e_acos.c

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/* @(#)e_acos.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/* acos(x)
 * 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 x itself;
 *  if |x|>1, return NaN with invalid signal.
 *
 * Function needed: sqrt
 */

#include <sys/cdefs.h>
#include <float.h>
#include <math.h>

#include "math_private.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 */

double
acos(double x)
{
    double z,p,q,r,w,s,c,df;
    int32_t hx,ix;
    GET_HIGH_WORD(hx,x);
    ix = hx&0x7fffffff;
    if(ix>=0x3ff00000) {    /* |x| >= 1 */
        u_int32_t lx;
        GET_LOW_WORD(lx,x);
        if(((ix-0x3ff00000)|lx)==0) {   /* |x|==1 */
        if(hx>0) return 0.0;        /* acos(1) = 0  */
        else return pi+2.0*pio2_lo; /* acos(-1)= pi */
        }
        return (x-x)/(x-x);     /* acos(|x|>1) is NaN */
    }
    if(ix<0x3fe00000) { /* |x| < 0.5 */
        if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
        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));
    } else  if (hx<0) {     /* x < -0.5 */
        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);
    } else {            /* 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);
    }
}

#if LDBL_MANT_DIG == 53
#ifdef __weak_alias
__weak_alias(acosl, acos);
#endif /* __weak_alias */
#endif /* LDBL_MANT_DIG == 53 */