--- /dev/null
+// Special functions -*- C++ -*-
+
+// Copyright (C) 2006, 2007, 2008, 2009
+// 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/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
+ / (_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
projects / msp430-gcc.git / blobdiff