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