special_functions.cpp

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/*
 *  SPDX-FileCopyrightText: Copyright 2021, Siavash Ameli <sameli@berkeley.edu>
 *  SPDX-License-Identifier: BSD-3-Clause
 *  SPDX-FileType: SOURCE
 *
 *  This program is free software: you can redistribute it and/or modify it
 *  under the terms of the license found in the LICENSE.txt file in the root
 *  directory of this source tree.
 */
 
 
// =======
// Headers
// =======
 
// Before including cmath, define _USE_MATH_DEFINES. This is only required to
// define the math constants like M_PI, etc, in win32 operating system.
#if defined(WIN32) || defined(_WIN32) || defined(__WIN32) && \
    !defined(__CYGWIN__)
    #define _USE_MATH_DEFINES
#endif
 
#include <cmath>  // sqrt, log, exp, erf, INFINITY, M_PI
#include "./special_functions.h"
#include "../_c_arithmetics/c_arithmetics.h"  // c_arithmetics
 
 
// ====
// sign
// ====
 
 
double _sign(const double x)
{
    return (x > 0) - (x < 0);
}
 
 
// =======
// erf inv
// =======
 
 
double erf_inv(const double x)
{
    // Output
    double r;
 
    // Check extreme values
    if (c_arithmetics::is_equal(x, 1.0) || c_arithmetics::is_equal(x, -1.0))
    {
        r = _sign(x) * INFINITY;
        return r;
    }
 
    double a[4] = {0.886226899, -1.645349621, 0.914624893, -0.140543331};
    double b[5] = {1.0, -2.118377725, 1.442710462, -0.329097515, 0.012229801};
    double c[4] = {-1.970840454, -1.62490649, 3.429567803, 1.641345311};
    double d[3] = {1.0, 3.543889200, 1.637067800};
 
    double z = _sign(x) * x;
 
    if (z <= 0.7)
    {
        double x2 = z * z;
        r = z * (((a[3] * x2 + a[2]) * x2 + a[1]) * x2 + a[0]);
        r /= (((b[4] * x2 + b[3]) * x2 + b[2]) * x2 + b[1]) * x2 + b[0];
    }
    else
    {
        double y = sqrt(-log((1.0 - z) / 2.0));
        r = (((c[3] * y + c[2]) * y + c[1]) * y + c[0]);
        r /= ((d[2] * y + d[1]) * y + d[0]);
    }
 
    r = r * _sign(x);
    z = z * _sign(x);
 
    // These two lines below are identical and repeated for double refinement.
    // Comment one line below for a single refinement of the Newton method.
    r -= (erf(r) - z) / ((2.0 / sqrt(M_PI)) * exp(-r * r));
    r -= (erf(r) - z) / ((2.0 / sqrt(M_PI)) * exp(-r * r));
 
    return r;
}