e_asin.c revision 22993
1272343Sngie/* @(#)e_asin.c 5.1 93/09/24 */ 2272343Sngie/* 3272343Sngie * ==================================================== 4272343Sngie * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5272343Sngie * 6272343Sngie * Developed at SunPro, a Sun Microsystems, Inc. business. 7272343Sngie * Permission to use, copy, modify, and distribute this 8272343Sngie * software is freely granted, provided that this notice 9272343Sngie * is preserved. 10272343Sngie * ==================================================== 11272343Sngie */ 12272343Sngie 13272343Sngie#ifndef lint 14272343Sngiestatic char rcsid[] = "$Id$"; 15272343Sngie#endif 16272343Sngie 17272343Sngie/* __ieee754_asin(x) 18272343Sngie * Method : 19272343Sngie * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 20272343Sngie * we approximate asin(x) on [0,0.5] by 21272343Sngie * asin(x) = x + x*x^2*R(x^2) 22272343Sngie * where 23272343Sngie * R(x^2) is a rational approximation of (asin(x)-x)/x^3 24272343Sngie * and its remez error is bounded by 25272343Sngie * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 26272343Sngie * 27272343Sngie * For x in [0.5,1] 28272343Sngie * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 29272343Sngie * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 30272343Sngie * then for x>0.98 31272343Sngie * asin(x) = pi/2 - 2*(s+s*z*R(z)) 32272343Sngie * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 33272343Sngie * For x<=0.98, let pio4_hi = pio2_hi/2, then 34272343Sngie * f = hi part of s; 35272343Sngie * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 36272343Sngie * and 37272343Sngie * asin(x) = pi/2 - 2*(s+s*z*R(z)) 38272343Sngie * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 39272343Sngie * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 40272343Sngie * 41272343Sngie * Special cases: 42272343Sngie * if x is NaN, return x itself; 43272343Sngie * if |x|>1, return NaN with invalid signal. 44272343Sngie * 45272343Sngie */ 46272343Sngie 47272343Sngie 48272343Sngie#include "math.h" 49272343Sngie#include "math_private.h" 50272343Sngie 51272343Sngie#ifdef __STDC__ 52272343Sngiestatic const double 53272343Sngie#else 54272343Sngiestatic double 55272343Sngie#endif 56272343Sngieone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 57272343Sngiehuge = 1.000e+300, 58272343Sngiepio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 59272343Sngiepio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 60272343Sngiepio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 61272343Sngie /* coefficient for R(x^2) */ 62272343SngiepS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 63272343SngiepS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 64272343SngiepS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 65272343SngiepS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 66272343SngiepS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 67272343SngiepS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 68272343SngieqS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 69272343SngieqS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 70272343SngieqS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 71272343SngieqS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 72272343Sngie 73272343Sngie#ifdef __STDC__ 74272343Sngie double __generic___ieee754_asin(double x) 75272343Sngie#else 76272343Sngie double __generic___ieee754_asin(x) 77272343Sngie double x; 78272343Sngie#endif 79272343Sngie{ 80272343Sngie double t=0.0,w,p,q,c,r,s; 81272343Sngie int32_t hx,ix; 82272343Sngie GET_HIGH_WORD(hx,x); 83272343Sngie ix = hx&0x7fffffff; 84272343Sngie if(ix>= 0x3ff00000) { /* |x|>= 1 */ 85272343Sngie u_int32_t lx; 86272343Sngie GET_LOW_WORD(lx,x); 87272343Sngie if(((ix-0x3ff00000)|lx)==0) 88272343Sngie /* asin(1)=+-pi/2 with inexact */ 89272343Sngie return x*pio2_hi+x*pio2_lo; 90272343Sngie return (x-x)/(x-x); /* asin(|x|>1) is NaN */ 91272343Sngie } else if (ix<0x3fe00000) { /* |x|<0.5 */ 92272343Sngie if(ix<0x3e400000) { /* if |x| < 2**-27 */ 93272343Sngie if(huge+x>one) return x;/* return x with inexact if x!=0*/ 94272343Sngie } else 95272343Sngie t = x*x; 96272343Sngie p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 97272343Sngie q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 98272343Sngie w = p/q; 99272343Sngie return x+x*w; 100272343Sngie } 101272343Sngie /* 1> |x|>= 0.5 */ 102272343Sngie w = one-fabs(x); 103272343Sngie t = w*0.5; 104272343Sngie p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 105272343Sngie q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 106272343Sngie s = sqrt(t); 107272343Sngie if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ 108272343Sngie w = p/q; 109272343Sngie t = pio2_hi-(2.0*(s+s*w)-pio2_lo); 110272343Sngie } else { 111272343Sngie w = s; 112272343Sngie SET_LOW_WORD(w,0); 113272343Sngie c = (t-w*w)/(s+w); 114272343Sngie r = p/q; 115272343Sngie p = 2.0*s*r-(pio2_lo-2.0*c); 116272343Sngie q = pio4_hi-2.0*w; 117272343Sngie t = pio4_hi-(p-q); 118272343Sngie } 119272343Sngie if(hx>0) return t; else return -t; 120272343Sngie} 121272343Sngie