Boost C++ Libraries

boost/math/special_functions/detail/lgamma_small.hpp

//  (C) Copyright John Maddock 2006.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file
//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

#ifndef BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
#define BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL

#ifdef _MSC_VER
#pragma once
#endif

namespace boost{ namespace math{ namespace detail{

//
// lgamma for small arguments:
//
template <class T, class Policy, class L>
T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<64>&, const Policy& /* l */, const L&)
{
   // This version uses rational approximations for small
   // values of z accurate enough for 64-bit mantissas
   // (80-bit long doubles), works well for 53-bit doubles as well.
   // L is only used to select the Lanczos function.

   BOOST_MATH_STD_USING  // for ADL of std names
   T result = 0;
   if(z < tools::epsilon<T>())
   {
      result = -log(z);
   }
   else if((zm1 == 0) || (zm2 == 0))
   {
      // nothing to do, result is zero....
   }
   else if(z > 2)
   {
      //
      // Begin by performing argument reduction until
      // z is in [2,3):
      //
      if(z >= 3)
      {
         do
         {
            z -= 1;
            zm2 -= 1;
            result += log(z);
         }while(z >= 3);
         // Update zm2, we need it below:
         zm2 = z - 2;
      }

      //
      // Use the following form:
      //
      // lgamma(z) = (z-2)(z+1)(Y + R(z-2))
      //
      // where R(z-2) is a rational approximation optimised for
      // low absolute error - as long as it's absolute error
      // is small compared to the constant Y - then any rounding
      // error in it's computation will get wiped out.
      //
      // R(z-2) has the following properties:
      //
      // At double: Max error found:                    4.231e-18
      // At long double: Max error found:               1.987e-21
      // Maximum Deviation Found (approximation error): 5.900e-24
      //
      static const T P[] = {
         static_cast<T>(-0.180355685678449379109e-1L),
         static_cast<T>(0.25126649619989678683e-1L),
         static_cast<T>(0.494103151567532234274e-1L),
         static_cast<T>(0.172491608709613993966e-1L),
         static_cast<T>(-0.259453563205438108893e-3L),
         static_cast<T>(-0.541009869215204396339e-3L),
         static_cast<T>(-0.324588649825948492091e-4L)
      };
      static const T Q[] = {
         static_cast<T>(0.1e1),
         static_cast<T>(0.196202987197795200688e1L),
         static_cast<T>(0.148019669424231326694e1L),
         static_cast<T>(0.541391432071720958364e0L),
         static_cast<T>(0.988504251128010129477e-1L),
         static_cast<T>(0.82130967464889339326e-2L),
         static_cast<T>(0.224936291922115757597e-3L),
         static_cast<T>(-0.223352763208617092964e-6L)
      };

      static const float Y = 0.158963680267333984375e0f;

      T r = zm2 * (z + 1);
      T R = tools::evaluate_polynomial(P, zm2);
      R /= tools::evaluate_polynomial(Q, zm2);

      result +=  r * Y + r * R;
   }
   else
   {
      //
      // If z is less than 1 use recurrance to shift to
      // z in the interval [1,2]:
      //
      if(z < 1)
      {
         result += -log(z);
         zm2 = zm1;
         zm1 = z;
         z += 1;
      }
      //
      // Two approximations, on for z in [1,1.5] and
      // one for z in [1.5,2]:
      //
      if(z <= 1.5)
      {
         //
         // Use the following form:
         //
         // lgamma(z) = (z-1)(z-2)(Y + R(z-1))
         //
         // where R(z-1) is a rational approximation optimised for
         // low absolute error - as long as it's absolute error
         // is small compared to the constant Y - then any rounding
         // error in it's computation will get wiped out.
         //
         // R(z-1) has the following properties:
         //
         // At double precision: Max error found:                1.230011e-17
         // At 80-bit long double precision:   Max error found:  5.631355e-21
         // Maximum Deviation Found:                             3.139e-021
         // Expected Error Term:                                 3.139e-021

         //
         static const float Y = 0.52815341949462890625f;

         static const T P[] = {
            static_cast<T>(0.490622454069039543534e-1L),
            static_cast<T>(-0.969117530159521214579e-1L),
            static_cast<T>(-0.414983358359495381969e0L),
            static_cast<T>(-0.406567124211938417342e0L),
            static_cast<T>(-0.158413586390692192217e0L),
            static_cast<T>(-0.240149820648571559892e-1L),
            static_cast<T>(-0.100346687696279557415e-2L)
         };
         static const T Q[] = {
            static_cast<T>(0.1e1L),
            static_cast<T>(0.302349829846463038743e1L),
            static_cast<T>(0.348739585360723852576e1L),
            static_cast<T>(0.191415588274426679201e1L),
            static_cast<T>(0.507137738614363510846e0L),
            static_cast<T>(0.577039722690451849648e-1L),
            static_cast<T>(0.195768102601107189171e-2L)
         };

         T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
         T prefix = zm1 * zm2;

         result += prefix * Y + prefix * r;
      }
      else
      {
         //
         // Use the following form:
         //
         // lgamma(z) = (2-z)(1-z)(Y + R(2-z))
         //
         // where R(2-z) is a rational approximation optimised for
         // low absolute error - as long as it's absolute error
         // is small compared to the constant Y - then any rounding
         // error in it's computation will get wiped out.
         //
         // R(2-z) has the following properties:
         //
         // At double precision, max error found:              1.797565e-17
         // At 80-bit long double precision, max error found:  9.306419e-21
         // Maximum Deviation Found:                           2.151e-021
         // Expected Error Term:                               2.150e-021
         //
         static const float Y = 0.452017307281494140625f;

         static const T P[] = {
            static_cast<T>(-0.292329721830270012337e-1L), 
            static_cast<T>(0.144216267757192309184e0L),
            static_cast<T>(-0.142440390738631274135e0L),
            static_cast<T>(0.542809694055053558157e-1L),
            static_cast<T>(-0.850535976868336437746e-2L),
            static_cast<T>(0.431171342679297331241e-3L)
         };
         static const T Q[] = {
            static_cast<T>(0.1e1),
            static_cast<T>(-0.150169356054485044494e1L),
            static_cast<T>(0.846973248876495016101e0L),
            static_cast<T>(-0.220095151814995745555e0L),
            static_cast<T>(0.25582797155975869989e-1L),
            static_cast<T>(-0.100666795539143372762e-2L),
            static_cast<T>(-0.827193521891290553639e-6L)
         };
         T r = zm2 * zm1;
         T R = tools::evaluate_polynomial(P, -zm2) / tools::evaluate_polynomial(Q, -zm2);

         result += r * Y + r * R;
      }
   }
   return result;
}
template <class T, class Policy, class L>
T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<113>&, const Policy& /* l */, const L&)
{
   //
   // This version uses rational approximations for small
   // values of z accurate enough for 113-bit mantissas
   // (128-bit long doubles).
   //
   BOOST_MATH_STD_USING  // for ADL of std names
   T result = 0;
   if(z < tools::epsilon<T>())
   {
      result = -log(z);
      BOOST_MATH_INSTRUMENT_CODE(result);
   }
   else if((zm1 == 0) || (zm2 == 0))
   {
      // nothing to do, result is zero....
   }
   else if(z > 2)
   {
      //
      // Begin by performing argument reduction until
      // z is in [2,3):
      //
      if(z >= 3)
      {
         do
         {
            z -= 1;
            result += log(z);
         }while(z >= 3);
         zm2 = z - 2;
      }
      BOOST_MATH_INSTRUMENT_CODE(zm2);
      BOOST_MATH_INSTRUMENT_CODE(z);
      BOOST_MATH_INSTRUMENT_CODE(result);

      //
      // Use the following form:
      //
      // lgamma(z) = (z-2)(z+1)(Y + R(z-2))
      //
      // where R(z-2) is a rational approximation optimised for
      // low absolute error - as long as it's absolute error
      // is small compared to the constant Y - then any rounding
      // error in it's computation will get wiped out.
      //
      // Maximum Deviation Found (approximation error)      3.73e-37

      static const T P[] = {
         -0.018035568567844937910504030027467476655L,
         0.013841458273109517271750705401202404195L,
         0.062031842739486600078866923383017722399L,
         0.052518418329052161202007865149435256093L,
         0.01881718142472784129191838493267755758L,
         0.0025104830367021839316463675028524702846L,
         -0.00021043176101831873281848891452678568311L,
         -0.00010249622350908722793327719494037981166L,
         -0.11381479670982006841716879074288176994e-4L,
         -0.49999811718089980992888533630523892389e-6L,
         -0.70529798686542184668416911331718963364e-8L
      };
      static const T Q[] = {
         1L,
         2.5877485070422317542808137697939233685L,
         2.8797959228352591788629602533153837126L,
         1.8030885955284082026405495275461180977L,
         0.69774331297747390169238306148355428436L,
         0.17261566063277623942044077039756583802L,
         0.02729301254544230229429621192443000121L,
         0.0026776425891195270663133581960016620433L,
         0.00015244249160486584591370355730402168106L,
         0.43997034032479866020546814475414346627e-5L,
         0.46295080708455613044541885534408170934e-7L,
         -0.93326638207459533682980757982834180952e-11L,
         0.42316456553164995177177407325292867513e-13L
      };

      T R = tools::evaluate_polynomial(P, zm2);
      R /= tools::evaluate_polynomial(Q, zm2);

      static const float Y = 0.158963680267333984375F;

      T r = zm2 * (z + 1);

      result +=  r * Y + r * R;
      BOOST_MATH_INSTRUMENT_CODE(result);
   }
   else
   {
      //
      // If z is less than 1 use recurrance to shift to
      // z in the interval [1,2]:
      //
      if(z < 1)
      {
         result += -log(z);
         zm2 = zm1;
         zm1 = z;
         z += 1;
      }
      BOOST_MATH_INSTRUMENT_CODE(result);
      BOOST_MATH_INSTRUMENT_CODE(z);
      BOOST_MATH_INSTRUMENT_CODE(zm2);
      //
      // Three approximations, on for z in [1,1.35], [1.35,1.625] and [1.625,1]
      //
      if(z <= 1.35)
      {
         //
         // Use the following form:
         //
         // lgamma(z) = (z-1)(z-2)(Y + R(z-1))
         //
         // where R(z-1) is a rational approximation optimised for
         // low absolute error - as long as it's absolute error
         // is small compared to the constant Y - then any rounding
         // error in it's computation will get wiped out.
         //
         // R(z-1) has the following properties:
         //
         // Maximum Deviation Found (approximation error)            1.659e-36
         // Expected Error Term (theoretical error)                  1.343e-36
         // Max error found at 128-bit long double precision         1.007e-35
         //
         static const float Y = 0.54076099395751953125f;

         static const T P[] = {
            0.036454670944013329356512090082402429697L,
            -0.066235835556476033710068679907798799959L,
            -0.67492399795577182387312206593595565371L,
            -1.4345555263962411429855341651960000166L,
            -1.4894319559821365820516771951249649563L,
            -0.87210277668067964629483299712322411566L,
            -0.29602090537771744401524080430529369136L,
            -0.0561832587517836908929331992218879676L,
            -0.0053236785487328044334381502530383140443L,
            -0.00018629360291358130461736386077971890789L,
            -0.10164985672213178500790406939467614498e-6L,
            0.13680157145361387405588201461036338274e-8L
         };
         static const T Q[] = {
            1,
            4.9106336261005990534095838574132225599L,
            10.258804800866438510889341082793078432L,
            11.88588976846826108836629960537466889L,
            8.3455000546999704314454891036700998428L,
            3.6428823682421746343233362007194282703L,
            0.97465989807254572142266753052776132252L,
            0.15121052897097822172763084966793352524L,
            0.012017363555383555123769849654484594893L,
            0.0003583032812720649835431669893011257277L
         };

         T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
         T prefix = zm1 * zm2;

         result += prefix * Y + prefix * r;
         BOOST_MATH_INSTRUMENT_CODE(result);
      }
      else if(z <= 1.625)
      {
         //
         // Use the following form:
         //
         // lgamma(z) = (2-z)(1-z)(Y + R(2-z))
         //
         // where R(2-z) is a rational approximation optimised for
         // low absolute error - as long as it's absolute error
         // is small compared to the constant Y - then any rounding
         // error in it's computation will get wiped out.
         //
         // R(2-z) has the following properties:
         //
         // Max error found at 128-bit long double precision  9.634e-36
         // Maximum Deviation Found (approximation error)     1.538e-37
         // Expected Error Term (theoretical error)           2.350e-38
         //
         static const float Y = 0.483787059783935546875f;

         static const T P[] = {
            -0.017977422421608624353488126610933005432L,
            0.18484528905298309555089509029244135703L,
            -0.40401251514859546989565001431430884082L,
            0.40277179799147356461954182877921388182L,
            -0.21993421441282936476709677700477598816L,
            0.069595742223850248095697771331107571011L,
            -0.012681481427699686635516772923547347328L,
            0.0012489322866834830413292771335113136034L,
            -0.57058739515423112045108068834668269608e-4L,
            0.8207548771933585614380644961342925976e-6L
         };
         static const T Q[] = {
            1,
            -2.9629552288944259229543137757200262073L,
            3.7118380799042118987185957298964772755L,
            -2.5569815272165399297600586376727357187L,
            1.0546764918220835097855665680632153367L,
            -0.26574021300894401276478730940980810831L,
            0.03996289731752081380552901986471233462L,
            -0.0033398680924544836817826046380586480873L,
            0.00013288854760548251757651556792598235735L,
            -0.17194794958274081373243161848194745111e-5L
         };
         T r = zm2 * zm1;
         T R = tools::evaluate_polynomial(P, 0.625 - zm1) / tools::evaluate_polynomial(Q, 0.625 - zm1);

         result += r * Y + r * R;
         BOOST_MATH_INSTRUMENT_CODE(result);
      }
      else
      {
         //
         // Same form as above.
         //
         // Max error found (at 128-bit long double precision) 1.831e-35
         // Maximum Deviation Found (approximation error)      8.588e-36
         // Expected Error Term (theoretical error)            1.458e-36
         //
         static const float Y = 0.443811893463134765625f;

         static const T P[] = {
            -0.021027558364667626231512090082402429494L,
            0.15128811104498736604523586803722368377L,
            -0.26249631480066246699388544451126410278L,
            0.21148748610533489823742352180628489742L,
            -0.093964130697489071999873506148104370633L,
            0.024292059227009051652542804957550866827L,
            -0.0036284453226534839926304745756906117066L,
            0.0002939230129315195346843036254392485984L,
            -0.11088589183158123733132268042570710338e-4L,
            0.13240510580220763969511741896361984162e-6L
         };
         static const T Q[] = {
            1,
            -2.4240003754444040525462170802796471996L,
            2.4868383476933178722203278602342786002L,
            -1.4047068395206343375520721509193698547L,
            0.47583809087867443858344765659065773369L,
            -0.09865724264554556400463655444270700132L,
            0.012238223514176587501074150988445109735L,
            -0.00084625068418239194670614419707491797097L,
            0.2796574430456237061420839429225710602e-4L,
            -0.30202973883316730694433702165188835331e-6L
         };
         // (2 - x) * (1 - x) * (c + R(2 - x))
         T r = zm2 * zm1;
         T R = tools::evaluate_polynomial(P, -zm2) / tools::evaluate_polynomial(Q, -zm2);

         result += r * Y + r * R;
         BOOST_MATH_INSTRUMENT_CODE(result);
      }
   }
   BOOST_MATH_INSTRUMENT_CODE(result);
   return result;
}
template <class T, class Policy, class L>
T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<0>&, const Policy& pol, const L&)
{
   //
   // No rational approximations are available because either
   // T has no numeric_limits support (so we can't tell how
   // many digits it has), or T has more digits than we know
   // what to do with.... we do have a Lanczos approximation
   // though, and that can be used to keep errors under control.
   //
   BOOST_MATH_STD_USING  // for ADL of std names
   T result = 0;
   if(z < tools::epsilon<T>())
   {
      result = -log(z);
   }
   else if(z < 0.5)
   {
      // taking the log of tgamma reduces the error, no danger of overflow here:
      result = log(gamma_imp(z, pol, L()));
   }
   else if(z >= 3)
   {
      // taking the log of tgamma reduces the error, no danger of overflow here:
      result = log(gamma_imp(z, pol, L()));
   }
   else if(z >= 1.5)
   {
      // special case near 2:
      T dz = zm2;
      result = dz * log((z + L::g() - T(0.5)) / boost::math::constants::e<T>());
      result += boost::math::log1p(dz / (L::g() + T(1.5)), pol) * T(1.5);
      result += boost::math::log1p(L::lanczos_sum_near_2(dz), pol);
   }
   else
   {
      // special case near 1:
      T dz = zm1;
      result = dz * log((z + L::g() - T(0.5)) / boost::math::constants::e<T>());
      result += boost::math::log1p(dz / (L::g() + T(0.5)), pol) / 2;
      result += boost::math::log1p(L::lanczos_sum_near_1(dz), pol);
   }
   return result;
}

}}} // namespaces

#endif // BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL