_MATH_BIG_CONSTANT(T, 113, -0.220327406578552089820753181821115181e-5),
1066       BOOST_MATH_BIG_CONSTANT(T, 113, -0.189483157545587592043421445645377439e-6),
1067       BOOST_MATH_BIG_CONSTANT(T, 113, -0.122426571518570587750898968123803867e-7),
1068       BOOST_MATH_BIG_CONSTANT(T, 113, -0.635187358949437991465353268374523944e-9),
1069       BOOST_MATH_BIG_CONSTANT(T, 113, -0.203015132965870311935118337194860863e-10),
1070       BOOST_MATH_BIG_CONSTANT(T, 113, -0.384276705503357655108096065452950822e-12)
1071    };
1072    static const T Q[15] = {
1073       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1074       BOOST_MATH_BIG_CONSTANT(T, 113, 1.58784732785354597996617046880946257),
1075       BOOST_MATH_BIG_CONSTANT(T, 113, 1.18550755302279446339364262338114098),
1076       BOOST_MATH_BIG_CONSTANT(T, 113, 0.55598993549661368604527040349702836),
1077       BOOST_MATH_BIG_CONSTANT(T, 113, 0.184290888380564236919107835030984453),
1078       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0459658051803613282360464632326866113),
1079       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0089505064268613225167835599456014705),
1080       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139042673882987693424772855926289077),
1081       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000174210708041584097450805790176479012),
1082       BOOST_MATH_BIG_CONSTANT(T, 113, 0.176324034009707558089086875136647376e-4),
1083       BOOST_MATH_BIG_CONSTANT(T, 113, 0.142935845999505649273084545313710581e-5),
1084       BOOST_MATH_BIG_CONSTANT(T, 113, 0.907502324487057260675816233312747784e-7),
1085       BOOST_MATH_BIG_CONSTANT(T, 113, 0.431044337808893270797934621235918418e-8),
1086       BOOST_MATH_BIG_CONSTANT(T, 113, 0.139007266881450521776529705677086902e-9),
1087       BOOST_MATH_BIG_CONSTANT(T, 113, 0.234715286125516430792452741830364672e-11)
1088    };
1089    T t = z / 2 - 4;
1090    result = Y + tools::evaluate_polynomial(P, t)
1091       / tools::evaluate_polynomial(Q, t);
1092    result *= exp(z) / z;
1093    result += z;
1094 }
1095
1096 template <class T>
1097 void expint_i_113c(T& result, const T& z)
1098 {
1099    BOOST_MATH_STD_USING
1100    // Maximum Deviation Found:                     1.082e-34
1101    // Expected Error Term:                         1.080e-34
1102    // Max Error found at long double precision =   Poly: 1.958294e-34   Cheb: 2.472261e-34
1103
1104
1105    static const T Y = 1.091579437255859375F;
1106    static const T P[17] = {
1107       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00685089599550151282724924894258520532),
1108       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0443313550253580053324487059748497467),
1109       BOOST_MATH_BIG_CONSTANT(T, 113, -0.071538561252424027443296958795814874),
1110       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0622923153354102682285444067843300583),
1111       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0361631270264607478205393775461208794),
1112       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0153192826839624850298106509601033261),
1113       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00496967904961260031539602977748408242),
1114       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126989079663425780800919171538920589),
1115       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000258933143097125199914724875206326698),
1116       BOOST_MATH_BIG_CONSTANT(T, 113, -0.422110326689204794443002330541441956e-4),
1117       BOOST_MATH_BIG_CONSTANT(T, 113, -0.546004547590412661451073996127115221e-5),
1118       BOOST_MATH_BIG_CONSTANT(T, 113, -0.546775260262202177131068692199272241e-6),
1119       BOOST_MATH_BIG_CONSTANT(T, 113, -0.404157632825805803833379568956559215e-7),
1120       BOOST_MATH_BIG_CONSTANT(T, 113, -0.200612596196561323832327013027419284e-8),
1121       BOOST_MATH_BIG_CONSTANT(T, 113, -0.502538501472133913417609379765434153e-10),
1122       BOOST_MATH_BIG_CONSTANT(T, 113, -0.326283053716799774936661568391296584e-13),
1123       BOOST_MATH_BIG_CONSTANT(T, 113, 0.869226483473172853557775877908693647e-15)
1124    };
1125    static const T Q[15] = {
1126       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1127       BOOST_MATH_BIG_CONSTANT(T, 113, 2.23227220874479061894038229141871087),
1128       BOOST_MATH_BIG_CONSTANT(T, 113, 2.40221000361027971895657505660959863),
1129       BOOST_MATH_BIG_CONSTANT(T, 113, 1.65476320985936174728238416007084214),
1130       BOOST_MATH_BIG_CONSTANT(T, 113, 0.816828602963895720369875535001248227),
1131       BOOST_MATH_BIG_CONSTANT(T, 113, 0.306337922909446903672123418670921066),
1132       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0902400121654409267774593230720600752),
1133       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0212708882169429206498765100993228086),
1134       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00404442626252467471957713495828165491),
1135       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0006195601618842253612635241404054589),
1136       BOOST_MATH_BIG_CONSTANT(T, 113, 0.755930932686543009521454653994321843e-4),
1137       BOOST_MATH_BIG_CONSTANT(T, 113, 0.716004532773778954193609582677482803e-5),
1138       BOOST_MATH_BIG_CONSTANT(T, 113, 0.500881663076471627699290821742924233e-6),
1139       BOOST_MATH_BIG_CONSTANT(T, 113, 0.233593219218823384508105943657387644e-7),
1140       BOOST_MATH_BIG_CONSTANT(T, 113, 0.554900353169148897444104962034267682e-9)
1141    };
1142    T t = z / 4 - 3.5;
1143    result = Y + tools::evaluate_polynomial(P, t)
1144       / tools::evaluate_polynomial(Q, t);
1145    result *= exp(z) / z;
1146    result += z;
1147 }
1148
1149 template <class T>
1150 void expint_i_113d(T& result, const T& z)
1151 {
1152    BOOST_MATH_STD_USING
1153    // Maximum Deviation Found:                     3.163e-35
1154    // Expected Error Term:                         3.163e-35
1155    // Max Error found at long double precision =   Poly: 4.158110e-35   Cheb: 5.385532e-35
1156
1157    static const T Y = 1.051731109619140625F;
1158    static const T P[14] = {
1159       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00144552494420652573815404828020593565),
1160       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0126747451594545338365684731262912741),
1161       BOOST_MATH_BIG_CONSTANT(T, 113, -0.01757394877502366717526779263438073),
1162       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0126838952395506921945756139424722588),
1163       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0060045057928894974954756789352443522),
1164       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00205349237147226126653803455793107903),
1165       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000532606040579654887676082220195624207),
1166       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000107344687098019891474772069139014662),
1167       BOOST_MATH_BIG_CONSTANT(T, 113, -0.169536802705805811859089949943435152e-4),
1168       BOOST_MATH_BIG_CONSTANT(T, 113, -0.20863311729206543881826553010120078e-5),
1169       BOOST_MATH_BIG_CONSTANT(T, 113, -0.195670358542116256713560296776654385e-6),
1170       BOOST_MATH_BIG_CONSTANT(T, 113, -0.133291168587253145439184028259772437e-7),
1171       BOOST_MATH_BIG_CONSTANT(T, 113, -0.595500337089495614285777067722823397e-9),
1172       BOOST_MATH_BIG_CONSTANT(T, 113, -0.133141358866324100955927979606981328e-10)
1173    };
1174    static const T Q[14] = {
1175       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1176       BOOST_MATH_BIG_CONSTANT(T, 113, 1.72490783907582654629537013560044682),
1177       BOOST_MATH_BIG_CONSTANT(T, 113, 1.44524329516800613088375685659759765),
1178       BOOST_MATH_BIG_CONSTANT(T, 113, 0.778241785539308257585068744978050181),
1179       BOOST_MATH_BIG_CONSTANT(T, 113, 0.300520486589206605184097270225725584),
1180       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0879346899691339661394537806057953957),
1181       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0200802415843802892793583043470125006),
1182       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00362842049172586254520256100538273214),
1183       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000519731362862955132062751246769469957),
1184       BOOST_MATH_BIG_CONSTANT(T, 113, 0.584092147914050999895178697392282665e-4),
1185       BOOST_MATH_BIG_CONSTANT(T, 113, 0.501851497707855358002773398333542337e-5),
1186       BOOST_MATH_BIG_CONSTANT(T, 113, 0.313085677467921096644895738538865537e-6),
1187       BOOST_MATH_BIG_CONSTANT(T, 113, 0.127552010539733113371132321521204458e-7),
1188       BOOST_MATH_BIG_CONSTANT(T, 113, 0.25737310826983451144405899970774587e-9)
1189    };
1190    T t = z / 4 - 5.5;
1191    result = Y + tools::evaluate_polynomial(P, t)
1192       / tools::evaluate_polynomial(Q, t);
1193    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1194    result *= exp(z) / z;
1195    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1196    result += z;
1197    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1198 }
1199
1200 template <class T>
1201 void expint_i_113e(T& result, const T& z)
1202 {
1203    BOOST_MATH_STD_USING
1204    // Maximum Deviation Found:                     7.972e-36
1205    // Expected Error Term:                         7.962e-36
1206    // Max Error found at long double precision =   Poly: 1.711721e-34   Cheb: 3.100018e-34
1207
1208    static const T Y = 1.032726287841796875F;
1209    static const T P[15] = {
1210       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00141056919297307534690895009969373233),
1211       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0123384175302540291339020257071411437),
1212       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0298127270706864057791526083667396115),
1213       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0390686759471630584626293670260768098),
1214       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0338226792912607409822059922949035589),
1215       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0211659736179834946452561197559654582),
1216       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0100428887460879377373158821400070313),
1217       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00370717396015165148484022792801682932),
1218       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0010768667551001624764329000496561659),
1219       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000246127328761027039347584096573123531),
1220       BOOST_MATH_BIG_CONSTANT(T, 113, -0.437318110527818613580613051861991198e-4),
1221       BOOST_MATH_BIG_CONSTANT(T, 113, -0.587532682329299591501065482317771497e-5),
1222       BOOST_MATH_BIG_CONSTANT(T, 113, -0.565697065670893984610852937110819467e-6),
1223       BOOST_MATH_BIG_CONSTANT(T, 113, -0.350233957364028523971768887437839573e-7),
1224       BOOST_MATH_BIG_CONSTANT(T, 113, -0.105428907085424234504608142258423505e-8)
1225    };
1226    static const T Q[16] = {
1227       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1228       BOOST_MATH_BIG_CONSTANT(T, 113, 3.17261315255467581204685605414005525),
1229       BOOST_MATH_BIG_CONSTANT(T, 113, 4.85267952971640525245338392887217426),
1230       BOOST_MATH_BIG_CONSTANT(T, 113, 4.74341914912439861451492872946725151),
1231       BOOST_MATH_BIG_CONSTANT(T, 113, 3.31108463283559911602405970817931801),
1232       BOOST_MATH_BIG_CONSTANT(T, 113, 1.74657006336994649386607925179848899),
1233       BOOST_MATH_BIG_CONSTANT(T, 113, 0.718255607416072737965933040353653244),
1234       BOOST_MATH_BIG_CONSTANT(T, 113, 0.234037553177354542791975767960643864),
1235       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0607470145906491602476833515412605389),
1236       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0125048143774226921434854172947548724),
1237       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00201034366420433762935768458656609163),
1238       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000244823338417452367656368849303165721),
1239       BOOST_MATH_BIG_CONSTANT(T, 113, 0.213511655166983177960471085462540807e-4),
1240       BOOST_MATH_BIG_CONSTANT(T, 113, 0.119323998465870686327170541547982932e-5),
1241       BOOST_MATH_BIG_CONSTANT(T, 113, 0.322153582559488797803027773591727565e-7),
1242       BOOST_MATH_BIG_CONSTANT(T, 113, -0.161635525318683508633792845159942312e-16)
1243    };
1244    T t = z / 8 - 4.25;
1245    result = Y + tools::evaluate_polynomial(P, t)
1246       / tools::evaluate_polynomial(Q, t);
1247    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1248    result *= exp(z) / z;
1249    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1250    result += z;
1251    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1252 }
1253
1254 template <class T>
1255 void expint_i_113f(T& result, const T& z)
1256 {
1257    BOOST_MATH_STD_USING
1258    // Maximum Deviation Found:                     4.469e-36
1259    // Expected Error Term:                         4.468e-36
1260    // Max Error found at long double precision =   Poly: 1.288958e-35   Cheb: 2.304586e-35
1261
1262    static const T Y = 1.0216197967529296875F;
1263    static const T P[12] = {
1264       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000322999116096627043476023926572650045),
1265       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00385606067447365187909164609294113346),
1266       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00686514524727568176735949971985244415),
1267       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00606260649593050194602676772589601799),
1268       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00334382362017147544335054575436194357),
1269       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126108534260253075708625583630318043),
1270       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000337881489347846058951220431209276776),
1271       BOOST_MATH_BIG_CONSTANT(T, 113, -0.648480902304640018785370650254018022e-4),
1272       BOOST_MATH_BIG_CONSTANT(T, 113, -0.87652644082970492211455290209092766e-5),
1273       BOOST_MATH_BIG_CONSTANT(T, 113, -0.794712243338068631557849449519994144e-6),
1274       BOOST_MATH_BIG_CONSTANT(T, 113, -0.434084023639508143975983454830954835e-7),
1275       BOOST_MATH_BIG_CONSTANT(T, 113, -0.107839681938752337160494412638656696e-8)
1276    };
1277    static const T Q[12] = {
1278       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1279       BOOST_MATH_BIG_CONSTANT(T, 113, 2.09913805456661084097134805151524958),
1280       BOOST_MATH_BIG_CONSTANT(T, 113, 2.07041755535439919593503171320431849),
1281       BOOST_MATH_BIG_CONSTANT(T, 113, 1.26406517226052371320416108604874734),
1282       BOOST_MATH_BIG_CONSTANT(T, 113, 0.529689923703770353961553223973435569),
1283       BOOST_MATH_BIG_CONSTANT(T, 113, 0.159578150879536711042269658656115746),
1284       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0351720877642000691155202082629857131),
1285       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00565313621289648752407123620997063122),
1286       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000646920278540515480093843570291218295),
1287       BOOST_MATH_BIG_CONSTANT(T, 113, 0.499904084850091676776993523323213591e-4),
1288       BOOST_MATH_BIG_CONSTANT(T, 113, 0.233740058688179614344680531486267142e-5),
1289       BOOST_MATH_BIG_CONSTANT(T, 113, 0.498800627828842754845418576305379469e-7)
1290    };
1291    T t = z / 7 - 7;
1292    result = Y + tools::evaluate_polynomial(P, t)
1293       / tools::evaluate_polynomial(Q, t);
1294    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1295    result *= exp(z) / z;
1296    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1297    result += z;
1298    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1299 }
1300
1301 template <class T>
1302 void expint_i_113g(T& result, const T& z)
1303 {
1304    BOOST_MATH_STD_USING
1305    // Maximum Deviation Found:                     5.588e-35
1306    // Expected Error Term:                         -5.566e-35
1307    // Max Error found at long double precision =   Poly: 9.976345e-35   Cheb: 8.358865e-35
1308
1309    static const T Y = 1.015148162841796875F;
1310    static const T P[11] = {
1311       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000435714784725086961464589957142615216),
1312       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00432114324353830636009453048419094314),
1313       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0100740363285526177522819204820582424),
1314       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0116744115827059174392383504427640362),
1315       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00816145387784261141360062395898644652),
1316       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00371380272673500791322744465394211508),
1317       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00112958263488611536502153195005736563),
1318       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000228316462389404645183269923754256664),
1319       BOOST_MATH_BIG_CONSTANT(T, 113, -0.29462181955852860250359064291292577e-4),
1320       BOOST_MATH_BIG_CONSTANT(T, 113, -0.21972450610957417963227028788460299e-5),
1321       BOOST_MATH_BIG_CONSTANT(T, 113, -0.720558173805289167524715527536874694e-7)
1322    };
1323    static const T Q[11] = {
1324       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1325       BOOST_MATH_BIG_CONSTANT(T, 113, 2.95918362458402597039366979529287095),
1326       BOOST_MATH_BIG_CONSTANT(T, 113, 3.96472247520659077944638411856748924),
1327       BOOST_MATH_BIG_CONSTANT(T, 113, 3.15563251550528513747923714884142131),
1328       BOOST_MATH_BIG_CONSTANT(T, 113, 1.64674612007093983894215359287448334),
1329       BOOST_MATH_BIG_CONSTANT(T, 113, 0.58695020129846594405856226787156424),
1330       BOOST_MATH_BIG_CONSTANT(T, 113, 0.144358385319329396231755457772362793),
1331       BOOST_MATH_BIG_CONSTANT(T, 113, 0.024146911506411684815134916238348063),
1332       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0026257132337460784266874572001650153),
1333       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000167479843750859222348869769094711093),
1334       BOOST_MATH_BIG_CONSTANT(T, 113, 0.475673638665358075556452220192497036e-5)
1335    };
1336    T t = z / 14 - 5;
1337    result = Y + tools::evaluate_polynomial(P, t)
1338       / tools::evaluate_polynomial(Q, t);
1339    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1340    result *= exp(z) / z;
1341    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1342    result += z;
1343    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1344 }
1345
1346 template <class T>
1347 void expint_i_113h(T& result, const T& z)
1348 {
1349    BOOST_MATH_STD_USING
1350    // Maximum Deviation Found:                     4.448e-36
1351    // Expected Error Term:                         4.445e-36
1352    // Max Error found at long double precision =   Poly: 2.058532e-35   Cheb: 2.165465e-27
1353
1354    static const T Y= 1.00849151611328125F;
1355    static const T P[9] = {
1356       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0084915161132812500000001440233607358),
1357       BOOST_MATH_BIG_CONSTANT(T, 113, 1.84479378737716028341394223076147872),
1358       BOOST_MATH_BIG_CONSTANT(T, 113, -130.431146923726715674081563022115568),
1359       BOOST_MATH_BIG_CONSTANT(T, 113, 4336.26945491571504885214176203512015),
1360       BOOST_MATH_BIG_CONSTANT(T, 113, -76279.0031974974730095170437591004177),
1361       BOOST_MATH_BIG_CONSTANT(T, 113, 729577.956271997673695191455111727774),
1362       BOOST_MATH_BIG_CONSTANT(T, 113, -3661928.69330208734947103004900349266),
1363       BOOST_MATH_BIG_CONSTANT(T, 113, 8570600.041606912735872059184527855),
1364       BOOST_MATH_BIG_CONSTANT(T, 113, -6758379.93672362080947905580906028645)
1365    };
1366    static const T Q[10] = {
1367       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1368       BOOST_MATH_BIG_CONSTANT(T, 113, -99.4868026047611434569541483506091713),
1369       BOOST_MATH_BIG_CONSTANT(T, 113, 3879.67753690517114249705089803055473),
1370       BOOST_MATH_BIG_CONSTANT(T, 113, -76495.82413252517165830203774900806),
1371       BOOST_MATH_BIG_CONSTANT(T, 113, 820773.726408311894342553758526282667),
1372       BOOST_MATH_BIG_CONSTANT(T, 113, -4803087.64956923577571031564909646579),
1373       BOOST_MATH_BIG_CONSTANT(T, 113, 14521246.227703545012713173740895477),
1374       BOOST_MATH_BIG_CONSTANT(T, 113, -19762752.0196769712258527849159393044),
1375       BOOST_MATH_BIG_CONSTANT(T, 113, 8354144.67882768405803322344185185517),
1376       BOOST_MATH_BIG_CONSTANT(T, 113, 355076.853106511136734454134915432571)
1377    };
1378    T t = 1 / z;
1379    result = Y + tools::evaluate_polynomial(P, t)
1380       / tools::evaluate_polynomial(Q, t);
1381    result *= exp(z) / z;
1382    result += z;
1383 }
1384
1385 template <class T, class Policy>
1386 T expint_i_imp(T z, const Policy& pol, const boost::integral_constant<int, 113>& tag)
1387 {
1388    BOOST_MATH_STD_USING
1389    static const char* function = "boost::math::expint<%1%>(%1%)";
1390    if(z < 0)
1391       return -expint_imp(1, T(-z), pol, tag);
1392    if(z == 0)
1393       return -policies::raise_overflow_error<T>(function, 0, pol);
1394
1395    T result;
1396
1397    if(z <= 6)
1398    {
1399       expint_i_imp_113a(result, z, pol);
1400    }
1401    else if (z <= 10)
1402    {
1403       expint_i_113b(result, z);
1404    }
1405    else if(z <= 18)
1406    {
1407       expint_i_113c(result, z);
1408    }
1409    else if(z <= 26)
1410    {
1411       expint_i_113d(result, z);
1412    }
1413    else if(z <= 42)
1414    {
1415       expint_i_113e(result, z);
1416    }
1417    else if(z <= 56)
1418    {
1419       expint_i_113f(result, z);
1420    }
1421    else if(z <= 84)
1422    {
1423       expint_i_113g(result, z);
1424    }
1425    else if(z <= 210)
1426    {
1427       expint_i_113h(result, z);
1428    }
1429    else // z > 210
1430    {
1431       // Maximum Deviation Found:                     3.963e-37
1432       // Expected Error Term:                         3.963e-37
1433       // Max Error found at long double precision =   Poly: 1.248049e-36   Cheb: 2.843486e-29
1434
1435       static const T exp40 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2.35385266837019985407899910749034804508871617254555467236651e17));
1436       static const T Y= 1.00252532958984375F;
1437       static const T P[8] = {
1438          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00252532958984375000000000000000000085),
1439          BOOST_MATH_BIG_CONSTANT(T, 113, 1.16591386866059087390621952073890359),
1440          BOOST_MATH_BIG_CONSTANT(T, 113, -67.8483431314018462417456828499277579),
1441          BOOST_MATH_BIG_CONSTANT(T, 113, 1567.68688154683822956359536287575892),
1442          BOOST_MATH_BIG_CONSTANT(T, 113, -17335.4683325819116482498725687644986),
1443          BOOST_MATH_BIG_CONSTANT(T, 113, 93632.6567462673524739954389166550069),
1444          BOOST_MATH_BIG_CONSTANT(T, 113, -225025.189335919133214440347510936787),
1445          BOOST_MATH_BIG_CONSTANT(T, 113, 175864.614717440010942804684741336853)
1446       };
1447       static const T Q[9] = {
1448          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1449          BOOST_MATH_BIG_CONSTANT(T, 113, -65.6998869881600212224652719706425129),
1450          BOOST_MATH_BIG_CONSTANT(T, 113, 1642.73850032324014781607859416890077),
1451          BOOST_MATH_BIG_CONSTANT(T, 113, -19937.2610222467322481947237312818575),
1452          BOOST_MATH_BIG_CONSTANT(T, 113, 124136.267326632742667972126625064538),
1453          BOOST_MATH_BIG_CONSTANT(T, 113, -384614.251466704550678760562965502293),
1454          BOOST_MATH_BIG_CONSTANT(T, 113, 523355.035910385688578278384032026998),
1455          BOOST_MATH_BIG_CONSTANT(T, 113, -217809.552260834025885677791936351294),
1456          BOOST_MATH_BIG_CONSTANT(T, 113, -8555.81719551123640677261226549550872)
1457       };
1458       T t = 1 / z;
1459       result = Y + tools::evaluate_polynomial(P, t)
1460          / tools::evaluate_polynomial(Q, t);
1461       if(z < 41)
1462          result *= exp(z) / z;
1463       else
1464       {
1465          // Avoid premature overflow if we can:
1466          t = z - 40;
1467          if(t > tools::log_max_value<T>())
1468          {
1469             result = policies::raise_overflow_error<T>(function, 0, pol);
1470          }
1471          else
1472          {
1473             result *= exp(z - 40) / z;
1474             if(result > tools::max_value<T>() / exp40)
1475             {
1476                result = policies::raise_overflow_error<T>(function, 0, pol);
1477             }
1478             else
1479             {
1480                result *= exp40;
1481             }
1482          }
1483       }
1484       result += z;
1485    }
1486    return result;
1487 }
1488
1489 template <class T, class Policy, class tag>
1490 struct expint_i_initializer
1491 {
1492    struct init
1493    {
1494       init()
1495       {
1496          do_init(tag());
1497       }
1498       static void do_init(const boost::integral_constant<int, 0>&){}
1499       static void do_init(const boost::integral_constant<int, 53>&)
1500       {
1501          boost::math::expint(T(5));
1502          boost::math::expint(T(7));
1503          boost::math::expint(T(18));
1504          boost::math::expint(T(38));
1505          boost::math::expint(T(45));
1506       }
1507       static void do_init(const boost::integral_constant<int, 64>&)
1508       {
1509          boost::math::expint(T(5));
1510          boost::math::expint(T(7));
1511          boost::math::expint(T(18));
1512          boost::math::expint(T(38));
1513          boost::math::expint(T(45));
1514       }
1515       static void do_init(const boost::integral_constant<int, 113>&)
1516       {
1517          boost::math::expint(T(5));
1518          boost::math::expint(T(7));
1519          boost::math::expint(T(17));
1520          boost::math::expint(T(25));
1521          boost::math::expint(T(40));
1522          boost::math::expint(T(50));
1523          boost::math::expint(T(80));
1524          boost::math::expint(T(200));
1525          boost::math::expint(T(220));
1526       }
1527       void force_instantiate()const{}
1528    };
1529    static const init initializer;
1530    static void force_instantiate()
1531    {
1532       initializer.force_instantiate();
1533    }
1534 };
1535
1536 template <class T, class Policy, class tag>
1537 const typename expint_i_initializer<T, Policy, tag>::init expint_i_initializer<T, Policy, tag>::initializer;
1538
1539 template <class T, class Policy, class tag>
1540 struct expint_1_initializer
1541 {
1542    struct init
1543    {
1544       init()
1545       {
1546          do_init(tag());
1547       }
1548       static void do_init(const boost::integral_constant<int, 0>&){}
1549       static void do_init(const boost::integral_constant<int, 53>&)
1550       {
1551          boost::math::expint(1, T(0.5));
1552          boost::math::expint(1, T(2));
1553       }
1554       static void do_init(const boost::integral_constant<int, 64>&)
1555       {
1556          boost::math::expint(1, T(0.5));
1557          boost::math::expint(1, T(2));
1558       }
1559       static void do_init(const boost::integral_constant<int, 113>&)
1560       {
1561          boost::math::expint(1, T(0.5));
1562          boost::math::expint(1, T(2));
1563          boost::math::expint(1, T(6));
1564       }
1565       void force_instantiate()const{}
1566    };
1567    static const init initializer;
1568    static void force_instantiate()
1569    {
1570       initializer.force_instantiate();
1571    }
1572 };
1573
1574 template <class T, class Policy, class tag>
1575 const typename expint_1_initializer<T, Policy, tag>::init expint_1_initializer<T, Policy, tag>::initializer;
1576
1577 template <class T, class Policy>
1578 inline typename tools::promote_args<T>::type
1579    expint_forwarder(T z, const Policy& /*pol*/, boost::true_type const&)
1580 {
1581    typedef typename tools::promote_args<T>::type result_type;
1582    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1583    typedef typename policies::precision<result_type, Policy>::type precision_type;
1584    typedef typename policies::normalise<
1585       Policy,
1586       policies::promote_float<false>,
1587       policies::promote_double<false>,
1588       policies::discrete_quantile<>,
1589       policies::assert_undefined<> >::type forwarding_policy;
1590    typedef boost::integral_constant<int,
1591       precision_type::value <= 0 ? 0 :
1592       precision_type::value <= 53 ? 53 :
1593       precision_type::value <= 64 ? 64 :
1594       precision_type::value <= 113 ? 113 : 0
1595    > tag_type;
1596
1597    expint_i_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1598
1599    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::expint_i_imp(
1600       static_cast<value_type>(z),
1601       forwarding_policy(),
1602       tag_type()), "boost::math::expint<%1%>(%1%)");
1603 }
1604
1605 template <class T>
1606 inline typename tools::promote_args<T>::type
1607 expint_forwarder(unsigned n, T z, const boost::false_type&)
1608 {
1609    return boost::math::expint(n, z, policies::policy<>());
1610 }
1611
1612 } // namespace detail
1613
1614 template <class T, class Policy>
1615 inline typename tools::promote_args<T>::type
1616    expint(unsigned n, T z, const Policy& /*pol*/)
1617 {
1618    typedef typename tools::promote_args<T>::type result_type;
1619    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1620    typedef typename policies::precision<result_type, Policy>::type precision_type;
1621    typedef typename policies::normalise<
1622       Policy,
1623       policies::promote_float<false>,
1624       policies::promote_double<false>,
1625       policies::discrete_quantile<>,
1626       policies::assert_undefined<> >::type forwarding_policy;
1627    typedef boost::integral_constant<int,
1628       precision_type::value <= 0 ? 0 :
1629       precision_type::value <= 53 ? 53 :
1630       precision_type::value <= 64 ? 64 :
1631       precision_type::value <= 113 ? 113 : 0
1632    > tag_type;
1633
1634    detail::expint_1_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1635
1636    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::expint_imp(
1637       n,
1638       static_cast<value_type>(z),
1639       forwarding_policy(),
1640       tag_type()), "boost::math::expint<%1%>(unsigned, %1%)");
1641 }
1642
1643 template <class T, class U>
1644 inline typename detail::expint_result<T, U>::type
1645    expint(T const z, U const u)
1646 {
1647    typedef typename policies::is_policy<U>::type tag_type;
1648    return detail::expint_forwarder(z, u, tag_type());
1649 }
1650
1651 template <class T>
1652 inline typename tools::promote_args<T>::type
1653    expint(T z)
1654 {
1655    return expint(z, policies::policy<>());
1656 }
1657
1658 }} // namespaces
1659
1660 #ifdef _MSC_VER
1661 #pragma warning(pop)
1662 #endif
1663
1664 #endif // BOOST_MATH_EXPINT_HPP
1665
1666