49 lines
1.7 KiB
C++
49 lines
1.7 KiB
C++
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// Copyright 2022 Hans Dembinski, Jay Gohil
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//
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// Distributed under the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_HISTOGRAM_DETAIL_ERF_INF_HPP
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#define BOOST_HISTOGRAM_DETAIL_ERF_INF_HPP
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#include <cmath>
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namespace boost {
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namespace histogram {
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namespace detail {
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// Simple implementation of erf_inv so that we do not depend on boost::math.
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// If you happen to discover this, prefer the boost::math implementation,
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// it is more accurate for x very close to -1 or 1 and faster.
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// The only virtue of this implementation is its simplicity.
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template <int Iterate = 3>
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double erf_inv(double x) noexcept {
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// Strategy: solve f(y) = x - erf(y) = 0 for given x with Newton's method.
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// f'(y) = -erf'(y) = -2/sqrt(pi) e^(-y^2)
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// Has quadratic convergence. Since erf_inv<0> is accurate to 1e-3,
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// we should have machine precision after three iterations.
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const double x0 = erf_inv<Iterate - 1>(x); // recursion
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const double fx0 = x - std::erf(x0);
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const double pi = std::acos(-1);
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double fpx0 = -2.0 / std::sqrt(pi) * std::exp(-x0 * x0);
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return x0 - fx0 / fpx0; // = x1
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}
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template <>
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inline double erf_inv<0>(double x) noexcept {
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// Specialization to get initial estimate.
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// This formula is accurate to about 1e-3.
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// Based on https://stackoverflow.com/questions/27229371/inverse-error-function-in-c
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const double a = std::log((1 - x) * (1 + x));
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const double b = std::fma(0.5, a, 4.120666747961526);
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const double c = 6.47272819164 * a;
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return std::copysign(std::sqrt(-b + std::sqrt(std::fma(b, b, -c))), x);
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}
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} // namespace detail
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} // namespace histogram
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} // namespace boost
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#endif
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