// (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> namespace boost::math { template<typename Real> inline Real rsqrt(Real const & x) { using std::sqrt; if constexpr (std::is_same_v<Real, float> || std::is_same_v<Real, double> || std::is_same_v<Real, long double>) { 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