acos.c

Go to the documentation of this file.

/*-
 * 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/e_acos.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  acos.c
 * @brief Inverse cosine function
 */

#include "prim.h"

static const double
    one = 1.00000000000000000000e+00,     /* 0x3FF00000, 0x00000000 */
    pi = 3.14159265358979311600e+00,      /* 0x400921FB, 0x54442D18 */
    pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
    pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
    pS0 = 1.66666666666666657415e-01,     /* 0x3FC55555, 0x55555555 */
    pS1 = -3.25565818622400915405e-01,    /* 0xBFD4D612, 0x03EB6F7D */
    pS2 = 2.01212532134862925881e-01,     /* 0x3FC9C155, 0x0E884455 */
    pS3 = -4.00555345006794114027e-02,    /* 0xBFA48228, 0xB5688F3B */
    pS4 = 7.91534994289814532176e-04,     /* 0x3F49EFE0, 0x7501B288 */
    pS5 = 3.47933107596021167570e-05,     /* 0x3F023DE1, 0x0DFDF709 */
    qS1 = -2.40339491173441421878e+00,    /* 0xC0033A27, 0x1C8A2D4B */
    qS2 = 2.02094576023350569471e+00,     /* 0x40002AE5, 0x9C598AC8 */
    qS3 = -6.88283971605453293030e-01,    /* 0xBFE6066C, 0x1B8D0159 */
    qS4 = 7.70381505559019352791e-02;     /* 0x3FB3B8C5, 0xB12E9282 */

/**
 * @brief   Inverse cosine function
 * @details
 * <pre>
 * Method
 *     acos(x)  = pi/2 - asin(x)
 *     acos(-x) = pi/2 + asin(x)
 * 
 *     For |x|<=0.5
 *     acos(x)  = pi/2 - (x + x*x^2*R(x^2))        (see asin.c)
 * 
 *     For x>0.5
 *     acos(x)  = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
 *              = 2asin(sqrt((1-x)/2))
 *              = 2s + 2s*z*R(z)    ...z=(1-x)/2, s=sqrt(z)
 *              = 2f + (2c + 2s*z*R(z))
 *     where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
 *     for f so that f+c ~ sqrt(z).
 * 
 *     For x<-0.5
 *     acos(x)  = pi - 2asin(sqrt((1-|x|)/2))
 *              = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
 *
 * Special cases
 *     if x is NaN, return NaN
 *     if |x|>1, return NaN
 * </pre>
 */
double acos(double x)
{
    double z, p, q, r, w, s, c, df;
    int32_t hx, ix;
    uint32_t lx;

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

    // |x| >= 1
    if (ix >= 0x3FF00000)
    {
        GET_LOW_WORD(lx, x);

        // |x|==1
        if (((ix - 0x3FF00000) | lx) == 0)
        {
            if (hx > 0)
            {
                // acos(1) = 0
                return 0.0;
            }

            // acos(-1) = pi
            return pi + 2.0 * pio2_lo;
        }

        // acos(|x|>1) is NaN
        return NAN;
    }

    // |x| < 0.5
    if (ix < 0x3FE00000)
    {
        // if |x|<2**-57
        if (ix <= 0x3C600000)
        {
            return pio2_hi + pio2_lo;
        }
        z = x * x;
        p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
        q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
        r = p / q;
        return pio2_hi - (x - (pio2_lo - x * r));
    }

    // x < -0.5
    if (hx < 0)
    {
        z = (one + x) * 0.5;
        p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
        q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
        s = sqrt(z);
        r = p / q;
        w = r * s - pio2_lo;
        return pi - 2.0 * (s + w);
    }

    // x > 0.5
    z = (one - x) * 0.5;
    s = sqrt(z);
    df = s;
    SET_LOW_WORD(df, 0);
    c = (z - df * df) / (s + df);
    p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
    q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
    r = p / q;
    w = r * s + c;
    return 2.0 * (df + w);
}