amd-builtins/math64/sincosD_piby4.h

/*
 * Copyright (c) 2014 Advanced Micro Devices, Inc.
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in
 * all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
 * THE SOFTWARE.
 */

static inline double2
sincos_piby4(double x, double xx)
{
    // Taylor series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! ...
    //                      = x * (1 - x^2/3! + x^4/5! - x^6/7! ...
    //                      = x * f(w)
    // where w = x*x and f(w) = (1 - w/3! + w^2/5! - w^3/7! ...
    // We use a minimax approximation of (f(w) - 1) / w
    // because this produces an expansion in even powers of x.
    // If xx (the tail of x) is non-zero, we add a correction
    // term g(x,xx) = (1-x*x/2)*xx to the result, where g(x,xx)
    // is an approximation to cos(x)*sin(xx) valid because
    // xx is tiny relative to x.

    // Taylor series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! ...
    //                      = f(w)
    // where w = x*x and f(w) = (1 - w/2! + w^2/4! - w^3/6! ...
    // We use a minimax approximation of (f(w) - 1 + w/2) / (w*w)
    // because this produces an expansion in even powers of x.
    // If xx (the tail of x) is non-zero, we subtract a correction
    // term g(x,xx) = x*xx to the result, where g(x,xx)
    // is an approximation to sin(x)*sin(xx) valid because
    // xx is tiny relative to x.

    const double sc1 = -0.166666666666666646259241729;
    const double sc2 =  0.833333333333095043065222816e-2;
    const double sc3 = -0.19841269836761125688538679e-3;
    const double sc4 =  0.275573161037288022676895908448e-5;
    const double sc5 = -0.25051132068021699772257377197e-7;
    const double sc6 =  0.159181443044859136852668200e-9;

    const double cc1 =  0.41666666666666665390037e-1;
    const double cc2 = -0.13888888888887398280412e-2;
    const double cc3 =  0.248015872987670414957399e-4;
    const double cc4 = -0.275573172723441909470836e-6;
    const double cc5 =  0.208761463822329611076335e-8;
    const double cc6 = -0.113826398067944859590880e-10;

    double x2 = x * x;
    double x3 = x2 * x;
    double r = 0.5 * x2;
    double t = 1.0 - r;

    double sp = fma(fma(fma(fma(sc6, x2, sc5), x2, sc4), x2, sc3), x2, sc2);

    double cp = t + fma(fma(fma(fma(fma(fma(cc6, x2, cc5), x2, cc4), x2, cc3), x2, cc2), x2, cc1),
                        x2*x2, fma(x, xx, (1.0 - t) - r));

    double2 ret;
    ret.lo = x - fma(-x3, sc1, fma(fma(-x3, sp, 0.5*xx), x2, -xx));
    ret.hi = cp;

    return ret;
}