special_functions/detail/lgamma_small.hpp

0001 //  (C) Copyright John Maddock 2006.
0002 //  (C) Copyright Matt Borland 2024.
0003 //  Use, modification and distribution are subject to the
0004 //  Boost Software License, Version 1.0. (See accompanying file
0005 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
0006
0007 #ifndef BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
0008 #define BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
0009
0010 #ifdef _MSC_VER
0011 #pragma once
0012 #endif
0013
0014 #include <boost/math/tools/config.hpp>
0015 #include <boost/math/tools/big_constant.hpp>
0016 #include <boost/math/tools/type_traits.hpp>
0017 #include <boost/math/tools/precision.hpp>
0018 #include <boost/math/special_functions/lanczos.hpp>
0019
0020 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
0021 //
0022 // This is the only way we can avoid
0023 // warning: non-standard suffix on floating constant [-Wpedantic]
0024 // when building with -Wall -pedantic.  Neither __extension__
0025 // nor #pragma diagnostic ignored work :(
0026 //
0027 #pragma GCC system_header
0028 #endif
0029
0030 namespace boost{ namespace math{ namespace detail{
0031
0032 //
0033 // These need forward declaring to keep GCC happy:
0034 //
0035 template <class T, class Policy, class Lanczos>
0036 BOOST_MATH_GPU_ENABLED T gamma_imp(T z, const Policy& pol, const Lanczos& l);
0037 template <class T, class Policy>
0038 BOOST_MATH_GPU_ENABLED T gamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos& l);
0039
0040 //
0041 // lgamma for small arguments:
0042 //
0043 template <class T, class Policy, class Lanczos>
0044 BOOST_MATH_GPU_ENABLED T lgamma_small_imp(T z, T zm1, T zm2, const boost::math::integral_constant<int, 64>&, const Policy& /* l */, const Lanczos&)
0045 {
0046    // This version uses rational approximations for small
0047    // values of z accurate enough for 64-bit mantissas
0048    // (80-bit long doubles), works well for 53-bit doubles as well.
0049    // Lanczos is only used to select the Lanczos function.
0050
0051    BOOST_MATH_STD_USING  // for ADL of std names
0052    T result = 0;
0053    if(z < tools::epsilon<T>())
0054    {
0055       result = -log(z);
0056    }
0057    else if((zm1 == 0) || (zm2 == 0))
0058    {
0059       // nothing to do, result is zero....
0060    }
0061    else if(z > 2)
0062    {
0063       //
0064       // Begin by performing argument reduction until
0065       // z is in [2,3):
0066       //
0067       if(z >= 3)
0068       {
0069          do
0070          {
0071             z -= 1;
0072             zm2 -= 1;
0073             result += log(z);
0074          }while(z >= 3);
0075          // Update zm2, we need it below:
0076          zm2 = z - 2;
0077       }
0078
0079       //
0080       // Use the following form:
0081       //
0082       // lgamma(z) = (z-2)(z+1)(Y + R(z-2))
0083       //
0084       // where R(z-2) is a rational approximation optimised for
0085       // low absolute error - as long as it's absolute error
0086       // is small compared to the constant Y - then any rounding
0087       // error in it's computation will get wiped out.
0088       //
0089       // R(z-2) has the following properties:
0090       //
0091       // At double: Max error found:                    4.231e-18
0092       // At long double: Max error found:               1.987e-21
0093       // Maximum Deviation Found (approximation error): 5.900e-24
0094       //
0095       BOOST_MATH_STATIC const T P[] = {
0096          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.180355685678449379109e-1)),
0097          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.25126649619989678683e-1)),
0098          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.494103151567532234274e-1)),
0099          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.172491608709613993966e-1)),
0100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.259453563205438108893e-3)),
0101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.541009869215204396339e-3)),
0102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.324588649825948492091e-4))
0103       };
0104       BOOST_MATH_STATIC const T Q[] = {
0105          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
0106          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.196202987197795200688e1)),
0107          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.148019669424231326694e1)),
0108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.541391432071720958364e0)),
0109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.988504251128010129477e-1)),
0110          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.82130967464889339326e-2)),
0111          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.224936291922115757597e-3)),
0112          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.223352763208617092964e-6))
0113       };
0114
0115       constexpr float Y = 0.158963680267333984375e0f;
0116
0117       T r = zm2 * (z + 1);
0118       T R = tools::evaluate_polynomial(P, zm2);
0119       R /= tools::evaluate_polynomial(Q, zm2);
0120
0121       result +=  r * Y + r * R;
0122    }
0123    else
0124    {
0125       //
0126       // If z is less than 1 use recurrence to shift to
0127       // z in the interval [1,2]:
0128       //
0129       if(z < 1)
0130       {
0131          result += -log(z);
0132          zm2 = zm1;
0133          zm1 = z;
0134          z += 1;
0135       }
0136       //
0137       // Two approximations, on for z in [1,1.5] and
0138       // one for z in [1.5,2]:
0139       //
0140       if(z <= T(1.5))
0141       {
0142          //
0143          // Use the following form:
0144          //
0145          // lgamma(z) = (z-1)(z-2)(Y + R(z-1))
0146          //
0147          // where R(z-1) is a rational approximation optimised for
0148          // low absolute error - as long as it's absolute error
0149          // is small compared to the constant Y - then any rounding
0150          // error in it's computation will get wiped out.
0151          //
0152          // R(z-1) has the following properties:
0153          //
0154          // At double precision: Max error found:                1.230011e-17
0155          // At 80-bit long double precision:   Max error found:  5.631355e-21
0156          // Maximum Deviation Found:                             3.139e-021
0157          // Expected Error Term:                                 3.139e-021
0158
0159          //
0160          constexpr float Y = 0.52815341949462890625f;
0161
0162          BOOST_MATH_STATIC const T P[] = {
0163             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.490622454069039543534e-1)),
0164             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.969117530159521214579e-1)),
0165             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.414983358359495381969e0)),
0166             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.406567124211938417342e0)),
0167             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.158413586390692192217e0)),
0168             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.240149820648571559892e-1)),
0169             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.100346687696279557415e-2))
0170          };
0171          BOOST_MATH_STATIC const T Q[] = {
0172             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
0173             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.302349829846463038743e1)),
0174             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.348739585360723852576e1)),
0175             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.191415588274426679201e1)),
0176             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.507137738614363510846e0)),
0177             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.577039722690451849648e-1)),
0178             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.195768102601107189171e-2))
0179          };
0180
0181          T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
0182          T prefix = zm1 * zm2;
0183
0184          result += prefix * Y + prefix * r;
0185       }
0186       else
0187       {
0188          //
0189          // Use the following form:
0190          //
0191          // lgamma(z) = (2-z)(1-z)(Y + R(2-z))
0192          //
0193          // where R(2-z) is a rational approximation optimised for
0194          // low absolute error - as long as it's absolute error
0195          // is small compared to the constant Y - then any rounding
0196          // error in it's computation will get wiped out.
0197          //
0198          // R(2-z) has the following properties:
0199          //
0200          // At double precision, max error found:              1.797565e-17
0201          // At 80-bit long double precision, max error found:  9.306419e-21
0202          // Maximum Deviation Found:                           2.151e-021
0203          // Expected Error Term:                               2.150e-021
0204          //
0205          constexpr float Y = 0.452017307281494140625f;
0206
0207          BOOST_MATH_STATIC const T P[] = {
0208             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.292329721830270012337e-1)),
0209             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.144216267757192309184e0)),
0210             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.142440390738631274135e0)),
0211             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.542809694055053558157e-1)),
0212             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.850535976868336437746e-2)),
0213             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.431171342679297331241e-3))
0214          };
0215          BOOST_MATH_STATIC const T Q[] = {
0216             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
0217             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.150169356054485044494e1)),
0218             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.846973248876495016101e0)),
0219             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.220095151814995745555e0)),
0220             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.25582797155975869989e-1)),
0221             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.100666795539143372762e-2)),
0222             static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.827193521891290553639e-6))
0223          };
0224          T r = zm2 * zm1;
0225          T R = tools::evaluate_polynomial(P, T(-zm2)) / tools::evaluate_polynomial(Q, T(-zm2));
0226
0227          result += r * Y + r * R;
0228       }
0229    }
0230    return result;
0231 }
0232
0233 #ifndef BOOST_MATH_HAS_GPU_SUPPORT
0234 template <class T, class Policy, class Lanczos>
0235 T lgamma_small_imp(T z, T zm1, T zm2, const boost::math::integral_constant<int, 113>&, const Policy& /* l */, const Lanczos&)
0236 {
0237    //
0238    // This version uses rational approximations for small
0239    // values of z accurate enough for 113-bit mantissas
0240    // (128-bit long doubles).
0241    //
0242    BOOST_MATH_STD_USING  // for ADL of std names
0243    T result = 0;
0244    if(z < tools::epsilon<T>())
0245    {
0246       result = -log(z);
0247       BOOST_MATH_INSTRUMENT_CODE(result);
0248    }
0249    else if((zm1 == 0) || (zm2 == 0))
0250    {
0251       // nothing to do, result is zero....
0252    }
0253    else if(z > 2)
0254    {
0255       //
0256       // Begin by performing argument reduction until
0257       // z is in [2,3):
0258       //
0259       if(z >= 3)
0260       {
0261          do
0262          {
0263             z -= 1;
0264             result += log(z);
0265          }while(z >= 3);
0266          zm2 = z - 2;
0267       }
0268       BOOST_MATH_INSTRUMENT_CODE(zm2);
0269       BOOST_MATH_INSTRUMENT_CODE(z);
0270       BOOST_MATH_INSTRUMENT_CODE(result);
0271
0272       //
0273       // Use the following form:
0274       //
0275       // lgamma(z) = (z-2)(z+1)(Y + R(z-2))
0276       //
0277       // where R(z-2) is a rational approximation optimised for
0278       // low absolute error - as long as it's absolute error
0279       // is small compared to the constant Y - then any rounding
0280       // error in it's computation will get wiped out.
0281       //
0282       // Maximum Deviation Found (approximation error)      3.73e-37
0283
0284       static const T P[] = {
0285          BOOST_MATH_BIG_CONSTANT(T, 113, -0.018035568567844937910504030027467476655),
0286          BOOST_MATH_BIG_CONSTANT(T, 113, 0.013841458273109517271750705401202404195),
0287          BOOST_MATH_BIG_CONSTANT(T, 113, 0.062031842739486600078866923383017722399),
0288          BOOST_MATH_BIG_CONSTANT(T, 113, 0.052518418329052161202007865149435256093),
0289          BOOST_MATH_BIG_CONSTANT(T, 113, 0.01881718142472784129191838493267755758),
0290          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0025104830367021839316463675028524702846),
0291          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00021043176101831873281848891452678568311),
0292          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00010249622350908722793327719494037981166),
0293          BOOST_MATH_BIG_CONSTANT(T, 113, -0.11381479670982006841716879074288176994e-4),
0294          BOOST_MATH_BIG_CONSTANT(T, 113, -0.49999811718089980992888533630523892389e-6),
0295          BOOST_MATH_BIG_CONSTANT(T, 113, -0.70529798686542184668416911331718963364e-8)
0296       };
0297       static const T Q[] = {
0298          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
0299          BOOST_MATH_BIG_CONSTANT(T, 113, 2.5877485070422317542808137697939233685),
0300          BOOST_MATH_BIG_CONSTANT(T, 113, 2.8797959228352591788629602533153837126),
0301          BOOST_MATH_BIG_CONSTANT(T, 113, 1.8030885955284082026405495275461180977),
0302          BOOST_MATH_BIG_CONSTANT(T, 113, 0.69774331297747390169238306148355428436),
0303          BOOST_MATH_BIG_CONSTANT(T, 113, 0.17261566063277623942044077039756583802),
0304          BOOST_MATH_BIG_CONSTANT(T, 113, 0.02729301254544230229429621192443000121),
0305          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0026776425891195270663133581960016620433),
0306          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00015244249160486584591370355730402168106),
0307          BOOST_MATH_BIG_CONSTANT(T, 113, 0.43997034032479866020546814475414346627e-5),
0308          BOOST_MATH_BIG_CONSTANT(T, 113, 0.46295080708455613044541885534408170934e-7),
0309          BOOST_MATH_BIG_CONSTANT(T, 113, -0.93326638207459533682980757982834180952e-11),
0310          BOOST_MATH_BIG_CONSTANT(T, 113, 0.42316456553164995177177407325292867513e-13)
0311       };
0312
0313       T R = tools::evaluate_polynomial(P, zm2);
0314       R /= tools::evaluate_polynomial(Q, zm2);
0315
0316       static const float Y = 0.158963680267333984375F;
0317
0318       T r = zm2 * (z + 1);
0319
0320       result +=  r * Y + r * R;
0321       BOOST_MATH_INSTRUMENT_CODE(result);
0322    }
0323    else
0324    {
0325       //
0326       // If z is less than 1 use recurrence to shift to
0327       // z in the interval [1,2]:
0328       //
0329       if(z < 1)
0330       {
0331          result += -log(z);
0332          zm2 = zm1;
0333          zm1 = z;
0334          z += 1;
0335       }
0336       BOOST_MATH_INSTRUMENT_CODE(result);
0337       BOOST_MATH_INSTRUMENT_CODE(z);
0338       BOOST_MATH_INSTRUMENT_CODE(zm2);
0339       //
0340       // Three approximations, on for z in [1,1.35], [1.35,1.625] and [1.625,1]
0341       //
0342       if(z <= 1.35)
0343       {
0344          //
0345          // Use the following form:
0346          //
0347          // lgamma(z) = (z-1)(z-2)(Y + R(z-1))
0348          //
0349          // where R(z-1) is a rational approximation optimised for
0350          // low absolute error - as long as it's absolute error
0351          // is small compared to the constant Y - then any rounding
0352          // error in it's computation will get wiped out.
0353          //
0354          // R(z-1) has the following properties:
0355          //
0356          // Maximum Deviation Found (approximation error)            1.659e-36
0357          // Expected Error Term (theoretical error)                  1.343e-36
0358          // Max error found at 128-bit long double precision         1.007e-35
0359          //
0360          static const float Y = 0.54076099395751953125f;
0361
0362          static const T P[] = {
0363             BOOST_MATH_BIG_CONSTANT(T, 113, 0.036454670944013329356512090082402429697),
0364             BOOST_MATH_BIG_CONSTANT(T, 113, -0.066235835556476033710068679907798799959),
0365             BOOST_MATH_BIG_CONSTANT(T, 113, -0.67492399795577182387312206593595565371),
0366             BOOST_MATH_BIG_CONSTANT(T, 113, -1.4345555263962411429855341651960000166),
0367             BOOST_MATH_BIG_CONSTANT(T, 113, -1.4894319559821365820516771951249649563),
0368             BOOST_MATH_BIG_CONSTANT(T, 113, -0.87210277668067964629483299712322411566),
0369             BOOST_MATH_BIG_CONSTANT(T, 113, -0.29602090537771744401524080430529369136),
0370             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0561832587517836908929331992218879676),
0371             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0053236785487328044334381502530383140443),
0372             BOOST_MATH_BIG_CONSTANT(T, 113, -0.00018629360291358130461736386077971890789),
0373             BOOST_MATH_BIG_CONSTANT(T, 113, -0.10164985672213178500790406939467614498e-6),
0374             BOOST_MATH_BIG_CONSTANT(T, 113, 0.13680157145361387405588201461036338274e-8)
0375          };
0376          static const T Q[] = {
0377             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
0378             BOOST_MATH_BIG_CONSTANT(T, 113, 4.9106336261005990534095838574132225599),
0379             BOOST_MATH_BIG_CONSTANT(T, 113, 10.258804800866438510889341082793078432),
0380             BOOST_MATH_BIG_CONSTANT(T, 113, 11.88588976846826108836629960537466889),
0381             BOOST_MATH_BIG_CONSTANT(T, 113, 8.3455000546999704314454891036700998428),
0382             BOOST_MATH_BIG_CONSTANT(T, 113, 3.6428823682421746343233362007194282703),
0383             BOOST_MATH_BIG_CONSTANT(T, 113, 0.97465989807254572142266753052776132252),
0384             BOOST_MATH_BIG_CONSTANT(T, 113, 0.15121052897097822172763084966793352524),
0385             BOOST_MATH_BIG_CONSTANT(T, 113, 0.012017363555383555123769849654484594893),
0386             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0003583032812720649835431669893011257277)
0387          };
0388
0389          T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
0390          T prefix = zm1 * zm2;
0391
0392          result += prefix * Y + prefix * r;
0393          BOOST_MATH_INSTRUMENT_CODE(result);
0394       }
0395       else if(z <= 1.625)
0396       {
0397          //
0398          // Use the following form:
0399          //
0400          // lgamma(z) = (2-z)(1-z)(Y + R(2-z))
0401          //
0402          // where R(2-z) is a rational approximation optimised for
0403          // low absolute error - as long as it's absolute error
0404          // is small compared to the constant Y - then any rounding
0405          // error in it's computation will get wiped out.
0406          //
0407          // R(2-z) has the following properties:
0408          //
0409          // Max error found at 128-bit long double precision  9.634e-36
0410          // Maximum Deviation Found (approximation error)     1.538e-37
0411          // Expected Error Term (theoretical error)           2.350e-38
0412          //
0413          static const float Y = 0.483787059783935546875f;
0414
0415          static const T P[] = {
0416             BOOST_MATH_BIG_CONSTANT(T, 113, -0.017977422421608624353488126610933005432),
0417             BOOST_MATH_BIG_CONSTANT(T, 113, 0.18484528905298309555089509029244135703),
0418             BOOST_MATH_BIG_CONSTANT(T, 113, -0.40401251514859546989565001431430884082),
0419             BOOST_MATH_BIG_CONSTANT(T, 113, 0.40277179799147356461954182877921388182),
0420             BOOST_MATH_BIG_CONSTANT(T, 113, -0.21993421441282936476709677700477598816),
0421             BOOST_MATH_BIG_CONSTANT(T, 113, 0.069595742223850248095697771331107571011),
0422             BOOST_MATH_BIG_CONSTANT(T, 113, -0.012681481427699686635516772923547347328),
0423             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0012489322866834830413292771335113136034),
0424             BOOST_MATH_BIG_CONSTANT(T, 113, -0.57058739515423112045108068834668269608e-4),
0425             BOOST_MATH_BIG_CONSTANT(T, 113, 0.8207548771933585614380644961342925976e-6)
0426          };
0427          static const T Q[] = {
0428             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
0429             BOOST_MATH_BIG_CONSTANT(T, 113, -2.9629552288944259229543137757200262073),
0430             BOOST_MATH_BIG_CONSTANT(T, 113, 3.7118380799042118987185957298964772755),
0431             BOOST_MATH_BIG_CONSTANT(T, 113, -2.5569815272165399297600586376727357187),
0432             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0546764918220835097855665680632153367),
0433             BOOST_MATH_BIG_CONSTANT(T, 113, -0.26574021300894401276478730940980810831),
0434             BOOST_MATH_BIG_CONSTANT(T, 113, 0.03996289731752081380552901986471233462),
0435             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0033398680924544836817826046380586480873),
0436             BOOST_MATH_BIG_CONSTANT(T, 113, 0.00013288854760548251757651556792598235735),
0437             BOOST_MATH_BIG_CONSTANT(T, 113, -0.17194794958274081373243161848194745111e-5)
0438          };
0439          T r = zm2 * zm1;
0440          T R = tools::evaluate_polynomial(P, T(0.625 - zm1)) / tools::evaluate_polynomial(Q, T(0.625 - zm1));
0441
0442          result += r * Y + r * R;
0443          BOOST_MATH_INSTRUMENT_CODE(result);
0444       }
0445       else
0446       {
0447          //
0448          // Same form as above.
0449          //
0450          // Max error found (at 128-bit long double precision) 1.831e-35
0451          // Maximum Deviation Found (approximation error)      8.588e-36
0452          // Expected Error Term (theoretical error)            1.458e-36
0453          //
0454          static const float Y = 0.443811893463134765625f;
0455
0456          static const T P[] = {
0457             BOOST_MATH_BIG_CONSTANT(T, 113, -0.021027558364667626231512090082402429494),
0458             BOOST_MATH_BIG_CONSTANT(T, 113, 0.15128811104498736604523586803722368377),
0459             BOOST_MATH_BIG_CONSTANT(T, 113, -0.26249631480066246699388544451126410278),
0460             BOOST_MATH_BIG_CONSTANT(T, 113, 0.21148748610533489823742352180628489742),
0461             BOOST_MATH_BIG_CONSTANT(T, 113, -0.093964130697489071999873506148104370633),
0462             BOOST_MATH_BIG_CONSTANT(T, 113, 0.024292059227009051652542804957550866827),
0463             BOOST_MATH_BIG_CONSTANT(T, 113, -0.0036284453226534839926304745756906117066),
0464             BOOST_MATH_BIG_CONSTANT(T, 113, 0.0002939230129315195346843036254392485984),
0465             BOOST_MATH_BIG_CONSTANT(T, 113, -0.11088589183158123733132268042570710338e-4),
0466             BOOST_MATH_BIG_CONSTANT(T, 113, 0.13240510580220763969511741896361984162e-6)
0467          };
0468          static const T Q[] = {
0469             BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
0470             BOOST_MATH_BIG_CONSTANT(T, 113, -2.4240003754444040525462170802796471996),
0471             BOOST_MATH_BIG_CONSTANT(T, 113, 2.4868383476933178722203278602342786002),
0472             BOOST_MATH_BIG_CONSTANT(T, 113, -1.4047068395206343375520721509193698547),
0473             BOOST_MATH_BIG_CONSTANT(T, 113, 0.47583809087867443858344765659065773369),
0474             BOOST_MATH_BIG_CONSTANT(T, 113, -0.09865724264554556400463655444270700132),
0475             BOOST_MATH_BIG_CONSTANT(T, 113, 0.012238223514176587501074150988445109735),
0476             BOOST_MATH_BIG_CONSTANT(T, 113, -0.00084625068418239194670614419707491797097),
0477             BOOST_MATH_BIG_CONSTANT(T, 113, 0.2796574430456237061420839429225710602e-4),
0478             BOOST_MATH_BIG_CONSTANT(T, 113, -0.30202973883316730694433702165188835331e-6)
0479          };
0480          // (2 - x) * (1 - x) * (c + R(2 - x))
0481          T r = zm2 * zm1;
0482          T R = tools::evaluate_polynomial(P, T(-zm2)) / tools::evaluate_polynomial(Q, T(-zm2));
0483
0484          result += r * Y + r * R;
0485          BOOST_MATH_INSTRUMENT_CODE(result);
0486       }
0487    }
0488    BOOST_MATH_INSTRUMENT_CODE(result);
0489    return result;
0490 }
0491 template <class T, class Policy, class Lanczos>
0492 BOOST_MATH_GPU_ENABLED T lgamma_small_imp(T z, T zm1, T zm2, const boost::math::integral_constant<int, 0>&, const Policy& pol, const Lanczos& l)
0493 {
0494    //
0495    // No rational approximations are available because either
0496    // T has no numeric_limits support (so we can't tell how
0497    // many digits it has), or T has more digits than we know
0498    // what to do with.... we do have a Lanczos approximation
0499    // though, and that can be used to keep errors under control.
0500    //
0501    BOOST_MATH_STD_USING  // for ADL of std names
0502    T result = 0;
0503    if(z < tools::epsilon<T>())
0504    {
0505       result = -log(z);
0506    }
0507    else if(z < 0.5)
0508    {
0509       // taking the log of tgamma reduces the error, no danger of overflow here:
0510       result = log(gamma_imp(z, pol, Lanczos()));
0511    }
0512    else if(z >= 3)
0513    {
0514       // taking the log of tgamma reduces the error, no danger of overflow here:
0515       result = log(gamma_imp(z, pol, Lanczos()));
0516    }
0517    else if(z >= 1.5)
0518    {
0519       // special case near 2:
0520       T dz = zm2;
0521       result = dz * log((z + lanczos_g_near_1_and_2(l) - T(0.5)) / boost::math::constants::e<T>());
0522       result += boost::math::log1p(dz / (lanczos_g_near_1_and_2(l) + T(1.5)), pol) * T(1.5);
0523       result += boost::math::log1p(Lanczos::lanczos_sum_near_2(dz), pol);
0524    }
0525    else
0526    {
0527       // special case near 1:
0528       T dz = zm1;
0529       result = dz * log((z + lanczos_g_near_1_and_2(l) - T(0.5)) / boost::math::constants::e<T>());
0530       result += boost::math::log1p(dz / (lanczos_g_near_1_and_2(l) + T(0.5)), pol) / 2;
0531       result += boost::math::log1p(Lanczos::lanczos_sum_near_1(dz), pol);
0532    }
0533    return result;
0534 }
0535
0536 #endif // BOOST_MATH_HAS_GPU_SUPPORT
0537
0538 }}} // namespaces
0539
0540 #endif // BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
0541