kcos.c

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/*-
 * Copyright (c) 2014-2018 Carsten Sonne Larsen <cs@innolan.net>
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 *
 * Project homepage:
 * https://amath.innolan.net
 *
 * The original source code can be obtained from:
 * http://www.netlib.org/fdlibm/k_cos.c
 * 
 * =================================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * =================================================================
 */

/**
 * @file  kcos.c
 * @brief Kernel cosine function
 */

#include "prim.h"

static const double
    one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
    C1 = 4.16666666666666019037e-02,  /* 0x3FA55555, 0x5555554C */
    C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
    C3 = 2.48015872894767294178e-05,  /* 0x3EFA01A0, 0x19CB1590 */
    C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
    C5 = 2.08757232129817482790e-09,  /* 0x3E21EE9E, 0xBDB4B1C4 */
    C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */

/**
 * @brief   Kernel cosine function
 * @details
 * <pre>
 * Kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 *
 * Algorithm
 *  1. Since cos(-x) = cos(x), we need only to consider positive x.
 *  2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
 *  3. cos(x) is approximated by a polynomial of degree 14 on
 *     [0,pi/4]
 *                           4            14
 *      cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
 *     where the Remes error is
 *
 *  |              2     4     6     8     10    12     14 |     -58
 *  |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
 *  |                                      |
 *
 *                 4     6     8     10    12     14
 *  4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
 *         cos(x) = 1 - x*x/2 + r
 *     since cos(x+y) ~ cos(x) - sin(x)*y
 *            ~ cos(x) - x*y,
 *     a correction term is necessary in cos(x) and hence
 *      cos(x+y) = 1 - (x*x/2 - (r - x*y))
 *     For better accuracy when x > 0.3, let qx = |x|/4 with
 *     the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
 *     Then
 *      cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
 *     Note that 1-qx and (x*x/2-qx) is EXACT here, and the
 *     magnitude of the latter is at least a quarter of x*x/2,
 *     thus, reducing the rounding error in the subtraction.
 * </pre>
 */
double __kernel_cos(double x, double y)
{
    double a, hz, z, r, qx;
    int32_t ix;

    GET_HIGH_WORD(ix, x);
    ix &= 0x7FFFFFFF;

    // if x < 2**27
    if (ix < 0x3E400000)
    {
        // generate inexact
        if ((int)x == 0)
        {
            return one;
        }
    }

    z = x * x;
    r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));

    // |x| < 0.3
    if (ix < 0x3FD33333)
    {
        return one - (0.5 * z - (z * r - x * y));
    }

    // x > 0.78125
    if (ix > 0x3FE90000)
    {
        qx = 0.28125;
    }
    else
    {
        INSERT_WORDS(qx, ix - 0x00200000, 0);
    }

    hz = 0.5 * z - qx;
    a = one - qx;
    return a - (hz - (z * r - x * y));
}