// Copyright John Maddock 2012.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_JACOBI_ELLIPTIC_HPP
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#define BOOST_MATH_JACOBI_ELLIPTIC_HPP
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#include <boost/math/tools/precision.hpp>
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#include <boost/math/tools/promotion.hpp>
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#include <boost/math/policies/error_handling.hpp>
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#include <boost/math/special_functions/math_fwd.hpp>
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namespace boost{ namespace math{
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namespace detail{
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template <class T, class Policy>
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T jacobi_recurse(const T& x, const T& k, T anm1, T bnm1, unsigned N, T* pTn, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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++N;
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T Tn;
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T cn = (anm1 - bnm1) / 2;
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T an = (anm1 + bnm1) / 2;
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if(cn < policies::get_epsilon<T, Policy>())
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{
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Tn = ldexp(T(1), (int)N) * x * an;
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}
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else
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Tn = jacobi_recurse<T>(x, k, an, sqrt(anm1 * bnm1), N, 0, pol);
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if(pTn)
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*pTn = Tn;
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return (Tn + asin((cn / an) * sin(Tn))) / 2;
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}
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template <class T, class Policy>
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T jacobi_imp(const T& x, const T& k, T* cn, T* dn, const Policy& pol, const char* function)
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{
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BOOST_MATH_STD_USING
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if(k < 0)
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{
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*cn = policies::raise_domain_error<T>(function, "Modulus k must be positive but got %1%.", k, pol);
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*dn = *cn;
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return *cn;
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}
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if(k > 1)
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{
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T xp = x * k;
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T kp = 1 / k;
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T snp, cnp, dnp;
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snp = jacobi_imp(xp, kp, &cnp, &dnp, pol, function);
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*cn = dnp;
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*dn = cnp;
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return snp * kp;
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}
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//
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// Special cases first:
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//
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if(x == 0)
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{
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*cn = *dn = 1;
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return 0;
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}
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if(k == 0)
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{
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*cn = cos(x);
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*dn = 1;
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return sin(x);
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}
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if(k == 1)
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{
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*cn = *dn = 1 / cosh(x);
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return tanh(x);
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}
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//
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// Asymptotic forms from A&S 16.13:
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//
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if(k < tools::forth_root_epsilon<T>())
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{
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T su = sin(x);
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T cu = cos(x);
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T m = k * k;
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*dn = 1 - m * su * su / 2;
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*cn = cu + m * (x - su * cu) * su / 4;
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return su - m * (x - su * cu) * cu / 4;
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}
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/* Can't get this to work to adequate precision - disabled for now...
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//
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// Asymptotic forms from A&S 16.15:
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//
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if(k > 1 - tools::root_epsilon<T>())
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{
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T tu = tanh(x);
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T su = sinh(x);
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T cu = cosh(x);
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T sec = 1 / cu;
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T kp = 1 - k;
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T m1 = 2 * kp - kp * kp;
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*dn = sec + m1 * (su * cu + x) * tu * sec / 4;
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*cn = sec - m1 * (su * cu - x) * tu * sec / 4;
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T sn = tu;
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T sn2 = m1 * (x * sec * sec - tu) / 4;
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T sn3 = (72 * x * cu + 4 * (8 * x * x - 5) * su - 19 * sinh(3 * x) + sinh(5 * x)) * sec * sec * sec * m1 * m1 / 512;
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return sn + sn2 - sn3;
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}*/
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T T1;
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T kc = 1 - k;
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T k_prime = k < 0.5 ? T(sqrt(1 - k * k)) : T(sqrt(2 * kc - kc * kc));
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T T0 = jacobi_recurse(x, k, T(1), k_prime, 0, &T1, pol);
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*cn = cos(T0);
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*dn = cos(T0) / cos(T1 - T0);
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return sin(T0);
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}
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} // namespace detail
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template <class T, class U, class V, class Policy>
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inline typename tools::promote_args<T, U, V>::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn, const Policy&)
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{
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BOOST_FPU_EXCEPTION_GUARD
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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typedef typename policies::normalise<
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Policy,
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policies::promote_float<false>,
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policies::promote_double<false>,
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policies::discrete_quantile<>,
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policies::assert_undefined<> >::type forwarding_policy;
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static const char* function = "boost::math::jacobi_elliptic<%1%>(%1%)";
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value_type sn, cn, dn;
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sn = detail::jacobi_imp<value_type>(static_cast<value_type>(theta), static_cast<value_type>(k), &cn, &dn, forwarding_policy(), function);
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if(pcn)
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*pcn = policies::checked_narrowing_cast<result_type, Policy>(cn, function);
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if(pdn)
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*pdn = policies::checked_narrowing_cast<result_type, Policy>(dn, function);
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return policies::checked_narrowing_cast<result_type, Policy>(sn, function);
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}
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template <class T, class U, class V>
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inline typename tools::promote_args<T, U, V>::type jacobi_elliptic(T k, U theta, V* pcn, V* pdn)
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{
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return jacobi_elliptic(k, theta, pcn, pdn, policies::policy<>());
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}
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template <class U, class T, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_sn(U k, T theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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return jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), static_cast<result_type*>(0), pol);
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}
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template <class U, class T>
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inline typename tools::promote_args<T, U>::type jacobi_sn(U k, T theta)
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{
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return jacobi_sn(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_cn(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type cn;
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jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol);
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return cn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_cn(T k, U theta)
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{
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return jacobi_cn(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_dn(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type dn;
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jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol);
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return dn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_dn(T k, U theta)
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{
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return jacobi_dn(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_cd(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type cn, dn;
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jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, &dn, pol);
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return cn / dn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_cd(T k, U theta)
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{
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return jacobi_cd(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_dc(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type cn, dn;
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jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, &dn, pol);
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return dn / cn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_dc(T k, U theta)
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{
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return jacobi_dc(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_ns(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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return 1 / jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), static_cast<result_type*>(0), pol);
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_ns(T k, U theta)
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{
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return jacobi_ns(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_sd(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type sn, dn;
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sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol);
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return sn / dn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_sd(T k, U theta)
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{
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return jacobi_sd(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_ds(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type sn, dn;
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sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), static_cast<result_type*>(0), &dn, pol);
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return dn / sn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_ds(T k, U theta)
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{
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return jacobi_ds(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_nc(T k, U theta, const Policy& pol)
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{
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return 1 / jacobi_cn(k, theta, pol);
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_nc(T k, U theta)
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{
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return jacobi_nc(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_nd(T k, U theta, const Policy& pol)
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{
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return 1 / jacobi_dn(k, theta, pol);
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_nd(T k, U theta)
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{
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return jacobi_nd(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_sc(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type sn, cn;
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sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol);
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return sn / cn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_sc(T k, U theta)
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{
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return jacobi_sc(k, theta, policies::policy<>());
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}
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template <class T, class U, class Policy>
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inline typename tools::promote_args<T, U>::type jacobi_cs(T k, U theta, const Policy& pol)
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{
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typedef typename tools::promote_args<T, U>::type result_type;
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result_type sn, cn;
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sn = jacobi_elliptic(static_cast<result_type>(k), static_cast<result_type>(theta), &cn, static_cast<result_type*>(0), pol);
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return cn / sn;
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}
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template <class T, class U>
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inline typename tools::promote_args<T, U>::type jacobi_cs(T k, U theta)
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{
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return jacobi_cs(k, theta, policies::policy<>());
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}
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}} // namespaces
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#endif // BOOST_MATH_JACOBI_ELLIPTIC_HPP
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