amd-builtins/math32/sincospiF_piby4.h

/*
 * Copyright (c) 2014 Advanced Micro Devices, Inc.
 *
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 * of this software and associated documentation files (the "Software"), to deal
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 * 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
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 * THE SOFTWARE.
 */

// Evaluate single precisions in and cos of value in interval [-pi/4, pi/4]
static inline float2
sincosf_piby4(float x)
{
    // 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.

    // 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.

    const float sc1 = -0.166666666638608441788607926e0F;
    const float sc2 =  0.833333187633086262120839299e-2F;
    const float sc3 = -0.198400874359527693921333720e-3F;
    const float sc4 =  0.272500015145584081596826911e-5F;

    const float cc1 =  0.41666666664325175238031e-1F;
    const float cc2 = -0.13888887673175665567647e-2F;
    const float cc3 =  0.24800600878112441958053e-4F;
    const float cc4 = -0.27301013343179832472841e-6F;

    float x2 = x * x;

    float2 ret;
    ret.x = mad(x*x2, mad(x2, mad(x2, mad(x2, sc4, sc3), sc2), sc1), x);
    ret.y = mad(x2*x2, mad(x2, mad(x2, mad(x2, cc4, cc3), cc2), cc1), mad(x2, -0.5f, 1.0f));
    return ret;
}