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Annotation of src/lib/libm/src/e_asin.c, Revision 1.6

1.1       jtc         1: /* @(#)e_asin.c 5.1 93/09/24 */
                      2: /*
                      3:  * ====================================================
                      4:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
                      5:  *
                      6:  * Developed at SunPro, a Sun Microsystems, Inc. business.
                      7:  * Permission to use, copy, modify, and distribute this
                      8:  * software is freely granted, provided that this notice
                      9:  * is preserved.
                     10:  * ====================================================
                     11:  */
1.3       jtc        12:
                     13: #ifndef lint
1.6     ! jtc        14: static char rcsid[] = "$Id: e_asin.c,v 1.5 1994/08/10 20:30:33 jtc Exp $";
1.3       jtc        15: #endif
1.1       jtc        16:
                     17: /* __ieee754_asin(x)
                     18:  * Method :
                     19:  *     Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
                     20:  *     we approximate asin(x) on [0,0.5] by
                     21:  *             asin(x) = x + x*x^2*R(x^2)
                     22:  *     where
                     23:  *             R(x^2) is a rational approximation of (asin(x)-x)/x^3
                     24:  *     and its remez error is bounded by
                     25:  *             |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
                     26:  *
                     27:  *     For x in [0.5,1]
                     28:  *             asin(x) = pi/2-2*asin(sqrt((1-x)/2))
                     29:  *     Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
                     30:  *     then for x>0.98
                     31:  *             asin(x) = pi/2 - 2*(s+s*z*R(z))
                     32:  *                     = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
                     33:  *     For x<=0.98, let pio4_hi = pio2_hi/2, then
                     34:  *             f = hi part of s;
                     35:  *             c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
                     36:  *     and
                     37:  *             asin(x) = pi/2 - 2*(s+s*z*R(z))
                     38:  *                     = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
                     39:  *                     = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
                     40:  *
                     41:  * Special cases:
                     42:  *     if x is NaN, return x itself;
                     43:  *     if |x|>1, return NaN with invalid signal.
                     44:  *
                     45:  */
                     46:
                     47:
1.5       jtc        48: #include "math.h"
                     49: #include "math_private.h"
1.1       jtc        50:
                     51: #ifdef __STDC__
                     52: static const double
                     53: #else
                     54: static double
                     55: #endif
                     56: one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
                     57: huge =  1.000e+300,
                     58: pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
                     59: pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
                     60: pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
                     61:        /* coefficient for R(x^2) */
                     62: pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
                     63: pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
                     64: pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
                     65: pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
                     66: pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
                     67: pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
                     68: qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
                     69: qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
                     70: qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
                     71: qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
                     72:
                     73: #ifdef __STDC__
                     74:        double __ieee754_asin(double x)
                     75: #else
                     76:        double __ieee754_asin(x)
                     77:        double x;
                     78: #endif
                     79: {
                     80:        double t,w,p,q,c,r,s;
1.6     ! jtc        81:        int32_t hx,ix;
1.5       jtc        82:        GET_HIGH_WORD(hx,x);
1.1       jtc        83:        ix = hx&0x7fffffff;
                     84:        if(ix>= 0x3ff00000) {           /* |x|>= 1 */
1.6     ! jtc        85:            u_int32_t lx;
1.5       jtc        86:            GET_LOW_WORD(lx,x);
                     87:            if(((ix-0x3ff00000)|lx)==0)
1.1       jtc        88:                    /* asin(1)=+-pi/2 with inexact */
                     89:                return x*pio2_hi+x*pio2_lo;
                     90:            return (x-x)/(x-x);         /* asin(|x|>1) is NaN */
                     91:        } else if (ix<0x3fe00000) {     /* |x|<0.5 */
                     92:            if(ix<0x3e400000) {         /* if |x| < 2**-27 */
                     93:                if(huge+x>one) return x;/* return x with inexact if x!=0*/
                     94:            } else
                     95:                t = x*x;
                     96:                p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
                     97:                q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
                     98:                w = p/q;
                     99:                return x+x*w;
                    100:        }
                    101:        /* 1> |x|>= 0.5 */
                    102:        w = one-fabs(x);
                    103:        t = w*0.5;
                    104:        p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
                    105:        q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
                    106:        s = sqrt(t);
                    107:        if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
                    108:            w = p/q;
                    109:            t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
                    110:        } else {
                    111:            w  = s;
1.5       jtc       112:            SET_LOW_WORD(w,0);
1.1       jtc       113:            c  = (t-w*w)/(s+w);
                    114:            r  = p/q;
                    115:            p  = 2.0*s*r-(pio2_lo-2.0*c);
                    116:            q  = pio4_hi-2.0*w;
                    117:            t  = pio4_hi-(p-q);
                    118:        }
                    119:        if(hx>0) return t; else return -t;
                    120: }

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