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 <https://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 #include <math-underflow.h>
65 #include <libm-alias-finite.h>
66
67 static const long double
68 one = 1.0L,
69 huge = 1.0e+4932L,
70 pio2_hi = 0x1.921fb54442d1846ap+0L, /* pi/2 rounded to nearest to 64
71 bits. */
72 pio2_lo = -0x7.6733ae8fe47c65d8p-68L, /* pi/2 - pio2_hi rounded to
73 nearest to 64 bits. */
74 pio4_hi = 0xc.90fdaa22168c235p-4L, /* pi/4 rounded to nearest to 64
75 bits. */
76
77 /* coefficient for R(x^2) */
78
79 /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
80 0 <= x <= 0.5
81 peak relative error 1.9e-21 */
82 pS0 = -1.008714657938491626019651170502036851607E1L,
83 pS1 = 2.331460313214179572063441834101394865259E1L,
84 pS2 = -1.863169762159016144159202387315381830227E1L,
85 pS3 = 5.930399351579141771077475766877674661747E0L,
86 pS4 = -6.121291917696920296944056882932695185001E-1L,
87 pS5 = 3.776934006243367487161248678019350338383E-3L,
88
89 qS0 = -6.052287947630949712886794360635592886517E1L,
90 qS1 = 1.671229145571899593737596543114258558503E2L,
91 qS2 = -1.707840117062586426144397688315411324388E2L,
92 qS3 = 7.870295154902110425886636075950077640623E1L,
93 qS4 = -1.568433562487314651121702982333303458814E1L;
94 /* 1.000000000000000000000000000000000000000E0 */
95
96 long double
97 __ieee754_asinl (long double x)
98 {
99 long double t, w, p, q, c, r, s;
100 int32_t ix;
101 uint32_t se, i0, i1, k;
102
103 GET_LDOUBLE_WORDS (se, i0, i1, x);
104 ix = se & 0x7fff;
105 ix = (ix << 16) | (i0 >> 16);
106 if (ix >= 0x3fff8000)
107 { /* |x|>= 1 */
108 if (ix == 0x3fff8000 && ((i0 - 0x80000000) | i1) == 0)
109 /* asin(1)=+-pi/2 with inexact */
110 return x * pio2_hi + x * pio2_lo;
111 return (x - x) / (x - x); /* asin(|x|>1) is NaN */
112 }
113 else if (ix < 0x3ffe8000)
114 { /* |x|<0.5 */
115 if (ix < 0x3fde8000)
116 { /* if |x| < 2**-33 */
117 math_check_force_underflow (x);
118 if (huge + x > one)
119 return x; /* return x with inexact if x!=0 */
120 }
121 else
122 {
123 t = x * x;
124 p =
125 t * (pS0 +
126 t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
127 q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
128 w = p / q;
129 return x + x * w;
130 }
131 }
132 /* 1> |x|>= 0.5 */
133 w = one - fabsl (x);
134 t = w * 0.5;
135 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
136 q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
137 s = sqrtl (t);
138 if (ix >= 0x3ffef999)
139 { /* if |x| > 0.975 */
140 w = p / q;
141 t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
142 }
143 else
144 {
145 GET_LDOUBLE_WORDS (k, i0, i1, s);
146 i1 = 0;
147 SET_LDOUBLE_WORDS (w,k,i0,i1);
148 c = (t - w * w) / (s + w);
149 r = p / q;
150 p = 2.0 * s * r - (pio2_lo - 2.0 * c);
151 q = pio4_hi - 2.0 * w;
152 t = pio4_hi - (p - q);
153 }
154 if ((se & 0x8000) == 0)
155 return t;
156 else
157 return -t;
158 }
159 libm_alias_finite (__ieee754_asinl, __asinl)