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 | <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 | * 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-barriers.h> |
65 | #include <math_private.h> |
66 | #include <math-underflow.h> |
67 | #include <libm-alias-finite.h> |
68 | |
69 | static const long double |
70 | one = 1.0L, |
71 | huge = 1.0e+300L, |
72 | pio2_hi = 1.5707963267948966192313216916397514420986L, |
73 | pio2_lo = 4.3359050650618905123985220130216759843812E-35L, |
74 | pio4_hi = 7.8539816339744830961566084581987569936977E-1L, |
75 | |
76 | /* coefficient for R(x^2) */ |
77 | |
78 | /* asin(x) = x + x^3 pS(x^2) / qS(x^2) |
79 | 0 <= x <= 0.5 |
80 | peak relative error 1.9e-35 */ |
81 | pS0 = -8.358099012470680544198472400254596543711E2L, |
82 | pS1 = 3.674973957689619490312782828051860366493E3L, |
83 | pS2 = -6.730729094812979665807581609853656623219E3L, |
84 | pS3 = 6.643843795209060298375552684423454077633E3L, |
85 | pS4 = -3.817341990928606692235481812252049415993E3L, |
86 | pS5 = 1.284635388402653715636722822195716476156E3L, |
87 | pS6 = -2.410736125231549204856567737329112037867E2L, |
88 | pS7 = 2.219191969382402856557594215833622156220E1L, |
89 | pS8 = -7.249056260830627156600112195061001036533E-1L, |
90 | pS9 = 1.055923570937755300061509030361395604448E-3L, |
91 | |
92 | qS0 = -5.014859407482408326519083440151745519205E3L, |
93 | qS1 = 2.430653047950480068881028451580393430537E4L, |
94 | qS2 = -4.997904737193653607449250593976069726962E4L, |
95 | qS3 = 5.675712336110456923807959930107347511086E4L, |
96 | qS4 = -3.881523118339661268482937768522572588022E4L, |
97 | qS5 = 1.634202194895541569749717032234510811216E4L, |
98 | qS6 = -4.151452662440709301601820849901296953752E3L, |
99 | qS7 = 5.956050864057192019085175976175695342168E2L, |
100 | qS8 = -4.175375777334867025769346564600396877176E1L, |
101 | /* 1.000000000000000000000000000000000000000E0 */ |
102 | |
103 | /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x) |
104 | -0.0625 <= x <= 0.0625 |
105 | peak relative error 3.3e-35 */ |
106 | rS0 = -5.619049346208901520945464704848780243887E0L, |
107 | rS1 = 4.460504162777731472539175700169871920352E1L, |
108 | rS2 = -1.317669505315409261479577040530751477488E2L, |
109 | rS3 = 1.626532582423661989632442410808596009227E2L, |
110 | rS4 = -3.144806644195158614904369445440583873264E1L, |
111 | rS5 = -9.806674443470740708765165604769099559553E1L, |
112 | rS6 = 5.708468492052010816555762842394927806920E1L, |
113 | rS7 = 1.396540499232262112248553357962639431922E1L, |
114 | rS8 = -1.126243289311910363001762058295832610344E1L, |
115 | rS9 = -4.956179821329901954211277873774472383512E-1L, |
116 | rS10 = 3.313227657082367169241333738391762525780E-1L, |
117 | |
118 | sS0 = -4.645814742084009935700221277307007679325E0L, |
119 | sS1 = 3.879074822457694323970438316317961918430E1L, |
120 | sS2 = -1.221986588013474694623973554726201001066E2L, |
121 | sS3 = 1.658821150347718105012079876756201905822E2L, |
122 | sS4 = -4.804379630977558197953176474426239748977E1L, |
123 | sS5 = -1.004296417397316948114344573811562952793E2L, |
124 | sS6 = 7.530281592861320234941101403870010111138E1L, |
125 | sS7 = 1.270735595411673647119592092304357226607E1L, |
126 | sS8 = -1.815144839646376500705105967064792930282E1L, |
127 | sS9 = -7.821597334910963922204235247786840828217E-2L, |
128 | /* 1.000000000000000000000000000000000000000E0 */ |
129 | |
130 | asinr5625 = 5.9740641664535021430381036628424864397707E-1L; |
131 | |
132 | |
133 | |
134 | long double |
135 | __ieee754_asinl (long double x) |
136 | { |
137 | long double a, t, w, p, q, c, r, s; |
138 | int flag; |
139 | |
140 | if (__glibc_unlikely (isnan (x))) |
141 | return x + x; |
142 | flag = 0; |
143 | a = __builtin_fabsl (x); |
144 | if (a == 1.0L) /* |x|>= 1 */ |
145 | return x * pio2_hi + x * pio2_lo; /* asin(1)=+-pi/2 with inexact */ |
146 | else if (a >= 1.0L) |
147 | return (x - x) / (x - x); /* asin(|x|>1) is NaN */ |
148 | else if (a < 0.5L) |
149 | { |
150 | if (a < 6.938893903907228e-18L) /* |x| < 2**-57 */ |
151 | { |
152 | math_check_force_underflow (x); |
153 | long double force_inexact = huge + x; |
154 | math_force_eval (force_inexact); |
155 | return x; /* return x with inexact if x!=0 */ |
156 | } |
157 | else |
158 | { |
159 | t = x * x; |
160 | /* Mark to use pS, qS later on. */ |
161 | flag = 1; |
162 | } |
163 | } |
164 | else if (a < 0.625L) |
165 | { |
166 | t = a - 0.5625; |
167 | p = ((((((((((rS10 * t |
168 | + rS9) * t |
169 | + rS8) * t |
170 | + rS7) * t |
171 | + rS6) * t |
172 | + rS5) * t |
173 | + rS4) * t |
174 | + rS3) * t |
175 | + rS2) * t |
176 | + rS1) * t |
177 | + rS0) * t; |
178 | |
179 | q = ((((((((( t |
180 | + sS9) * t |
181 | + sS8) * t |
182 | + sS7) * t |
183 | + sS6) * t |
184 | + sS5) * t |
185 | + sS4) * t |
186 | + sS3) * t |
187 | + sS2) * t |
188 | + sS1) * t |
189 | + sS0; |
190 | t = asinr5625 + p / q; |
191 | if (x > 0.0L) |
192 | return t; |
193 | else |
194 | return -t; |
195 | } |
196 | else |
197 | { |
198 | /* 1 > |x| >= 0.625 */ |
199 | w = one - a; |
200 | t = w * 0.5; |
201 | } |
202 | |
203 | p = (((((((((pS9 * t |
204 | + pS8) * t |
205 | + pS7) * t |
206 | + pS6) * t |
207 | + pS5) * t |
208 | + pS4) * t |
209 | + pS3) * t |
210 | + pS2) * t |
211 | + pS1) * t |
212 | + pS0) * t; |
213 | |
214 | q = (((((((( t |
215 | + qS8) * t |
216 | + qS7) * t |
217 | + qS6) * t |
218 | + qS5) * t |
219 | + qS4) * t |
220 | + qS3) * t |
221 | + qS2) * t |
222 | + qS1) * t |
223 | + qS0; |
224 | |
225 | if (flag) /* 2^-57 < |x| < 0.5 */ |
226 | { |
227 | w = p / q; |
228 | return x + x * w; |
229 | } |
230 | |
231 | s = sqrtl (t); |
232 | if (a > 0.975L) |
233 | { |
234 | w = p / q; |
235 | t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); |
236 | } |
237 | else |
238 | { |
239 | w = ldbl_high (s); |
240 | c = (t - w * w) / (s + w); |
241 | r = p / q; |
242 | p = 2.0 * s * r - (pio2_lo - 2.0 * c); |
243 | q = pio4_hi - 2.0 * w; |
244 | t = pio4_hi - (p - q); |
245 | } |
246 | |
247 | if (x > 0.0L) |
248 | return t; |
249 | else |
250 | return -t; |
251 | } |
252 | libm_alias_finite (__ieee754_asinl, __asinl) |
253 | |