1 /* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
14 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
15 * we approximate asin(x) on [0,0.5] by
16 * asin(x) = x + x*x^2*R(x^2)
18 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
19 * and its remez error is bounded by
20 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
23 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
24 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
26 * asin(x) = pi/2 - 2*(s+s*z*R(z))
27 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
28 * For x<=0.98, let pio4_hi = pio2_hi/2, then
30 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
32 * asin(x) = pi/2 - 2*(s+s*z*R(z))
33 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
34 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
37 * if x is NaN, return x itself;
38 * if |x|>1, return NaN with invalid signal.
45 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
47 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
48 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
49 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
50 /* coefficients for R(x^2) */
51 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
52 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
53 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
54 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
55 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
56 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
57 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
58 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
59 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
60 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
64 double t=0.0,w,p,q,c,r,s;
69 if (ix >= 0x3ff00000) { /* |x|>= 1 */
73 if ((ix-0x3ff00000 | lx) == 0)
74 /* asin(1) = +-pi/2 with inexact */
75 return x*pio2_hi + x*pio2_lo;
76 return (x-x)/(x-x); /* asin(|x|>1) is NaN */
77 } else if (ix < 0x3fe00000) { /* |x|<0.5 */
78 if (ix < 0x3e500000) { /* if |x| < 2**-26 */
80 return x; /* return x with inexact if x!=0*/
83 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
84 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
91 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
92 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
94 if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
96 t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
102 p = 2.0*s*r-(pio2_lo-2.0*c);