///////////////////////////////////////////////////////////////////////////////
// Copyright Christopher Kormanyos 2014.
// Copyright John Maddock 2014.
// Copyright Paul Bristow 2014.
// 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)
//

// Implement quadruple-precision <cmath> support.

#ifndef _BOOST_CSTDFLOAT_CMATH_2014_02_15_HPP_
  #define _BOOST_CSTDFLOAT_CMATH_2014_02_15_HPP_

  #include <boost/math/cstdfloat/cstdfloat_types.hpp>
  #include <boost/math/cstdfloat/cstdfloat_limits.hpp>

  #if defined(BOOST_CSTDFLOAT_HAS_INTERNAL_FLOAT128_T) && defined(BOOST_MATH_USE_FLOAT128) && !defined(BOOST_CSTDFLOAT_NO_LIBQUADMATH_SUPPORT)

  #include <cmath>
  #include <stdexcept>
  #include <boost/cstdint.hpp>
  #include <boost/static_assert.hpp>
  #include <boost/throw_exception.hpp>

  #if defined(_WIN32) && defined(__GNUC__)
    // Several versions of Mingw and probably cygwin too have broken
    // libquadmath implementations that segfault as soon as you call
    // expq or any function that depends on it.
    #define BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS
  #endif

  // Here is a helper function used for raising the value of a given
  // floating-point type to the power of n, where n has integral type.
  namespace boost { namespace math { namespace cstdfloat { namespace detail {

  template<class float_type, class integer_type>
  inline float_type pown(const float_type& x, const integer_type p)
  {
    const bool isneg  = (x < 0);
    const bool isnan  = (x != x);
    const bool isinf  = ((!isneg) ? bool(+x > (std::numeric_limits<float_type>::max)())
                                  : bool(-x > (std::numeric_limits<float_type>::max)()));

    if(isnan) { return x; }

    if(isinf) { return std::numeric_limits<float_type>::quiet_NaN(); }

    const bool       x_is_neg = (x < 0);
    const float_type abs_x    = (x_is_neg ? -x : x);

    if(p < static_cast<integer_type>(0))
    {
      if(abs_x < (std::numeric_limits<float_type>::min)())
      {
        return (x_is_neg ? -std::numeric_limits<float_type>::infinity()
                         : +std::numeric_limits<float_type>::infinity());
      }
      else
      {
        return float_type(1) / pown(x, static_cast<integer_type>(-p));
      }
    }

    if(p == static_cast<integer_type>(0))
    {
      return float_type(1);
    }
    else
    {
      if(p == static_cast<integer_type>(1)) { return x; }

      if(abs_x > (std::numeric_limits<float_type>::max)())
      {
        return (x_is_neg ? -std::numeric_limits<float_type>::infinity()
                         : +std::numeric_limits<float_type>::infinity());
      }

      if     (p == static_cast<integer_type>(2)) { return  (x * x); }
      else if(p == static_cast<integer_type>(3)) { return ((x * x) * x); }
      else if(p == static_cast<integer_type>(4)) { const float_type x2 = (x * x); return (x2 * x2); }
      else
      {
        // The variable xn stores the binary powers of x.
        float_type result(((p % integer_type(2)) != integer_type(0)) ? x : float_type(1));
        float_type xn    (x);

        integer_type p2 = p;

        while(integer_type(p2 /= 2) != integer_type(0))
        {
          // Square xn for each binary power.
          xn *= xn;

          const bool has_binary_power = (integer_type(p2 % integer_type(2)) != integer_type(0));

          if(has_binary_power)
          {
            // Multiply the result with each binary power contained in the exponent.
            result *= xn;
          }
        }

        return result;
      }
    }
  }

  } } } } // boost::math::cstdfloat::detail

  // We will now define preprocessor symbols representing quadruple-precision <cmath> functions.
  #if defined(BOOST_INTEL)
    #define BOOST_CSTDFLOAT_FLOAT128_LDEXP  __ldexpq
    #define BOOST_CSTDFLOAT_FLOAT128_FREXP  __frexpq
    #define BOOST_CSTDFLOAT_FLOAT128_FABS   __fabsq
    #define BOOST_CSTDFLOAT_FLOAT128_FLOOR  __floorq
    #define BOOST_CSTDFLOAT_FLOAT128_CEIL   __ceilq
    #if !defined(BOOST_CSTDFLOAT_FLOAT128_SQRT)
    #define BOOST_CSTDFLOAT_FLOAT128_SQRT   __sqrtq
    #endif
    #define BOOST_CSTDFLOAT_FLOAT128_TRUNC  __truncq
    #define BOOST_CSTDFLOAT_FLOAT128_EXP    __expq
    #define BOOST_CSTDFLOAT_FLOAT128_EXPM1  __expm1q
    #define BOOST_CSTDFLOAT_FLOAT128_POW    __powq
    #define BOOST_CSTDFLOAT_FLOAT128_LOG    __logq
    #define BOOST_CSTDFLOAT_FLOAT128_LOG10  __log10q
    #define BOOST_CSTDFLOAT_FLOAT128_SIN    __sinq
    #define BOOST_CSTDFLOAT_FLOAT128_COS    __cosq
    #define BOOST_CSTDFLOAT_FLOAT128_TAN    __tanq
    #define BOOST_CSTDFLOAT_FLOAT128_ASIN   __asinq
    #define BOOST_CSTDFLOAT_FLOAT128_ACOS   __acosq
    #define BOOST_CSTDFLOAT_FLOAT128_ATAN   __atanq
    #define BOOST_CSTDFLOAT_FLOAT128_SINH   __sinhq
    #define BOOST_CSTDFLOAT_FLOAT128_COSH   __coshq
    #define BOOST_CSTDFLOAT_FLOAT128_TANH   __tanhq
    #define BOOST_CSTDFLOAT_FLOAT128_ASINH  __asinhq
    #define BOOST_CSTDFLOAT_FLOAT128_ACOSH  __acoshq
    #define BOOST_CSTDFLOAT_FLOAT128_ATANH  __atanhq
    #define BOOST_CSTDFLOAT_FLOAT128_FMOD   __fmodq
    #define BOOST_CSTDFLOAT_FLOAT128_ATAN2  __atan2q
    #define BOOST_CSTDFLOAT_FLOAT128_LGAMMA __lgammaq
    #define BOOST_CSTDFLOAT_FLOAT128_TGAMMA __tgammaq
  #elif defined(__GNUC__)
    #define BOOST_CSTDFLOAT_FLOAT128_LDEXP  ldexpq
    #define BOOST_CSTDFLOAT_FLOAT128_FREXP  frexpq
    #define BOOST_CSTDFLOAT_FLOAT128_FABS   fabsq
    #define BOOST_CSTDFLOAT_FLOAT128_FLOOR  floorq
    #define BOOST_CSTDFLOAT_FLOAT128_CEIL   ceilq
    #if !defined(BOOST_CSTDFLOAT_FLOAT128_SQRT)
    #define BOOST_CSTDFLOAT_FLOAT128_SQRT   sqrtq
    #endif
    #define BOOST_CSTDFLOAT_FLOAT128_TRUNC  truncq
    #define BOOST_CSTDFLOAT_FLOAT128_POW    powq
    #define BOOST_CSTDFLOAT_FLOAT128_LOG    logq
    #define BOOST_CSTDFLOAT_FLOAT128_LOG10  log10q
    #define BOOST_CSTDFLOAT_FLOAT128_SIN    sinq
    #define BOOST_CSTDFLOAT_FLOAT128_COS    cosq
    #define BOOST_CSTDFLOAT_FLOAT128_TAN    tanq
    #define BOOST_CSTDFLOAT_FLOAT128_ASIN   asinq
    #define BOOST_CSTDFLOAT_FLOAT128_ACOS   acosq
    #define BOOST_CSTDFLOAT_FLOAT128_ATAN   atanq
    #define BOOST_CSTDFLOAT_FLOAT128_FMOD   fmodq
    #define BOOST_CSTDFLOAT_FLOAT128_ATAN2  atan2q
    #define BOOST_CSTDFLOAT_FLOAT128_LGAMMA lgammaq
    #if !defined(BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS)
    #define BOOST_CSTDFLOAT_FLOAT128_EXP    expq
    #define BOOST_CSTDFLOAT_FLOAT128_EXPM1  expm1q_internal
    #define BOOST_CSTDFLOAT_FLOAT128_SINH   sinhq
    #define BOOST_CSTDFLOAT_FLOAT128_COSH   coshq
    #define BOOST_CSTDFLOAT_FLOAT128_TANH   tanhq
    #define BOOST_CSTDFLOAT_FLOAT128_ASINH  asinhq
    #define BOOST_CSTDFLOAT_FLOAT128_ACOSH  acoshq
    #define BOOST_CSTDFLOAT_FLOAT128_ATANH  atanhq
    #define BOOST_CSTDFLOAT_FLOAT128_TGAMMA tgammaq
    #else // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS
    #define BOOST_CSTDFLOAT_FLOAT128_EXP    expq_patch
    #define BOOST_CSTDFLOAT_FLOAT128_SINH   sinhq_patch
    #define BOOST_CSTDFLOAT_FLOAT128_COSH   coshq_patch
    #define BOOST_CSTDFLOAT_FLOAT128_TANH   tanhq_patch
    #define BOOST_CSTDFLOAT_FLOAT128_ASINH  asinhq_patch
    #define BOOST_CSTDFLOAT_FLOAT128_ACOSH  acoshq_patch
    #define BOOST_CSTDFLOAT_FLOAT128_ATANH  atanhq_patch
    #define BOOST_CSTDFLOAT_FLOAT128_TGAMMA tgammaq_patch
    #endif // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS
  #endif

  // Implement quadruple-precision <cmath> functions in the namespace
  // boost::math::cstdfloat::detail. Subsequently inject these into the
  // std namespace via *using* directive.

  // Begin with some forward function declarations. Also implement patches
  // for compilers that have broken float128 exponential functions.

  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LDEXP (boost::math::cstdfloat::detail::float_internal128_t, int)  throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FREXP (boost::math::cstdfloat::detail::float_internal128_t, int*) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FABS  (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FLOOR (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_CEIL  (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SQRT  (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TRUNC (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_POW   (boost::math::cstdfloat::detail::float_internal128_t, boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LOG   (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LOG10 (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SIN   (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_COS   (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TAN   (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ASIN  (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ACOS  (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATAN  (boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FMOD  (boost::math::cstdfloat::detail::float_internal128_t, boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATAN2 (boost::math::cstdfloat::detail::float_internal128_t, boost::math::cstdfloat::detail::float_internal128_t) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LGAMMA(boost::math::cstdfloat::detail::float_internal128_t) throw();

  #if !defined(BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS)

  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXP   (boost::math::cstdfloat::detail::float_internal128_t x) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SINH  (boost::math::cstdfloat::detail::float_internal128_t x) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_COSH  (boost::math::cstdfloat::detail::float_internal128_t x) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TANH  (boost::math::cstdfloat::detail::float_internal128_t x) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ASINH (boost::math::cstdfloat::detail::float_internal128_t x) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ACOSH (boost::math::cstdfloat::detail::float_internal128_t x) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATANH (boost::math::cstdfloat::detail::float_internal128_t x) throw();
  extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TGAMMA(boost::math::cstdfloat::detail::float_internal128_t x) throw();

  #else // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS

  // Forward declaration of the patched exponent function, exp(x).
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXP   (boost::math::cstdfloat::detail::float_internal128_t x);

  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXPM1 (boost::math::cstdfloat::detail::float_internal128_t x)
  {
    // Compute exp(x) - 1 for x small.

    // Use an order-36 polynomial approximation of the exponential function
    // in the range of (-ln2 < x < ln2). Scale the argument to this range
    // and subsequently multiply the result by 2^n accordingly.

    // Derive the polynomial coefficients with Mathematica(R) by generating
    // a table of high-precision values of exp(x) in the range (-ln2 < x < ln2)
    // and subsequently applying the built-in *Fit* function.

    // Table[{x, Exp[x] - 1}, {x, -Log[2], Log[2], 1/180}]
    // N[%, 120]
    // Fit[%, {x, x^2, x^3, x^4, x^5, x^6, x^7, x^8, x^9, x^10, x^11, x^12,
    //         x^13, x^14, x^15, x^16, x^17, x^18, x^19, x^20, x^21, x^22,
    //         x^23, x^24, x^25, x^26, x^27, x^28, x^29, x^30, x^31, x^32,
    //         x^33, x^34, x^35, x^36}, x]

    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;

    float_type sum;

    if(x > BOOST_FLOAT128_C(0.693147180559945309417232121458176568075500134360255))
    {
      sum = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x) - float_type(1);
    }
    else
    {
      // Compute the polynomial approximation of exp(alpha).
      sum = ((((((((((((((((((((((((((((((((((((  float_type(BOOST_FLOAT128_C(2.69291698127774166063293705964720493864630783729857438187365E-42))  * x
                                                + float_type(BOOST_FLOAT128_C(9.70937085471487654794114679403710456028986572118859594614033E-41))) * x
                                                + float_type(BOOST_FLOAT128_C(3.38715585158055097155585505318085512156885389014410753080500E-39))) * x
                                                + float_type(BOOST_FLOAT128_C(1.15162718532861050809222658798662695267019717760563645440433E-37))) * x
                                                + float_type(BOOST_FLOAT128_C(3.80039074689434663295873584133017767349635602413675471702393E-36))) * x
                                                + float_type(BOOST_FLOAT128_C(1.21612504934087520075905434734158045947460467096773246215239E-34))) * x
                                                + float_type(BOOST_FLOAT128_C(3.76998762883139753126119821241037824830069851253295480396224E-33))) * x
                                                + float_type(BOOST_FLOAT128_C(1.13099628863830344684998293828608215735777107850991029729440E-31))) * x
                                                + float_type(BOOST_FLOAT128_C(3.27988923706982293204067897468714277771890104022419696770352E-30))) * x
                                                + float_type(BOOST_FLOAT128_C(9.18368986379558482800593745627556950089950023355628325088207E-29))) * x
                                                + float_type(BOOST_FLOAT128_C(2.47959626322479746949155352659617642905315302382639380521497E-27))) * x
                                                + float_type(BOOST_FLOAT128_C(6.44695028438447337900255966737803112935639344283098705091949E-26))) * x
                                                + float_type(BOOST_FLOAT128_C(1.61173757109611834904452725462599961406036904573072897122957E-24))) * x
                                                + float_type(BOOST_FLOAT128_C(3.86817017063068403772269360016918092488847584660382953555804E-23))) * x
                                                + float_type(BOOST_FLOAT128_C(8.89679139245057328674891109315654704307721758924206107351744E-22))) * x
                                                + float_type(BOOST_FLOAT128_C(1.95729410633912612308475595397946731738088422488032228717097E-20))) * x
                                                + float_type(BOOST_FLOAT128_C(4.11031762331216485847799061511674191805055663711439605760231E-19))) * x
                                                + float_type(BOOST_FLOAT128_C(8.22063524662432971695598123977873600603370758794431071426640E-18))) * x
                                                + float_type(BOOST_FLOAT128_C(1.56192069685862264622163643500633782667263448653185159383285E-16))) * x
                                                + float_type(BOOST_FLOAT128_C(2.81145725434552076319894558300988749849555291507956994126835E-15))) * x
                                                + float_type(BOOST_FLOAT128_C(4.77947733238738529743820749111754320727153728139716409114011E-14))) * x
                                                + float_type(BOOST_FLOAT128_C(7.64716373181981647590113198578807092707697416852226691068627E-13))) * x
                                                + float_type(BOOST_FLOAT128_C(1.14707455977297247138516979786821056670509688396295740818677E-11))) * x
                                                + float_type(BOOST_FLOAT128_C(1.60590438368216145993923771701549479323291461578567184216302E-10))) * x
                                                + float_type(BOOST_FLOAT128_C(2.08767569878680989792100903212014323125428376052986408239620E-09))) * x
                                                + float_type(BOOST_FLOAT128_C(2.50521083854417187750521083854417187750523408006206780016659E-08))) * x
                                                + float_type(BOOST_FLOAT128_C(2.75573192239858906525573192239858906525573195144226062684604E-07))) * x
                                                + float_type(BOOST_FLOAT128_C(2.75573192239858906525573192239858906525573191310049321957902E-06))) * x
                                                + float_type(BOOST_FLOAT128_C(0.00002480158730158730158730158730158730158730158730149317774)))     * x
                                                + float_type(BOOST_FLOAT128_C(0.00019841269841269841269841269841269841269841269841293575920)))     * x
                                                + float_type(BOOST_FLOAT128_C(0.00138888888888888888888888888888888888888888888888889071045)))     * x
                                                + float_type(BOOST_FLOAT128_C(0.00833333333333333333333333333333333333333333333333332986595)))     * x
                                                + float_type(BOOST_FLOAT128_C(0.04166666666666666666666666666666666666666666666666666664876)))     * x
                                                + float_type(BOOST_FLOAT128_C(0.16666666666666666666666666666666666666666666666666666669048)))     * x
                                                + float_type(BOOST_FLOAT128_C(0.50000000000000000000000000000000000000000000000000000000006)))     * x
                                                + float_type(BOOST_FLOAT128_C(0.99999999999999999999999999999999999999999999999999999999995)))     * x);
    }

    return sum;
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXP   (boost::math::cstdfloat::detail::float_internal128_t x)
  {
    // Patch the expq() function for a subset of broken GCC compilers
    // like GCC 4.7, 4.8 on MinGW.

    // Use an order-36 polynomial approximation of the exponential function
    // in the range of (-ln2 < x < ln2). Scale the argument to this range
    // and subsequently multiply the result by 2^n accordingly.

    // Derive the polynomial coefficients with Mathematica(R) by generating
    // a table of high-precision values of exp(x) in the range (-ln2 < x < ln2)
    // and subsequently applying the built-in *Fit* function.

    // Table[{x, Exp[x] - 1}, {x, -Log[2], Log[2], 1/180}]
    // N[%, 120]
    // Fit[%, {x, x^2, x^3, x^4, x^5, x^6, x^7, x^8, x^9, x^10, x^11, x^12,
    //         x^13, x^14, x^15, x^16, x^17, x^18, x^19, x^20, x^21, x^22,
    //         x^23, x^24, x^25, x^26, x^27, x^28, x^29, x^30, x^31, x^32,
    //         x^33, x^34, x^35, x^36}, x]

    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;

    // Scale the argument x to the range (-ln2 < x < ln2).
    BOOST_CONSTEXPR_OR_CONST float_type one_over_ln2 = float_type(BOOST_FLOAT128_C(1.44269504088896340735992468100189213742664595415299));
    const float_type x_over_ln2   = x * one_over_ln2;

    boost::int_fast32_t n;

    if(x != x)
    {
      // The argument is NaN.
      return std::numeric_limits<float_type>::quiet_NaN();
    }
    else if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x) > BOOST_FLOAT128_C(+0.693147180559945309417232121458176568075500134360255))
    {
      // The absolute value of the argument exceeds ln2.
      n = static_cast<boost::int_fast32_t>(::BOOST_CSTDFLOAT_FLOAT128_FLOOR(x_over_ln2));
    }
    else if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x) < BOOST_FLOAT128_C(+0.693147180559945309417232121458176568075500134360255))
    {
      // The absolute value of the argument is less than ln2.
      n = static_cast<boost::int_fast32_t>(0);
    }
    else
    {
      // The absolute value of the argument is exactly equal to ln2 (in the sense of floating-point equality).
      return float_type(2);
    }

    // Check if the argument is very near an integer.
    const float_type floor_of_x = ::BOOST_CSTDFLOAT_FLOAT128_FLOOR(x);

    if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x - floor_of_x) < float_type(BOOST_CSTDFLOAT_FLOAT128_EPS))
    {
      // Return e^n for arguments very near an integer.
      return boost::math::cstdfloat::detail::pown(BOOST_FLOAT128_C(2.71828182845904523536028747135266249775724709369996), static_cast<boost::int_fast32_t>(floor_of_x));
    }

    // Compute the scaled argument alpha.
    const float_type alpha = x - (n * BOOST_FLOAT128_C(0.693147180559945309417232121458176568075500134360255));

    // Compute the polynomial approximation of expm1(alpha) and add to it
    // in order to obtain the scaled result.
    const float_type scaled_result = ::BOOST_CSTDFLOAT_FLOAT128_EXPM1(alpha) + float_type(1);

    // Rescale the result and return it.
    return scaled_result * boost::math::cstdfloat::detail::pown(float_type(2), n);
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SINH  (boost::math::cstdfloat::detail::float_internal128_t x)
  {
    // Patch the sinhq() function for a subset of broken GCC compilers
    // like GCC 4.7, 4.8 on MinGW.
    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;

    // Here, we use the following:
    // Set: ex  = exp(x)
    // Set: em1 = expm1(x)
    // Then
    // sinh(x) = (ex - 1/ex) / 2         ; for |x| >= 1
    // sinh(x) = (2em1 + em1^2) / (2ex)  ; for |x| < 1

    const float_type ex = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x);

    if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x) < float_type(+1))
    {
      const float_type em1 = ::BOOST_CSTDFLOAT_FLOAT128_EXPM1(x);

      return ((em1 * 2) + (em1 * em1)) / (ex * 2);
    }
    else
    {
      return (ex - (float_type(1) / ex)) / 2;
    }
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_COSH  (boost::math::cstdfloat::detail::float_internal128_t x)
  {
    // Patch the coshq() function for a subset of broken GCC compilers
    // like GCC 4.7, 4.8 on MinGW.
    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;
    const float_type ex = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x);
    return (ex + (float_type(1) / ex)) / 2;
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TANH  (boost::math::cstdfloat::detail::float_internal128_t x)
  {
    // Patch the tanhq() function for a subset of broken GCC compilers
    // like GCC 4.7, 4.8 on MinGW.
    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;
    const float_type ex_plus  = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x);
    const float_type ex_minus = (float_type(1) / ex_plus);
    return (ex_plus - ex_minus) / (ex_plus + ex_minus);
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ASINH(boost::math::cstdfloat::detail::float_internal128_t x) throw()
  {
    // Patch the asinh() function since quadmath does not have it.
    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;
    return ::BOOST_CSTDFLOAT_FLOAT128_LOG(x + ::BOOST_CSTDFLOAT_FLOAT128_SQRT((x * x) + float_type(1)));
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ACOSH(boost::math::cstdfloat::detail::float_internal128_t x) throw()
  {
    // Patch the acosh() function since quadmath does not have it.
    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;
    const float_type zp(x + float_type(1));
    const float_type zm(x - float_type(1));

    return ::BOOST_CSTDFLOAT_FLOAT128_LOG(x + (zp * ::BOOST_CSTDFLOAT_FLOAT128_SQRT(zm / zp)));
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATANH(boost::math::cstdfloat::detail::float_internal128_t x) throw()
  {
    // Patch the atanh() function since quadmath does not have it.
    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;
    return (  ::BOOST_CSTDFLOAT_FLOAT128_LOG(float_type(1) + x)
            - ::BOOST_CSTDFLOAT_FLOAT128_LOG(float_type(1) - x)) / 2;
  }
  inline     boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TGAMMA(boost::math::cstdfloat::detail::float_internal128_t x) throw()
  {
    // Patch the tgammaq() function for a subset of broken GCC compilers
    // like GCC 4.7, 4.8 on MinGW.
    typedef boost::math::cstdfloat::detail::float_internal128_t float_type;

    if(x > float_type(0))
    {
      return ::BOOST_CSTDFLOAT_FLOAT128_EXP(::BOOST_CSTDFLOAT_FLOAT128_LGAMMA(x));
    }
    else if(x < float_type(0))
    {
      // For x < 0, compute tgamma(-x) and use the reflection formula.
      const float_type positive_x          = -x;
            float_type gamma_value         = ::BOOST_CSTDFLOAT_FLOAT128_TGAMMA(positive_x);
      const float_type floor_of_positive_x = ::BOOST_CSTDFLOAT_FLOAT128_FLOOR (positive_x);

      // Take the reflection checks (slightly adapted) from <boost/math/gamma.hpp>.
      const bool floor_of_z_is_equal_to_z = (positive_x == ::BOOST_CSTDFLOAT_FLOAT128_FLOOR(positive_x));

      BOOST_CONSTEXPR_OR_CONST float_type my_pi = BOOST_FLOAT128_C(3.14159265358979323846264338327950288419716939937511);

      if(floor_of_z_is_equal_to_z)
      {
        const bool is_odd = ((boost::int32_t(floor_of_positive_x) % boost::int32_t(2)) != boost::int32_t(0));

        return (is_odd ? -std::numeric_limits<float_type>::infinity()
                       : +std::numeric_limits<float_type>::infinity());
      }

      const float_type sinpx_value = x * ::BOOST_CSTDFLOAT_FLOAT128_SIN(my_pi * x);

      gamma_value *= sinpx_value;

      const bool result_is_too_large_to_represent = (   (::BOOST_CSTDFLOAT_FLOAT128_FABS(gamma_value) < float_type(1))
                                                     && (((std::numeric_limits<float_type>::max)() * ::BOOST_CSTDFLOAT_FLOAT128_FABS(gamma_value)) < my_pi));

      if(result_is_too_large_to_represent)
      {
        const bool is_odd = ((boost::int32_t(floor_of_positive_x) % boost::int32_t(2)) != boost::int32_t(0));

        return (is_odd ? -std::numeric_limits<float_type>::infinity()
                       : +std::numeric_limits<float_type>::infinity());
      }

      gamma_value = -my_pi / gamma_value;

      if((gamma_value > float_type(0)) || (gamma_value < float_type(0)))
      {
        return gamma_value;
      }
      else
      {
        // The value of gamma is too small to represent. Return 0.0 here.
        return float_type(0);
      }
    }
    else
    {
      // Gamma of zero is complex infinity. Return NaN here.
      return std::numeric_limits<float_type>::quiet_NaN();
    }
  }
  #endif // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS

  // Define the quadruple-precision <cmath> functions in the namespace boost::math::cstdfloat::detail.

  namespace boost { namespace math { namespace cstdfloat { namespace detail {
  inline   boost::math::cstdfloat::detail::float_internal128_t ldexp (boost::math::cstdfloat::detail::float_internal128_t x, int n)                                                 { return ::BOOST_CSTDFLOAT_FLOAT128_LDEXP (x, n); }
  inline   boost::math::cstdfloat::detail::float_internal128_t frexp (boost::math::cstdfloat::detail::float_internal128_t x, int* pn)                                               { return ::BOOST_CSTDFLOAT_FLOAT128_FREXP (x, pn); }
  inline   boost::math::cstdfloat::detail::float_internal128_t fabs  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_FABS  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t abs   (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_FABS  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t floor (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_FLOOR (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t ceil  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_CEIL  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t sqrt  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_SQRT  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t trunc (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_TRUNC (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t exp   (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_EXP   (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t pow   (boost::math::cstdfloat::detail::float_internal128_t x, boost::math::cstdfloat::detail::float_internal128_t a) { return ::BOOST_CSTDFLOAT_FLOAT128_POW   (x, a); }
  inline   boost::math::cstdfloat::detail::float_internal128_t pow   (boost::math::cstdfloat::detail::float_internal128_t x, int a)                                                 { return ::BOOST_CSTDFLOAT_FLOAT128_POW   (x, boost::math::cstdfloat::detail::float_internal128_t(a)); }
  inline   boost::math::cstdfloat::detail::float_internal128_t log   (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_LOG   (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t log10 (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_LOG10 (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t sin   (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_SIN   (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t cos   (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_COS   (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t tan   (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_TAN   (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t asin  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_ASIN  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t acos  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_ACOS  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t atan  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_ATAN  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t sinh  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_SINH  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t cosh  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_COSH  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t tanh  (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_TANH  (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t asinh (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_ASINH (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t acosh (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_ACOSH (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t atanh (boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_ATANH (x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t fmod  (boost::math::cstdfloat::detail::float_internal128_t a, boost::math::cstdfloat::detail::float_internal128_t b) { return ::BOOST_CSTDFLOAT_FLOAT128_FMOD  (a, b); }
  inline   boost::math::cstdfloat::detail::float_internal128_t atan2 (boost::math::cstdfloat::detail::float_internal128_t y, boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ATAN2 (y, x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t lgamma(boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_LGAMMA(x); }
  inline   boost::math::cstdfloat::detail::float_internal128_t tgamma(boost::math::cstdfloat::detail::float_internal128_t x)                                                        { return ::BOOST_CSTDFLOAT_FLOAT128_TGAMMA(x); }
  } } } } // boost::math::cstdfloat::detail

  // We will now inject the quadruple-precision <cmath> functions
  // into the std namespace. This is done via *using* directive.
  namespace std
  {
    using boost::math::cstdfloat::detail::ldexp;
    using boost::math::cstdfloat::detail::frexp;
    using boost::math::cstdfloat::detail::fabs;

#if !(defined(_GLIBCXX_USE_FLOAT128) && defined(__GNUC__) && (__GNUC__ >= 7))
    using boost::math::cstdfloat::detail::abs;
#endif

    using boost::math::cstdfloat::detail::floor;
    using boost::math::cstdfloat::detail::ceil;
    using boost::math::cstdfloat::detail::sqrt;
    using boost::math::cstdfloat::detail::trunc;
    using boost::math::cstdfloat::detail::exp;
    using boost::math::cstdfloat::detail::pow;
    using boost::math::cstdfloat::detail::log;
    using boost::math::cstdfloat::detail::log10;
    using boost::math::cstdfloat::detail::sin;
    using boost::math::cstdfloat::detail::cos;
    using boost::math::cstdfloat::detail::tan;
    using boost::math::cstdfloat::detail::asin;
    using boost::math::cstdfloat::detail::acos;
    using boost::math::cstdfloat::detail::atan;
    using boost::math::cstdfloat::detail::sinh;
    using boost::math::cstdfloat::detail::cosh;
    using boost::math::cstdfloat::detail::tanh;
    using boost::math::cstdfloat::detail::asinh;
    using boost::math::cstdfloat::detail::acosh;
    using boost::math::cstdfloat::detail::atanh;
    using boost::math::cstdfloat::detail::fmod;
    using boost::math::cstdfloat::detail::atan2;
    using boost::math::cstdfloat::detail::lgamma;
    using boost::math::cstdfloat::detail::tgamma;
  } // namespace std

  // We will now remove the preprocessor symbols representing quadruple-precision <cmath>
  // functions from the preprocessor.

  #undef BOOST_CSTDFLOAT_FLOAT128_LDEXP
  #undef BOOST_CSTDFLOAT_FLOAT128_FREXP
  #undef BOOST_CSTDFLOAT_FLOAT128_FABS
  #undef BOOST_CSTDFLOAT_FLOAT128_FLOOR
  #undef BOOST_CSTDFLOAT_FLOAT128_CEIL
  #undef BOOST_CSTDFLOAT_FLOAT128_SQRT
  #undef BOOST_CSTDFLOAT_FLOAT128_TRUNC
  #undef BOOST_CSTDFLOAT_FLOAT128_EXP
  #undef BOOST_CSTDFLOAT_FLOAT128_EXPM1
  #undef BOOST_CSTDFLOAT_FLOAT128_POW
  #undef BOOST_CSTDFLOAT_FLOAT128_LOG
  #undef BOOST_CSTDFLOAT_FLOAT128_LOG10
  #undef BOOST_CSTDFLOAT_FLOAT128_SIN
  #undef BOOST_CSTDFLOAT_FLOAT128_COS
  #undef BOOST_CSTDFLOAT_FLOAT128_TAN
  #undef BOOST_CSTDFLOAT_FLOAT128_ASIN
  #undef BOOST_CSTDFLOAT_FLOAT128_ACOS
  #undef BOOST_CSTDFLOAT_FLOAT128_ATAN
  #undef BOOST_CSTDFLOAT_FLOAT128_SINH
  #undef BOOST_CSTDFLOAT_FLOAT128_COSH
  #undef BOOST_CSTDFLOAT_FLOAT128_TANH
  #undef BOOST_CSTDFLOAT_FLOAT128_ASINH
  #undef BOOST_CSTDFLOAT_FLOAT128_ACOSH
  #undef BOOST_CSTDFLOAT_FLOAT128_ATANH
  #undef BOOST_CSTDFLOAT_FLOAT128_FMOD
  #undef BOOST_CSTDFLOAT_FLOAT128_ATAN2
  #undef BOOST_CSTDFLOAT_FLOAT128_LGAMMA
  #undef BOOST_CSTDFLOAT_FLOAT128_TGAMMA

  #endif // Not BOOST_CSTDFLOAT_NO_LIBQUADMATH_SUPPORT (i.e., the user would like to have libquadmath support)

#endif // _BOOST_CSTDFLOAT_CMATH_2014_02_15_HPP_
