1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #pragma weak asin = __asin
31
32 /* INDENT OFF */
33 /*
34 * asin(x)
35 * Method :
36 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
37 * we approximate asin(x) on [0,0.5] by
38 * asin(x) = x + x*x^2*R(x^2)
39 * where
40 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
41 * and its remez error is bounded by
42 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
43 *
44 * For x in [0.5,1]
45 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
46 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
47 * then for x>0.98
48 * asin(x) = pi/2 - 2*(s+s*z*R(z))
49 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
50 * For x<=0.98, let pio4_hi = pio2_hi/2, then
51 * f = hi part of s;
52 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
53 * and
54 * asin(x) = pi/2 - 2*(s+s*z*R(z))
55 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
56 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
57 *
58 * Special cases:
59 * if x is NaN, return x itself;
60 * if |x|>1, return NaN with invalid signal.
61 *
62 */
63 /* INDENT ON */
64
65 #include "libm_synonyms.h" /* __asin, __sqrt, __isnan */
66 #include "libm_protos.h" /* _SVID_libm_error */
67 #include "libm_macros.h"
68 #include <math.h>
69
70 /* INDENT OFF */
71 static const double xxx[] = {
72 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
73 /* huge */ 1.000e+300,
74 /* pio2_hi */ 1.57079632679489655800e+00, /* 3FF921FB, 54442D18 */
75 /* pio2_lo */ 6.12323399573676603587e-17, /* 3C91A626, 33145C07 */
76 /* pio4_hi */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
77 /* coefficient for R(x^2) */
78 /* pS0 */ 1.66666666666666657415e-01, /* 3FC55555, 55555555 */
79 /* pS1 */ -3.25565818622400915405e-01, /* BFD4D612, 03EB6F7D */
80 /* pS2 */ 2.01212532134862925881e-01, /* 3FC9C155, 0E884455 */
81 /* pS3 */ -4.00555345006794114027e-02, /* BFA48228, B5688F3B */
82 /* pS4 */ 7.91534994289814532176e-04, /* 3F49EFE0, 7501B288 */
83 /* pS5 */ 3.47933107596021167570e-05, /* 3F023DE1, 0DFDF709 */
84 /* qS1 */ -2.40339491173441421878e+00, /* C0033A27, 1C8A2D4B */
85 /* qS2 */ 2.02094576023350569471e+00, /* 40002AE5, 9C598AC8 */
86 /* qS3 */ -6.88283971605453293030e-01, /* BFE6066C, 1B8D0159 */
87 /* qS4 */ 7.70381505559019352791e-02 /* 3FB3B8C5, B12E9282 */
88 };
89 #define one xxx[0]
90 #define huge xxx[1]
91 #define pio2_hi xxx[2]
92 #define pio2_lo xxx[3]
93 #define pio4_hi xxx[4]
94 #define pS0 xxx[5]
95 #define pS1 xxx[6]
96 #define pS2 xxx[7]
97 #define pS3 xxx[8]
98 #define pS4 xxx[9]
99 #define pS5 xxx[10]
100 #define qS1 xxx[11]
101 #define qS2 xxx[12]
102 #define qS3 xxx[13]
103 #define qS4 xxx[14]
104 /* INDENT ON */
105
106 double
107 asin(double x) {
108 double t, w, p, q, c, r, s;
109 int hx, ix, i;
110
111 hx = ((int *) &x)[HIWORD];
112 ix = hx & 0x7fffffff;
113 if (ix >= 0x3ff00000) { /* |x| >= 1 */
114 if (((ix - 0x3ff00000) | ((int *) &x)[LOWORD]) == 0)
115 /* asin(1)=+-pi/2 with inexact */
116 return (x * pio2_hi + x * pio2_lo);
117 else if (isnan(x))
118 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
119 return (ix >= 0x7ff80000 ? x : (x - x) / (x - x));
120 /* assumes sparc-like QNaN */
121 #else
122 return (x - x) / (x - x); /* asin(|x|>1) is NaN */
123 #endif
124 else
125 return (_SVID_libm_err(x, x, 2));
126 } else if (ix < 0x3fe00000) { /* |x| < 0.5 */
127 if (ix < 0x3e400000) { /* if |x| < 2**-27 */
128 if ((i = (int) x) == 0)
129 /* return x with inexact if x != 0 */
130 return (x);
131 }
132 t = x * x;
133 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 +
134 t * (pS4 + t * pS5)))));
135 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
136 w = p / q;
137 return (x + x * w);
138 }
139 /* 1 > |x| >= 0.5 */
140 w = one - fabs(x);
141 t = w * 0.5;
142 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
143 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
144 s = sqrt(t);
145 if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
146 w = p / q;
147 t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
148 } else {
149 w = s;
150 ((int *) &w)[LOWORD] = 0;
151 c = (t - w * w) / (s + w);
152 r = p / q;
153 p = 2.0 * s * r - (pio2_lo - 2.0 * c);
154 q = pio4_hi - 2.0 * w;
155 t = pio4_hi - (p - q);
156 }
157 return (hx > 0 ? t : -t);
158 }