1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
|
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
Long double expansions are
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
and are incorporated herein by permission of the author. The author
reserves the right to distribute this material elsewhere under different
copying permissions. These modifications are distributed here under
the following terms:
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see
<http://www.gnu.org/licenses/>. */
/* __ieee754_asin(x)
* Method :
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we approximate asin(x) on [0,0.5] by
* asin(x) = x + x*x^2*R(x^2)
*
* For x in [0.5,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
* then for x>0.98
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
* For x<=0.98, let pio4_hi = pio2_hi/2, then
* f = hi part of s;
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
* and
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
*/
#include <math.h>
#include <math_private.h>
static const long double
one = 1.0L,
huge = 1.0e+4932L,
pio2_hi = 0x1.921fb54442d1846ap+0L, /* pi/2 rounded to nearest to 64
bits. */
pio2_lo = -0x7.6733ae8fe47c65d8p-68L, /* pi/2 - pio2_hi rounded to
nearest to 64 bits. */
pio4_hi = 0xc.90fdaa22168c235p-4L, /* pi/4 rounded to nearest to 64
bits. */
/* coefficient for R(x^2) */
/* asin(x) = x + x^3 pS(x^2) / qS(x^2)
0 <= x <= 0.5
peak relative error 1.9e-21 */
pS0 = -1.008714657938491626019651170502036851607E1L,
pS1 = 2.331460313214179572063441834101394865259E1L,
pS2 = -1.863169762159016144159202387315381830227E1L,
pS3 = 5.930399351579141771077475766877674661747E0L,
pS4 = -6.121291917696920296944056882932695185001E-1L,
pS5 = 3.776934006243367487161248678019350338383E-3L,
qS0 = -6.052287947630949712886794360635592886517E1L,
qS1 = 1.671229145571899593737596543114258558503E2L,
qS2 = -1.707840117062586426144397688315411324388E2L,
qS3 = 7.870295154902110425886636075950077640623E1L,
qS4 = -1.568433562487314651121702982333303458814E1L;
/* 1.000000000000000000000000000000000000000E0 */
long double
__ieee754_asinl (long double x)
{
long double t, w, p, q, c, r, s;
int32_t ix;
u_int32_t se, i0, i1, k;
GET_LDOUBLE_WORDS (se, i0, i1, x);
ix = se & 0x7fff;
ix = (ix << 16) | (i0 >> 16);
if (ix >= 0x3fff8000)
{ /* |x|>= 1 */
if (ix == 0x3fff8000 && ((i0 - 0x80000000) | i1) == 0)
/* asin(1)=+-pi/2 with inexact */
return x * pio2_hi + x * pio2_lo;
return (x - x) / (x - x); /* asin(|x|>1) is NaN */
}
else if (ix < 0x3ffe8000)
{ /* |x|<0.5 */
if (ix < 0x3fde8000)
{ /* if |x| < 2**-33 */
if (huge + x > one)
return x; /* return x with inexact if x!=0 */
}
else
{
t = x * x;
p =
t * (pS0 +
t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
w = p / q;
return x + x * w;
}
}
/* 1> |x|>= 0.5 */
w = one - fabsl (x);
t = w * 0.5;
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
s = __ieee754_sqrtl (t);
if (ix >= 0x3ffef999)
{ /* if |x| > 0.975 */
w = p / q;
t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
}
else
{
GET_LDOUBLE_WORDS (k, i0, i1, s);
i1 = 0;
SET_LDOUBLE_WORDS (w,k,i0,i1);
c = (t - w * w) / (s + w);
r = p / q;
p = 2.0 * s * r - (pio2_lo - 2.0 * c);
q = pio4_hi - 2.0 * w;
t = pio4_hi - (p - q);
}
if ((se & 0x8000) == 0)
return t;
else
return -t;
}
strong_alias (__ieee754_asinl, __asinl_finite)
|
