e_asin.c

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/* @(#)e_asin.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.
 * ====================================================
 */

/* asin(x)
 * Method :                  
 *  Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
 *  we approximate asin(x) on [0,0.5] by
 *      asin(x) = x + x*x^2*R(x^2)
 *  where
 *      R(x^2) is a rational approximation of (asin(x)-x)/x^3 
 *  and its Remes error is bounded by
 *      |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
 *
 *  For x in [0.5,1]
 *      asin(x) = pi/2-2*asin(sqrt((1-x)/2))
 *  Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
 *  then for x>0.98
 *      asin(x) = pi/2 - 2*(s+s*z*R(z))
 *          = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
 *  For x<=0.98, let pio4_hi = pio2_hi/2, then
 *      f = hi part of s;
 *      c = sqrt(z) - f = (z-f*f)/(s+f)     ...f+c=sqrt(z)
 *  and
 *      asin(x) = pi/2 - 2*(s+s*z*R(z))
 *          = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
 *          = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
 *
 * Special cases:
 *  if x is NaN, return x itself;
 *  if |x|>1, return NaN with invalid signal.
 *
 */

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

#include "math_private.h"

static const double 
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
huge =  1.000e+300,
pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
    /* coefficient for R(x^2) */
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
asin(double x)
{
    double t,w,p,q,c,r,s;
    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)
            /* asin(1)=+-pi/2 with inexact */
        return x*pio2_hi+x*pio2_lo; 
        return (x-x)/(x-x);     /* asin(|x|>1) is NaN */   
    } else if (ix<0x3fe00000) { /* |x|<0.5 */
        if(ix<0x3e400000) {     /* if |x| < 2**-27 */
        if(huge+x>one) return x;/* return x with inexact if x!=0*/
        } else 
        t = x*x;
        p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
        q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
        w = p/q;
        return x+x*w;
    }
    /* 1> |x|>= 0.5 */
    w = one-fabs(x);
    t = w*0.5;
    p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
    q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
    s = sqrt(t);
    if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
        w = p/q;
        t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
    } else {
        w  = s;
        SET_LOW_WORD(w,0);
        c  = (t-w*w)/(s+w);
        r  = p/q;
        p  = 2.0*s*r-(pio2_lo-2.0*c);
        q  = pio4_hi-2.0*w;
        t  = pio4_hi-(p-q);
    }    
    if(hx>0) return t; else return -t;    
}

#if LDBL_MANT_DIG == 53
#ifdef __weak_alias
__weak_alias(asinl, asin);
#endif /* __weak_alias */
#endif /* LDBL_MANT_DIG == 53 */