golden hour

/usr/include/c++/4.8.2/tr1

File/Folder	Size	Actions
array	6.8 KB
bessel_function.tcc	21.6 KB
beta_function.tcc	5.47 KB
ccomplex	1.23 KB
cctype	1.38 KB
cfenv	1.96 KB
cfloat	1.35 KB
cinttypes	2.2 KB
climits	1.42 KB
cmath	36.55 KB
complex	12.04 KB
complex.h	1.23 KB
cstdarg	1.22 KB
cstdbool	1.31 KB
cstdint	2.56 KB
cstdio	1.44 KB
cstdlib	1.74 KB
ctgmath	1.22 KB
ctime	1.21 KB
ctype.h	1.18 KB
cwchar	1.67 KB
cwctype	1.42 KB
ell_integral.tcc	26.85 KB
exp_integral.tcc	15.41 KB
fenv.h	1.18 KB
float.h	1.18 KB
functional	69.15 KB
functional_hash.h	5.7 KB
gamma.tcc	13.97 KB
hashtable.h	40.56 KB
hashtable_policy.h	24.64 KB
hypergeometric.tcc	27.07 KB
inttypes.h	1.24 KB
legendre_function.tcc	10.32 KB
limits.h	1.19 KB
math.h	4.45 KB
memory	1.75 KB
modified_bessel_func.tcc	15.35 KB
poly_hermite.tcc	3.61 KB
poly_laguerre.tcc	11.08 KB
random	1.55 KB
random.h	71.48 KB
random.tcc	52.73 KB
regex	90.77 KB
riemann_zeta.tcc	13.34 KB
shared_ptr.h	31.91 KB
special_function_util.h	4.71 KB
stdarg.h	1.19 KB
stdbool.h	1.19 KB
stdint.h	1.19 KB
stdio.h	1.18 KB
stdlib.h	1.45 KB
tgmath.h	1.23 KB
tuple	11.83 KB
type_traits	18.57 KB
unordered_map	1.54 KB
unordered_map.h	9.98 KB
unordered_set	1.54 KB
unordered_set.h	9.32 KB
utility	3.15 KB
wchar.h	1.22 KB
wctype.h	1.23 KB

Edit: gamma.tcc

// Special functions -*- C++ -*-

// Copyright (C) 2006-2013 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.

// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
// <http://www.gnu.org/licenses/>.

/** @file tr1/gamma.tcc
 *  This is an internal header file, included by other library headers.
 *  Do not attempt to use it directly. @headername{tr1/cmath}
 */

//
// ISO C++ 14882 TR1: 5.2  Special functions
//

// Written by Edward Smith-Rowland based on:
//   (1) Handbook of Mathematical Functions,
//       ed. Milton Abramowitz and Irene A. Stegun,
//       Dover Publications,
//       Section 6, pp. 253-266
//   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
//   (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
//       W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
//       2nd ed, pp. 213-216
//   (4) Gamma, Exploring Euler's Constant, Julian Havil,
//       Princeton, 2003.

#ifndef _GLIBCXX_TR1_GAMMA_TCC
#define _GLIBCXX_TR1_GAMMA_TCC 1

#include "special_function_util.h"

namespace std _GLIBCXX_VISIBILITY(default)
{
namespace tr1
{
  // Implementation-space details.
  namespace __detail
  {
  _GLIBCXX_BEGIN_NAMESPACE_VERSION

/**
     *   @brief This returns Bernoulli numbers from a table or by summation
     *          for larger values.
     *
     *   Recursion is unstable.
     *
     *   @param __n the order n of the Bernoulli number.
     *   @return  The Bernoulli number of order n.
     */
    template <typename _Tp>
    _Tp
    __bernoulli_series(unsigned int __n)
    {

static const _Tp __num[28] = {
        _Tp(1UL),                        -_Tp(1UL) / _Tp(2UL),
        _Tp(1UL) / _Tp(6UL),             _Tp(0UL),
        -_Tp(1UL) / _Tp(30UL),           _Tp(0UL),
        _Tp(1UL) / _Tp(42UL),            _Tp(0UL),
        -_Tp(1UL) / _Tp(30UL),           _Tp(0UL),
        _Tp(5UL) / _Tp(66UL),            _Tp(0UL),
        -_Tp(691UL) / _Tp(2730UL),       _Tp(0UL),
        _Tp(7UL) / _Tp(6UL),             _Tp(0UL),
        -_Tp(3617UL) / _Tp(510UL),       _Tp(0UL),
        _Tp(43867UL) / _Tp(798UL),       _Tp(0UL),
        -_Tp(174611) / _Tp(330UL),       _Tp(0UL),
        _Tp(854513UL) / _Tp(138UL),      _Tp(0UL),
        -_Tp(236364091UL) / _Tp(2730UL), _Tp(0UL),
        _Tp(8553103UL) / _Tp(6UL),       _Tp(0UL)
      };

if (__n == 0)
        return _Tp(1);

if (__n == 1)
        return -_Tp(1) / _Tp(2);

//  Take care of the rest of the odd ones.
      if (__n % 2 == 1)
        return _Tp(0);

//  Take care of some small evens that are painful for the series.
      if (__n < 28)
        return __num[__n];

_Tp __fact = _Tp(1);
      if ((__n / 2) % 2 == 0)
        __fact *= _Tp(-1);
      for (unsigned int __k = 1; __k <= __n; ++__k)
        __fact *= __k / (_Tp(2) * __numeric_constants<_Tp>::__pi());
      __fact *= _Tp(2);

_Tp __sum = _Tp(0);
      for (unsigned int __i = 1; __i < 1000; ++__i)
        {
          _Tp __term = std::pow(_Tp(__i), -_Tp(__n));
          if (__term < std::numeric_limits<_Tp>::epsilon())
            break;
          __sum += __term;
        }

return __fact * __sum;
    }

/**
     *   @brief This returns Bernoulli number \f$B_n\f$.
     *
     *   @param __n the order n of the Bernoulli number.
     *   @return  The Bernoulli number of order n.
     */
    template<typename _Tp>
    inline _Tp
    __bernoulli(int __n)
    { return __bernoulli_series<_Tp>(__n); }

/**
     *   @brief Return \f$log(\Gamma(x))\f$ by asymptotic expansion
     *          with Bernoulli number coefficients.  This is like
     *          Sterling's approximation.
     *
     *   @param __x The argument of the log of the gamma function.
     *   @return  The logarithm of the gamma function.
     */
    template<typename _Tp>
    _Tp
    __log_gamma_bernoulli(_Tp __x)
    {
      _Tp __lg = (__x - _Tp(0.5L)) * std::log(__x) - __x
               + _Tp(0.5L) * std::log(_Tp(2)
               * __numeric_constants<_Tp>::__pi());

const _Tp __xx = __x * __x;
      _Tp __help = _Tp(1) / __x;
      for ( unsigned int __i = 1; __i < 20; ++__i )
        {
          const _Tp __2i = _Tp(2 * __i);
          __help /= __2i * (__2i - _Tp(1)) * __xx;
          __lg += __bernoulli<_Tp>(2 * __i) * __help;
        }

return __lg;
    }

/**
     *   @brief Return \f$log(\Gamma(x))\f$ by the Lanczos method.
     *          This method dominates all others on the positive axis I think.
     *
     *   @param __x The argument of the log of the gamma function.
     *   @return  The logarithm of the gamma function.
     */
    template<typename _Tp>
    _Tp
    __log_gamma_lanczos(_Tp __x)
    {
      const _Tp __xm1 = __x - _Tp(1);

static const _Tp __lanczos_cheb_7[9] = {
       _Tp( 0.99999999999980993227684700473478L),
       _Tp( 676.520368121885098567009190444019L),
       _Tp(-1259.13921672240287047156078755283L),
       _Tp( 771.3234287776530788486528258894L),
       _Tp(-176.61502916214059906584551354L),
       _Tp( 12.507343278686904814458936853L),
       _Tp(-0.13857109526572011689554707L),
       _Tp( 9.984369578019570859563e-6L),
       _Tp( 1.50563273514931155834e-7L)
      };

static const _Tp __LOGROOT2PI
          = _Tp(0.9189385332046727417803297364056176L);

_Tp __sum = __lanczos_cheb_7[0];
      for(unsigned int __k = 1; __k < 9; ++__k)
        __sum += __lanczos_cheb_7[__k] / (__xm1 + __k);

const _Tp __term1 = (__xm1 + _Tp(0.5L))
                        * std::log((__xm1 + _Tp(7.5L))
                       / __numeric_constants<_Tp>::__euler());
      const _Tp __term2 = __LOGROOT2PI + std::log(__sum);
      const _Tp __result = __term1 + (__term2 - _Tp(7));

return __result;
    }

/**
     *   @brief Return \f$ log(|\Gamma(x)|) \f$.
     *          This will return values even for \f$ x < 0 \f$.
     *          To recover the sign of \f$ \Gamma(x) \f$ for
     *          any argument use @a __log_gamma_sign.
     *
     *   @param __x The argument of the log of the gamma function.
     *   @return  The logarithm of the gamma function.
     */
    template<typename _Tp>
    _Tp
    __log_gamma(_Tp __x)
    {
      if (__x > _Tp(0.5L))
        return __log_gamma_lanczos(__x);
      else
        {
          const _Tp __sin_fact
                 = std::abs(std::sin(__numeric_constants<_Tp>::__pi() * __x));
          if (__sin_fact == _Tp(0))
            std::__throw_domain_error(__N("Argument is nonpositive integer "
                                          "in __log_gamma"));
          return __numeric_constants<_Tp>::__lnpi()
                     - std::log(__sin_fact)
                     - __log_gamma_lanczos(_Tp(1) - __x);
        }
    }

/**
     *   @brief Return the sign of \f$ \Gamma(x) \f$.
     *          At nonpositive integers zero is returned.
     *
     *   @param __x The argument of the gamma function.
     *   @return  The sign of the gamma function.
     */
    template<typename _Tp>
    _Tp
    __log_gamma_sign(_Tp __x)
    {
      if (__x > _Tp(0))
        return _Tp(1);
      else
        {
          const _Tp __sin_fact
                  = std::sin(__numeric_constants<_Tp>::__pi() * __x);
          if (__sin_fact > _Tp(0))
            return (1);
          else if (__sin_fact < _Tp(0))
            return -_Tp(1);
          else
            return _Tp(0);
        }
    }

/**
     *   @brief Return the logarithm of the binomial coefficient.
     *   The binomial coefficient is given by:
     *   @f[
     *   \left(  \right) = \frac{n!}{(n-k)! k!}
     *   @f]
     *
     *   @param __n The first argument of the binomial coefficient.
     *   @param __k The second argument of the binomial coefficient.
     *   @return  The binomial coefficient.
     */
    template<typename _Tp>
    _Tp
    __log_bincoef(unsigned int __n, unsigned int __k)
    {
      //  Max e exponent before overflow.
      static const _Tp __max_bincoeff
                      = std::numeric_limits<_Tp>::max_exponent10
                      * std::log(_Tp(10)) - _Tp(1);
#if _GLIBCXX_USE_C99_MATH_TR1
      _Tp __coeff =  std::tr1::lgamma(_Tp(1 + __n))
                  - std::tr1::lgamma(_Tp(1 + __k))
                  - std::tr1::lgamma(_Tp(1 + __n - __k));
#else
      _Tp __coeff =  __log_gamma(_Tp(1 + __n))
                  - __log_gamma(_Tp(1 + __k))
                  - __log_gamma(_Tp(1 + __n - __k));
#endif
    }

/**
     *   @brief Return the binomial coefficient.
     *   The binomial coefficient is given by:
     *   @f[
     *   \left(  \right) = \frac{n!}{(n-k)! k!}
     *   @f]
     *
     *   @param __n The first argument of the binomial coefficient.
     *   @param __k The second argument of the binomial coefficient.
     *   @return  The binomial coefficient.
     */
    template<typename _Tp>
    _Tp
    __bincoef(unsigned int __n, unsigned int __k)
    {
      //  Max e exponent before overflow.
      static const _Tp __max_bincoeff
                      = std::numeric_limits<_Tp>::max_exponent10
                      * std::log(_Tp(10)) - _Tp(1);

const _Tp __log_coeff = __log_bincoef<_Tp>(__n, __k);
      if (__log_coeff > __max_bincoeff)
        return std::numeric_limits<_Tp>::quiet_NaN();
      else
        return std::exp(__log_coeff);
    }

/**
     *   @brief Return \f$ \Gamma(x) \f$.
     *
     *   @param __x The argument of the gamma function.
     *   @return  The gamma function.
     */
    template<typename _Tp>
    inline _Tp
    __gamma(_Tp __x)
    { return std::exp(__log_gamma(__x)); }

/**
     *   @brief  Return the digamma function by series expansion.
     *   The digamma or @f$ \psi(x) @f$ function is defined by
     *   @f[
     *     \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
     *   @f]
     *
     *   The series is given by:
     *   @f[
     *     \psi(x) = -\gamma_E - \frac{1}{x}
     *              \sum_{k=1}^{\infty} \frac{x}{k(x + k)}
     *   @f]
     */
    template<typename _Tp>
    _Tp
    __psi_series(_Tp __x)
    {
      _Tp __sum = -__numeric_constants<_Tp>::__gamma_e() - _Tp(1) / __x;
      const unsigned int __max_iter = 100000;
      for (unsigned int __k = 1; __k < __max_iter; ++__k)
        {
          const _Tp __term = __x / (__k * (__k + __x));
          __sum += __term;
          if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon())
            break;
        }
      return __sum;
    }

/**
     *   @brief  Return the digamma function for large argument.
     *   The digamma or @f$ \psi(x) @f$ function is defined by
     *   @f[
     *     \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
     *   @f]
     *
     *   The asymptotic series is given by:
     *   @f[
     *     \psi(x) = \ln(x) - \frac{1}{2x}
     *             - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}}
     *   @f]
     */
    template<typename _Tp>
    _Tp
    __psi_asymp(_Tp __x)
    {
      _Tp __sum = std::log(__x) - _Tp(0.5L) / __x;
      const _Tp __xx = __x * __x;
      _Tp __xp = __xx;
      const unsigned int __max_iter = 100;
      for (unsigned int __k = 1; __k < __max_iter; ++__k)
        {
          const _Tp __term = __bernoulli<_Tp>(2 * __k) / (2 * __k * __xp);
          __sum -= __term;
          if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon())
            break;
          __xp *= __xx;
        }
      return __sum;
    }

/**
     *   @brief  Return the digamma function.
     *   The digamma or @f$ \psi(x) @f$ function is defined by
     *   @f[
     *     \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
     *   @f]
     *   For negative argument the reflection formula is used:
     *   @f[
     *     \psi(x) = \psi(1-x) - \pi \cot(\pi x)
     *   @f]
     */
    template<typename _Tp>
    _Tp
    __psi(_Tp __x)
    {
      const int __n = static_cast<int>(__x + 0.5L);
      const _Tp __eps = _Tp(4) * std::numeric_limits<_Tp>::epsilon();
      if (__n <= 0 && std::abs(__x - _Tp(__n)) < __eps)
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__x < _Tp(0))
        {
          const _Tp __pi = __numeric_constants<_Tp>::__pi();
          return __psi(_Tp(1) - __x)
               - __pi * std::cos(__pi * __x) / std::sin(__pi * __x);
        }
      else if (__x > _Tp(100))
        return __psi_asymp(__x);
      else
        return __psi_series(__x);
    }

/**
     *   @brief  Return the polygamma function @f$ \psi^{(n)}(x) @f$.
     * 
     *   The polygamma function is related to the Hurwitz zeta function:
     *   @f[
     *     \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x)
     *   @f]
     */
    template<typename _Tp>
    _Tp
    __psi(unsigned int __n, _Tp __x)
    {
      if (__x <= _Tp(0))
        std::__throw_domain_error(__N("Argument out of range "
                                      "in __psi"));
      else if (__n == 0)
        return __psi(__x);
      else
        {
          const _Tp __hzeta = __hurwitz_zeta(_Tp(__n + 1), __x);
#if _GLIBCXX_USE_C99_MATH_TR1
          const _Tp __ln_nfact = std::tr1::lgamma(_Tp(__n + 1));
#else
          const _Tp __ln_nfact = __log_gamma(_Tp(__n + 1));
#endif
          _Tp __result = std::exp(__ln_nfact) * __hzeta;
          if (__n % 2 == 1)
            __result = -__result;
          return __result;
        }
    }

_GLIBCXX_END_NAMESPACE_VERSION
  } // namespace std::tr1::__detail
}
}

#endif // _GLIBCXX_TR1_GAMMA_TCC