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[/] [or1k/] [trunk/] [newlib-1.10.0/] [newlib/] [libm/] [math/] [e_asin.c] - Blame information for rev 1765

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1 1010 ivang
 
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/* @(#)e_asin.c 5.1 93/09/24 */
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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/* __ieee754_asin(x)
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 * Method :
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 *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
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 *      we approximate asin(x) on [0,0.5] by
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 *              asin(x) = x + x*x^2*R(x^2)
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 *      where
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 *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
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 *      and its remez error is bounded by
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 *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
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 *
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 *      For x in [0.5,1]
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 *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
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 *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
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 *      then for x>0.98
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 *              asin(x) = pi/2 - 2*(s+s*z*R(z))
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 *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
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 *      For x<=0.98, let pio4_hi = pio2_hi/2, then
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 *              f = hi part of s;
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 *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
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 *      and
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 *              asin(x) = pi/2 - 2*(s+s*z*R(z))
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 *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
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 *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
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 *
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 * Special cases:
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 *      if x is NaN, return x itself;
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 *      if |x|>1, return NaN with invalid signal.
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 *
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 */
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#include "fdlibm.h"
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#ifndef _DOUBLE_IS_32BITS
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#ifdef __STDC__
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static const double
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#else
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static double
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#endif
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one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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huge =  1.000e+300,
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pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
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pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
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pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
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        /* coefficient for R(x^2) */
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pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
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pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
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pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
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pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
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pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
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pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
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qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
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qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
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qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
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qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
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#ifdef __STDC__
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        double __ieee754_asin(double x)
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#else
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        double __ieee754_asin(x)
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        double x;
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#endif
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{
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        double t,w,p,q,c,r,s;
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        __int32_t hx,ix;
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        GET_HIGH_WORD(hx,x);
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        ix = hx&0x7fffffff;
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        if(ix>= 0x3ff00000) {           /* |x|>= 1 */
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            __uint32_t lx;
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            GET_LOW_WORD(lx,x);
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            if(((ix-0x3ff00000)|lx)==0)
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                    /* asin(1)=+-pi/2 with inexact */
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                return x*pio2_hi+x*pio2_lo;
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            return (x-x)/(x-x);         /* asin(|x|>1) is NaN */
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        } else if (ix<0x3fe00000) {     /* |x|<0.5 */
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            if(ix<0x3e400000) {         /* if |x| < 2**-27 */
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                if(huge+x>one) return x;/* return x with inexact if x!=0*/
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          } else {
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                t = x*x;
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                p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
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                q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
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                w = p/q;
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                return x+x*w;
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          }
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        }
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        /* 1> |x|>= 0.5 */
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        w = one-fabs(x);
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        t = w*0.5;
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        p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
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        q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
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        s = __ieee754_sqrt(t);
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        if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
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            w = p/q;
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            t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
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        } else {
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            w  = s;
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            SET_LOW_WORD(w,0);
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            c  = (t-w*w)/(s+w);
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            r  = p/q;
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            p  = 2.0*s*r-(pio2_lo-2.0*c);
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            q  = pio4_hi-2.0*w;
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            t  = pio4_hi-(p-q);
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        }
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        if(hx>0) return t; else return -t;
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}
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#endif /* defined(_DOUBLE_IS_32BITS) */

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