//  Copyright (c) 2006 Xiaogang Zhang

//  Copyright (c) 2006 John Maddock

//  Use, modification and distribution are subject to 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)

//

//  History:

//  XZ wrote the original of this file as part of the Google

//  Summer of Code 2006.  JM modified it to fit into the

//  Boost.Math conceptual framework better, and to ensure

//  that the code continues to work no matter how many digits

//  type T has.



#ifndef BOOST_MATH_ELLINT_2_HPP

#define BOOST_MATH_ELLINT_2_HPP



#include <boost/math/special_functions/ellint_rf.hpp>

#include <boost/math/special_functions/ellint_rd.hpp>

#include <boost/math/constants/constants.hpp>

#include <boost/math/policies/error_handling.hpp>

#include <boost/math/tools/workaround.hpp>



// Elliptic integrals (complete and incomplete) of the second kind

// Carlson, Numerische Mathematik, vol 33, 1 (1979)



namespace boost { namespace math { 

   

template <class T1, class T2, class Policy>

typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol);

   

namespace detail{



template <typename T, typename Policy>

T ellint_e_imp(T k, const Policy& pol);



// Elliptic integral (Legendre form) of the second kind

template <typename T, typename Policy>

T ellint_e_imp(T phi, T k, const Policy& pol)

{

    BOOST_MATH_STD_USING

    using namespace boost::math::tools;

    using namespace boost::math::constants;



    bool invert = false;

    if(phi < 0)

    {

       phi = fabs(phi);

       invert = true;

    }



    T result;



    if(phi >= tools::max_value<T>())

    {

       // Need to handle infinity as a special case:

       result = policies::raise_overflow_error<T>("boost::math::ellint_e<%1%>(%1%,%1%)", 0, pol);

    }

    else if(phi > 1 / tools::epsilon<T>())

    {

       // Phi is so large that phi%pi is necessarily zero (or garbage),

       // just return the second part of the duplication formula:

       result = 2 * phi * ellint_e_imp(k, pol) / constants::pi<T>();

    }

    else

    {

       // Carlson's algorithm works only for |phi| <= pi/2,

       // use the integrand's periodicity to normalize phi

       //

       // Xiaogang's original code used a cast to long long here

       // but that fails if T has more digits than a long long,

       // so rewritten to use fmod instead:

       //

       T rphi = boost::math::tools::fmod_workaround(phi, constants::pi<T>() / 2);

       T m = 2 * (phi - rphi) / constants::pi<T>();

       int s = 1;

       if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5)

       {

          m += 1;

          s = -1;

          rphi = constants::pi<T>() / 2 - rphi;

       }

       T sinp = sin(rphi);

       T cosp = cos(rphi);

       T x = cosp * cosp;

       T t = k * k * sinp * sinp;

       T y = 1 - t;

       T z = 1;

       result = s * sinp * (ellint_rf_imp(x, y, z, pol) - t * ellint_rd_imp(x, y, z, pol) / 3);

       if(m != 0)

          result += m * ellint_e_imp(k, pol);

    }

    return invert ? -result : result;

}



// Complete elliptic integral (Legendre form) of the second kind

template <typename T, typename Policy>

T ellint_e_imp(T k, const Policy& pol)

{

    BOOST_MATH_STD_USING

    using namespace boost::math::tools;



    if (abs(k) > 1)

    {

       return policies::raise_domain_error<T>("boost::math::ellint_e<%1%>(%1%)",

            "Got k = %1%, function requires |k| <= 1", k, pol);

    }

    if (abs(k) == 1)

    {

        return static_cast<T>(1);

    }



    T x = 0;

    T t = k * k;

    T y = 1 - t;

    T z = 1;

    T value = ellint_rf_imp(x, y, z, pol) - t * ellint_rd_imp(x, y, z, pol) / 3;



    return value;

}



template <typename T, typename Policy>

inline typename tools::promote_args<T>::type ellint_2(T k, const Policy& pol, const mpl::true_&)

{

   typedef typename tools::promote_args<T>::type result_type;

   typedef typename policies::evaluation<result_type, Policy>::type value_type;

   return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%)");

}



// Elliptic integral (Legendre form) of the second kind

template <class T1, class T2>

inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const mpl::false_&)

{

   return boost::math::ellint_2(k, phi, policies::policy<>());

}



} // detail



// Complete elliptic integral (Legendre form) of the second kind

template <typename T>

inline typename tools::promote_args<T>::type ellint_2(T k)

{

   return ellint_2(k, policies::policy<>());

}



// Elliptic integral (Legendre form) of the second kind

template <class T1, class T2>

inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi)

{

   typedef typename policies::is_policy<T2>::type tag_type;

   return detail::ellint_2(k, phi, tag_type());

}



template <class T1, class T2, class Policy>

inline typename tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi, const Policy& pol)

{

   typedef typename tools::promote_args<T1, T2>::type result_type;

   typedef typename policies::evaluation<result_type, Policy>::type value_type;

   return policies::checked_narrowing_cast<result_type, Policy>(detail::ellint_e_imp(static_cast<value_type>(phi), static_cast<value_type>(k), pol), "boost::math::ellint_2<%1%>(%1%,%1%)");

}



}} // namespaces



#endif // BOOST_MATH_ELLINT_2_HPP

