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// (C) Copyright Nick Thompson 2020.
// 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)
#ifndef BOOST_MATH_SPECIAL_FUNCTIONS_RSQRT_HPP
#define BOOST_MATH_SPECIAL_FUNCTIONS_RSQRT_HPP
#include <cmath>
#include <type_traits>
#include <limits>
#include <boost/math/tools/is_standalone.hpp>
#ifndef BOOST_MATH_STANDALONE
# include <boost/config.hpp>
# ifdef BOOST_MATH_NO_CXX17_IF_CONSTEXPR
# error "The header <boost/math/rqrt.hpp> can only be used in C++17 and later."
# endif
#endif
namespace boost::math {
template<typename Real>
inline Real rsqrt(Real const & x)
{
using std::sqrt;
if constexpr (std::is_arithmetic_v<Real> && !std::is_integral_v<Real>)
{
return 1/sqrt(x);
}
else
{
// if it's so tiny it rounds to 0 as long double,
// no performance gains are possible:
if (x < std::numeric_limits<long double>::denorm_min() || x > (std::numeric_limits<long double>::max)()) {
return 1/sqrt(x);
}
Real x0 = 1/sqrt(static_cast<long double>(x));
// Divide by 512 for leeway:
Real s = sqrt(std::numeric_limits<Real>::epsilon())*x0/512;
Real x1 = x0 + x0*(1-x*x0*x0)/2;
while(abs(x1 - x0) > s) {
x0 = x1;
x1 = x0 + x0*(1-x*x0*x0)/2;
}
// Final iteration get ~2ULPs:
return x1 + x1*(1-x*x1*x1)/2;;
}
}
}
#endif
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