/* @(#)e_asin.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, 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 remez 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 #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=0.0,w,p,q,c,r,s; int32_t hx,ix; GET_HIGH_WORD(hx,x); ix = hx&0x7fffffff; if(ix>= 0x3ff00000) { /* |x|>= 1 */ uint32_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; }