523 lines
23 KiB
Plaintext
523 lines
23 KiB
Plaintext
// (C) Copyright John Maddock 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
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#define BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/tools/big_constant.hpp>
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namespace boost{ namespace math{ namespace detail{
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//
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// These need forward declaring to keep GCC happy:
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//
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template <class T, class Policy, class Lanczos>
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T gamma_imp(T z, const Policy& pol, const Lanczos& l);
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template <class T, class Policy>
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T gamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos& l);
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//
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// lgamma for small arguments:
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//
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template <class T, class Policy, class Lanczos>
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T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<64>&, const Policy& /* l */, const Lanczos&)
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{
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// This version uses rational approximations for small
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// values of z accurate enough for 64-bit mantissas
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// (80-bit long doubles), works well for 53-bit doubles as well.
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// Lanczos is only used to select the Lanczos function.
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BOOST_MATH_STD_USING // for ADL of std names
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T result = 0;
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if(z < tools::epsilon<T>())
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{
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result = -log(z);
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}
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else if((zm1 == 0) || (zm2 == 0))
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{
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// nothing to do, result is zero....
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}
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else if(z > 2)
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{
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//
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// Begin by performing argument reduction until
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// z is in [2,3):
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//
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if(z >= 3)
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{
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do
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{
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z -= 1;
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zm2 -= 1;
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result += log(z);
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}while(z >= 3);
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// Update zm2, we need it below:
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zm2 = z - 2;
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}
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//
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// Use the following form:
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//
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// lgamma(z) = (z-2)(z+1)(Y + R(z-2))
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//
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// where R(z-2) is a rational approximation optimised for
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// low absolute error - as long as it's absolute error
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// is small compared to the constant Y - then any rounding
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// error in it's computation will get wiped out.
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//
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// R(z-2) has the following properties:
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//
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// At double: Max error found: 4.231e-18
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// At long double: Max error found: 1.987e-21
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// Maximum Deviation Found (approximation error): 5.900e-24
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//
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static const T P[] = {
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.180355685678449379109e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.25126649619989678683e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.494103151567532234274e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.172491608709613993966e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.259453563205438108893e-3)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.541009869215204396339e-3)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.324588649825948492091e-4))
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};
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static const T Q[] = {
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.196202987197795200688e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.148019669424231326694e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.541391432071720958364e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.988504251128010129477e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.82130967464889339326e-2)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.224936291922115757597e-3)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.223352763208617092964e-6))
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};
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static const float Y = 0.158963680267333984375e0f;
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T r = zm2 * (z + 1);
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T R = tools::evaluate_polynomial(P, zm2);
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R /= tools::evaluate_polynomial(Q, zm2);
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result += r * Y + r * R;
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}
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else
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{
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//
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// If z is less than 1 use recurrance to shift to
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// z in the interval [1,2]:
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//
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if(z < 1)
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{
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result += -log(z);
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zm2 = zm1;
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zm1 = z;
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z += 1;
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}
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//
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// Two approximations, on for z in [1,1.5] and
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// one for z in [1.5,2]:
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//
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if(z <= 1.5)
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{
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//
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// Use the following form:
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//
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// lgamma(z) = (z-1)(z-2)(Y + R(z-1))
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//
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// where R(z-1) is a rational approximation optimised for
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// low absolute error - as long as it's absolute error
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// is small compared to the constant Y - then any rounding
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// error in it's computation will get wiped out.
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//
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// R(z-1) has the following properties:
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//
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// At double precision: Max error found: 1.230011e-17
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// At 80-bit long double precision: Max error found: 5.631355e-21
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// Maximum Deviation Found: 3.139e-021
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// Expected Error Term: 3.139e-021
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//
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static const float Y = 0.52815341949462890625f;
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static const T P[] = {
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.490622454069039543534e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.969117530159521214579e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.414983358359495381969e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.406567124211938417342e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.158413586390692192217e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.240149820648571559892e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.100346687696279557415e-2))
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};
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static const T Q[] = {
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.302349829846463038743e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.348739585360723852576e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.191415588274426679201e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.507137738614363510846e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.577039722690451849648e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.195768102601107189171e-2))
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};
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T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
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T prefix = zm1 * zm2;
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result += prefix * Y + prefix * r;
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}
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else
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{
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//
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// Use the following form:
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//
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// lgamma(z) = (2-z)(1-z)(Y + R(2-z))
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//
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// where R(2-z) is a rational approximation optimised for
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// low absolute error - as long as it's absolute error
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// is small compared to the constant Y - then any rounding
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// error in it's computation will get wiped out.
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//
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// R(2-z) has the following properties:
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//
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// At double precision, max error found: 1.797565e-17
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// At 80-bit long double precision, max error found: 9.306419e-21
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// Maximum Deviation Found: 2.151e-021
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// Expected Error Term: 2.150e-021
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//
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static const float Y = 0.452017307281494140625f;
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static const T P[] = {
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.292329721830270012337e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.144216267757192309184e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.142440390738631274135e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.542809694055053558157e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.850535976868336437746e-2)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.431171342679297331241e-3))
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};
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static const T Q[] = {
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.150169356054485044494e1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.846973248876495016101e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.220095151814995745555e0)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.25582797155975869989e-1)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.100666795539143372762e-2)),
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static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.827193521891290553639e-6))
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};
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T r = zm2 * zm1;
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T R = tools::evaluate_polynomial(P, T(-zm2)) / tools::evaluate_polynomial(Q, T(-zm2));
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result += r * Y + r * R;
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}
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}
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return result;
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}
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template <class T, class Policy, class Lanczos>
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T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<113>&, const Policy& /* l */, const Lanczos&)
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{
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//
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// This version uses rational approximations for small
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// values of z accurate enough for 113-bit mantissas
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// (128-bit long doubles).
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//
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BOOST_MATH_STD_USING // for ADL of std names
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T result = 0;
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if(z < tools::epsilon<T>())
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{
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result = -log(z);
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BOOST_MATH_INSTRUMENT_CODE(result);
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}
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else if((zm1 == 0) || (zm2 == 0))
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{
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// nothing to do, result is zero....
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}
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else if(z > 2)
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{
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//
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// Begin by performing argument reduction until
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// z is in [2,3):
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//
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if(z >= 3)
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{
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do
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{
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z -= 1;
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result += log(z);
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}while(z >= 3);
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zm2 = z - 2;
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}
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BOOST_MATH_INSTRUMENT_CODE(zm2);
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BOOST_MATH_INSTRUMENT_CODE(z);
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BOOST_MATH_INSTRUMENT_CODE(result);
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//
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// Use the following form:
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//
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// lgamma(z) = (z-2)(z+1)(Y + R(z-2))
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//
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// where R(z-2) is a rational approximation optimised for
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// low absolute error - as long as it's absolute error
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// is small compared to the constant Y - then any rounding
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// error in it's computation will get wiped out.
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//
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// Maximum Deviation Found (approximation error) 3.73e-37
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static const T P[] = {
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.018035568567844937910504030027467476655),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.013841458273109517271750705401202404195),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.062031842739486600078866923383017722399),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.052518418329052161202007865149435256093),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.01881718142472784129191838493267755758),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.0025104830367021839316463675028524702846),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.00021043176101831873281848891452678568311),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.00010249622350908722793327719494037981166),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.11381479670982006841716879074288176994e-4),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.49999811718089980992888533630523892389e-6),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.70529798686542184668416911331718963364e-8)
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};
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static const T Q[] = {
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BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
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BOOST_MATH_BIG_CONSTANT(T, 113, 2.5877485070422317542808137697939233685),
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BOOST_MATH_BIG_CONSTANT(T, 113, 2.8797959228352591788629602533153837126),
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BOOST_MATH_BIG_CONSTANT(T, 113, 1.8030885955284082026405495275461180977),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.69774331297747390169238306148355428436),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.17261566063277623942044077039756583802),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.02729301254544230229429621192443000121),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.0026776425891195270663133581960016620433),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.00015244249160486584591370355730402168106),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.43997034032479866020546814475414346627e-5),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.46295080708455613044541885534408170934e-7),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.93326638207459533682980757982834180952e-11),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.42316456553164995177177407325292867513e-13)
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};
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T R = tools::evaluate_polynomial(P, zm2);
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R /= tools::evaluate_polynomial(Q, zm2);
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static const float Y = 0.158963680267333984375F;
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T r = zm2 * (z + 1);
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result += r * Y + r * R;
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BOOST_MATH_INSTRUMENT_CODE(result);
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}
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else
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{
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//
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// If z is less than 1 use recurrance to shift to
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// z in the interval [1,2]:
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//
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if(z < 1)
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{
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result += -log(z);
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zm2 = zm1;
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zm1 = z;
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z += 1;
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}
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BOOST_MATH_INSTRUMENT_CODE(result);
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BOOST_MATH_INSTRUMENT_CODE(z);
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BOOST_MATH_INSTRUMENT_CODE(zm2);
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//
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// Three approximations, on for z in [1,1.35], [1.35,1.625] and [1.625,1]
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//
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if(z <= 1.35)
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{
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//
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// Use the following form:
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//
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// lgamma(z) = (z-1)(z-2)(Y + R(z-1))
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//
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// where R(z-1) is a rational approximation optimised for
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// low absolute error - as long as it's absolute error
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// is small compared to the constant Y - then any rounding
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// error in it's computation will get wiped out.
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//
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// R(z-1) has the following properties:
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//
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// Maximum Deviation Found (approximation error) 1.659e-36
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// Expected Error Term (theoretical error) 1.343e-36
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// Max error found at 128-bit long double precision 1.007e-35
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//
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static const float Y = 0.54076099395751953125f;
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static const T P[] = {
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.036454670944013329356512090082402429697),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.066235835556476033710068679907798799959),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.67492399795577182387312206593595565371),
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BOOST_MATH_BIG_CONSTANT(T, 113, -1.4345555263962411429855341651960000166),
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BOOST_MATH_BIG_CONSTANT(T, 113, -1.4894319559821365820516771951249649563),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.87210277668067964629483299712322411566),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.29602090537771744401524080430529369136),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.0561832587517836908929331992218879676),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.0053236785487328044334381502530383140443),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.00018629360291358130461736386077971890789),
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BOOST_MATH_BIG_CONSTANT(T, 113, -0.10164985672213178500790406939467614498e-6),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.13680157145361387405588201461036338274e-8)
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};
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static const T Q[] = {
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BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
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BOOST_MATH_BIG_CONSTANT(T, 113, 4.9106336261005990534095838574132225599),
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BOOST_MATH_BIG_CONSTANT(T, 113, 10.258804800866438510889341082793078432),
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BOOST_MATH_BIG_CONSTANT(T, 113, 11.88588976846826108836629960537466889),
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BOOST_MATH_BIG_CONSTANT(T, 113, 8.3455000546999704314454891036700998428),
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BOOST_MATH_BIG_CONSTANT(T, 113, 3.6428823682421746343233362007194282703),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.97465989807254572142266753052776132252),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.15121052897097822172763084966793352524),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.012017363555383555123769849654484594893),
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BOOST_MATH_BIG_CONSTANT(T, 113, 0.0003583032812720649835431669893011257277)
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};
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T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
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T prefix = zm1 * zm2;
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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[] = {
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.017977422421608624353488126610933005432),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.18484528905298309555089509029244135703),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.40401251514859546989565001431430884082),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.40277179799147356461954182877921388182),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.21993421441282936476709677700477598816),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.069595742223850248095697771331107571011),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.012681481427699686635516772923547347328),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0012489322866834830413292771335113136034),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.57058739515423112045108068834668269608e-4),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.8207548771933585614380644961342925976e-6)
|
|
};
|
|
static const T Q[] = {
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -2.9629552288944259229543137757200262073),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 3.7118380799042118987185957298964772755),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -2.5569815272165399297600586376727357187),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0546764918220835097855665680632153367),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.26574021300894401276478730940980810831),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.03996289731752081380552901986471233462),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0033398680924544836817826046380586480873),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00013288854760548251757651556792598235735),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.17194794958274081373243161848194745111e-5)
|
|
};
|
|
T r = zm2 * zm1;
|
|
T R = tools::evaluate_polynomial(P, T(0.625 - zm1)) / tools::evaluate_polynomial(Q, T(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[] = {
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.021027558364667626231512090082402429494),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.15128811104498736604523586803722368377),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.26249631480066246699388544451126410278),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.21148748610533489823742352180628489742),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.093964130697489071999873506148104370633),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.024292059227009051652542804957550866827),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0036284453226534839926304745756906117066),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0002939230129315195346843036254392485984),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.11088589183158123733132268042570710338e-4),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.13240510580220763969511741896361984162e-6)
|
|
};
|
|
static const T Q[] = {
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -2.4240003754444040525462170802796471996),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2.4868383476933178722203278602342786002),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -1.4047068395206343375520721509193698547),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.47583809087867443858344765659065773369),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.09865724264554556400463655444270700132),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.012238223514176587501074150988445109735),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.00084625068418239194670614419707491797097),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.2796574430456237061420839429225710602e-4),
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.30202973883316730694433702165188835331e-6)
|
|
};
|
|
// (2 - x) * (1 - x) * (c + R(2 - x))
|
|
T r = zm2 * zm1;
|
|
T R = tools::evaluate_polynomial(P, T(-zm2)) / tools::evaluate_polynomial(Q, T(-zm2));
|
|
|
|
result += r * Y + r * R;
|
|
BOOST_MATH_INSTRUMENT_CODE(result);
|
|
}
|
|
}
|
|
BOOST_MATH_INSTRUMENT_CODE(result);
|
|
return result;
|
|
}
|
|
template <class T, class Policy, class Lanczos>
|
|
T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<0>&, const Policy& pol, const Lanczos&)
|
|
{
|
|
//
|
|
// 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, Lanczos()));
|
|
}
|
|
else if(z >= 3)
|
|
{
|
|
// taking the log of tgamma reduces the error, no danger of overflow here:
|
|
result = log(gamma_imp(z, pol, Lanczos()));
|
|
}
|
|
else if(z >= 1.5)
|
|
{
|
|
// special case near 2:
|
|
T dz = zm2;
|
|
result = dz * log((z + Lanczos::g() - T(0.5)) / boost::math::constants::e<T>());
|
|
result += boost::math::log1p(dz / (Lanczos::g() + T(1.5)), pol) * T(1.5);
|
|
result += boost::math::log1p(Lanczos::lanczos_sum_near_2(dz), pol);
|
|
}
|
|
else
|
|
{
|
|
// special case near 1:
|
|
T dz = zm1;
|
|
result = dz * log((z + Lanczos::g() - T(0.5)) / boost::math::constants::e<T>());
|
|
result += boost::math::log1p(dz / (Lanczos::g() + T(0.5)), pol) / 2;
|
|
result += boost::math::log1p(Lanczos::lanczos_sum_near_1(dz), pol);
|
|
}
|
|
return result;
|
|
}
|
|
|
|
}}} // namespaces
|
|
|
|
#endif // BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
|
|
|