FM / tr1

/usr/include/c++/8/tr1

Name	Size	Mode	Actions
array	6964	0644	edit dl rm
bessel_function.tcc	22473	0644	edit dl rm
beta_function.tcc	5995	0644	edit dl rm
ccomplex	1255	0644	edit dl rm
cctype	1412	0644	edit dl rm
cfenv	2004	0644	edit dl rm
cfloat	1380	0644	edit dl rm
cinttypes	2256	0644	edit dl rm
climits	1454	0644	edit dl rm
cmath	43806	0644	edit dl rm
complex	12384	0644	edit dl rm
complex.h	1261	0644	edit dl rm
cstdarg	1246	0644	edit dl rm
cstdbool	1344	0644	edit dl rm
cstdint	2623	0644	edit dl rm
cstdio	1482	0644	edit dl rm
cstdlib	1796	0644	edit dl rm
ctgmath	1248	0644	edit dl rm
ctime	1234	0644	edit dl rm
ctype.h	1209	0644	edit dl rm
cwchar	1718	0644	edit dl rm
cwctype	1459	0644	edit dl rm
ell_integral.tcc	27724	0644	edit dl rm
exp_integral.tcc	16009	0644	edit dl rm
fenv.h	1204	0644	edit dl rm
float.h	1209	0644	edit dl rm
functional	70545	0644	edit dl rm
functional_hash.h	6043	0644	edit dl rm
gamma.tcc	14682	0644	edit dl rm
hashtable.h	41537	0644	edit dl rm
hashtable_policy.h	25086	0644	edit dl rm
hypergeometric.tcc	28066	0644	edit dl rm
inttypes.h	1267	0644	edit dl rm
legendre_function.tcc	10909	0644	edit dl rm
limits.h	1214	0644	edit dl rm
math.h	4553	0644	edit dl rm
memory	1791	0644	edit dl rm
modified_bessel_func.tcc	16320	0644	edit dl rm
poly_hermite.tcc	3925	0644	edit dl rm
poly_laguerre.tcc	11676	0644	edit dl rm
random	1589	0644	edit dl rm
random.h	73123	0644	edit dl rm
random.tcc	53927	0644	edit dl rm
regex	92880	0644	edit dl rm
riemann_zeta.tcc	14063	0644	edit dl rm
shared_ptr.h	32608	0644	edit dl rm
special_function_util.h	5055	0644	edit dl rm
stdarg.h	1214	0644	edit dl rm
stdbool.h	1219	0644	edit dl rm
stdint.h	1214	0644	edit dl rm
stdio.h	1209	0644	edit dl rm
stdlib.h	1487	0644	edit dl rm
tgmath.h	1255	0644	edit dl rm
tuple	12119	0644	edit dl rm
type_traits	19019	0644	edit dl rm
unordered_map	1574	0644	edit dl rm
unordered_map.h	10216	0644	edit dl rm
unordered_set	1574	0644	edit dl rm
unordered_set.h	9540	0644	edit dl rm
utility	3225	0644	edit dl rm
wchar.h	1249	0644	edit dl rm
wctype.h	1255	0644	edit dl rm

Edit: /usr/include/c++/8/tr1/ell_integral.tcc (27724B)

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

// Copyright (C) 2006-2018 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.

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

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

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

// Written by Edward Smith-Rowland based on:
//   (1)  B. C. Carlson Numer. Math. 33, 1 (1979)
//   (2)  B. C. Carlson, Special Functions of Applied Mathematics (1977)
//   (3)  The Gnu Scientific Library, http://www.gnu.org/software/gsl
//   (4)  Numerical Recipes in C, 2nd ed, by W. H. Press, S. A. Teukolsky,
//        W. T. Vetterling, B. P. Flannery, Cambridge University Press
//        (1992), pp. 261-269

#ifndef _GLIBCXX_TR1_ELL_INTEGRAL_TCC
#define _GLIBCXX_TR1_ELL_INTEGRAL_TCC 1

namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION

#if _GLIBCXX_USE_STD_SPEC_FUNCS
#elif defined(_GLIBCXX_TR1_CMATH)
namespace tr1
{
#else
# error do not include this header directly, use <cmath> or <tr1/cmath>
#endif
  // [5.2] Special functions

// Implementation-space details.
  namespace __detail
  {
    /**
     *   @brief Return the Carlson elliptic function @f$ R_F(x,y,z) @f$
     *          of the first kind.
     * 
     *   The Carlson elliptic function of the first kind is defined by:
     *   @f[
     *       R_F(x,y,z) = \frac{1}{2} \int_0^\infty
     *                 \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}}
     *   @f]
     *
     *   @param  __x  The first of three symmetric arguments.
     *   @param  __y  The second of three symmetric arguments.
     *   @param  __z  The third of three symmetric arguments.
     *   @return  The Carlson elliptic function of the first kind.
     */
    template<typename _Tp>
    _Tp
    __ellint_rf(_Tp __x, _Tp __y, _Tp __z)
    {
      const _Tp __min = std::numeric_limits<_Tp>::min();
      const _Tp __max = std::numeric_limits<_Tp>::max();
      const _Tp __lolim = _Tp(5) * __min;
      const _Tp __uplim = __max / _Tp(5);

if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0))
        std::__throw_domain_error(__N("Argument less than zero "
                                      "in __ellint_rf."));
      else if (__x + __y < __lolim || __x + __z < __lolim
            || __y + __z < __lolim)
        std::__throw_domain_error(__N("Argument too small in __ellint_rf"));
      else
        {
          const _Tp __c0 = _Tp(1) / _Tp(4);
          const _Tp __c1 = _Tp(1) / _Tp(24);
          const _Tp __c2 = _Tp(1) / _Tp(10);
          const _Tp __c3 = _Tp(3) / _Tp(44);
          const _Tp __c4 = _Tp(1) / _Tp(14);

_Tp __xn = __x;
          _Tp __yn = __y;
          _Tp __zn = __z;

const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
          const _Tp __errtol = std::pow(__eps, _Tp(1) / _Tp(6));
          _Tp __mu;
          _Tp __xndev, __yndev, __zndev;

const unsigned int __max_iter = 100;
          for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
            {
              __mu = (__xn + __yn + __zn) / _Tp(3);
              __xndev = 2 - (__mu + __xn) / __mu;
              __yndev = 2 - (__mu + __yn) / __mu;
              __zndev = 2 - (__mu + __zn) / __mu;
              _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
              __epsilon = std::max(__epsilon, std::abs(__zndev));
              if (__epsilon < __errtol)
                break;
              const _Tp __xnroot = std::sqrt(__xn);
              const _Tp __ynroot = std::sqrt(__yn);
              const _Tp __znroot = std::sqrt(__zn);
              const _Tp __lambda = __xnroot * (__ynroot + __znroot)
                                 + __ynroot * __znroot;
              __xn = __c0 * (__xn + __lambda);
              __yn = __c0 * (__yn + __lambda);
              __zn = __c0 * (__zn + __lambda);
            }

const _Tp __e2 = __xndev * __yndev - __zndev * __zndev;
          const _Tp __e3 = __xndev * __yndev * __zndev;
          const _Tp __s  = _Tp(1) + (__c1 * __e2 - __c2 - __c3 * __e3) * __e2
                   + __c4 * __e3;

return __s / std::sqrt(__mu);
        }
    }

/**
     *   @brief Return the complete elliptic integral of the first kind
     *          @f$ K(k) @f$ by series expansion.
     * 
     *   The complete elliptic integral of the first kind is defined as
     *   @f[
     *     K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
     *                              {\sqrt{1 - k^2sin^2\theta}}
     *   @f]
     * 
     *   This routine is not bad as long as |k| is somewhat smaller than 1
     *   but is not is good as the Carlson elliptic integral formulation.
     * 
     *   @param  __k  The argument of the complete elliptic function.
     *   @return  The complete elliptic function of the first kind.
     */
    template<typename _Tp>
    _Tp
    __comp_ellint_1_series(_Tp __k)
    {

const _Tp __kk = __k * __k;

_Tp __term = __kk / _Tp(4);
      _Tp __sum = _Tp(1) + __term;

const unsigned int __max_iter = 1000;
      for (unsigned int __i = 2; __i < __max_iter; ++__i)
        {
          __term *= (2 * __i - 1) * __kk / (2 * __i);
          if (__term < std::numeric_limits<_Tp>::epsilon())
            break;
          __sum += __term;
        }

return __numeric_constants<_Tp>::__pi_2() * __sum;
    }

/**
     *   @brief  Return the complete elliptic integral of the first kind
     *           @f$ K(k) @f$ using the Carlson formulation.
     * 
     *   The complete elliptic integral of the first kind is defined as
     *   @f[
     *     K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
     *                                           {\sqrt{1 - k^2 sin^2\theta}}
     *   @f]
     *   where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the
     *   first kind.
     * 
     *   @param  __k  The argument of the complete elliptic function.
     *   @return  The complete elliptic function of the first kind.
     */
    template<typename _Tp>
    _Tp
    __comp_ellint_1(_Tp __k)
    {

if (__isnan(__k))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (std::abs(__k) >= _Tp(1))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else
        return __ellint_rf(_Tp(0), _Tp(1) - __k * __k, _Tp(1));
    }

/**
     *   @brief  Return the incomplete elliptic integral of the first kind
     *           @f$ F(k,\phi) @f$ using the Carlson formulation.
     * 
     *   The incomplete elliptic integral of the first kind is defined as
     *   @f[
     *     F(k,\phi) = \int_0^{\phi}\frac{d\theta}
     *                                   {\sqrt{1 - k^2 sin^2\theta}}
     *   @f]
     * 
     *   @param  __k  The argument of the elliptic function.
     *   @param  __phi  The integral limit argument of the elliptic function.
     *   @return  The elliptic function of the first kind.
     */
    template<typename _Tp>
    _Tp
    __ellint_1(_Tp __k, _Tp __phi)
    {

if (__isnan(__k) || __isnan(__phi))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (std::abs(__k) > _Tp(1))
        std::__throw_domain_error(__N("Bad argument in __ellint_1."));
      else
        {
          //  Reduce phi to -pi/2 < phi < +pi/2.
          const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
                                   + _Tp(0.5L));
          const _Tp __phi_red = __phi
                              - __n * __numeric_constants<_Tp>::__pi();

const _Tp __s = std::sin(__phi_red);
          const _Tp __c = std::cos(__phi_red);

const _Tp __F = __s
                        * __ellint_rf(__c * __c,
                                _Tp(1) - __k * __k * __s * __s, _Tp(1));

if (__n == 0)
            return __F;
          else
            return __F + _Tp(2) * __n * __comp_ellint_1(__k);
        }
    }

/**
     *   @brief Return the complete elliptic integral of the second kind
     *          @f$ E(k) @f$ by series expansion.
     * 
     *   The complete elliptic integral of the second kind is defined as
     *   @f[
     *     E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
     *   @f]
     * 
     *   This routine is not bad as long as |k| is somewhat smaller than 1
     *   but is not is good as the Carlson elliptic integral formulation.
     * 
     *   @param  __k  The argument of the complete elliptic function.
     *   @return  The complete elliptic function of the second kind.
     */
    template<typename _Tp>
    _Tp
    __comp_ellint_2_series(_Tp __k)
    {

const _Tp __kk = __k * __k;

_Tp __term = __kk;
      _Tp __sum = __term;

const unsigned int __max_iter = 1000;
      for (unsigned int __i = 2; __i < __max_iter; ++__i)
        {
          const _Tp __i2m = 2 * __i - 1;
          const _Tp __i2 = 2 * __i;
          __term *= __i2m * __i2m * __kk / (__i2 * __i2);
          if (__term < std::numeric_limits<_Tp>::epsilon())
            break;
          __sum += __term / __i2m;
        }

return __numeric_constants<_Tp>::__pi_2() * (_Tp(1) - __sum);
    }

/**
     *   @brief  Return the Carlson elliptic function of the second kind
     *           @f$ R_D(x,y,z) = R_J(x,y,z,z) @f$ where
     *           @f$ R_J(x,y,z,p) @f$ is the Carlson elliptic function
     *           of the third kind.
     * 
     *   The Carlson elliptic function of the second kind is defined by:
     *   @f[
     *       R_D(x,y,z) = \frac{3}{2} \int_0^\infty
     *                 \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{3/2}}
     *   @f]
     *
     *   Based on Carlson's algorithms:
     *   -  B. C. Carlson Numer. Math. 33, 1 (1979)
     *   -  B. C. Carlson, Special Functions of Applied Mathematics (1977)
     *   -  Numerical Recipes in C, 2nd ed, pp. 261-269,
     *      by Press, Teukolsky, Vetterling, Flannery (1992)
     *
     *   @param  __x  The first of two symmetric arguments.
     *   @param  __y  The second of two symmetric arguments.
     *   @param  __z  The third argument.
     *   @return  The Carlson elliptic function of the second kind.
     */
    template<typename _Tp>
    _Tp
    __ellint_rd(_Tp __x, _Tp __y, _Tp __z)
    {
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6));
      const _Tp __min = std::numeric_limits<_Tp>::min();
      const _Tp __max = std::numeric_limits<_Tp>::max();
      const _Tp __lolim = _Tp(2) / std::pow(__max, _Tp(2) / _Tp(3));
      const _Tp __uplim = std::pow(_Tp(0.1L) * __errtol / __min, _Tp(2) / _Tp(3));

if (__x < _Tp(0) || __y < _Tp(0))
        std::__throw_domain_error(__N("Argument less than zero "
                                      "in __ellint_rd."));
      else if (__x + __y < __lolim || __z < __lolim)
        std::__throw_domain_error(__N("Argument too small "
                                      "in __ellint_rd."));
      else
        {
          const _Tp __c0 = _Tp(1) / _Tp(4);
          const _Tp __c1 = _Tp(3) / _Tp(14);
          const _Tp __c2 = _Tp(1) / _Tp(6);
          const _Tp __c3 = _Tp(9) / _Tp(22);
          const _Tp __c4 = _Tp(3) / _Tp(26);

_Tp __xn = __x;
          _Tp __yn = __y;
          _Tp __zn = __z;
          _Tp __sigma = _Tp(0);
          _Tp __power4 = _Tp(1);

_Tp __mu;
          _Tp __xndev, __yndev, __zndev;

const unsigned int __max_iter = 100;
          for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
            {
              __mu = (__xn + __yn + _Tp(3) * __zn) / _Tp(5);
              __xndev = (__mu - __xn) / __mu;
              __yndev = (__mu - __yn) / __mu;
              __zndev = (__mu - __zn) / __mu;
              _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
              __epsilon = std::max(__epsilon, std::abs(__zndev));
              if (__epsilon < __errtol)
                break;
              _Tp __xnroot = std::sqrt(__xn);
              _Tp __ynroot = std::sqrt(__yn);
              _Tp __znroot = std::sqrt(__zn);
              _Tp __lambda = __xnroot * (__ynroot + __znroot)
                           + __ynroot * __znroot;
              __sigma += __power4 / (__znroot * (__zn + __lambda));
              __power4 *= __c0;
              __xn = __c0 * (__xn + __lambda);
              __yn = __c0 * (__yn + __lambda);
              __zn = __c0 * (__zn + __lambda);
            }

// Note: __ea is an SPU badname.
          _Tp __eaa = __xndev * __yndev;
          _Tp __eb = __zndev * __zndev;
          _Tp __ec = __eaa - __eb;
          _Tp __ed = __eaa - _Tp(6) * __eb;
          _Tp __ef = __ed + __ec + __ec;
          _Tp __s1 = __ed * (-__c1 + __c3 * __ed
                                   / _Tp(3) - _Tp(3) * __c4 * __zndev * __ef
                                   / _Tp(2));
          _Tp __s2 = __zndev
                   * (__c2 * __ef
                    + __zndev * (-__c3 * __ec - __zndev * __c4 - __eaa));

return _Tp(3) * __sigma + __power4 * (_Tp(1) + __s1 + __s2)
                                        / (__mu * std::sqrt(__mu));
        }
    }

/**
     *   @brief  Return the complete elliptic integral of the second kind
     *           @f$ E(k) @f$ using the Carlson formulation.
     * 
     *   The complete elliptic integral of the second kind is defined as
     *   @f[
     *     E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
     *   @f]
     * 
     *   @param  __k  The argument of the complete elliptic function.
     *   @return  The complete elliptic function of the second kind.
     */
    template<typename _Tp>
    _Tp
    __comp_ellint_2(_Tp __k)
    {

if (__isnan(__k))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (std::abs(__k) == 1)
        return _Tp(1);
      else if (std::abs(__k) > _Tp(1))
        std::__throw_domain_error(__N("Bad argument in __comp_ellint_2."));
      else
        {
          const _Tp __kk = __k * __k;

return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1))
               - __kk * __ellint_rd(_Tp(0), _Tp(1) - __kk, _Tp(1)) / _Tp(3);
        }
    }

/**
     *   @brief  Return the incomplete elliptic integral of the second kind
     *           @f$ E(k,\phi) @f$ using the Carlson formulation.
     * 
     *   The incomplete elliptic integral of the second kind is defined as
     *   @f[
     *     E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta}
     *   @f]
     * 
     *   @param  __k  The argument of the elliptic function.
     *   @param  __phi  The integral limit argument of the elliptic function.
     *   @return  The elliptic function of the second kind.
     */
    template<typename _Tp>
    _Tp
    __ellint_2(_Tp __k, _Tp __phi)
    {

if (__isnan(__k) || __isnan(__phi))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (std::abs(__k) > _Tp(1))
        std::__throw_domain_error(__N("Bad argument in __ellint_2."));
      else
        {
          //  Reduce phi to -pi/2 < phi < +pi/2.
          const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
                                   + _Tp(0.5L));
          const _Tp __phi_red = __phi
                              - __n * __numeric_constants<_Tp>::__pi();

const _Tp __kk = __k * __k;
          const _Tp __s = std::sin(__phi_red);
          const _Tp __ss = __s * __s;
          const _Tp __sss = __ss * __s;
          const _Tp __c = std::cos(__phi_red);
          const _Tp __cc = __c * __c;

const _Tp __E = __s
                        * __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1))
                        - __kk * __sss
                        * __ellint_rd(__cc, _Tp(1) - __kk * __ss, _Tp(1))
                        / _Tp(3);

if (__n == 0)
            return __E;
          else
            return __E + _Tp(2) * __n * __comp_ellint_2(__k);
        }
    }

/**
     *   @brief  Return the Carlson elliptic function
     *           @f$ R_C(x,y) = R_F(x,y,y) @f$ where @f$ R_F(x,y,z) @f$
     *           is the Carlson elliptic function of the first kind.
     * 
     *   The Carlson elliptic function is defined by:
     *   @f[
     *       R_C(x,y) = \frac{1}{2} \int_0^\infty
     *                 \frac{dt}{(t + x)^{1/2}(t + y)}
     *   @f]
     *
     *   Based on Carlson's algorithms:
     *   -  B. C. Carlson Numer. Math. 33, 1 (1979)
     *   -  B. C. Carlson, Special Functions of Applied Mathematics (1977)
     *   -  Numerical Recipes in C, 2nd ed, pp. 261-269,
     *      by Press, Teukolsky, Vetterling, Flannery (1992)
     *
     *   @param  __x  The first argument.
     *   @param  __y  The second argument.
     *   @return  The Carlson elliptic function.
     */
    template<typename _Tp>
    _Tp
    __ellint_rc(_Tp __x, _Tp __y)
    {
      const _Tp __min = std::numeric_limits<_Tp>::min();
      const _Tp __max = std::numeric_limits<_Tp>::max();
      const _Tp __lolim = _Tp(5) * __min;
      const _Tp __uplim = __max / _Tp(5);

if (__x < _Tp(0) || __y < _Tp(0) || __x + __y < __lolim)
        std::__throw_domain_error(__N("Argument less than zero "
                                      "in __ellint_rc."));
      else
        {
          const _Tp __c0 = _Tp(1) / _Tp(4);
          const _Tp __c1 = _Tp(1) / _Tp(7);
          const _Tp __c2 = _Tp(9) / _Tp(22);
          const _Tp __c3 = _Tp(3) / _Tp(10);
          const _Tp __c4 = _Tp(3) / _Tp(8);

_Tp __xn = __x;
          _Tp __yn = __y;

const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
          const _Tp __errtol = std::pow(__eps / _Tp(30), _Tp(1) / _Tp(6));
          _Tp __mu;
          _Tp __sn;

const unsigned int __max_iter = 100;
          for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
            {
              __mu = (__xn + _Tp(2) * __yn) / _Tp(3);
              __sn = (__yn + __mu) / __mu - _Tp(2);
              if (std::abs(__sn) < __errtol)
                break;
              const _Tp __lambda = _Tp(2) * std::sqrt(__xn) * std::sqrt(__yn)
                             + __yn;
              __xn = __c0 * (__xn + __lambda);
              __yn = __c0 * (__yn + __lambda);
            }

_Tp __s = __sn * __sn
                  * (__c3 + __sn*(__c1 + __sn * (__c4 + __sn * __c2)));

return (_Tp(1) + __s) / std::sqrt(__mu);
        }
    }

/**
     *   @brief  Return the Carlson elliptic function @f$ R_J(x,y,z,p) @f$
     *           of the third kind.
     * 
     *   The Carlson elliptic function of the third kind is defined by:
     *   @f[
     *       R_J(x,y,z,p) = \frac{3}{2} \int_0^\infty
     *       \frac{dt}{(t + x)^{1/2}(t + y)^{1/2}(t + z)^{1/2}(t + p)}
     *   @f]
     *
     *   Based on Carlson's algorithms:
     *   -  B. C. Carlson Numer. Math. 33, 1 (1979)
     *   -  B. C. Carlson, Special Functions of Applied Mathematics (1977)
     *   -  Numerical Recipes in C, 2nd ed, pp. 261-269,
     *      by Press, Teukolsky, Vetterling, Flannery (1992)
     *
     *   @param  __x  The first of three symmetric arguments.
     *   @param  __y  The second of three symmetric arguments.
     *   @param  __z  The third of three symmetric arguments.
     *   @param  __p  The fourth argument.
     *   @return  The Carlson elliptic function of the fourth kind.
     */
    template<typename _Tp>
    _Tp
    __ellint_rj(_Tp __x, _Tp __y, _Tp __z, _Tp __p)
    {
      const _Tp __min = std::numeric_limits<_Tp>::min();
      const _Tp __max = std::numeric_limits<_Tp>::max();
      const _Tp __lolim = std::pow(_Tp(5) * __min, _Tp(1)/_Tp(3));
      const _Tp __uplim = _Tp(0.3L)
                        * std::pow(_Tp(0.2L) * __max, _Tp(1)/_Tp(3));

if (__x < _Tp(0) || __y < _Tp(0) || __z < _Tp(0))
        std::__throw_domain_error(__N("Argument less than zero "
                                      "in __ellint_rj."));
      else if (__x + __y < __lolim || __x + __z < __lolim
            || __y + __z < __lolim || __p < __lolim)
        std::__throw_domain_error(__N("Argument too small "
                                      "in __ellint_rj"));
      else
        {
          const _Tp __c0 = _Tp(1) / _Tp(4);
          const _Tp __c1 = _Tp(3) / _Tp(14);
          const _Tp __c2 = _Tp(1) / _Tp(3);
          const _Tp __c3 = _Tp(3) / _Tp(22);
          const _Tp __c4 = _Tp(3) / _Tp(26);

_Tp __xn = __x;
          _Tp __yn = __y;
          _Tp __zn = __z;
          _Tp __pn = __p;
          _Tp __sigma = _Tp(0);
          _Tp __power4 = _Tp(1);

const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
          const _Tp __errtol = std::pow(__eps / _Tp(8), _Tp(1) / _Tp(6));

_Tp __lambda, __mu;
          _Tp __xndev, __yndev, __zndev, __pndev;

const unsigned int __max_iter = 100;
          for (unsigned int __iter = 0; __iter < __max_iter; ++__iter)
            {
              __mu = (__xn + __yn + __zn + _Tp(2) * __pn) / _Tp(5);
              __xndev = (__mu - __xn) / __mu;
              __yndev = (__mu - __yn) / __mu;
              __zndev = (__mu - __zn) / __mu;
              __pndev = (__mu - __pn) / __mu;
              _Tp __epsilon = std::max(std::abs(__xndev), std::abs(__yndev));
              __epsilon = std::max(__epsilon, std::abs(__zndev));
              __epsilon = std::max(__epsilon, std::abs(__pndev));
              if (__epsilon < __errtol)
                break;
              const _Tp __xnroot = std::sqrt(__xn);
              const _Tp __ynroot = std::sqrt(__yn);
              const _Tp __znroot = std::sqrt(__zn);
              const _Tp __lambda = __xnroot * (__ynroot + __znroot)
                                 + __ynroot * __znroot;
              const _Tp __alpha1 = __pn * (__xnroot + __ynroot + __znroot)
                                + __xnroot * __ynroot * __znroot;
              const _Tp __alpha2 = __alpha1 * __alpha1;
              const _Tp __beta = __pn * (__pn + __lambda)
                                      * (__pn + __lambda);
              __sigma += __power4 * __ellint_rc(__alpha2, __beta);
              __power4 *= __c0;
              __xn = __c0 * (__xn + __lambda);
              __yn = __c0 * (__yn + __lambda);
              __zn = __c0 * (__zn + __lambda);
              __pn = __c0 * (__pn + __lambda);
            }

// Note: __ea is an SPU badname.
          _Tp __eaa = __xndev * (__yndev + __zndev) + __yndev * __zndev;
          _Tp __eb = __xndev * __yndev * __zndev;
          _Tp __ec = __pndev * __pndev;
          _Tp __e2 = __eaa - _Tp(3) * __ec;
          _Tp __e3 = __eb + _Tp(2) * __pndev * (__eaa - __ec);
          _Tp __s1 = _Tp(1) + __e2 * (-__c1 + _Tp(3) * __c3 * __e2 / _Tp(4)
                            - _Tp(3) * __c4 * __e3 / _Tp(2));
          _Tp __s2 = __eb * (__c2 / _Tp(2)
                   + __pndev * (-__c3 - __c3 + __pndev * __c4));
          _Tp __s3 = __pndev * __eaa * (__c2 - __pndev * __c3)
                   - __c2 * __pndev * __ec;

return _Tp(3) * __sigma + __power4 * (__s1 + __s2 + __s3)
                                             / (__mu * std::sqrt(__mu));
        }
    }

/**
     *   @brief Return the complete elliptic integral of the third kind
     *          @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ using the
     *          Carlson formulation.
     * 
     *   The complete elliptic integral of the third kind is defined as
     *   @f[
     *     \Pi(k,\nu) = \int_0^{\pi/2}
     *                   \frac{d\theta}
     *                 {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}
     *   @f]
     * 
     *   @param  __k  The argument of the elliptic function.
     *   @param  __nu  The second argument of the elliptic function.
     *   @return  The complete elliptic function of the third kind.
     */
    template<typename _Tp>
    _Tp
    __comp_ellint_3(_Tp __k, _Tp __nu)
    {

if (__isnan(__k) || __isnan(__nu))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__nu == _Tp(1))
        return std::numeric_limits<_Tp>::infinity();
      else if (std::abs(__k) > _Tp(1))
        std::__throw_domain_error(__N("Bad argument in __comp_ellint_3."));
      else
        {
          const _Tp __kk = __k * __k;

return __ellint_rf(_Tp(0), _Tp(1) - __kk, _Tp(1))
               + __nu
               * __ellint_rj(_Tp(0), _Tp(1) - __kk, _Tp(1), _Tp(1) - __nu)
               / _Tp(3);
        }
    }

/**
     *   @brief Return the incomplete elliptic integral of the third kind
     *          @f$ \Pi(k,\nu,\phi) @f$ using the Carlson formulation.
     * 
     *   The incomplete elliptic integral of the third kind is defined as
     *   @f[
     *     \Pi(k,\nu,\phi) = \int_0^{\phi}
     *                       \frac{d\theta}
     *                            {(1 - \nu \sin^2\theta)
     *                             \sqrt{1 - k^2 \sin^2\theta}}
     *   @f]
     * 
     *   @param  __k  The argument of the elliptic function.
     *   @param  __nu  The second argument of the elliptic function.
     *   @param  __phi  The integral limit argument of the elliptic function.
     *   @return  The elliptic function of the third kind.
     */
    template<typename _Tp>
    _Tp
    __ellint_3(_Tp __k, _Tp __nu, _Tp __phi)
    {

if (__isnan(__k) || __isnan(__nu) || __isnan(__phi))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (std::abs(__k) > _Tp(1))
        std::__throw_domain_error(__N("Bad argument in __ellint_3."));
      else
        {
          //  Reduce phi to -pi/2 < phi < +pi/2.
          const int __n = std::floor(__phi / __numeric_constants<_Tp>::__pi()
                                   + _Tp(0.5L));
          const _Tp __phi_red = __phi
                              - __n * __numeric_constants<_Tp>::__pi();

const _Tp __Pi = __s
                         * __ellint_rf(__cc, _Tp(1) - __kk * __ss, _Tp(1))
                         + __nu * __sss
                         * __ellint_rj(__cc, _Tp(1) - __kk * __ss, _Tp(1),
                                       _Tp(1) - __nu * __ss) / _Tp(3);

if (__n == 0)
            return __Pi;
          else
            return __Pi + _Tp(2) * __n * __comp_ellint_3(__k, __nu);
        }
    }
  } // namespace __detail
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
} // namespace tr1
#endif

_GLIBCXX_END_NAMESPACE_VERSION
}

#endif // _GLIBCXX_TR1_ELL_INTEGRAL_TCC

cmd: