summaryrefslogtreecommitdiff
path: root/src/third_party/boost-1.69.0/boost/math/special_functions/gamma.hpp
diff options
context:
space:
mode:
Diffstat (limited to 'src/third_party/boost-1.69.0/boost/math/special_functions/gamma.hpp')
-rw-r--r--src/third_party/boost-1.69.0/boost/math/special_functions/gamma.hpp2080
1 files changed, 2080 insertions, 0 deletions
diff --git a/src/third_party/boost-1.69.0/boost/math/special_functions/gamma.hpp b/src/third_party/boost-1.69.0/boost/math/special_functions/gamma.hpp
new file mode 100644
index 00000000000..1903b997cbf
--- /dev/null
+++ b/src/third_party/boost-1.69.0/boost/math/special_functions/gamma.hpp
@@ -0,0 +1,2080 @@
+
+// Copyright John Maddock 2006-7, 2013-14.
+// Copyright Paul A. Bristow 2007, 2013-14.
+// Copyright Nikhar Agrawal 2013-14
+// Copyright Christopher Kormanyos 2013-14
+
+// 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_SF_GAMMA_HPP
+#define BOOST_MATH_SF_GAMMA_HPP
+
+#ifdef _MSC_VER
+#pragma once
+#endif
+
+#include <boost/config.hpp>
+#include <boost/math/tools/series.hpp>
+#include <boost/math/tools/fraction.hpp>
+#include <boost/math/tools/precision.hpp>
+#include <boost/math/tools/promotion.hpp>
+#include <boost/math/policies/error_handling.hpp>
+#include <boost/math/constants/constants.hpp>
+#include <boost/math/special_functions/math_fwd.hpp>
+#include <boost/math/special_functions/log1p.hpp>
+#include <boost/math/special_functions/trunc.hpp>
+#include <boost/math/special_functions/powm1.hpp>
+#include <boost/math/special_functions/sqrt1pm1.hpp>
+#include <boost/math/special_functions/lanczos.hpp>
+#include <boost/math/special_functions/fpclassify.hpp>
+#include <boost/math/special_functions/detail/igamma_large.hpp>
+#include <boost/math/special_functions/detail/unchecked_factorial.hpp>
+#include <boost/math/special_functions/detail/lgamma_small.hpp>
+#include <boost/math/special_functions/bernoulli.hpp>
+#include <boost/math/special_functions/zeta.hpp>
+#include <boost/type_traits/is_convertible.hpp>
+#include <boost/assert.hpp>
+#include <boost/mpl/greater.hpp>
+#include <boost/mpl/equal_to.hpp>
+#include <boost/mpl/greater.hpp>
+
+#include <boost/config/no_tr1/cmath.hpp>
+#include <algorithm>
+
+#ifdef BOOST_MSVC
+# pragma warning(push)
+# pragma warning(disable: 4702) // unreachable code (return after domain_error throw).
+# pragma warning(disable: 4127) // conditional expression is constant.
+# pragma warning(disable: 4100) // unreferenced formal parameter.
+// Several variables made comments,
+// but some difficulty as whether referenced on not may depend on macro values.
+// So to be safe, 4100 warnings suppressed.
+// TODO - revisit this?
+#endif
+
+namespace boost{ namespace math{
+
+namespace detail{
+
+template <class T>
+inline bool is_odd(T v, const boost::true_type&)
+{
+ int i = static_cast<int>(v);
+ return i&1;
+}
+template <class T>
+inline bool is_odd(T v, const boost::false_type&)
+{
+ // Oh dear can't cast T to int!
+ BOOST_MATH_STD_USING
+ T modulus = v - 2 * floor(v/2);
+ return static_cast<bool>(modulus != 0);
+}
+template <class T>
+inline bool is_odd(T v)
+{
+ return is_odd(v, ::boost::is_convertible<T, int>());
+}
+
+template <class T>
+T sinpx(T z)
+{
+ // Ad hoc function calculates x * sin(pi * x),
+ // taking extra care near when x is near a whole number.
+ BOOST_MATH_STD_USING
+ int sign = 1;
+ if(z < 0)
+ {
+ z = -z;
+ }
+ T fl = floor(z);
+ T dist;
+ if(is_odd(fl))
+ {
+ fl += 1;
+ dist = fl - z;
+ sign = -sign;
+ }
+ else
+ {
+ dist = z - fl;
+ }
+ BOOST_ASSERT(fl >= 0);
+ if(dist > 0.5)
+ dist = 1 - dist;
+ T result = sin(dist*boost::math::constants::pi<T>());
+ return sign*z*result;
+} // template <class T> T sinpx(T z)
+//
+// tgamma(z), with Lanczos support:
+//
+template <class T, class Policy, class Lanczos>
+T gamma_imp(T z, const Policy& pol, const Lanczos& l)
+{
+ BOOST_MATH_STD_USING
+
+ T result = 1;
+
+#ifdef BOOST_MATH_INSTRUMENT
+ static bool b = false;
+ if(!b)
+ {
+ std::cout << "tgamma_imp called with " << typeid(z).name() << " " << typeid(l).name() << std::endl;
+ b = true;
+ }
+#endif
+ static const char* function = "boost::math::tgamma<%1%>(%1%)";
+
+ if(z <= 0)
+ {
+ if(floor(z) == z)
+ return policies::raise_pole_error<T>(function, "Evaluation of tgamma at a negative integer %1%.", z, pol);
+ if(z <= -20)
+ {
+ result = gamma_imp(T(-z), pol, l) * sinpx(z);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ if((fabs(result) < 1) && (tools::max_value<T>() * fabs(result) < boost::math::constants::pi<T>()))
+ return -boost::math::sign(result) * policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol);
+ result = -boost::math::constants::pi<T>() / result;
+ if(result == 0)
+ return policies::raise_underflow_error<T>(function, "Result of tgamma is too small to represent.", pol);
+ if((boost::math::fpclassify)(result) == (int)FP_SUBNORMAL)
+ return policies::raise_denorm_error<T>(function, "Result of tgamma is denormalized.", result, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ return result;
+ }
+
+ // shift z to > 1:
+ while(z < 0)
+ {
+ result /= z;
+ z += 1;
+ }
+ }
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ if((floor(z) == z) && (z < max_factorial<T>::value))
+ {
+ result *= unchecked_factorial<T>(itrunc(z, pol) - 1);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ else if (z < tools::root_epsilon<T>())
+ {
+ if (z < 1 / tools::max_value<T>())
+ result = policies::raise_overflow_error<T>(function, 0, pol);
+ result *= 1 / z - constants::euler<T>();
+ }
+ else
+ {
+ result *= Lanczos::lanczos_sum(z);
+ T zgh = (z + static_cast<T>(Lanczos::g()) - boost::math::constants::half<T>());
+ T lzgh = log(zgh);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ BOOST_MATH_INSTRUMENT_VARIABLE(tools::log_max_value<T>());
+ if(z * lzgh > tools::log_max_value<T>())
+ {
+ // we're going to overflow unless this is done with care:
+ BOOST_MATH_INSTRUMENT_VARIABLE(zgh);
+ if(lzgh * z / 2 > tools::log_max_value<T>())
+ return boost::math::sign(result) * policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol);
+ T hp = pow(zgh, (z / 2) - T(0.25));
+ BOOST_MATH_INSTRUMENT_VARIABLE(hp);
+ result *= hp / exp(zgh);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ if(tools::max_value<T>() / hp < result)
+ return boost::math::sign(result) * policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol);
+ result *= hp;
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ else
+ {
+ BOOST_MATH_INSTRUMENT_VARIABLE(zgh);
+ BOOST_MATH_INSTRUMENT_VARIABLE(pow(zgh, z - boost::math::constants::half<T>()));
+ BOOST_MATH_INSTRUMENT_VARIABLE(exp(zgh));
+ result *= pow(zgh, z - boost::math::constants::half<T>()) / exp(zgh);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ }
+ return result;
+}
+//
+// lgamma(z) with Lanczos support:
+//
+template <class T, class Policy, class Lanczos>
+T lgamma_imp(T z, const Policy& pol, const Lanczos& l, int* sign = 0)
+{
+#ifdef BOOST_MATH_INSTRUMENT
+ static bool b = false;
+ if(!b)
+ {
+ std::cout << "lgamma_imp called with " << typeid(z).name() << " " << typeid(l).name() << std::endl;
+ b = true;
+ }
+#endif
+
+ BOOST_MATH_STD_USING
+
+ static const char* function = "boost::math::lgamma<%1%>(%1%)";
+
+ T result = 0;
+ int sresult = 1;
+ if(z <= -tools::root_epsilon<T>())
+ {
+ // reflection formula:
+ if(floor(z) == z)
+ return policies::raise_pole_error<T>(function, "Evaluation of lgamma at a negative integer %1%.", z, pol);
+
+ T t = sinpx(z);
+ z = -z;
+ if(t < 0)
+ {
+ t = -t;
+ }
+ else
+ {
+ sresult = -sresult;
+ }
+ result = log(boost::math::constants::pi<T>()) - lgamma_imp(z, pol, l) - log(t);
+ }
+ else if (z < tools::root_epsilon<T>())
+ {
+ if (0 == z)
+ return policies::raise_pole_error<T>(function, "Evaluation of lgamma at %1%.", z, pol);
+ if (fabs(z) < 1 / tools::max_value<T>())
+ result = -log(fabs(z));
+ else
+ result = log(fabs(1 / z - constants::euler<T>()));
+ if (z < 0)
+ sresult = -1;
+ }
+ else if(z < 15)
+ {
+ typedef typename policies::precision<T, Policy>::type precision_type;
+ typedef typename mpl::if_<
+ mpl::and_<
+ mpl::less_equal<precision_type, mpl::int_<64> >,
+ mpl::greater<precision_type, mpl::int_<0> >
+ >,
+ mpl::int_<64>,
+ typename mpl::if_<
+ mpl::and_<
+ mpl::less_equal<precision_type, mpl::int_<113> >,
+ mpl::greater<precision_type, mpl::int_<0> >
+ >,
+ mpl::int_<113>, mpl::int_<0> >::type
+ >::type tag_type;
+ result = lgamma_small_imp<T>(z, T(z - 1), T(z - 2), tag_type(), pol, l);
+ }
+ else if((z >= 3) && (z < 100) && (std::numeric_limits<T>::max_exponent >= 1024))
+ {
+ // taking the log of tgamma reduces the error, no danger of overflow here:
+ result = log(gamma_imp(z, pol, l));
+ }
+ else
+ {
+ // regular evaluation:
+ T zgh = static_cast<T>(z + Lanczos::g() - boost::math::constants::half<T>());
+ result = log(zgh) - 1;
+ result *= z - 0.5f;
+ result += log(Lanczos::lanczos_sum_expG_scaled(z));
+ }
+
+ if(sign)
+ *sign = sresult;
+ return result;
+}
+
+//
+// Incomplete gamma functions follow:
+//
+template <class T>
+struct upper_incomplete_gamma_fract
+{
+private:
+ T z, a;
+ int k;
+public:
+ typedef std::pair<T,T> result_type;
+
+ upper_incomplete_gamma_fract(T a1, T z1)
+ : z(z1-a1+1), a(a1), k(0)
+ {
+ }
+
+ result_type operator()()
+ {
+ ++k;
+ z += 2;
+ return result_type(k * (a - k), z);
+ }
+};
+
+template <class T>
+inline T upper_gamma_fraction(T a, T z, T eps)
+{
+ // Multiply result by z^a * e^-z to get the full
+ // upper incomplete integral. Divide by tgamma(z)
+ // to normalise.
+ upper_incomplete_gamma_fract<T> f(a, z);
+ return 1 / (z - a + 1 + boost::math::tools::continued_fraction_a(f, eps));
+}
+
+template <class T>
+struct lower_incomplete_gamma_series
+{
+private:
+ T a, z, result;
+public:
+ typedef T result_type;
+ lower_incomplete_gamma_series(T a1, T z1) : a(a1), z(z1), result(1){}
+
+ T operator()()
+ {
+ T r = result;
+ a += 1;
+ result *= z/a;
+ return r;
+ }
+};
+
+template <class T, class Policy>
+inline T lower_gamma_series(T a, T z, const Policy& pol, T init_value = 0)
+{
+ // Multiply result by ((z^a) * (e^-z) / a) to get the full
+ // lower incomplete integral. Then divide by tgamma(a)
+ // to get the normalised value.
+ lower_incomplete_gamma_series<T> s(a, z);
+ boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
+ T factor = policies::get_epsilon<T, Policy>();
+ T result = boost::math::tools::sum_series(s, factor, max_iter, init_value);
+ policies::check_series_iterations<T>("boost::math::detail::lower_gamma_series<%1%>(%1%)", max_iter, pol);
+ return result;
+}
+
+//
+// Fully generic tgamma and lgamma use Stirling's approximation
+// with Bernoulli numbers.
+//
+template<class T>
+std::size_t highest_bernoulli_index()
+{
+ const float digits10_of_type = (std::numeric_limits<T>::is_specialized
+ ? static_cast<float>(std::numeric_limits<T>::digits10)
+ : static_cast<float>(boost::math::tools::digits<T>() * 0.301F));
+
+ // Find the high index n for Bn to produce the desired precision in Stirling's calculation.
+ return static_cast<std::size_t>(18.0F + (0.6F * digits10_of_type));
+}
+
+template<class T>
+T minimum_argument_for_bernoulli_recursion()
+{
+ const float digits10_of_type = (std::numeric_limits<T>::is_specialized
+ ? static_cast<float>(std::numeric_limits<T>::digits10)
+ : static_cast<float>(boost::math::tools::digits<T>() * 0.301F));
+
+ return T(digits10_of_type * 1.7F);
+}
+
+// Forward declaration of the lgamma_imp template specialization.
+template <class T, class Policy>
+T lgamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos&, int* sign = 0);
+
+template <class T, class Policy>
+T gamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos&)
+{
+ BOOST_MATH_STD_USING
+
+ static const char* function = "boost::math::tgamma<%1%>(%1%)";
+
+ // Check if the argument of tgamma is identically zero.
+ const bool is_at_zero = (z == 0);
+
+ if((is_at_zero) || ((boost::math::isinf)(z) && (z < 0)))
+ return policies::raise_domain_error<T>(function, "Evaluation of tgamma at %1%.", z, pol);
+
+ const bool b_neg = (z < 0);
+
+ const bool floor_of_z_is_equal_to_z = (floor(z) == z);
+
+ // Special case handling of small factorials:
+ if((!b_neg) && floor_of_z_is_equal_to_z && (z < boost::math::max_factorial<T>::value))
+ {
+ return boost::math::unchecked_factorial<T>(itrunc(z) - 1);
+ }
+
+ // Make a local, unsigned copy of the input argument.
+ T zz((!b_neg) ? z : -z);
+
+ // Special case for ultra-small z:
+ if(zz < tools::cbrt_epsilon<T>())
+ {
+ const T a0(1);
+ const T a1(boost::math::constants::euler<T>());
+ const T six_euler_squared((boost::math::constants::euler<T>() * boost::math::constants::euler<T>()) * 6);
+ const T a2((six_euler_squared - boost::math::constants::pi_sqr<T>()) / 12);
+
+ const T inverse_tgamma_series = z * ((a2 * z + a1) * z + a0);
+
+ return 1 / inverse_tgamma_series;
+ }
+
+ // Scale the argument up for the calculation of lgamma,
+ // and use downward recursion later for the final result.
+ const T min_arg_for_recursion = minimum_argument_for_bernoulli_recursion<T>();
+
+ int n_recur;
+
+ if(zz < min_arg_for_recursion)
+ {
+ n_recur = boost::math::itrunc(min_arg_for_recursion - zz) + 1;
+
+ zz += n_recur;
+ }
+ else
+ {
+ n_recur = 0;
+ }
+
+ const T log_gamma_value = lgamma_imp(zz, pol, lanczos::undefined_lanczos());
+
+ if(log_gamma_value > tools::log_max_value<T>())
+ return policies::raise_overflow_error<T>(function, 0, pol);
+
+ T gamma_value = exp(log_gamma_value);
+
+ // Rescale the result using downward recursion if necessary.
+ if(n_recur)
+ {
+ // The order of divides is important, if we keep subtracting 1 from zz
+ // we DO NOT get back to z (cancellation error). Further if z < epsilon
+ // we would end up dividing by zero. Also in order to prevent spurious
+ // overflow with the first division, we must save dividing by |z| till last,
+ // so the optimal order of divides is z+1, z+2, z+3...z+n_recur-1,z.
+ zz = fabs(z) + 1;
+ for(int k = 1; k < n_recur; ++k)
+ {
+ gamma_value /= zz;
+ zz += 1;
+ }
+ gamma_value /= fabs(z);
+ }
+
+ // Return the result, accounting for possible negative arguments.
+ if(b_neg)
+ {
+ // Provide special error analysis for:
+ // * arguments in the neighborhood of a negative integer
+ // * arguments exactly equal to a negative integer.
+
+ // Check if the argument of tgamma is exactly equal to a negative integer.
+ if(floor_of_z_is_equal_to_z)
+ return policies::raise_pole_error<T>(function, "Evaluation of tgamma at a negative integer %1%.", z, pol);
+
+ gamma_value *= sinpx(z);
+
+ BOOST_MATH_INSTRUMENT_VARIABLE(gamma_value);
+
+ const bool result_is_too_large_to_represent = ( (abs(gamma_value) < 1)
+ && ((tools::max_value<T>() * abs(gamma_value)) < boost::math::constants::pi<T>()));
+
+ if(result_is_too_large_to_represent)
+ return policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol);
+
+ gamma_value = -boost::math::constants::pi<T>() / gamma_value;
+ BOOST_MATH_INSTRUMENT_VARIABLE(gamma_value);
+
+ if(gamma_value == 0)
+ return policies::raise_underflow_error<T>(function, "Result of tgamma is too small to represent.", pol);
+
+ if((boost::math::fpclassify)(gamma_value) == static_cast<int>(FP_SUBNORMAL))
+ return policies::raise_denorm_error<T>(function, "Result of tgamma is denormalized.", gamma_value, pol);
+ }
+
+ return gamma_value;
+}
+
+template <class T, class Policy>
+inline T log_gamma_near_1(const T& z, Policy const& pol)
+{
+ //
+ // This is for the multiprecision case where there is
+ // no lanczos support...
+ //
+ BOOST_MATH_STD_USING // ADL of std names
+
+ BOOST_ASSERT(fabs(z) < 1);
+
+ T result = -constants::euler<T>() * z;
+
+ T power_term = z * z;
+ T term;
+ unsigned j = 0;
+
+ do
+ {
+ term = boost::math::zeta<T>(j + 2, pol) * power_term / (j + 2);
+ if(j & 1)
+ result -= term;
+ else
+ result += term;
+ power_term *= z;
+ ++j;
+ } while(fabs(result) * tools::epsilon<T>() < fabs(term));
+
+ return result;
+}
+
+template <class T, class Policy>
+T lgamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos&, int* sign)
+{
+ BOOST_MATH_STD_USING
+
+ static const char* function = "boost::math::lgamma<%1%>(%1%)";
+
+ // Check if the argument of lgamma is identically zero.
+ const bool is_at_zero = (z == 0);
+
+ if(is_at_zero)
+ return policies::raise_domain_error<T>(function, "Evaluation of lgamma at zero %1%.", z, pol);
+ if((boost::math::isnan)(z))
+ return policies::raise_domain_error<T>(function, "Evaluation of lgamma at %1%.", z, pol);
+ if((boost::math::isinf)(z))
+ return policies::raise_overflow_error<T>(function, 0, pol);
+
+ const bool b_neg = (z < 0);
+
+ const bool floor_of_z_is_equal_to_z = (floor(z) == z);
+
+ // Special case handling of small factorials:
+ if((!b_neg) && floor_of_z_is_equal_to_z && (z < boost::math::max_factorial<T>::value))
+ {
+ return log(boost::math::unchecked_factorial<T>(itrunc(z) - 1));
+ }
+
+ // Make a local, unsigned copy of the input argument.
+ T zz((!b_neg) ? z : -z);
+
+ const T min_arg_for_recursion = minimum_argument_for_bernoulli_recursion<T>();
+
+ T log_gamma_value;
+
+ if (zz < min_arg_for_recursion)
+ {
+ // Here we simply take the logarithm of tgamma(). This is somewhat
+ // inefficient, but simple. The rationale is that the argument here
+ // is relatively small and overflow is not expected to be likely.
+ if(fabs(z - 1) < 0.25)
+ {
+ return log_gamma_near_1(T(zz - 1), pol);
+ }
+ else if(fabs(z - 2) < 0.25)
+ {
+ return log_gamma_near_1(T(zz - 2), pol) + log(zz - 1);
+ }
+ else if (z > -tools::root_epsilon<T>())
+ {
+ // Reflection formula may fail if z is very close to zero, let the series
+ // expansion for tgamma close to zero do the work:
+ log_gamma_value = log(abs(gamma_imp(z, pol, lanczos::undefined_lanczos())));
+ if (sign)
+ {
+ *sign = z < 0 ? -1 : 1;
+ }
+ return log_gamma_value;
+ }
+ else
+ {
+ // No issue with spurious overflow in reflection formula,
+ // just fall through to regular code:
+ log_gamma_value = log(abs(gamma_imp(zz, pol, lanczos::undefined_lanczos())));
+ }
+ }
+ else
+ {
+ // Perform the Bernoulli series expansion of Stirling's approximation.
+
+ const std::size_t number_of_bernoullis_b2n = highest_bernoulli_index<T>();
+
+ T one_over_x_pow_two_n_minus_one = 1 / zz;
+ const T one_over_x2 = one_over_x_pow_two_n_minus_one * one_over_x_pow_two_n_minus_one;
+ T sum = (boost::math::bernoulli_b2n<T>(1) / 2) * one_over_x_pow_two_n_minus_one;
+ const T target_epsilon_to_break_loop = (sum * boost::math::tools::epsilon<T>()) * T(1.0E-10F);
+
+ for(std::size_t n = 2U; n < number_of_bernoullis_b2n; ++n)
+ {
+ one_over_x_pow_two_n_minus_one *= one_over_x2;
+
+ const std::size_t n2 = static_cast<std::size_t>(n * 2U);
+
+ const T term = (boost::math::bernoulli_b2n<T>(static_cast<int>(n)) * one_over_x_pow_two_n_minus_one) / (n2 * (n2 - 1U));
+
+ if((n >= 8U) && (abs(term) < target_epsilon_to_break_loop))
+ {
+ // We have reached the desired precision in Stirling's expansion.
+ // Adding additional terms to the sum of this divergent asymptotic
+ // expansion will not improve the result.
+
+ // Break from the loop.
+ break;
+ }
+
+ sum += term;
+ }
+
+ // Complete Stirling's approximation.
+ const T half_ln_two_pi = log(boost::math::constants::two_pi<T>()) / 2;
+
+ log_gamma_value = ((((zz - boost::math::constants::half<T>()) * log(zz)) - zz) + half_ln_two_pi) + sum;
+ }
+
+ int sign_of_result = 1;
+
+ if(b_neg)
+ {
+ // Provide special error analysis if the argument is exactly
+ // equal to a negative integer.
+
+ // Check if the argument of lgamma is exactly equal to a negative integer.
+ if(floor_of_z_is_equal_to_z)
+ return policies::raise_pole_error<T>(function, "Evaluation of lgamma at a negative integer %1%.", z, pol);
+
+ T t = sinpx(z);
+
+ if(t < 0)
+ {
+ t = -t;
+ }
+ else
+ {
+ sign_of_result = -sign_of_result;
+ }
+
+ log_gamma_value = - log_gamma_value
+ + log(boost::math::constants::pi<T>())
+ - log(t);
+ }
+
+ if(sign != static_cast<int*>(0U)) { *sign = sign_of_result; }
+
+ return log_gamma_value;
+}
+
+//
+// This helper calculates tgamma(dz+1)-1 without cancellation errors,
+// used by the upper incomplete gamma with z < 1:
+//
+template <class T, class Policy, class Lanczos>
+T tgammap1m1_imp(T dz, Policy const& pol, const Lanczos& l)
+{
+ BOOST_MATH_STD_USING
+
+ typedef typename policies::precision<T,Policy>::type precision_type;
+
+ typedef typename mpl::if_<
+ mpl::or_<
+ mpl::less_equal<precision_type, mpl::int_<0> >,
+ mpl::greater<precision_type, mpl::int_<113> >
+ >,
+ typename mpl::if_<
+ mpl::and_<is_same<Lanczos, lanczos::lanczos24m113>, mpl::greater<precision_type, mpl::int_<0> > >,
+ mpl::int_<113>,
+ mpl::int_<0>
+ >::type,
+ typename mpl::if_<
+ mpl::less_equal<precision_type, mpl::int_<64> >,
+ mpl::int_<64>, mpl::int_<113> >::type
+ >::type tag_type;
+
+ T result;
+ if(dz < 0)
+ {
+ if(dz < -0.5)
+ {
+ // Best method is simply to subtract 1 from tgamma:
+ result = boost::math::tgamma(1+dz, pol) - 1;
+ BOOST_MATH_INSTRUMENT_CODE(result);
+ }
+ else
+ {
+ // Use expm1 on lgamma:
+ result = boost::math::expm1(-boost::math::log1p(dz, pol)
+ + lgamma_small_imp<T>(dz+2, dz + 1, dz, tag_type(), pol, l));
+ BOOST_MATH_INSTRUMENT_CODE(result);
+ }
+ }
+ else
+ {
+ if(dz < 2)
+ {
+ // Use expm1 on lgamma:
+ result = boost::math::expm1(lgamma_small_imp<T>(dz+1, dz, dz-1, tag_type(), pol, l), pol);
+ BOOST_MATH_INSTRUMENT_CODE(result);
+ }
+ else
+ {
+ // Best method is simply to subtract 1 from tgamma:
+ result = boost::math::tgamma(1+dz, pol) - 1;
+ BOOST_MATH_INSTRUMENT_CODE(result);
+ }
+ }
+
+ return result;
+}
+
+template <class T, class Policy>
+inline T tgammap1m1_imp(T z, Policy const& pol,
+ const ::boost::math::lanczos::undefined_lanczos&)
+{
+ BOOST_MATH_STD_USING // ADL of std names
+
+ if(fabs(z) < 0.55)
+ {
+ return boost::math::expm1(log_gamma_near_1(z, pol));
+ }
+ return boost::math::expm1(boost::math::lgamma(1 + z, pol));
+}
+
+//
+// Series representation for upper fraction when z is small:
+//
+template <class T>
+struct small_gamma2_series
+{
+ typedef T result_type;
+
+ small_gamma2_series(T a_, T x_) : result(-x_), x(-x_), apn(a_+1), n(1){}
+
+ T operator()()
+ {
+ T r = result / (apn);
+ result *= x;
+ result /= ++n;
+ apn += 1;
+ return r;
+ }
+
+private:
+ T result, x, apn;
+ int n;
+};
+//
+// calculate power term prefix (z^a)(e^-z) used in the non-normalised
+// incomplete gammas:
+//
+template <class T, class Policy>
+T full_igamma_prefix(T a, T z, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ T prefix;
+ T alz = a * log(z);
+
+ if(z >= 1)
+ {
+ if((alz < tools::log_max_value<T>()) && (-z > tools::log_min_value<T>()))
+ {
+ prefix = pow(z, a) * exp(-z);
+ }
+ else if(a >= 1)
+ {
+ prefix = pow(z / exp(z/a), a);
+ }
+ else
+ {
+ prefix = exp(alz - z);
+ }
+ }
+ else
+ {
+ if(alz > tools::log_min_value<T>())
+ {
+ prefix = pow(z, a) * exp(-z);
+ }
+ else if(z/a < tools::log_max_value<T>())
+ {
+ prefix = pow(z / exp(z/a), a);
+ }
+ else
+ {
+ prefix = exp(alz - z);
+ }
+ }
+ //
+ // This error handling isn't very good: it happens after the fact
+ // rather than before it...
+ //
+ if((boost::math::fpclassify)(prefix) == (int)FP_INFINITE)
+ return policies::raise_overflow_error<T>("boost::math::detail::full_igamma_prefix<%1%>(%1%, %1%)", "Result of incomplete gamma function is too large to represent.", pol);
+
+ return prefix;
+}
+//
+// Compute (z^a)(e^-z)/tgamma(a)
+// most if the error occurs in this function:
+//
+template <class T, class Policy, class Lanczos>
+T regularised_gamma_prefix(T a, T z, const Policy& pol, const Lanczos& l)
+{
+ BOOST_MATH_STD_USING
+ T agh = a + static_cast<T>(Lanczos::g()) - T(0.5);
+ T prefix;
+ T d = ((z - a) - static_cast<T>(Lanczos::g()) + T(0.5)) / agh;
+
+ if(a < 1)
+ {
+ //
+ // We have to treat a < 1 as a special case because our Lanczos
+ // approximations are optimised against the factorials with a > 1,
+ // and for high precision types especially (128-bit reals for example)
+ // very small values of a can give rather eroneous results for gamma
+ // unless we do this:
+ //
+ // TODO: is this still required? Lanczos approx should be better now?
+ //
+ if(z <= tools::log_min_value<T>())
+ {
+ // Oh dear, have to use logs, should be free of cancellation errors though:
+ return exp(a * log(z) - z - lgamma_imp(a, pol, l));
+ }
+ else
+ {
+ // direct calculation, no danger of overflow as gamma(a) < 1/a
+ // for small a.
+ return pow(z, a) * exp(-z) / gamma_imp(a, pol, l);
+ }
+ }
+ else if((fabs(d*d*a) <= 100) && (a > 150))
+ {
+ // special case for large a and a ~ z.
+ prefix = a * boost::math::log1pmx(d, pol) + z * static_cast<T>(0.5 - Lanczos::g()) / agh;
+ prefix = exp(prefix);
+ }
+ else
+ {
+ //
+ // general case.
+ // direct computation is most accurate, but use various fallbacks
+ // for different parts of the problem domain:
+ //
+ T alz = a * log(z / agh);
+ T amz = a - z;
+ if(((std::min)(alz, amz) <= tools::log_min_value<T>()) || ((std::max)(alz, amz) >= tools::log_max_value<T>()))
+ {
+ T amza = amz / a;
+ if(((std::min)(alz, amz)/2 > tools::log_min_value<T>()) && ((std::max)(alz, amz)/2 < tools::log_max_value<T>()))
+ {
+ // compute square root of the result and then square it:
+ T sq = pow(z / agh, a / 2) * exp(amz / 2);
+ prefix = sq * sq;
+ }
+ else if(((std::min)(alz, amz)/4 > tools::log_min_value<T>()) && ((std::max)(alz, amz)/4 < tools::log_max_value<T>()) && (z > a))
+ {
+ // compute the 4th root of the result then square it twice:
+ T sq = pow(z / agh, a / 4) * exp(amz / 4);
+ prefix = sq * sq;
+ prefix *= prefix;
+ }
+ else if((amza > tools::log_min_value<T>()) && (amza < tools::log_max_value<T>()))
+ {
+ prefix = pow((z * exp(amza)) / agh, a);
+ }
+ else
+ {
+ prefix = exp(alz + amz);
+ }
+ }
+ else
+ {
+ prefix = pow(z / agh, a) * exp(amz);
+ }
+ }
+ prefix *= sqrt(agh / boost::math::constants::e<T>()) / Lanczos::lanczos_sum_expG_scaled(a);
+ return prefix;
+}
+//
+// And again, without Lanczos support:
+//
+template <class T, class Policy>
+T regularised_gamma_prefix(T a, T z, const Policy& pol, const lanczos::undefined_lanczos&)
+{
+ BOOST_MATH_STD_USING
+
+ T limit = (std::max)(T(10), a);
+ T sum = detail::lower_gamma_series(a, limit, pol) / a;
+ sum += detail::upper_gamma_fraction(a, limit, ::boost::math::policies::get_epsilon<T, Policy>());
+
+ if(a < 10)
+ {
+ // special case for small a:
+ T prefix = pow(z / 10, a);
+ prefix *= exp(10-z);
+ if(0 == prefix)
+ {
+ prefix = pow((z * exp((10-z)/a)) / 10, a);
+ }
+ prefix /= sum;
+ return prefix;
+ }
+
+ T zoa = z / a;
+ T amz = a - z;
+ T alzoa = a * log(zoa);
+ T prefix;
+ if(((std::min)(alzoa, amz) <= tools::log_min_value<T>()) || ((std::max)(alzoa, amz) >= tools::log_max_value<T>()))
+ {
+ T amza = amz / a;
+ if((amza <= tools::log_min_value<T>()) || (amza >= tools::log_max_value<T>()))
+ {
+ prefix = exp(alzoa + amz);
+ }
+ else
+ {
+ prefix = pow(zoa * exp(amza), a);
+ }
+ }
+ else
+ {
+ prefix = pow(zoa, a) * exp(amz);
+ }
+ prefix /= sum;
+ return prefix;
+}
+//
+// Upper gamma fraction for very small a:
+//
+template <class T, class Policy>
+inline T tgamma_small_upper_part(T a, T x, const Policy& pol, T* pgam = 0, bool invert = false, T* pderivative = 0)
+{
+ BOOST_MATH_STD_USING // ADL of std functions.
+ //
+ // Compute the full upper fraction (Q) when a is very small:
+ //
+ T result;
+ result = boost::math::tgamma1pm1(a, pol);
+ if(pgam)
+ *pgam = (result + 1) / a;
+ T p = boost::math::powm1(x, a, pol);
+ result -= p;
+ result /= a;
+ detail::small_gamma2_series<T> s(a, x);
+ boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>() - 10;
+ p += 1;
+ if(pderivative)
+ *pderivative = p / (*pgam * exp(x));
+ T init_value = invert ? *pgam : 0;
+ result = -p * tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, (init_value - result) / p);
+ policies::check_series_iterations<T>("boost::math::tgamma_small_upper_part<%1%>(%1%, %1%)", max_iter, pol);
+ if(invert)
+ result = -result;
+ return result;
+}
+//
+// Upper gamma fraction for integer a:
+//
+template <class T, class Policy>
+inline T finite_gamma_q(T a, T x, Policy const& pol, T* pderivative = 0)
+{
+ //
+ // Calculates normalised Q when a is an integer:
+ //
+ BOOST_MATH_STD_USING
+ T e = exp(-x);
+ T sum = e;
+ if(sum != 0)
+ {
+ T term = sum;
+ for(unsigned n = 1; n < a; ++n)
+ {
+ term /= n;
+ term *= x;
+ sum += term;
+ }
+ }
+ if(pderivative)
+ {
+ *pderivative = e * pow(x, a) / boost::math::unchecked_factorial<T>(itrunc(T(a - 1), pol));
+ }
+ return sum;
+}
+//
+// Upper gamma fraction for half integer a:
+//
+template <class T, class Policy>
+T finite_half_gamma_q(T a, T x, T* p_derivative, const Policy& pol)
+{
+ //
+ // Calculates normalised Q when a is a half-integer:
+ //
+ BOOST_MATH_STD_USING
+ T e = boost::math::erfc(sqrt(x), pol);
+ if((e != 0) && (a > 1))
+ {
+ T term = exp(-x) / sqrt(constants::pi<T>() * x);
+ term *= x;
+ static const T half = T(1) / 2;
+ term /= half;
+ T sum = term;
+ for(unsigned n = 2; n < a; ++n)
+ {
+ term /= n - half;
+ term *= x;
+ sum += term;
+ }
+ e += sum;
+ if(p_derivative)
+ {
+ *p_derivative = 0;
+ }
+ }
+ else if(p_derivative)
+ {
+ // We'll be dividing by x later, so calculate derivative * x:
+ *p_derivative = sqrt(x) * exp(-x) / constants::root_pi<T>();
+ }
+ return e;
+}
+//
+// Main incomplete gamma entry point, handles all four incomplete gamma's:
+//
+template <class T, class Policy>
+T gamma_incomplete_imp(T a, T x, bool normalised, bool invert,
+ const Policy& pol, T* p_derivative)
+{
+ static const char* function = "boost::math::gamma_p<%1%>(%1%, %1%)";
+ if(a <= 0)
+ return policies::raise_domain_error<T>(function, "Argument a to the incomplete gamma function must be greater than zero (got a=%1%).", a, pol);
+ if(x < 0)
+ return policies::raise_domain_error<T>(function, "Argument x to the incomplete gamma function must be >= 0 (got x=%1%).", x, pol);
+
+ BOOST_MATH_STD_USING
+
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+
+ T result = 0; // Just to avoid warning C4701: potentially uninitialized local variable 'result' used
+
+ if(a >= max_factorial<T>::value && !normalised)
+ {
+ //
+ // When we're computing the non-normalized incomplete gamma
+ // and a is large the result is rather hard to compute unless
+ // we use logs. There are really two options - if x is a long
+ // way from a in value then we can reliably use methods 2 and 4
+ // below in logarithmic form and go straight to the result.
+ // Otherwise we let the regularized gamma take the strain
+ // (the result is unlikely to unerflow in the central region anyway)
+ // and combine with lgamma in the hopes that we get a finite result.
+ //
+ if(invert && (a * 4 < x))
+ {
+ // This is method 4 below, done in logs:
+ result = a * log(x) - x;
+ if(p_derivative)
+ *p_derivative = exp(result);
+ result += log(upper_gamma_fraction(a, x, policies::get_epsilon<T, Policy>()));
+ }
+ else if(!invert && (a > 4 * x))
+ {
+ // This is method 2 below, done in logs:
+ result = a * log(x) - x;
+ if(p_derivative)
+ *p_derivative = exp(result);
+ T init_value = 0;
+ result += log(detail::lower_gamma_series(a, x, pol, init_value) / a);
+ }
+ else
+ {
+ result = gamma_incomplete_imp(a, x, true, invert, pol, p_derivative);
+ if(result == 0)
+ {
+ if(invert)
+ {
+ // Try http://functions.wolfram.com/06.06.06.0039.01
+ result = 1 + 1 / (12 * a) + 1 / (288 * a * a);
+ result = log(result) - a + (a - 0.5f) * log(a) + log(boost::math::constants::root_two_pi<T>());
+ if(p_derivative)
+ *p_derivative = exp(a * log(x) - x);
+ }
+ else
+ {
+ // This is method 2 below, done in logs, we're really outside the
+ // range of this method, but since the result is almost certainly
+ // infinite, we should probably be OK:
+ result = a * log(x) - x;
+ if(p_derivative)
+ *p_derivative = exp(result);
+ T init_value = 0;
+ result += log(detail::lower_gamma_series(a, x, pol, init_value) / a);
+ }
+ }
+ else
+ {
+ result = log(result) + boost::math::lgamma(a, pol);
+ }
+ }
+ if(result > tools::log_max_value<T>())
+ return policies::raise_overflow_error<T>(function, 0, pol);
+ return exp(result);
+ }
+
+ BOOST_ASSERT((p_derivative == 0) || (normalised == true));
+
+ bool is_int, is_half_int;
+ bool is_small_a = (a < 30) && (a <= x + 1) && (x < tools::log_max_value<T>());
+ if(is_small_a)
+ {
+ T fa = floor(a);
+ is_int = (fa == a);
+ is_half_int = is_int ? false : (fabs(fa - a) == 0.5f);
+ }
+ else
+ {
+ is_int = is_half_int = false;
+ }
+
+ int eval_method;
+
+ if(is_int && (x > 0.6))
+ {
+ // calculate Q via finite sum:
+ invert = !invert;
+ eval_method = 0;
+ }
+ else if(is_half_int && (x > 0.2))
+ {
+ // calculate Q via finite sum for half integer a:
+ invert = !invert;
+ eval_method = 1;
+ }
+ else if((x < tools::root_epsilon<T>()) && (a > 1))
+ {
+ eval_method = 6;
+ }
+ else if(x < 0.5)
+ {
+ //
+ // Changeover criterion chosen to give a changeover at Q ~ 0.33
+ //
+ if(-0.4 / log(x) < a)
+ {
+ eval_method = 2;
+ }
+ else
+ {
+ eval_method = 3;
+ }
+ }
+ else if(x < 1.1)
+ {
+ //
+ // Changover here occurs when P ~ 0.75 or Q ~ 0.25:
+ //
+ if(x * 0.75f < a)
+ {
+ eval_method = 2;
+ }
+ else
+ {
+ eval_method = 3;
+ }
+ }
+ else
+ {
+ //
+ // Begin by testing whether we're in the "bad" zone
+ // where the result will be near 0.5 and the usual
+ // series and continued fractions are slow to converge:
+ //
+ bool use_temme = false;
+ if(normalised && std::numeric_limits<T>::is_specialized && (a > 20))
+ {
+ T sigma = fabs((x-a)/a);
+ if((a > 200) && (policies::digits<T, Policy>() <= 113))
+ {
+ //
+ // This limit is chosen so that we use Temme's expansion
+ // only if the result would be larger than about 10^-6.
+ // Below that the regular series and continued fractions
+ // converge OK, and if we use Temme's method we get increasing
+ // errors from the dominant erfc term as it's (inexact) argument
+ // increases in magnitude.
+ //
+ if(20 / a > sigma * sigma)
+ use_temme = true;
+ }
+ else if(policies::digits<T, Policy>() <= 64)
+ {
+ // Note in this zone we can't use Temme's expansion for
+ // types longer than an 80-bit real:
+ // it would require too many terms in the polynomials.
+ if(sigma < 0.4)
+ use_temme = true;
+ }
+ }
+ if(use_temme)
+ {
+ eval_method = 5;
+ }
+ else
+ {
+ //
+ // Regular case where the result will not be too close to 0.5.
+ //
+ // Changeover here occurs at P ~ Q ~ 0.5
+ // Note that series computation of P is about x2 faster than continued fraction
+ // calculation of Q, so try and use the CF only when really necessary, especially
+ // for small x.
+ //
+ if(x - (1 / (3 * x)) < a)
+ {
+ eval_method = 2;
+ }
+ else
+ {
+ eval_method = 4;
+ invert = !invert;
+ }
+ }
+ }
+
+ switch(eval_method)
+ {
+ case 0:
+ {
+ result = finite_gamma_q(a, x, pol, p_derivative);
+ if(normalised == false)
+ result *= boost::math::tgamma(a, pol);
+ break;
+ }
+ case 1:
+ {
+ result = finite_half_gamma_q(a, x, p_derivative, pol);
+ if(normalised == false)
+ result *= boost::math::tgamma(a, pol);
+ if(p_derivative && (*p_derivative == 0))
+ *p_derivative = regularised_gamma_prefix(a, x, pol, lanczos_type());
+ break;
+ }
+ case 2:
+ {
+ // Compute P:
+ result = normalised ? regularised_gamma_prefix(a, x, pol, lanczos_type()) : full_igamma_prefix(a, x, pol);
+ if(p_derivative)
+ *p_derivative = result;
+ if(result != 0)
+ {
+ //
+ // If we're going to be inverting the result then we can
+ // reduce the number of series evaluations by quite
+ // a few iterations if we set an initial value for the
+ // series sum based on what we'll end up subtracting it from
+ // at the end.
+ // Have to be careful though that this optimization doesn't
+ // lead to spurious numberic overflow. Note that the
+ // scary/expensive overflow checks below are more often
+ // than not bypassed in practice for "sensible" input
+ // values:
+ //
+ T init_value = 0;
+ bool optimised_invert = false;
+ if(invert)
+ {
+ init_value = (normalised ? 1 : boost::math::tgamma(a, pol));
+ if(normalised || (result >= 1) || (tools::max_value<T>() * result > init_value))
+ {
+ init_value /= result;
+ if(normalised || (a < 1) || (tools::max_value<T>() / a > init_value))
+ {
+ init_value *= -a;
+ optimised_invert = true;
+ }
+ else
+ init_value = 0;
+ }
+ else
+ init_value = 0;
+ }
+ result *= detail::lower_gamma_series(a, x, pol, init_value) / a;
+ if(optimised_invert)
+ {
+ invert = false;
+ result = -result;
+ }
+ }
+ break;
+ }
+ case 3:
+ {
+ // Compute Q:
+ invert = !invert;
+ T g;
+ result = tgamma_small_upper_part(a, x, pol, &g, invert, p_derivative);
+ invert = false;
+ if(normalised)
+ result /= g;
+ break;
+ }
+ case 4:
+ {
+ // Compute Q:
+ result = normalised ? regularised_gamma_prefix(a, x, pol, lanczos_type()) : full_igamma_prefix(a, x, pol);
+ if(p_derivative)
+ *p_derivative = result;
+ if(result != 0)
+ result *= upper_gamma_fraction(a, x, policies::get_epsilon<T, Policy>());
+ break;
+ }
+ case 5:
+ {
+ //
+ // Use compile time dispatch to the appropriate
+ // Temme asymptotic expansion. This may be dead code
+ // if T does not have numeric limits support, or has
+ // too many digits for the most precise version of
+ // these expansions, in that case we'll be calling
+ // an empty function.
+ //
+ typedef typename policies::precision<T, Policy>::type precision_type;
+
+ typedef typename mpl::if_<
+ mpl::or_<mpl::equal_to<precision_type, mpl::int_<0> >,
+ mpl::greater<precision_type, mpl::int_<113> > >,
+ mpl::int_<0>,
+ typename mpl::if_<
+ mpl::less_equal<precision_type, mpl::int_<53> >,
+ mpl::int_<53>,
+ typename mpl::if_<
+ mpl::less_equal<precision_type, mpl::int_<64> >,
+ mpl::int_<64>,
+ mpl::int_<113>
+ >::type
+ >::type
+ >::type tag_type;
+
+ result = igamma_temme_large(a, x, pol, static_cast<tag_type const*>(0));
+ if(x >= a)
+ invert = !invert;
+ if(p_derivative)
+ *p_derivative = regularised_gamma_prefix(a, x, pol, lanczos_type());
+ break;
+ }
+ case 6:
+ {
+ // x is so small that P is necessarily very small too,
+ // use http://functions.wolfram.com/GammaBetaErf/GammaRegularized/06/01/05/01/01/
+ result = !normalised ? pow(x, a) / (a) : pow(x, a) / boost::math::tgamma(a + 1, pol);
+ result *= 1 - a * x / (a + 1);
+ }
+ }
+
+ if(normalised && (result > 1))
+ result = 1;
+ if(invert)
+ {
+ T gam = normalised ? 1 : boost::math::tgamma(a, pol);
+ result = gam - result;
+ }
+ if(p_derivative)
+ {
+ //
+ // Need to convert prefix term to derivative:
+ //
+ if((x < 1) && (tools::max_value<T>() * x < *p_derivative))
+ {
+ // overflow, just return an arbitrarily large value:
+ *p_derivative = tools::max_value<T>() / 2;
+ }
+
+ *p_derivative /= x;
+ }
+
+ return result;
+}
+
+//
+// Ratios of two gamma functions:
+//
+template <class T, class Policy, class Lanczos>
+T tgamma_delta_ratio_imp_lanczos(T z, T delta, const Policy& pol, const Lanczos& l)
+{
+ BOOST_MATH_STD_USING
+ if(z < tools::epsilon<T>())
+ {
+ //
+ // We get spurious numeric overflow unless we're very careful, this
+ // can occur either inside Lanczos::lanczos_sum(z) or in the
+ // final combination of terms, to avoid this, split the product up
+ // into 2 (or 3) parts:
+ //
+ // G(z) / G(L) = 1 / (z * G(L)) ; z < eps, L = z + delta = delta
+ // z * G(L) = z * G(lim) * (G(L)/G(lim)) ; lim = largest factorial
+ //
+ if(boost::math::max_factorial<T>::value < delta)
+ {
+ T ratio = tgamma_delta_ratio_imp_lanczos(delta, T(boost::math::max_factorial<T>::value - delta), pol, l);
+ ratio *= z;
+ ratio *= boost::math::unchecked_factorial<T>(boost::math::max_factorial<T>::value - 1);
+ return 1 / ratio;
+ }
+ else
+ {
+ return 1 / (z * boost::math::tgamma(z + delta, pol));
+ }
+ }
+ T zgh = static_cast<T>(z + Lanczos::g() - constants::half<T>());
+ T result;
+ if(z + delta == z)
+ {
+ if(fabs(delta) < 10)
+ result = exp((constants::half<T>() - z) * boost::math::log1p(delta / zgh, pol));
+ else
+ result = 1;
+ }
+ else
+ {
+ if(fabs(delta) < 10)
+ {
+ result = exp((constants::half<T>() - z) * boost::math::log1p(delta / zgh, pol));
+ }
+ else
+ {
+ result = pow(zgh / (zgh + delta), z - constants::half<T>());
+ }
+ // Split the calculation up to avoid spurious overflow:
+ result *= Lanczos::lanczos_sum(z) / Lanczos::lanczos_sum(T(z + delta));
+ }
+ result *= pow(constants::e<T>() / (zgh + delta), delta);
+ return result;
+}
+//
+// And again without Lanczos support this time:
+//
+template <class T, class Policy>
+T tgamma_delta_ratio_imp_lanczos(T z, T delta, const Policy& pol, const lanczos::undefined_lanczos&)
+{
+ BOOST_MATH_STD_USING
+ //
+ // The upper gamma fraction is *very* slow for z < 6, actually it's very
+ // slow to converge everywhere but recursing until z > 6 gets rid of the
+ // worst of it's behaviour.
+ //
+ T prefix = 1;
+ T zd = z + delta;
+ while((zd < 6) && (z < 6))
+ {
+ prefix /= z;
+ prefix *= zd;
+ z += 1;
+ zd += 1;
+ }
+ if(delta < 10)
+ {
+ prefix *= exp(-z * boost::math::log1p(delta / z, pol));
+ }
+ else
+ {
+ prefix *= pow(z / zd, z);
+ }
+ prefix *= pow(constants::e<T>() / zd, delta);
+ T sum = detail::lower_gamma_series(z, z, pol) / z;
+ sum += detail::upper_gamma_fraction(z, z, ::boost::math::policies::get_epsilon<T, Policy>());
+ T sumd = detail::lower_gamma_series(zd, zd, pol) / zd;
+ sumd += detail::upper_gamma_fraction(zd, zd, ::boost::math::policies::get_epsilon<T, Policy>());
+ sum /= sumd;
+ if(fabs(tools::max_value<T>() / prefix) < fabs(sum))
+ return policies::raise_overflow_error<T>("boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)", "Result of tgamma is too large to represent.", pol);
+ return sum * prefix;
+}
+
+template <class T, class Policy>
+T tgamma_delta_ratio_imp(T z, T delta, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if((z <= 0) || (z + delta <= 0))
+ {
+ // This isn't very sofisticated, or accurate, but it does work:
+ return boost::math::tgamma(z, pol) / boost::math::tgamma(z + delta, pol);
+ }
+
+ if(floor(delta) == delta)
+ {
+ if(floor(z) == z)
+ {
+ //
+ // Both z and delta are integers, see if we can just use table lookup
+ // of the factorials to get the result:
+ //
+ if((z <= max_factorial<T>::value) && (z + delta <= max_factorial<T>::value))
+ {
+ return unchecked_factorial<T>((unsigned)itrunc(z, pol) - 1) / unchecked_factorial<T>((unsigned)itrunc(T(z + delta), pol) - 1);
+ }
+ }
+ if(fabs(delta) < 20)
+ {
+ //
+ // delta is a small integer, we can use a finite product:
+ //
+ if(delta == 0)
+ return 1;
+ if(delta < 0)
+ {
+ z -= 1;
+ T result = z;
+ while(0 != (delta += 1))
+ {
+ z -= 1;
+ result *= z;
+ }
+ return result;
+ }
+ else
+ {
+ T result = 1 / z;
+ while(0 != (delta -= 1))
+ {
+ z += 1;
+ result /= z;
+ }
+ return result;
+ }
+ }
+ }
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+ return tgamma_delta_ratio_imp_lanczos(z, delta, pol, lanczos_type());
+}
+
+template <class T, class Policy>
+T tgamma_ratio_imp(T x, T y, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if((x <= 0) || (boost::math::isinf)(x))
+ return policies::raise_domain_error<T>("boost::math::tgamma_ratio<%1%>(%1%, %1%)", "Gamma function ratios only implemented for positive arguments (got a=%1%).", x, pol);
+ if((y <= 0) || (boost::math::isinf)(y))
+ return policies::raise_domain_error<T>("boost::math::tgamma_ratio<%1%>(%1%, %1%)", "Gamma function ratios only implemented for positive arguments (got b=%1%).", y, pol);
+
+ if(x <= tools::min_value<T>())
+ {
+ // Special case for denorms...Ugh.
+ T shift = ldexp(T(1), tools::digits<T>());
+ return shift * tgamma_ratio_imp(T(x * shift), y, pol);
+ }
+
+ if((x < max_factorial<T>::value) && (y < max_factorial<T>::value))
+ {
+ // Rather than subtracting values, lets just call the gamma functions directly:
+ return boost::math::tgamma(x, pol) / boost::math::tgamma(y, pol);
+ }
+ T prefix = 1;
+ if(x < 1)
+ {
+ if(y < 2 * max_factorial<T>::value)
+ {
+ // We need to sidestep on x as well, otherwise we'll underflow
+ // before we get to factor in the prefix term:
+ prefix /= x;
+ x += 1;
+ while(y >= max_factorial<T>::value)
+ {
+ y -= 1;
+ prefix /= y;
+ }
+ return prefix * boost::math::tgamma(x, pol) / boost::math::tgamma(y, pol);
+ }
+ //
+ // result is almost certainly going to underflow to zero, try logs just in case:
+ //
+ return exp(boost::math::lgamma(x, pol) - boost::math::lgamma(y, pol));
+ }
+ if(y < 1)
+ {
+ if(x < 2 * max_factorial<T>::value)
+ {
+ // We need to sidestep on y as well, otherwise we'll overflow
+ // before we get to factor in the prefix term:
+ prefix *= y;
+ y += 1;
+ while(x >= max_factorial<T>::value)
+ {
+ x -= 1;
+ prefix *= x;
+ }
+ return prefix * boost::math::tgamma(x, pol) / boost::math::tgamma(y, pol);
+ }
+ //
+ // Result will almost certainly overflow, try logs just in case:
+ //
+ return exp(boost::math::lgamma(x, pol) - boost::math::lgamma(y, pol));
+ }
+ //
+ // Regular case, x and y both large and similar in magnitude:
+ //
+ return boost::math::tgamma_delta_ratio(x, y - x, pol);
+}
+
+template <class T, class Policy>
+T gamma_p_derivative_imp(T a, T x, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+ //
+ // Usual error checks first:
+ //
+ if(a <= 0)
+ return policies::raise_domain_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", "Argument a to the incomplete gamma function must be greater than zero (got a=%1%).", a, pol);
+ if(x < 0)
+ return policies::raise_domain_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", "Argument x to the incomplete gamma function must be >= 0 (got x=%1%).", x, pol);
+ //
+ // Now special cases:
+ //
+ if(x == 0)
+ {
+ return (a > 1) ? 0 :
+ (a == 1) ? 1 : policies::raise_overflow_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", 0, pol);
+ }
+ //
+ // Normal case:
+ //
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+ T f1 = detail::regularised_gamma_prefix(a, x, pol, lanczos_type());
+ if((x < 1) && (tools::max_value<T>() * x < f1))
+ {
+ // overflow:
+ return policies::raise_overflow_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", 0, pol);
+ }
+ if(f1 == 0)
+ {
+ // Underflow in calculation, use logs instead:
+ f1 = a * log(x) - x - lgamma(a, pol) - log(x);
+ f1 = exp(f1);
+ }
+ else
+ f1 /= x;
+
+ return f1;
+}
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type
+ tgamma(T z, const Policy& /* pol */, const mpl::true_)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::gamma_imp(static_cast<value_type>(z), forwarding_policy(), evaluation_type()), "boost::math::tgamma<%1%>(%1%)");
+}
+
+template <class T, class Policy>
+struct igamma_initializer
+{
+ struct init
+ {
+ init()
+ {
+ typedef typename policies::precision<T, Policy>::type precision_type;
+
+ typedef typename mpl::if_<
+ mpl::or_<mpl::equal_to<precision_type, mpl::int_<0> >,
+ mpl::greater<precision_type, mpl::int_<113> > >,
+ mpl::int_<0>,
+ typename mpl::if_<
+ mpl::less_equal<precision_type, mpl::int_<53> >,
+ mpl::int_<53>,
+ typename mpl::if_<
+ mpl::less_equal<precision_type, mpl::int_<64> >,
+ mpl::int_<64>,
+ mpl::int_<113>
+ >::type
+ >::type
+ >::type tag_type;
+
+ do_init(tag_type());
+ }
+ template <int N>
+ static void do_init(const mpl::int_<N>&)
+ {
+ // If std::numeric_limits<T>::digits is zero, we must not call
+ // our inituialization code here as the precision presumably
+ // varies at runtime, and will not have been set yet. Plus the
+ // code requiring initialization isn't called when digits == 0.
+ if(std::numeric_limits<T>::digits)
+ {
+ boost::math::gamma_p(static_cast<T>(400), static_cast<T>(400), Policy());
+ }
+ }
+ static void do_init(const mpl::int_<53>&){}
+ void force_instantiate()const{}
+ };
+ static const init initializer;
+ static void force_instantiate()
+ {
+ initializer.force_instantiate();
+ }
+};
+
+template <class T, class Policy>
+const typename igamma_initializer<T, Policy>::init igamma_initializer<T, Policy>::initializer;
+
+template <class T, class Policy>
+struct lgamma_initializer
+{
+ struct init
+ {
+ init()
+ {
+ typedef typename policies::precision<T, Policy>::type precision_type;
+ typedef typename mpl::if_<
+ mpl::and_<
+ mpl::less_equal<precision_type, mpl::int_<64> >,
+ mpl::greater<precision_type, mpl::int_<0> >
+ >,
+ mpl::int_<64>,
+ typename mpl::if_<
+ mpl::and_<
+ mpl::less_equal<precision_type, mpl::int_<113> >,
+ mpl::greater<precision_type, mpl::int_<0> >
+ >,
+ mpl::int_<113>, mpl::int_<0> >::type
+ >::type tag_type;
+ do_init(tag_type());
+ }
+ static void do_init(const mpl::int_<64>&)
+ {
+ boost::math::lgamma(static_cast<T>(2.5), Policy());
+ boost::math::lgamma(static_cast<T>(1.25), Policy());
+ boost::math::lgamma(static_cast<T>(1.75), Policy());
+ }
+ static void do_init(const mpl::int_<113>&)
+ {
+ boost::math::lgamma(static_cast<T>(2.5), Policy());
+ boost::math::lgamma(static_cast<T>(1.25), Policy());
+ boost::math::lgamma(static_cast<T>(1.5), Policy());
+ boost::math::lgamma(static_cast<T>(1.75), Policy());
+ }
+ static void do_init(const mpl::int_<0>&)
+ {
+ }
+ void force_instantiate()const{}
+ };
+ static const init initializer;
+ static void force_instantiate()
+ {
+ initializer.force_instantiate();
+ }
+};
+
+template <class T, class Policy>
+const typename lgamma_initializer<T, Policy>::init lgamma_initializer<T, Policy>::initializer;
+
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma(T1 a, T2 z, const Policy&, const mpl::false_)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ igamma_initializer<value_type, forwarding_policy>::force_instantiate();
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(
+ detail::gamma_incomplete_imp(static_cast<value_type>(a),
+ static_cast<value_type>(z), false, true,
+ forwarding_policy(), static_cast<value_type*>(0)), "boost::math::tgamma<%1%>(%1%, %1%)");
+}
+
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma(T1 a, T2 z, const mpl::false_ tag)
+{
+ return tgamma(a, z, policies::policy<>(), tag);
+}
+
+
+} // namespace detail
+
+template <class T>
+inline typename tools::promote_args<T>::type
+ tgamma(T z)
+{
+ return tgamma(z, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type
+ lgamma(T z, int* sign, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ detail::lgamma_initializer<value_type, forwarding_policy>::force_instantiate();
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::lgamma_imp(static_cast<value_type>(z), forwarding_policy(), evaluation_type(), sign), "boost::math::lgamma<%1%>(%1%)");
+}
+
+template <class T>
+inline typename tools::promote_args<T>::type
+ lgamma(T z, int* sign)
+{
+ return lgamma(z, sign, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type
+ lgamma(T x, const Policy& pol)
+{
+ return ::boost::math::lgamma(x, 0, pol);
+}
+
+template <class T>
+inline typename tools::promote_args<T>::type
+ lgamma(T x)
+{
+ return ::boost::math::lgamma(x, 0, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type
+ tgamma1pm1(T z, const Policy& /* pol */)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ return policies::checked_narrowing_cast<typename remove_cv<result_type>::type, forwarding_policy>(detail::tgammap1m1_imp(static_cast<value_type>(z), forwarding_policy(), evaluation_type()), "boost::math::tgamma1pm1<%!%>(%1%)");
+}
+
+template <class T>
+inline typename tools::promote_args<T>::type
+ tgamma1pm1(T z)
+{
+ return tgamma1pm1(z, policies::policy<>());
+}
+
+//
+// Full upper incomplete gamma:
+//
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma(T1 a, T2 z)
+{
+ //
+ // Type T2 could be a policy object, or a value, select the
+ // right overload based on T2:
+ //
+ typedef typename policies::is_policy<T2>::type maybe_policy;
+ return detail::tgamma(a, z, maybe_policy());
+}
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma(T1 a, T2 z, const Policy& pol)
+{
+ return detail::tgamma(a, z, pol, mpl::false_());
+}
+//
+// Full lower incomplete gamma:
+//
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma_lower(T1 a, T2 z, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ detail::igamma_initializer<value_type, forwarding_policy>::force_instantiate();
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(
+ detail::gamma_incomplete_imp(static_cast<value_type>(a),
+ static_cast<value_type>(z), false, false,
+ forwarding_policy(), static_cast<value_type*>(0)), "tgamma_lower<%1%>(%1%, %1%)");
+}
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma_lower(T1 a, T2 z)
+{
+ return tgamma_lower(a, z, policies::policy<>());
+}
+//
+// Regularised upper incomplete gamma:
+//
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ gamma_q(T1 a, T2 z, const Policy& /* pol */)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ detail::igamma_initializer<value_type, forwarding_policy>::force_instantiate();
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(
+ detail::gamma_incomplete_imp(static_cast<value_type>(a),
+ static_cast<value_type>(z), true, true,
+ forwarding_policy(), static_cast<value_type*>(0)), "gamma_q<%1%>(%1%, %1%)");
+}
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ gamma_q(T1 a, T2 z)
+{
+ return gamma_q(a, z, policies::policy<>());
+}
+//
+// Regularised lower incomplete gamma:
+//
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ gamma_p(T1 a, T2 z, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ detail::igamma_initializer<value_type, forwarding_policy>::force_instantiate();
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(
+ detail::gamma_incomplete_imp(static_cast<value_type>(a),
+ static_cast<value_type>(z), true, false,
+ forwarding_policy(), static_cast<value_type*>(0)), "gamma_p<%1%>(%1%, %1%)");
+}
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ gamma_p(T1 a, T2 z)
+{
+ return gamma_p(a, z, policies::policy<>());
+}
+
+// ratios of gamma functions:
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma_delta_ratio(T1 z, T2 delta, const Policy& /* pol */)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::tgamma_delta_ratio_imp(static_cast<value_type>(z), static_cast<value_type>(delta), forwarding_policy()), "boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)");
+}
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma_delta_ratio(T1 z, T2 delta)
+{
+ return tgamma_delta_ratio(z, delta, policies::policy<>());
+}
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma_ratio(T1 a, T2 b, const Policy&)
+{
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::tgamma_ratio_imp(static_cast<value_type>(a), static_cast<value_type>(b), forwarding_policy()), "boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)");
+}
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ tgamma_ratio(T1 a, T2 b)
+{
+ return tgamma_ratio(a, b, policies::policy<>());
+}
+
+template <class T1, class T2, class Policy>
+inline typename tools::promote_args<T1, T2>::type
+ gamma_p_derivative(T1 a, T2 x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T1, T2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::gamma_p_derivative_imp(static_cast<value_type>(a), static_cast<value_type>(x), forwarding_policy()), "boost::math::gamma_p_derivative<%1%>(%1%, %1%)");
+}
+template <class T1, class T2>
+inline typename tools::promote_args<T1, T2>::type
+ gamma_p_derivative(T1 a, T2 x)
+{
+ return gamma_p_derivative(a, x, policies::policy<>());
+}
+
+} // namespace math
+} // namespace boost
+
+#ifdef BOOST_MSVC
+# pragma warning(pop)
+#endif
+
+#include <boost/math/special_functions/detail/igamma_inverse.hpp>
+#include <boost/math/special_functions/detail/gamma_inva.hpp>
+#include <boost/math/special_functions/erf.hpp>
+
+#endif // BOOST_MATH_SF_GAMMA_HPP
+
+
+
+
View Lorry Controller Status