1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12/*
13 Long double expansions are
14 Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
15 and are incorporated herein by permission of the author. The author
16 reserves the right to distribute this material elsewhere under different
17 copying permissions. These modifications are distributed here under
18 the following terms:
19
20 This library is free software; you can redistribute it and/or
21 modify it under the terms of the GNU Lesser General Public
22 License as published by the Free Software Foundation; either
23 version 2.1 of the License, or (at your option) any later version.
24
25 This library is distributed in the hope that it will be useful,
26 but WITHOUT ANY WARRANTY; without even the implied warranty of
27 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
28 Lesser General Public License for more details.
29
30 You should have received a copy of the GNU Lesser General Public
31 License along with this library; if not, see
32 <http://www.gnu.org/licenses/>. */
33
34/* __ieee754_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 *
40 * For x in [0.5,1]
41 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
42 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
43 * then for x>0.98
44 * asin(x) = pi/2 - 2*(s+s*z*R(z))
45 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
46 * For x<=0.98, let pio4_hi = pio2_hi/2, then
47 * f = hi part of s;
48 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
49 * and
50 * asin(x) = pi/2 - 2*(s+s*z*R(z))
51 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
52 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
53 *
54 * Special cases:
55 * if x is NaN, return x itself;
56 * if |x|>1, return NaN with invalid signal.
57 *
58 */
59
60
61#include <float.h>
62#include <math.h>
63#include <math_private.h>
64
65static const long double
66 one = 1.0L,
67 huge = 1.0e+4932L,
68 pio2_hi = 0x1.921fb54442d1846ap+0L, /* pi/2 rounded to nearest to 64
69 bits. */
70 pio2_lo = -0x7.6733ae8fe47c65d8p-68L, /* pi/2 - pio2_hi rounded to
71 nearest to 64 bits. */
72 pio4_hi = 0xc.90fdaa22168c235p-4L, /* pi/4 rounded to nearest to 64
73 bits. */
74
75 /* coefficient for R(x^2) */
76
77 /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
78 0 <= x <= 0.5
79 peak relative error 1.9e-21 */
80 pS0 = -1.008714657938491626019651170502036851607E1L,
81 pS1 = 2.331460313214179572063441834101394865259E1L,
82 pS2 = -1.863169762159016144159202387315381830227E1L,
83 pS3 = 5.930399351579141771077475766877674661747E0L,
84 pS4 = -6.121291917696920296944056882932695185001E-1L,
85 pS5 = 3.776934006243367487161248678019350338383E-3L,
86
87 qS0 = -6.052287947630949712886794360635592886517E1L,
88 qS1 = 1.671229145571899593737596543114258558503E2L,
89 qS2 = -1.707840117062586426144397688315411324388E2L,
90 qS3 = 7.870295154902110425886636075950077640623E1L,
91 qS4 = -1.568433562487314651121702982333303458814E1L;
92 /* 1.000000000000000000000000000000000000000E0 */
93
94long double
95__ieee754_asinl (long double x)
96{
97 long double t, w, p, q, c, r, s;
98 int32_t ix;
99 uint32_t se, i0, i1, k;
100
101 GET_LDOUBLE_WORDS (se, i0, i1, x);
102 ix = se & 0x7fff;
103 ix = (ix << 16) | (i0 >> 16);
104 if (ix >= 0x3fff8000)
105 { /* |x|>= 1 */
106 if (ix == 0x3fff8000 && ((i0 - 0x80000000) | i1) == 0)
107 /* asin(1)=+-pi/2 with inexact */
108 return x * pio2_hi + x * pio2_lo;
109 return (x - x) / (x - x); /* asin(|x|>1) is NaN */
110 }
111 else if (ix < 0x3ffe8000)
112 { /* |x|<0.5 */
113 if (ix < 0x3fde8000)
114 { /* if |x| < 2**-33 */
115 math_check_force_underflow (x);
116 if (huge + x > one)
117 return x; /* return x with inexact if x!=0 */
118 }
119 else
120 {
121 t = x * x;
122 p =
123 t * (pS0 +
124 t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
125 q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
126 w = p / q;
127 return x + x * w;
128 }
129 }
130 /* 1> |x|>= 0.5 */
131 w = one - fabsl (x);
132 t = w * 0.5;
133 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
134 q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
135 s = __ieee754_sqrtl (t);
136 if (ix >= 0x3ffef999)
137 { /* if |x| > 0.975 */
138 w = p / q;
139 t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
140 }
141 else
142 {
143 GET_LDOUBLE_WORDS (k, i0, i1, s);
144 i1 = 0;
145 SET_LDOUBLE_WORDS (w,k,i0,i1);
146 c = (t - w * w) / (s + w);
147 r = p / q;
148 p = 2.0 * s * r - (pio2_lo - 2.0 * c);
149 q = pio4_hi - 2.0 * w;
150 t = pio4_hi - (p - q);
151 }
152 if ((se & 0x8000) == 0)
153 return t;
154 else
155 return -t;
156}
157strong_alias (__ieee754_asinl, __asinl_finite)
158