boost/boost/math/special_functions/sinc.hpp

//  boost sinc.hpp header file

//  (C) Copyright Hubert Holin 2001.
//  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)

// See http://www.boost.org for updates, documentation, and revision history.

#ifndef BOOST_SINC_HPP
#define BOOST_SINC_HPP


#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/tools/config.hpp>
#include <boost/math/tools/precision.hpp>
#include <boost/math/policies/policy.hpp>
#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/config/no_tr1/cmath.hpp>
#include <boost/limits.hpp>
#include <string>
#include <stdexcept>


#include <boost/config.hpp>


// These are the the "Sinus Cardinal" functions.

namespace boost
{
    namespace math
    {
       namespace detail
       {
        // This is the "Sinus Cardinal" of index Pi.

        template<typename T>
        inline T    sinc_pi_imp(const T x)
        {
            BOOST_MATH_STD_USING

            T const    taylor_0_bound = tools::epsilon<T>();
            T const    taylor_2_bound = tools::root_epsilon<T>();
            T const    taylor_n_bound = tools::forth_root_epsilon<T>();

            if    (abs(x) >= taylor_n_bound)
            {
                return(sin(x)/x);
            }
            else
            {
                // approximation by taylor series in x at 0 up to order 0
                T    result = static_cast<T>(1);

                if    (abs(x) >= taylor_0_bound)
                {
                    T    x2 = x*x;

                    // approximation by taylor series in x at 0 up to order 2
                    result -= x2/static_cast<T>(6);

                    if    (abs(x) >= taylor_2_bound)
                    {
                        // approximation by taylor series in x at 0 up to order 4
                        result += (x2*x2)/static_cast<T>(120);
                    }
                }

                return(result);
            }
        }

       } // namespace detail

       template <class T>
       inline typename tools::promote_args<T>::type sinc_pi(T x)
       {
          typedef typename tools::promote_args<T>::type result_type;
          return detail::sinc_pi_imp(static_cast<result_type>(x));
       }

       template <class T, class Policy>
       inline typename tools::promote_args<T>::type sinc_pi(T x, const Policy&)
       {
          typedef typename tools::promote_args<T>::type result_type;
          return detail::sinc_pi_imp(static_cast<result_type>(x));
       }

#ifndef    BOOST_NO_TEMPLATE_TEMPLATES
        template<typename T, template<typename> class U>
        inline U<T>    sinc_pi(const U<T> x)
        {
            BOOST_MATH_STD_USING
            using    ::std::numeric_limits;

            T const    taylor_0_bound = tools::epsilon<T>();
            T const    taylor_2_bound = tools::root_epsilon<T>();
            T const    taylor_n_bound = tools::forth_root_epsilon<T>();

            if    (abs(x) >= taylor_n_bound)
            {
                return(sin(x)/x);
            }
            else
            {
                // approximation by taylor series in x at 0 up to order 0
#ifdef __MWERKS__
                U<T>    result = static_cast<U<T> >(1);
#else
                U<T>    result = U<T>(1);
#endif

                if    (abs(x) >= taylor_0_bound)
                {
                    U<T>    x2 = x*x;

                    // approximation by taylor series in x at 0 up to order 2
                    result -= x2/static_cast<T>(6);

                    if    (abs(x) >= taylor_2_bound)
                    {
                        // approximation by taylor series in x at 0 up to order 4
                        result += (x2*x2)/static_cast<T>(120);
                    }
                }

                return(result);
            }
        }

        template<typename T, template<typename> class U, class Policy>
        inline U<T>    sinc_pi(const U<T> x, const Policy&)
        {
           return sinc_pi(x);
        }
#endif    /* BOOST_NO_TEMPLATE_TEMPLATES */
    }
}

#endif /* BOOST_SINC_HPP */