1 /* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
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
9 * is preserved.
10 * ====================================================
11 */
12 /* asin(x)
13 * Method :
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)
17 * where
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)
21 *
22 * For x in [0.5,1]
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;
25 * then for x>0.98
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
29 * f = hi part of s;
30 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
31 * and
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))
35 *
36 * Special cases:
37 * if x is NaN, return x itself;
38 * if |x|>1, return NaN with invalid signal.
39 *
40 */
41
42 #include "libm.h"
43
44 static const double
45 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
46 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
47 /* coefficients for R(x^2) */
48 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
49 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
50 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
51 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
52 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
53 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
54 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
55 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
56 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
57 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
58
R(double z)59 static double R(double z)
60 {
61 double_t p, q;
62 p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
63 q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
64 return p/q;
65 }
66
asin(double x)67 double asin(double x)
68 {
69 double z,r,s;
70 uint32_t hx,ix;
71
72 GET_HIGH_WORD(hx, x);
73 ix = hx & 0x7fffffff;
74 /* |x| >= 1 or nan */
75 if (ix >= 0x3ff00000) {
76 uint32_t lx;
77 GET_LOW_WORD(lx, x);
78 if ((ix-0x3ff00000 | lx) == 0)
79 /* asin(1) = +-pi/2 with inexact */
80 return x*pio2_hi + 0x1p-120f;
81 return 0/(x-x);
82 }
83 /* |x| < 0.5 */
84 if (ix < 0x3fe00000) {
85 /* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */
86 if (ix < 0x3e500000 && ix >= 0x00100000)
87 return x;
88 return x + x*R(x*x);
89 }
90 /* 1 > |x| >= 0.5 */
91 z = (1 - fabs(x))*0.5;
92 s = sqrt(z);
93 r = R(z);
94 if (ix >= 0x3fef3333) { /* if |x| > 0.975 */
95 x = pio2_hi-(2*(s+s*r)-pio2_lo);
96 } else {
97 double f,c;
98 /* f+c = sqrt(z) */
99 f = s;
100 SET_LOW_WORD(f,0);
101 c = (z-f*f)/(s+f);
102 x = 0.5*pio2_hi - (2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f));
103 }
104 if (hx >> 31)
105 return -x;
106 return x;
107 }
108