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

// Copyright (C) 2006, 2007, 2008
// 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 2, 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.
//
// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.
//
// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/riemann_zeta.tcc
 *  This is an internal header file, included by other library headers.
 *  You should not attempt to use it directly.
 */

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

// Written by Edward Smith-Rowland based on:
//   (1) Handbook of Mathematical Functions,
//       Ed. by Milton Abramowitz and Irene A. Stegun,
//       Dover Publications, New-York, Section 5, pp. 807-808.
//   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
//   (3) Gamma, Exploring Euler's Constant, Julian Havil,
//       Princeton, 2003.

#ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC
#define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1

#include "special_function_util.h"

namespace std
{
namespace tr1
{

  // [5.2] Special functions

  // Implementation-space details.
  namespace __detail
  {

    /**
     *   @brief  Compute the Riemann zeta function @f$ \zeta(s) @f$
     *           by summation for s > 1.
     * 
     *   The Riemann zeta function is defined by:
     *    \f[
     *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
     *    \f]
     *   For s < 1 use the reflection formula:
     *    \f[
     *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
     *    \f]
     */
    template<typename _Tp>
    _Tp
    __riemann_zeta_sum(const _Tp __s)
    {
      //  A user shouldn't get to this.
      if (__s < _Tp(1))
        std::__throw_domain_error(__N("Bad argument in zeta sum."));

      const unsigned int max_iter = 10000;
      _Tp __zeta = _Tp(0);
      for (unsigned int __k = 1; __k < max_iter; ++__k)
        {
          _Tp __term = std::pow(static_cast<_Tp>(__k), -__s);
          if (__term < std::numeric_limits<_Tp>::epsilon())
            {
              break;
            }
          __zeta += __term;
        }

      return __zeta;
    }


    /**
     *   @brief  Evaluate the Riemann zeta function @f$ \zeta(s) @f$
     *           by an alternate series for s > 0.
     * 
     *   The Riemann zeta function is defined by:
     *    \f[
     *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
     *    \f]
     *   For s < 1 use the reflection formula:
     *    \f[
     *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
     *    \f]
     */
    template<typename _Tp>
    _Tp
    __riemann_zeta_alt(const _Tp __s)
    {
      _Tp __sgn = _Tp(1);
      _Tp __zeta = _Tp(0);
      for (unsigned int __i = 1; __i < 10000000; ++__i)
        {
          _Tp __term = __sgn / std::pow(__i, __s);
          if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon())
            break;
          __zeta += __term;
          __sgn *= _Tp(-1);
        }
      __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s);

      return __zeta;
    }


    /**
     *   @brief  Evaluate the Riemann zeta function by series for all s != 1.
     *           Convergence is great until largish negative numbers.
     *           Then the convergence of the > 0 sum gets better.
     *
     *   The series is:
     *    \f[
     *      \zeta(s) = \frac{1}{1-2^{1-s}}
     *                 \sum_{n=0}^{\infty} \frac{1}{2^{n+1}}
     *                 \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s}
     *    \f]
     *   Havil 2003, p. 206.
     *
     *   The Riemann zeta function is defined by:
     *    \f[
     *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
     *    \f]
     *   For s < 1 use the reflection formula:
     *    \f[
     *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
     *    \f]
     */
    template<typename _Tp>
    _Tp
    __riemann_zeta_glob(const _Tp __s)
    {
      _Tp __zeta = _Tp(0);

      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      //  Max e exponent before overflow.
      const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10
                               * std::log(_Tp(10)) - _Tp(1);

      //  This series works until the binomial coefficient blows up
      //  so use reflection.
      if (__s < _Tp(0))
        {
#if _GLIBCXX_USE_C99_MATH_TR1
          if (std::tr1::fmod(__s,_Tp(2)) == _Tp(0))
            return _Tp(0);
          else
#endif
            {
              _Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s);
              __zeta *= std::pow(_Tp(2)
                     * __numeric_constants<_Tp>::__pi(), __s)
                     * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
#if _GLIBCXX_USE_C99_MATH_TR1
                     * std::exp(std::tr1::lgamma(_Tp(1) - __s))
#else
                     * std::exp(__log_gamma(_Tp(1) - __s))
#endif
                     / __numeric_constants<_Tp>::__pi();
              return __zeta;
            }
        }

      _Tp __num = _Tp(0.5L);
      const unsigned int __maxit = 10000;
      for (unsigned int __i = 0; __i < __maxit; ++__i)
        {
          bool __punt = false;
          _Tp __sgn = _Tp(1);
          _Tp __term = _Tp(0);
          for (unsigned int __j = 0; __j <= __i; ++__j)
            {
#if _GLIBCXX_USE_C99_MATH_TR1
              _Tp __bincoeff =  std::tr1::lgamma(_Tp(1 + __i))
                              - std::tr1::lgamma(_Tp(1 + __j))
                              - std::tr1::lgamma(_Tp(1 + __i - __j));
#else
              _Tp __bincoeff =  __log_gamma(_Tp(1 + __i))
                              - __log_gamma(_Tp(1 + __j))
                              - __log_gamma(_Tp(1 + __i - __j));
#endif
              if (__bincoeff > __max_bincoeff)
                {
                  //  This only gets hit for x << 0.
                  __punt = true;
                  break;
                }
              __bincoeff = std::exp(__bincoeff);
              __term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s);
              __sgn *= _Tp(-1);
            }
          if (__punt)
            break;
          __term *= __num;
          __zeta += __term;
          if (std::abs(__term/__zeta) < __eps)
            break;
          __num *= _Tp(0.5L);
        }

      __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s);

      return __zeta;
    }


    /**
     *   @brief  Compute the Riemann zeta function @f$ \zeta(s) @f$
     *           using the product over prime factors.
     *    \f[
     *      \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}}
     *    \f]
     *    where @f$ {p_i} @f$ are the prime numbers.
     * 
     *   The Riemann zeta function is defined by:
     *    \f[
     *      \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1
     *    \f]
     *   For s < 1 use the reflection formula:
     *    \f[
     *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
     *    \f]
     */
    template<typename _Tp>
    _Tp
    __riemann_zeta_product(const _Tp __s)
    {
      static const _Tp __prime[] = {
        _Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19),
        _Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47),
        _Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79),
        _Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109)
      };
      static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp);

      _Tp __zeta = _Tp(1);
      for (unsigned int __i = 0; __i < __num_primes; ++__i)
        {
          const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s);
          __zeta *= __fact;
          if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon())
            break;
        }

      __zeta = _Tp(1) / __zeta;

      return __zeta;
    }


    /**
     *   @brief  Return the Riemann zeta function @f$ \zeta(s) @f$.
     * 
     *   The Riemann zeta function is defined by:
     *    \f[
     *      \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1
     *                 \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2})
     *                 \Gamma (1 - s) \zeta (1 - s) for s < 1
     *    \f]
     *   For s < 1 use the reflection formula:
     *    \f[
     *      \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s)
     *    \f]
     */
    template<typename _Tp>
    _Tp
    __riemann_zeta(const _Tp __s)
    {
      if (__isnan(__s))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__s == _Tp(1))
        return std::numeric_limits<_Tp>::infinity();
      else if (__s < -_Tp(19))
        {
          _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s);
          __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s)
                 * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
#if _GLIBCXX_USE_C99_MATH_TR1
                 * std::exp(std::tr1::lgamma(_Tp(1) - __s))
#else
                 * std::exp(__log_gamma(_Tp(1) - __s))
#endif
                 / __numeric_constants<_Tp>::__pi();
          return __zeta;
        }
      else if (__s < _Tp(20))
        {
          //  Global double sum or McLaurin?
          bool __glob = true;
          if (__glob)
            return __riemann_zeta_glob(__s);
          else
            {
              if (__s > _Tp(1))
                return __riemann_zeta_sum(__s);
              else
                {
                  _Tp __zeta = std::pow(_Tp(2)
                                * __numeric_constants<_Tp>::__pi(), __s)
                         * std::sin(__numeric_constants<_Tp>::__pi_2() * __s)
#if _GLIBCXX_USE_C99_MATH_TR1
                             * std::tr1::tgamma(_Tp(1) - __s)
#else
                             * std::exp(__log_gamma(_Tp(1) - __s))
#endif
                             * __riemann_zeta_sum(_Tp(1) - __s);
                  return __zeta;
                }
            }
        }
      else
        return __riemann_zeta_product(__s);
    }


    /**
     *   @brief  Return the Hurwitz zeta function @f$ \zeta(x,s) @f$
     *           for all s != 1 and x > -1.
     * 
     *   The Hurwitz zeta function is defined by:
     *   @f[
     *     \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s}
     *   @f]
     *   The Riemann zeta function is a special case:
     *   @f[
     *     \zeta(s) = \zeta(1,s)
     *   @f]
     * 
     *   This functions uses the double sum that converges for s != 1
     *   and x > -1:
     *   @f[
     *     \zeta(x,s) = \frac{1}{s-1}
     *                \sum_{n=0}^{\infty} \frac{1}{n + 1}
     *                \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s}
     *   @f]
     */
    template<typename _Tp>
    _Tp
    __hurwitz_zeta_glob(const _Tp __a, const _Tp __s)
    {
      _Tp __zeta = _Tp(0);

      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      //  Max e exponent before overflow.
      const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10
                               * std::log(_Tp(10)) - _Tp(1);

      const unsigned int __maxit = 10000;
      for (unsigned int __i = 0; __i < __maxit; ++__i)
        {
          bool __punt = false;
          _Tp __sgn = _Tp(1);
          _Tp __term = _Tp(0);
          for (unsigned int _Í,Î,_j = 0; __j <= __i; ++__j)
            {
#if _GLIBCXX_USE_C99_MATH_TR1
              _Tp __bincoeff =  std::tr1::lgamma(_Tp(1 + __i))
                              - std::tr1::lgamma(_Tp(1 + __j))
                              - std::tr1::lgamma(_Tp(1 + __i - __j));
#else
              _Tp __bincoeff =  __log_gamma(_Tp(1 + __i))
                              - __log_gamma(_Tp(1 + __j))
                              - __log_gamma(_Tp(1 + __i - __j));
#endif
              if (__bincoeff > __max_bincoeff)
                {
                  //  This only gets hit for x << 0.
                  __punt = true;
                  break;
                }
              __bincoeff = std::exp(__bincoeff);
              __term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s);
              __sgn *= _Tp(-1);
            }
          if (__punt)
            break;
          __term /= _Tp(__i + 1);
          if (std::abs(__term / __zeta) < __eps)
            break;
          __zeta += __term;
        }

      __zeta /= __s - _Tp(1);

      return __zeta;
    }


    /**
     *   @brief  Return the Hurwitz zeta function @f$ \zeta(x,s) @f$
     *           for all s != 1 and x > -1.
     * 
     *   The Hurwitz zeta function is defined by:
     *   @f[
     *     \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s}
     *   @f]
     *   The Riemann zeta function is a special case:
     *   @f[
     *     \zeta(s) = \zeta(1,s)
     *   @f]
     */
    template<typename _Tp>
    inline _Tp
    __hurwitz_zeta(const _Tp __a, const _Tp __s)
    {
      return __hurwitz_zeta_glob(__a, __s);
    }

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

#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC
// TR1 wctype.h -*- C++ -*-

// Copyright (C) 2006, 2007 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/wctype.h
 *  This is a TR1 C++ Library header. 
 */

#ifndef _GLIBCXX_TR1_WCTYPE_H
#define _GLIBCXX_TR1_WCTYPE_H 1

#include <tr1/cwctype>

#endif // _GLIBCXX_TR1_WCTYPE_H
// TR1 utility -*- C++ -*-

// Copyright (C) 2004, 2005, 2006, 2007 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/utility
 *  This is a TR1 C++ Library header. 
 */

#ifndef _GLIBCXX_TR1_UTILITY
#define _GLIBCXX_TR1_UTILITY 1

#pragma GCC system_header

#if defined(_GLIBCXX_INCLUDE_AS_CXX0X)
#  error TR1 header cannot be included from C++0x header
#endif

#include <bits/c++config.h>
#include <bits/stl_relops.h>
#include <bits/stl_pair.h>

#if defined(_GLIBCXX_INCLUDE_AS_TR1)
#  include <tr1_impl/utility>
#else
#  define _GLIBCXX_INCLUDE_AS_TR1
#  define _GLIBCXX_BEGIN_NAMESPACE_TR1 namespace tr1 {
#  define _GLIBCXX_END_NAMESPACE_TR1 }
#  define _GLIBCXX_TR1 tr1::
#  include <tr1_impl/utility>
#  undef _GLIBCXX_TR1
#  undef _GLIBCXX_END_NAMESPACE_TR1
#  undef _GLIBCXX_BEGIN_NAMESPACE_TR1
#  undef _GLIBCXX_INCLUDE_AS_TR1
#endif

#endif // _GLIBCXX_TR1_UTILITY
// TR1 cfloat -*- C++ -*-

// Copyright (C) 2006, 2007 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/cfloat
 *  This is a TR1 C++ Library header. 
 */

#ifndef _GLIBCXX_TR1_CFLOAT
#define _GLIBCXX_TR1_CFLOAT 1

#include <cfloat>

#ifndef DECIMAL_DIG
#define DECIMAL_DIG __DECIMAL_DIG__
#endif

#ifndef FLT_EVAL_METHOD
#define FLT_EVAL_METHOD __FLT_EVAL_METHOD__
#endif

#endif //_GLIBCXX_TR1_CFLOAT 
// Special functions -*- C++ -*-

// Copyright (C) 2006, 2007, 2008
// 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 2, 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.
//
// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.
//
// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/hypergeometric.tcc
 *  This is an internal header file, included by other library headers.
 *  You should not attempt to use it directly.
 */

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

// Written by Edward Smith-Rowland based:
//   (1) Handbook of Mathematical Functions,
//       ed. Milton Abramowitz and Irene A. Stegun,
//       Dover Publications,
//       Section 6, pp. 555-566
//   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl

#ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
#define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1

namespace std
{
namespace tr1
{

  // [5.2] Special functions

  // Implementation-space details.
  namespace __detail
  {

    /**
     *   @brief This routine returns the confluent hypergeometric function
     *          by series expansion.
     * 
     *   @f[
     *     _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     * 
     *   If a and b are integers and a < 0 and either b > 0 or b < a then the
     *   series is a polynomial with a finite number of terms.  If b is an integer
     *   and b <= 0 the confluent hypergeometric function is undefined.
     *
     *   @param  __a  The "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    _Tp
    __conf_hyperg_series(const _Tp __a, const _Tp __c, const _Tp __x)
    {
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();

      _Tp __term = _Tp(1);
      _Tp __Fac = _Tp(1);
      const unsigned int __max_iter = 100000;
      unsigned int __i;
      for (__i = 0; __i < __max_iter; ++__i)
        {
          __term *= (__a + _Tp(__i)) * __x
                  / ((__c + _Tp(__i)) * _Tp(1 + __i));
          if (std::abs(__term) < __eps)
            {
              break;
            }
          __Fac += __term;
        }
      if (__i == __max_iter)
        std::__throw_runtime_error(__N("Series failed to converge "
                                       "in __conf_hyperg_series."));

      return __Fac;
    }


    /**
     *  @brief  Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
     *          by an iterative procedure described in
     *          Luke, Algorithms for the Computation of Mathematical Functions.
     *
     *  Like the case of the 2F1 rational approximations, these are 
     *  probably guaranteed to converge for x < 0, barring gross    
     *  numerical instability in the pre-asymptotic regime.         
     */
    template<typename _Tp>
    _Tp
    __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _Tp __xin)
    {
      const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
      const int __nmax = 20000;
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      const _Tp __x  = -__xin;
      const _Tp __x3 = __x * __x * __x;
      const _Tp __t0 = __a / __c;
      const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c);
      const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1)));
      _Tp __F = _Tp(1);
      _Tp __prec;

      _Tp __Bnm3 = _Tp(1);
      _Tp __Bnm2 = _Tp(1) + __t1 * __x;
      _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);

      _Tp __Anm3 = _Tp(1);
      _Tp __Anm2 = __Bnm2 - __t0 * __x;
      _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
                 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;

      int __n = 3;
      while(1)
        {
          _Tp __npam1 = _Tp(__n - 1) + __a;
          _Tp __npcm1 = _Tp(__n - 1) + __c;
          _Tp __npam2 = _Tp(__n - 2) + __a;
          _Tp __npcm2 = _Tp(__n - 2) + __c;
          _Tp __tnm1  = _Tp(2 * __n - 1);
          _Tp __tnm3  = _Tp(2 * __n - 3);
          _Tp __tnm5  = _Tp(2 * __n - 5);
          _Tp __F1 =  (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1);
          _Tp __F2 =  (_Tp(__n) + __a) * __npam1
                   / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
          _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a)
                   / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
                   * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
          _Tp __E  = -__npam1 * (_Tp(__n - 1) - __c)
                   / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);

          _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
                   + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
          _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
                   + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
          _Tp __r = __An / __Bn;

          __prec = std::abs((__F - __r) / __F);
          __F = __r;

          if (__prec < __eps || __n > __nmax)
            break;

          if (std::abs(__An) > __big || std::abs(__Bn) > __big)
            {
              __An   /= __big;
              __Bn   /= __big;
              __Anm1 /= __big;
              __Bnm1 /= __big;
              __Anm2 /= __big;
              __Bnm2 /= __big;
              __Anm3 /= __big;
              __Bnm3 /= __big;
            }
          else if (std::abs(__An) < _Tp(1) / __big
                || std::abs(__Bn) < _Tp(1) / __big)
            {
              __An   *= __big;
              __Bn   *= __big;
              __Anm1 *= __big;
              __Bnm1 *= __big;
              __Anm2 *= __big;
              __Bnm2 *= __big;
              __Anm3 *= __big;
              __Bnm3 *= __big;
            }

          ++__n;
          __Bnm3 = __Bnm2;
          __Bnm2 = __Bnm1;
          __Bnm1 = __Bn;
          __Anm3 = __Anm2;
          __Anm2 = __Anm1;
          __Anm1 = __An;
        }

      if (__n >= __nmax)
        std::__throw_runtime_error(__N("Iteration failed to converge "
                                       "in __conf_hyperg_luke."));

      return __F;
    }


    /**
     *   @brief  Return the confluent hypogeometric function
     *           @f$ _1F_1(a;c;x) @f$.
     * 
     *   @todo  Handle b == nonpositive integer blowup - return NaN.
     *
     *   @param  __a  The "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    inline _Tp
    __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x)
    {
#if _GLIBCXX_USE_C99_MATH_TR1
      const _Tp __c_nint = std::tr1::nearbyint(__c);
#else
      const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
#endif
      if (__isnan(__a) || __isnan(__c) || __isnan(__x))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__c_nint == __c && __c_nint <= 0)
        return std::numeric_limits<_Tp>::infinity();
      else if (__a == _Tp(0))
        return _Tp(1);
      else if (__c == __a)
        return std::exp(__x);
      else if (__x < _Tp(0))
        return __conf_hyperg_luke(__a, __c, __x);
      else
        return __conf_hyperg_series(__a, __c, __x);
    }


    /**
     *   @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
     *   by series expansion.
     * 
     *   The hypogeometric function is defined by
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     * 
     *   This works and it's pretty fast.
     *
     *   @param  __a  The first "numerator" parameter.
     *   @param  __a  The second "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    _Tp
    __hyperg_series(const _Tp __a, const _Tp __b,
                    const _Tp __c, const _Tp __x)
    {
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();

      _Tp __term = _Tp(1);
      _Tp __Fabc = _Tp(1);
      const unsigned int __max_iter = 100000;
      unsigned int __i;
      for (__i = 0; __i < __max_iter; ++__i)
        {
          __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x
                  / ((__c + _Tp(__i)) * _Tp(1 + __i));
          if (std::abs(__term) < __eps)
            {
              break;
            }
          __Fabc += __term;
        }
      if (__i == __max_iter)
        std::__throw_runtime_error(__N("Series failed to converge "
                                       "in __hyperg_series."));

      return __Fabc;
    }


    /**
     *   @brief  Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
     *           by an iterative procedure described in
     *           Luke, Algorithms for the Computation of Mathematical Functions.
     */
    template<typename _Tp>
    _Tp
    __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c,
                  const _Tp __xin)
    {
      const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
      const int __nmax = 20000;
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      const _Tp __x  = -__xin;
      const _Tp __x3 = __x * __x * __x;
      const _Tp __t0 = __a * __b / __c;
      const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c);
      const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2))
                     / (_Tp(2) * (__c + _Tp(1)));

      _Tp __F = _Tp(1);

      _Tp __Bnm3 = _Tp(1);
      _Tp __Bnm2 = _Tp(1) + __t1 * __x;
      _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);

      _Tp __Anm3 = _Tp(1);
      _Tp __Anm2 = __Bnm2 - __t0 * __x;
      _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
                 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;

      int __n = 3;
      while (1)
        {
          const _Tp __npam1 = _Tp(__n - 1) + __a;
          const _Tp __npbm1 = _Tp(__n - 1) + __b;
          const _Tp __npcm1 = _Tp(__n - 1) + __c;
          const _Tp __npam2 = _Tp(__n - 2) + __a;
          const _Tp __npbm2 = _Tp(__n - 2) + __b;
          const _Tp __npcm2 = _Tp(__n - 2) + __c;
          const _Tp __tnm1  = _Tp(2 * __n - 1);
          const _Tp __tnm3  = _Tp(2 * __n - 3);
          const _Tp __tnm5  = _Tp(2 * __n - 5);
          const _Tp __n2 = __n * __n;
          const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n
                         + _Tp(2) - __a * __b - _Tp(2) * (__a + __b))
                         / (_Tp(2) * __tnm3 * __npcm1);
          const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n
                         + _Tp(2) - __a * __b) * __npam1 * __npbm1
                  ã,ä,å,æ,ç,è,é,ê,ë,ì,í,î,ï,ð,ñ,ò,       / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
          const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1
                         * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b))
                         / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
                         * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
          const _Tp __E  = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c)
                         / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);

          _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
                   + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
          _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
                   + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
          const _Tp __r = __An / __Bn;

          const _Tp __prec = std::abs((__F - __r) / __F);
          __F = __r;

          if (__prec < __eps || __n > __nmax)
            break;

          if (std::abs(__An) > __big || std::abs(__Bn) > __big)
            {
              __An   /= __big;
              __Bn   /= __big;
              __Anm1 /= __big;
              __Bnm1 /= __big;
              __Anm2 /= __big;
              __Bnm2 /= __big;
              __Anm3 /= __big;
              __Bnm3 /= __big;
            }
          else if (std::abs(__An) < _Tp(1) / __big
                || std::abs(__Bn) < _Tp(1) / __big)
            {
              __An   *= __big;
              __Bn   *= __big;
              __Anm1 *= __big;
              __Bnm1 *= __big;
              __Anm2 *= __big;
              __Bnm2 *= __big;
              __Anm3 *= __big;
              __Bnm3 *= __big;
            }

          ++__n;
          __Bnm3 = __Bnm2;
          __Bnm2 = __Bnm1;
          __Bnm1 = __Bn;
          __Anm3 = __Anm2;
          __Anm2 = __Anm1;
          __Anm1 = __An;
        }

      if (__n >= __nmax)
        std::__throw_runtime_error(__N("Iteration failed to converge "
                                       "in __hyperg_luke."));

      return __F;
    }


    /**
     *  @brief  Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ by the reflection
     *          formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral
     *          and formula 15.3.11 for d = c - a - b integral.
     *          This assumes a, b, c != negative integer.
     *
     *   The hypogeometric function is defined by
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     *
     *   The reflection formula for nonintegral @f$ d = c - a - b @f$ is:
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)}
     *                            _2F_1(a,b;1-d;1-x)
     *                    + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)}
     *                            _2F_1(c-a,c-b;1+d;1-x)
     *   @f]
     *
     *   The reflection formula for integral @f$ m = c - a - b @f$ is:
     *   @f[
     *     _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)}
     *                        \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k}
     *                      - 
     *   @f]
     */
    template<typename _Tp>
    _Tp
    __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c,
                     const _Tp __x)
    {
      const _Tp __d = __c - __a - __b;
      const int __intd  = std::floor(__d + _Tp(0.5L));
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      const _Tp __toler = _Tp(1000) * __eps;
      const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max());
      const bool __d_integer = (std::abs(__d - __intd) < __toler);

      if (__d_integer)
        {
          const _Tp __ln_omx = std::log(_Tp(1) - __x);
          const _Tp __ad = std::abs(__d);
          _Tp __F1, __F2;

          _Tp __d1, __d2;
          if (__d >= _Tp(0))
            {
              __d1 = __d;
              __d2 = _Tp(0);
            }
          else
            {
              __d1 = _Tp(0);
              __d2 = __d;
            }

          const _Tp __lng_c = __log_gamma(__c);

          //  Evaluate F1.
          if (__ad < __eps)
            {
              //  d = c - a - b = 0.
              __F1 = _Tp(0);
            }
          else
            {

              bool __ok_d1 = true;
              _Tp __lng_ad, __lng_ad1, __lng_bd1;
              try
                {
                  __lng_ad = __log_gamma(__ad);
                  __lng_ad1 = __log_gamma(__a + __d1);
                  __lng_bd1 = __log_gamma(__b + __d1);
                }
              catch(...)
                {
                  __ok_d1 = false;
                }

              if (__ok_d1)
                {
                  /* Gamma functions in the denominator are ok.
                   * Proceed with evaluation.
                   */
                  _Tp __sum1 = _Tp(1);
                  _Tp __term = _Tp(1);
                  _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx
                                - __lng_ad1 - __lng_bd1;

                  /* Do F1 sum.
                   */
                  for (int __i = 1; __i < __ad; ++__i)
                    {
                      const int __j = __i - 1;
                      __term *= (__a + __d2 + __j) * (__b + __d2 + __j)
                              / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x);
                      __sum1 += __term;
                    }

                  if (__ln_pre1 > __log_max)
                    std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                   "in __hyperg_luke."));
                  else
                    __F1 = std::exp(__ln_pre1) * __sum1;
                }
              else
                {
                  //  Gamma functions in the denominator were not ok.
                  //  So the F1 term is zero.
                  __F1 = _Tp(0);
                }
            } // end F1 evaluation

          // Evaluate F2.
          bool __ok_d2 = true;
          _Tp __lng_ad2, __lng_bd2;
          try
            {
              __lng_ad2 = __log_gamma(__a + __d2);
              __lng_bd2 = __log_gamma(__b + __d2);
            }
          catch(...)
            {
              __ok_d2 = false;
            }

          if (__ok_d2)
            {
              //  Gamma functions in the denominator are ok.
              //  Proceed with evaluation.
              const int __maxiter = 2000;
              const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e();
              const _Tp __psi_1pd = __psi(_Tp(1) + __ad);
              const _Tp __psi_apd1 = __psi(__a + __d1);
              const _Tp __psi_bpd1 = __psi(__b + __d1);

              _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1
                             - __psi_bpd1 - __ln_omx;
              _Tp __fact = _Tp(1);
              _Tp __sum2 = __psi_term;
              _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx
                            - __lng_ad2 - __lng_bd2;

              // Do F2 sum.
              int __j;
              for (__j = 1; __j < __maxiter; ++__j)
                {
                  //  Values for psi functions use recurrence; Abramowitz & Stegun 6.3.5
                  const _Tp __term1 = _Tp(1) / _Tp(__j)
                                    + _Tp(1) / (__ad + __j);
                  const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1))
                                    + _Tp(1) / (__b + __d1 + _Tp(__j - 1));
                  __psi_term += __term1 - __term2;
                  __fact *= (__a + __d1 + _Tp(__j - 1))
                          * (__b + __d1 + _Tp(__j - 1))
                          / ((__ad + __j) * __j) * (_Tp(1) - __x);
                  const _Tp __delta = __fact * __psi_term;
                  __sum2 += __delta;
                  if (std::abs(__delta) < __eps * std::abs(__sum2))
                    break;
                }
              if (__j == __maxiter)
                std::__throw_runtime_error(__N("Sum F2 failed to converge "
                                               "in __hyperg_reflect"));

              if (__sum2 == _Tp(0))
                __F2 = _Tp(0);
              else
                __F2 = std::exp(__ln_pre2) * __sum2;
            }
          else
            {
              // Gamma functions in the denominator not ok.
              // So the F2 term is zero.
              __F2 = _Tp(0);
            } // end F2 evaluation

          const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
          const _Tp __F = __F1 + __sgn_2 * __F2;

          return __F;
        }
      else
        {
          //  d = c - a - b not an integer.

          //  These gamma functions appear in the denominator, so we
          //  catch their harmless domain errors and set the terms to zero.
          bool __ok1 = true;
          _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0);
          _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0);
          try
            {
              __sgn_g1ca = __log_gamma_sign(__c - __a);
              __ln_g1ca = __log_gamma(__c - __a);
              __sgn_g1cb = __log_gamma_sign(__c - __b);
              __ln_g1cb = __log_gamma(__c - __b);
            }
          catch(...)
            {
              __ok1 = false;
            }

          bool __ok2 = true;
          _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0);
          _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0);
          try
            {
              __sgn_g2a = __log_gamma_sign(__a);
              __ln_g2a = __log_gamma(__a);
              __sgn_g2b = __log_gamma_sign(__b);
              __ln_g2b = __log_gamma(__b);
            }
          catch(...)
            {
              __ok2 = false;
            }

          const _Tp __sgn_gc = __log_gamma_sign(__c);
          const _Tp __ln_gc = __log_gamma(__c);
          const _Tp __sgn_gd = __log_gamma_sign(__d);
          const _Tp __ln_gd = __log_gamma(__d);
          const _Tp __sgn_gmd = __log_gamma_sign(-__d);
          const _Tp __ln_gmd = __log_gamma(-__d);

          const _Tp __sgn1 = __sgn_gc * __sgn_gd  * __sgn_g1ca * __sgn_g1cb;
          const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a  * __sgn_g2b;

          _Tp __pre1, __pre2;
          if (__ok1 && __ok2)
            {
              _Tp __ln_pre1 = __ln_gc + __ln_gd  - __ln_g1ca - __ln_g1cb;
              _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a  - __ln_g2b
                            + __d * std::log(_Tp(1) - __x);
              if (__ln_pre1 < __log_max && __ln_pre2 < __log_max)
                {
                  __pre1 = std::exp(__ln_pre1);
                  __pre2 = std::exp(__ln_pre2);
                  __pre1 *= __sgn1;
                  __pre2 *= __sgn2;
                }
              else
                {
                  std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                 "in __hyperg_reflect"));
                }
            }
          else if (__ok1 && !__ok2)
            {
              _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
              if (__ln_pre1 < __log_max)
                {
                  __pre1 = std::exp(__ln_pre1);
                  __pre1 *= __sgn1;
                  __pre2 = _Tp(0);
                }
              else
                {
                  std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                 "in __hyperg_reflect"));
                }
            }
          else if (!__ok1 && __ok2)
            {
              _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
                            + __d * std::log(_Tp(1) - __x);
              if (__ln_pre2 < __log_max)
                {
                  __pre1 = _Tp(0);
                  __pre2 = std::exp(__ln_pre2);
                  __pre2 *= __sgn2;
                }
              else
                {
                  std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                 "in __hyperg_reflect"));
                }
            }
          else
            {
              __pre1 = _Tp(0);
              __pre2 = _Tp(0);
              std::__throw_runtime_error(__N("Underflow of gamma functions "
                                             "in __hyperg_reflect"));
            }

          const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d,
                                           _Tp(1) - __x);
          const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d,
                                           _Tp(1) - __x);

          const _Tp __F = __pre1 * __F1 + __pre2 * __F2;

          return __F;
        }
    }


    /**
     *   @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$.
     *
     *   The hypogeometric function is defined by
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     *
     *   @param  __a  The first "numerator" parameter.
     *   @param  __a  The second "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    inline _Tp
    __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x)
    {
#if _GLIBCXX_USE_C99_MATH_TR1
      const _Tp __a_nint = std::tr1::nearbyint(__a);
      const _Tp __b_nint = std::tr1::nearbyint(__b);
      const _Tp __c_nint = std::tr1::nearbyint(__c);
#else
      const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L));
      const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L));
      const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
#endif
      const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon();
      if (std::abs(__x) >= _Tp(1))
        std::__throw_domain_error(__N("Argument outside unit circle "
                                      "in __hyperg."));
      else if (__isnan(__a) || __isnan(__b)
            || __isnan(__c) || __isnan(__x))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__c_nint == __c && __c_nint <= _Tp(0))
        return std::numeric_limits<_Tp>::infinity();
      else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler)
        return std::pow(_Tp(1) - __x, __c - __a - __b);
      else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0)
            && __x >= _Tp(0) && __x < _Tp(0.995L))
        return __hyperg_series(__a, __b, __c, __x);
      else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10))
        {
          //  For integer a and b the hypergeometric function is a finite polynomial.
          if (__a < _Tp(0)  &&  std::abs(__a - __a_nint) < __toler)
            return __hyperg_series(__a_nint, __b, __c, __x);
          else if (__b < _Tp(0)  &&  std::abs(__b - __b_nint) < __toler)
            return __hyperg_series(__a, __b_nint, __c, __x);
          else if (__x < -_Tp(0.25L))
            return __hyperg_luke(__a, __b, __c, __x);
          else if (__x < _Tp(0.5L))
            return __hyperg_series(__a, __b, __c, __x);
          else
            if (std::abs(__c) > _Tp(10))
              return __hyperg_series(__a, __b, __c, __x);
            else
              return __hyperg_reflect(__a, __b, __c, __x);
        }
      else
        return __hyperg_luke(__a, __b, __c, __x);
    }

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

#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
// <tr1/boost_sp_shared_count.h> -*- C++ -*-

// Copyright (C) 2007 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

//  shared_count.hpp
//  Copyright (c) 2001, 2002, 2003 Peter Dimov and Multi Media Ltd.

//  shared_ptr.hpp
//  Copyright (C) 1998, 1999 Greg Colvin and Beman Dawes.
//  Copyright (C) 2001, 2002, 2003 Peter Dimov

//  weak_ptr.hpp
//  Copyright (C) 2001, 2002, 2003 Peter Dimov

//  enable_shared_from_this.hpp
//  Copyright (C) 2002 Peter Dimov

// Distributed under 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)

// GCC Note:  based on version 1.32.0 of the Boost library.

/** @file tr1/boost_sp_shared_count.h
 *  This is an internal header file, included by other library headers.
 *  You should not attempt to use it directly.
 */

#if defined(_GLIBCXX_INCLUDE_AS_CXX0X)
#  error TR1 header cannot be included from C++0x header
#endif

namespace std
{
namespace tr1
{

  template<typename _Ptr, typename _Deleter, _Lock_policy _Lp>
    class _Sp_counted_base_impl
    : public _Sp_counted_base<_Lp>
    {
    public:
      /**
       *  @brief   
       *  @pre     __d(__p) must not throw.
       */
      _Sp_counted_base_impl(_Ptr __p, _Deleter __d)
      : _M_ptr(__p), _M_del(__d) { }
    
      virtual void
      _M_dispose() // nothrow
      { _M_del(_M_ptr); }
      
      virtual void*
      _M_get_deleter(const std::type_info& __ti)
      { return __ti == typeid(_Deleter) ? &_M_del : 0; }
      
    private:
      _Sp_counted_base_impl(const _Sp_counted_base_impl&);
      _Sp_counted_base_impl& operator=(const _Sp_counted_base_impl&);
      
      _Ptr      _M_ptr;  // copy constructor must not throw
      _Deleter  _M_del;  // copy constructor must not throw
    };

  template<_Lock_policy _Lp = __default_lock_policy>
    class __weak_count;

  template<typename _Tp>
    struct _Sp_deleter
    {
      typedef void result_type;
      typedef _Tp* argument_type;
      void operator()(_Tp* __p) const { delete __p; }
    };

  template<_Lock_policy _Lp = __default_lock_policy>
    class __shared_count
    {
    public: 
      __shared_count()
      : _M_pi(0) // nothrow
      { }
  
      template<typename _Ptr>
        __shared_count(_Ptr __p) : _M_pi(0)
        {
	  try
	    {
	      typedef typename std::tr1::remove_pointer<_Ptr>::type _Tp;
	      _M_pi = new _Sp_counted_base_impl<_Ptr, _Sp_deleter<_Tp>, _Lp>(
	          __p, _Sp_deleter<_Tp>());
	    }
	  catch(...)
	    {
	      delete __p;
	      __throw_exception_again;
	    }
	}

      template<typename _Ptr, typename _Deleter>
        __shared_count(_Ptr __p, _Deleter __d) : _M_pi(0)
        {
	  try
	    {
	      _M_pi = new _Sp_counted_base_impl<_Ptr, _Deleter, _Lp>(__p, __d);
	    }
	  catch(...)
	    {
	      __d(__p); // Call _Deleter on __p.
	      __throw_exception_again;
	    }
	}

      // Special case for auto_ptr<_Tp> to provide the strong guarantee.
      template<typename _Tp>
        explicit
        __shared_count(std::auto_ptr<_Tp>& __r)
	: _M_pi(new _Sp_counted_base_impl<_Tp*,
		_Sp_deleter<_Tp>, _Lp >(__r.get(), _Sp_deleter<_Tp>()))
        { __r.release(); }

      // Throw bad_weak_ptr when __r._M_get_use_count() == 0.
      explicit
      __shared_count(const __weak_count<_Lp>& __r);
  
      ~__shared_count() // nothrow
      {
	if (_M_pi != 0)
	  _M_pi->_M_release();
      }
      
      __shared_count(const __shared_count& __r)
      : _M_pi(__r._M_pi) // nothrow
      {
	if (_M_pi != 0)
	  _M_pi->_M_add_ref_copy();
      }
  
      __shared_count&
      operator=(const __shared_count& __r) // nothrow
      {
	_Sp_counted_base<_Lp>* __tmp = __r._M_pi;
	if (__tmp != _M_pi)
	  {
	    if (__tmp != 0)
	      __tmp->_M_add_ref_copy();
	    if (_M_pi != 0)
	      _M_pi->_M_release();
	    _M_pi = __tmp;
	  }
	return *this;
      }
  
      void
      _M_swap(__shared_count& __r) // nothrow
      {
	_Sp_counted_base<_Lp>* __tmp = __r._M_pi;
	__r._M_pi = _M_pi;
	_M_pi = __tmp;
      }
  
      long
      _M_get_use_count() const // nothrow
      { return _M_pi != 0 ? _M_pi->_M_get_use_count() : 0; }

      bool
      _M_unique() const // nothrow
      { return this->_M_get_use_count() == 1; }
      
      friend inline bool
      operator==(const __shared_count& __a, const __shared_count& __b)
      { return __a._M_pi == __b._M_pi; }
  
      friend inline bool
      operator<(const __shared_count& __a, const __shared_count& __b)
      { return std::less<_Sp_counted_base<_Lp>*>()(__a._M_pi, __b._M_pi); }
  
      void*
      _M_get_deleter(const std::type_info& __ti) const
      { return _M_pi ? _M_pi->_M_get_deleter(__ti) : 0; }

    private:
      friend class __weak_count<_Lp>;

      _Sp_counted_base<_Lp>*  _M_pi;
    };
}
}
// TR1 climits -*- C++ -*-

// Copyright (C) 2006, 2007 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/climits
 *  This is a TR1 C++ Library header. 
 */

#ifndef _GLIBCXX_TR1_CLIMITS
#define _GLIBCXX_TR1_CLIMITS 1

#include <climits>

#ifndef LLONG_MIN
#define LLONG_MIN -__LONG_LONG_MAX__ - 1
#endif

#ifndef LLONG_MAX
#define LLONG_MAX __LONG_LONG_MAX__
#endif

#ifndef ULLONG_MAX
#define ULLONG_MAX __LONG_LONG_MAX__ * 2ULL + 1
#endif

#endif // _GLIBCXX_TR1_CLIMITS
// TR1 stdbool.h -*- C++ -*-

// Copyright (C) 2006 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/stdbool.h
 *  This is a TR1 C++ Library header. 
 */

#ifndef _TR1_STDBOOL_H
#define _TR1_STDBOOL_H 1

#include <tr1/cstdbool>

#endif
// TR1 ctgmath -*- C++ -*-

// Copyright (C) 2006, 2007 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

/** @file tr1/ctgmath
 *  This is a TR1 C++ Library header. 
 */

#ifndef _GLIBCXX_TR1_CTGMATH
#define _GLIBCXX_TR1_CTGMATH 1

#include <tr1/cmath>

#endif // _GLIBCXX_TR1_CTGMATH
// TR1 cstdio -*- C++ -*-

// Copyright (C) 2006 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 2, 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.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use m