Chris@16: //  (C) Copyright John Maddock 2006.
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_SF_CBRT_HPP
Chris@16: #define BOOST_MATH_SF_CBRT_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: #include <boost/math/tools/rational.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@16: #include <boost/math/special_functions/math_fwd.hpp>
Chris@16: #include <boost/math/special_functions/fpclassify.hpp>
Chris@16: #include <boost/mpl/divides.hpp>
Chris@16: #include <boost/mpl/plus.hpp>
Chris@16: #include <boost/mpl/if.hpp>
Chris@16: #include <boost/type_traits/is_convertible.hpp>
Chris@16: 
Chris@16: namespace boost{ namespace math{
Chris@16: 
Chris@16: namespace detail
Chris@16: {
Chris@16: 
Chris@16: struct big_int_type
Chris@16: {
Chris@16:    operator boost::uintmax_t()const;
Chris@16: };
Chris@16: 
Chris@16: template <class T>
Chris@16: struct largest_cbrt_int_type
Chris@16: {
Chris@16:    typedef typename mpl::if_<
Chris@16:       boost::is_convertible<big_int_type, T>,
Chris@16:       boost::uintmax_t,
Chris@16:       unsigned int
Chris@16:    >::type type;
Chris@16: };
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: T cbrt_imp(T z, const Policy& pol)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    //
Chris@16:    // cbrt approximation for z in the range [0.5,1]
Chris@16:    // It's hard to say what number of terms gives the optimum
Chris@16:    // trade off between precision and performance, this seems
Chris@16:    // to be about the best for double precision.
Chris@16:    //
Chris@16:    // Maximum Deviation Found:                     1.231e-006
Chris@16:    // Expected Error Term:                         -1.231e-006
Chris@16:    // Maximum Relative Change in Control Points:   5.982e-004
Chris@16:    //
Chris@16:    static const T P[] = { 
Chris@16:       static_cast<T>(0.37568269008611818),
Chris@16:       static_cast<T>(1.3304968705558024),
Chris@16:       static_cast<T>(-1.4897101632445036),
Chris@16:       static_cast<T>(1.2875573098219835),
Chris@16:       static_cast<T>(-0.6398703759826468),
Chris@16:       static_cast<T>(0.13584489959258635),
Chris@16:    };
Chris@16:    static const T correction[] = {
Chris@16:       static_cast<T>(0.62996052494743658238360530363911),  // 2^-2/3
Chris@16:       static_cast<T>(0.79370052598409973737585281963615),  // 2^-1/3
Chris@16:       static_cast<T>(1),
Chris@16:       static_cast<T>(1.2599210498948731647672106072782),   // 2^1/3
Chris@16:       static_cast<T>(1.5874010519681994747517056392723),   // 2^2/3
Chris@16:    };
Chris@16: 
Chris@16:    if(!(boost::math::isfinite)(z))
Chris@16:    {
Chris@16:       return policies::raise_domain_error("boost::math::cbrt<%1%>(%1%)", "Argument to function must be finite but got %1%.", z, pol);
Chris@16:    }
Chris@16: 
Chris@16:    int i_exp, sign(1);
Chris@16:    if(z < 0)
Chris@16:    {
Chris@16:       z = -z;
Chris@16:       sign = -sign;
Chris@16:    }
Chris@16:    if(z == 0)
Chris@16:       return 0;
Chris@16: 
Chris@16:    T guess = frexp(z, &i_exp);
Chris@16:    int original_i_exp = i_exp; // save for later
Chris@16:    guess = tools::evaluate_polynomial(P, guess);
Chris@16:    int i_exp3 = i_exp / 3;
Chris@16: 
Chris@16:    typedef typename largest_cbrt_int_type<T>::type shift_type;
Chris@16: 
Chris@16:    BOOST_STATIC_ASSERT( ::std::numeric_limits<shift_type>::radix == 2);
Chris@16: 
Chris@16:    if(abs(i_exp3) < std::numeric_limits<shift_type>::digits)
Chris@16:    {
Chris@16:       if(i_exp3 > 0)
Chris@16:          guess *= shift_type(1u) << i_exp3;
Chris@16:       else
Chris@16:          guess /= shift_type(1u) << -i_exp3;
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       guess = ldexp(guess, i_exp3);
Chris@16:    }
Chris@16:    i_exp %= 3;
Chris@16:    guess *= correction[i_exp + 2];
Chris@16:    //
Chris@16:    // Now inline Halley iteration.
Chris@16:    // We do this here rather than calling tools::halley_iterate since we can
Chris@16:    // simplify the expressions algebraically, and don't need most of the error
Chris@16:    // checking of the boilerplate version as we know in advance that the function
Chris@16:    // is well behaved...
Chris@16:    //
Chris@16:    typedef typename policies::precision<T, Policy>::type prec;
Chris@16:    typedef typename mpl::divides<prec, mpl::int_<3> >::type prec3;
Chris@16:    typedef typename mpl::plus<prec3, mpl::int_<3> >::type new_prec;
Chris@16:    typedef typename policies::normalise<Policy, policies::digits2<new_prec::value> >::type new_policy;
Chris@16:    //
Chris@16:    // Epsilon calculation uses compile time arithmetic when it's available for type T,
Chris@16:    // otherwise uses ldexp to calculate at runtime:
Chris@16:    //
Chris@16:    T eps = (new_prec::value > 3) ? policies::get_epsilon<T, new_policy>() : ldexp(T(1), -2 - tools::digits<T>() / 3);
Chris@16:    T diff;
Chris@16: 
Chris@16:    if(original_i_exp < std::numeric_limits<T>::max_exponent - 3)
Chris@16:    {
Chris@16:       //
Chris@16:       // Safe from overflow, use the fast method:
Chris@16:       //
Chris@16:       do
Chris@16:       {
Chris@16:          T g3 = guess * guess * guess;
Chris@16:          diff = (g3 + z + z) / (g3 + g3 + z);
Chris@16:          guess *= diff;
Chris@16:       }
Chris@16:       while(fabs(1 - diff) > eps);
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       //
Chris@16:       // Either we're ready to overflow, or we can't tell because numeric_limits isn't
Chris@16:       // available for type T:
Chris@16:       //
Chris@16:       do
Chris@16:       {
Chris@16:          T g2 = guess * guess;
Chris@16:          diff = (g2 - z / guess) / (2 * guess + z / g2);
Chris@16:          guess -= diff;
Chris@16:       }
Chris@16:       while((guess * eps) < fabs(diff));
Chris@16:    }
Chris@16: 
Chris@16:    return sign * guess;
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: inline typename tools::promote_args<T>::type cbrt(T z, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T>::type result_type;
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@16:    return static_cast<result_type>(detail::cbrt_imp(value_type(z), pol));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline typename tools::promote_args<T>::type cbrt(T z)
Chris@16: {
Chris@16:    return cbrt(z, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: #endif // BOOST_MATH_SF_CBRT_HPP
Chris@16: 
Chris@16: 
Chris@16: 
Chris@16: