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 the
18 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 * Between .5 and .625 the approximation is
40 * asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
41 * For x in [0.625,1]
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
61
62#include <float.h>
63#include <math.h>
64#include <math_private.h>
65
66static const _Float128
67 one = 1,
68 huge = L(1.0e+4932),
69 pio2_hi = L(1.5707963267948966192313216916397514420986),
70 pio2_lo = L(4.3359050650618905123985220130216759843812E-35),
71 pio4_hi = L(7.8539816339744830961566084581987569936977E-1),
72
73 /* coefficient for R(x^2) */
74
75 /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
76 0 <= x <= 0.5
77 peak relative error 1.9e-35 */
78 pS0 = L(-8.358099012470680544198472400254596543711E2),
79 pS1 = L(3.674973957689619490312782828051860366493E3),
80 pS2 = L(-6.730729094812979665807581609853656623219E3),
81 pS3 = L(6.643843795209060298375552684423454077633E3),
82 pS4 = L(-3.817341990928606692235481812252049415993E3),
83 pS5 = L(1.284635388402653715636722822195716476156E3),
84 pS6 = L(-2.410736125231549204856567737329112037867E2),
85 pS7 = L(2.219191969382402856557594215833622156220E1),
86 pS8 = L(-7.249056260830627156600112195061001036533E-1),
87 pS9 = L(1.055923570937755300061509030361395604448E-3),
88
89 qS0 = L(-5.014859407482408326519083440151745519205E3),
90 qS1 = L(2.430653047950480068881028451580393430537E4),
91 qS2 = L(-4.997904737193653607449250593976069726962E4),
92 qS3 = L(5.675712336110456923807959930107347511086E4),
93 qS4 = L(-3.881523118339661268482937768522572588022E4),
94 qS5 = L(1.634202194895541569749717032234510811216E4),
95 qS6 = L(-4.151452662440709301601820849901296953752E3),
96 qS7 = L(5.956050864057192019085175976175695342168E2),
97 qS8 = L(-4.175375777334867025769346564600396877176E1),
98 /* 1.000000000000000000000000000000000000000E0 */
99
100 /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x)
101 -0.0625 <= x <= 0.0625
102 peak relative error 3.3e-35 */
103 rS0 = L(-5.619049346208901520945464704848780243887E0),
104 rS1 = L(4.460504162777731472539175700169871920352E1),
105 rS2 = L(-1.317669505315409261479577040530751477488E2),
106 rS3 = L(1.626532582423661989632442410808596009227E2),
107 rS4 = L(-3.144806644195158614904369445440583873264E1),
108 rS5 = L(-9.806674443470740708765165604769099559553E1),
109 rS6 = L(5.708468492052010816555762842394927806920E1),
110 rS7 = L(1.396540499232262112248553357962639431922E1),
111 rS8 = L(-1.126243289311910363001762058295832610344E1),
112 rS9 = L(-4.956179821329901954211277873774472383512E-1),
113 rS10 = L(3.313227657082367169241333738391762525780E-1),
114
115 sS0 = L(-4.645814742084009935700221277307007679325E0),
116 sS1 = L(3.879074822457694323970438316317961918430E1),
117 sS2 = L(-1.221986588013474694623973554726201001066E2),
118 sS3 = L(1.658821150347718105012079876756201905822E2),
119 sS4 = L(-4.804379630977558197953176474426239748977E1),
120 sS5 = L(-1.004296417397316948114344573811562952793E2),
121 sS6 = L(7.530281592861320234941101403870010111138E1),
122 sS7 = L(1.270735595411673647119592092304357226607E1),
123 sS8 = L(-1.815144839646376500705105967064792930282E1),
124 sS9 = L(-7.821597334910963922204235247786840828217E-2),
125 /* 1.000000000000000000000000000000000000000E0 */
126
127 asinr5625 = L(5.9740641664535021430381036628424864397707E-1);
128
129
130
131_Float128
132__ieee754_asinl (_Float128 x)
133{
134 _Float128 t, w, p, q, c, r, s;
135 int32_t ix, sign, flag;
136 ieee854_long_double_shape_type u;
137
138 flag = 0;
139 u.value = x;
140 sign = u.parts32.w0;
141 ix = sign & 0x7fffffff;
142 u.parts32.w0 = ix; /* |x| */
143 if (ix >= 0x3fff0000) /* |x|>= 1 */
144 {
145 if (ix == 0x3fff0000
146 && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
147 /* asin(1)=+-pi/2 with inexact */
148 return x * pio2_hi + x * pio2_lo;
149 return (x - x) / (x - x); /* asin(|x|>1) is NaN */
150 }
151 else if (ix < 0x3ffe0000) /* |x| < 0.5 */
152 {
153 if (ix < 0x3fc60000) /* |x| < 2**-57 */
154 {
155 math_check_force_underflow (x);
156 _Float128 force_inexact = huge + x;
157 math_force_eval (force_inexact);
158 return x; /* return x with inexact if x!=0 */
159 }
160 else
161 {
162 t = x * x;
163 /* Mark to use pS, qS later on. */
164 flag = 1;
165 }
166 }
167 else if (ix < 0x3ffe4000) /* 0.625 */
168 {
169 t = u.value - 0.5625;
170 p = ((((((((((rS10 * t
171 + rS9) * t
172 + rS8) * t
173 + rS7) * t
174 + rS6) * t
175 + rS5) * t
176 + rS4) * t
177 + rS3) * t
178 + rS2) * t
179 + rS1) * t
180 + rS0) * t;
181
182 q = ((((((((( t
183 + sS9) * t
184 + sS8) * t
185 + sS7) * t
186 + sS6) * t
187 + sS5) * t
188 + sS4) * t
189 + sS3) * t
190 + sS2) * t
191 + sS1) * t
192 + sS0;
193 t = asinr5625 + p / q;
194 if ((sign & 0x80000000) == 0)
195 return t;
196 else
197 return -t;
198 }
199 else
200 {
201 /* 1 > |x| >= 0.625 */
202 w = one - u.value;
203 t = w * 0.5;
204 }
205
206 p = (((((((((pS9 * t
207 + pS8) * t
208 + pS7) * t
209 + pS6) * t
210 + pS5) * t
211 + pS4) * t
212 + pS3) * t
213 + pS2) * t
214 + pS1) * t
215 + pS0) * t;
216
217 q = (((((((( t
218 + qS8) * t
219 + qS7) * t
220 + qS6) * t
221 + qS5) * t
222 + qS4) * t
223 + qS3) * t
224 + qS2) * t
225 + qS1) * t
226 + qS0;
227
228 if (flag) /* 2^-57 < |x| < 0.5 */
229 {
230 w = p / q;
231 return x + x * w;
232 }
233
234 s = __ieee754_sqrtl (t);
235 if (ix >= 0x3ffef333) /* |x| > 0.975 */
236 {
237 w = p / q;
238 t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
239 }
240 else
241 {
242 u.value = s;
243 u.parts32.w3 = 0;
244 u.parts32.w2 = 0;
245 w = u.value;
246 c = (t - w * w) / (s + w);
247 r = p / q;
248 p = 2.0 * s * r - (pio2_lo - 2.0 * c);
249 q = pio4_hi - 2.0 * w;
250 t = pio4_hi - (p - q);
251 }
252
253 if ((sign & 0x80000000) == 0)
254 return t;
255 else
256 return -t;
257}
258strong_alias (__ieee754_asinl, __asinl_finite)
259