1 /* Copyright JS Foundation and other contributors, http://js.foundation
2 *
3 * Licensed under the Apache License, Version 2.0 (the "License");
4 * you may not use this file except in compliance with the License.
5 * You may obtain a copy of the License at
6 *
7 * http://www.apache.org/licenses/LICENSE-2.0
8 *
9 * Unless required by applicable law or agreed to in writing, software
10 * distributed under the License is distributed on an "AS IS" BASIS
11 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 * See the License for the specific language governing permissions and
13 * limitations under the License.
14 *
15 * This file is based on work under the following copyright and permission
16 * notice:
17 *
18 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
19 *
20 * Developed at SunSoft, a Sun Microsystems, Inc. business.
21 * Permission to use, copy, modify, and distribute this
22 * software is freely granted, provided that this notice
23 * is preserved.
24 *
25 * @(#)e_asin.c 1.3 95/01/18
26 */
27
28 #include "jerry-libm-internal.h"
29
30 /* asin(x)
31 *
32 * Method:
33 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
34 * we approximate asin(x) on [0,0.5] by
35 * asin(x) = x + x*x^2*R(x^2)
36 * where
37 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
38 * and its remez error is bounded by
39 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
40 *
41 * For x in [0.5,1]
42 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
43 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
44 * then for x>0.98
45 * asin(x) = pi/2 - 2*(s+s*z*R(z))
46 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
47 * For x<=0.98, let pio4_hi = pio2_hi/2, then
48 * f = hi part of s;
49 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
50 * and
51 * asin(x) = pi/2 - 2*(s+s*z*R(z))
52 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
53 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
54 *
55 * Special cases:
56 * if x is NaN, return x itself;
57 * if |x|>1, return NaN with invalid signal.
58 */
59
60 #define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */
61 #define huge 1.000e+300
62 #define pio2_hi 1.57079632679489655800e+00 /* 0x3FF921FB, 0x54442D18 */
63 #define pio2_lo 6.12323399573676603587e-17 /* 0x3C91A626, 0x33145C07 */
64 #define pio4_hi 7.85398163397448278999e-01 /* 0x3FE921FB, 0x54442D18 */
65 /* coefficient for R(x^2) */
66 #define pS0 1.66666666666666657415e-01 /* 0x3FC55555, 0x55555555 */
67 #define pS1 -3.25565818622400915405e-01 /* 0xBFD4D612, 0x03EB6F7D */
68 #define pS2 2.01212532134862925881e-01 /* 0x3FC9C155, 0x0E884455 */
69 #define pS3 -4.00555345006794114027e-02 /* 0xBFA48228, 0xB5688F3B */
70 #define pS4 7.91534994289814532176e-04 /* 0x3F49EFE0, 0x7501B288 */
71 #define pS5 3.47933107596021167570e-05 /* 0x3F023DE1, 0x0DFDF709 */
72 #define qS1 -2.40339491173441421878e+00 /* 0xC0033A27, 0x1C8A2D4B */
73 #define qS2 2.02094576023350569471e+00 /* 0x40002AE5, 0x9C598AC8 */
74 #define qS3 -6.88283971605453293030e-01 /* 0xBFE6066C, 0x1B8D0159 */
75 #define qS4 7.70381505559019352791e-02 /* 0x3FB3B8C5, 0xB12E9282 */
76
77 double
asin(double x)78 asin (double x)
79 {
80 double t, p, q, c, r, s;
81 double_accessor w;
82 int hx, ix;
83
84 hx = __HI (x);
85 ix = hx & 0x7fffffff;
86 if (ix >= 0x3ff00000) /* |x| >= 1 */
87 {
88 if (((ix - 0x3ff00000) | __LO (x)) == 0) /* asin(1) = +-pi/2 with inexact */
89 {
90 return x * pio2_hi + x * pio2_lo;
91 }
92 return NAN; /* asin(|x|>1) is NaN */
93 }
94 else if (ix < 0x3fe00000) /* |x| < 0.5 */
95 {
96 if (ix < 0x3e400000) /* if |x| < 2**-27 */
97 {
98 if (huge + x > one) /* return x with inexact if x != 0 */
99 {
100 return x;
101 }
102 }
103 t = x * x;
104 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
105 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
106 w.dbl = p / q;
107 return x + x * w.dbl;
108 }
109 /* 1 > |x| >= 0.5 */
110 w.dbl = one - fabs (x);
111 t = w.dbl * 0.5;
112 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
113 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
114 s = sqrt (t);
115 if (ix >= 0x3FEF3333) /* if |x| > 0.975 */
116 {
117 w.dbl = p / q;
118 t = pio2_hi - (2.0 * (s + s * w.dbl) - pio2_lo);
119 }
120 else
121 {
122 w.dbl = s;
123 w.as_int.lo = 0;
124 c = (t - w.dbl * w.dbl) / (s + w.dbl);
125 r = p / q;
126 p = 2.0 * s * r - (pio2_lo - 2.0 * c);
127 q = pio4_hi - 2.0 * w.dbl;
128 t = pio4_hi - (p - q);
129 }
130 if (hx > 0)
131 {
132 return t;
133 }
134 else
135 {
136 return -t;
137 }
138 } /* asin */
139
140 #undef one
141 #undef huge
142 #undef pio2_hi
143 #undef pio2_lo
144 #undef pio4_hi
145 #undef pS0
146 #undef pS1
147 #undef pS2
148 #undef pS3
149 #undef pS4
150 #undef pS5
151 #undef qS1
152 #undef qS2
153 #undef qS3
154 #undef qS4
155