1
2 /* @(#)e_asin.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, 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 /* asin(x)
15 * Method :
16 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
17 * we approximate 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 (asin(x)-x)/x^3
21 * and its remez error is bounded by
22 * |(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*asin(sqrt((1-x)/2))
26 * Let y = (1-x), z = y/2, s := 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 = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=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 #include "fdlibm.h"
45
46 #ifdef _NEED_FLOAT64
47
48 static const __float64
49 one = _F_64(1.00000000000000000000e+00), /* 0x3FF00000, 0x00000000 */
50 huge = _F_64(1.000e+300),
51 pio2_hi = _F_64(1.57079632679489655800e+00), /* 0x3FF921FB, 0x54442D18 */
52 pio2_lo = _F_64(6.12323399573676603587e-17), /* 0x3C91A626, 0x33145C07 */
53 pio4_hi = _F_64(7.85398163397448278999e-01), /* 0x3FE921FB, 0x54442D18 */
54 /* coefficient for R(x^2) */
55 pS0 = _F_64(1.66666666666666657415e-01), /* 0x3FC55555, 0x55555555 */
56 pS1 = _F_64(-3.25565818622400915405e-01), /* 0xBFD4D612, 0x03EB6F7D */
57 pS2 = _F_64(2.01212532134862925881e-01), /* 0x3FC9C155, 0x0E884455 */
58 pS3 = _F_64(-4.00555345006794114027e-02), /* 0xBFA48228, 0xB5688F3B */
59 pS4 = _F_64(7.91534994289814532176e-04), /* 0x3F49EFE0, 0x7501B288 */
60 pS5 = _F_64(3.47933107596021167570e-05), /* 0x3F023DE1, 0x0DFDF709 */
61 qS1 = _F_64(-2.40339491173441421878e+00), /* 0xC0033A27, 0x1C8A2D4B */
62 qS2 = _F_64(2.02094576023350569471e+00), /* 0x40002AE5, 0x9C598AC8 */
63 qS3 = _F_64(-6.88283971605453293030e-01), /* 0xBFE6066C, 0x1B8D0159 */
64 qS4 = _F_64(7.70381505559019352791e-02); /* 0x3FB3B8C5, 0xB12E9282 */
65
66 __float64
asin64(__float64 x)67 asin64(__float64 x)
68 {
69 __float64 t, w, p, q, c, r, s;
70 __int32_t hx, ix;
71 GET_HIGH_WORD(hx, x);
72 ix = hx & 0x7fffffff;
73 if (ix >= 0x3ff00000) { /* |x|>= 1 */
74 __uint32_t lx;
75 GET_LOW_WORD(lx, x);
76 if (((ix - 0x3ff00000) | lx) == 0)
77 /* asin(1)=+-pi/2 with inexact */
78 return x * pio2_hi + x * pio2_lo;
79 return __math_invalid(x); /* asin(|x|>1) is NaN */
80 } else if (ix < 0x3fe00000) { /* |x|<0.5 */
81 if (ix < 0x3e400000) { /* if |x| < 2**-27 */
82 if (huge + x > one)
83 return x; /* return x with inexact if x!=0*/
84 } else {
85 t = x * x;
86 p = t *
87 (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
88 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
89 w = p / q;
90 return x + x * w;
91 }
92 }
93 /* 1> |x|>= 0.5 */
94 w = one - fabs64(x);
95 t = w * _F_64(0.5);
96 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
97 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
98 s = sqrt(t);
99 if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
100 w = p / q;
101 t = pio2_hi - (_F_64(2.0) * (s + s * w) - pio2_lo);
102 } else {
103 w = s;
104 SET_LOW_WORD(w, 0);
105 c = (t - w * w) / (s + w);
106 r = p / q;
107 p = _F_64(2.0) * s * r - (pio2_lo - _F_64(2.0) * c);
108 q = pio4_hi - _F_64(2.0) * w;
109 t = pio4_hi - (p - q);
110 }
111 if (hx > 0)
112 return t;
113 else
114 return -t;
115 }
116
117 _MATH_ALIAS_d_d(asin)
118
119 #endif /* _NEED_FLOAT64 */
120
