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-barriers.h> |
65 | #include <math_private.h> |
66 | #include <math-underflow.h> |
67 | |
68 | static const _Float128 |
69 | one = 1, |
70 | huge = L(1.0e+4932), |
71 | pio2_hi = L(1.5707963267948966192313216916397514420986), |
72 | pio2_lo = L(4.3359050650618905123985220130216759843812E-35), |
73 | pio4_hi = L(7.8539816339744830961566084581987569936977E-1), |
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-35 */ |
80 | pS0 = L(-8.358099012470680544198472400254596543711E2), |
81 | pS1 = L(3.674973957689619490312782828051860366493E3), |
82 | pS2 = L(-6.730729094812979665807581609853656623219E3), |
83 | pS3 = L(6.643843795209060298375552684423454077633E3), |
84 | pS4 = L(-3.817341990928606692235481812252049415993E3), |
85 | pS5 = L(1.284635388402653715636722822195716476156E3), |
86 | pS6 = L(-2.410736125231549204856567737329112037867E2), |
87 | pS7 = L(2.219191969382402856557594215833622156220E1), |
88 | pS8 = L(-7.249056260830627156600112195061001036533E-1), |
89 | pS9 = L(1.055923570937755300061509030361395604448E-3), |
90 | |
91 | qS0 = L(-5.014859407482408326519083440151745519205E3), |
92 | qS1 = L(2.430653047950480068881028451580393430537E4), |
93 | qS2 = L(-4.997904737193653607449250593976069726962E4), |
94 | qS3 = L(5.675712336110456923807959930107347511086E4), |
95 | qS4 = L(-3.881523118339661268482937768522572588022E4), |
96 | qS5 = L(1.634202194895541569749717032234510811216E4), |
97 | qS6 = L(-4.151452662440709301601820849901296953752E3), |
98 | qS7 = L(5.956050864057192019085175976175695342168E2), |
99 | qS8 = L(-4.175375777334867025769346564600396877176E1), |
100 | /* 1.000000000000000000000000000000000000000E0 */ |
101 | |
102 | /* asin(0.5625 + x) = asin(0.5625) + x rS(x) / sS(x) |
103 | -0.0625 <= x <= 0.0625 |
104 | peak relative error 3.3e-35 */ |
105 | rS0 = L(-5.619049346208901520945464704848780243887E0), |
106 | rS1 = L(4.460504162777731472539175700169871920352E1), |
107 | rS2 = L(-1.317669505315409261479577040530751477488E2), |
108 | rS3 = L(1.626532582423661989632442410808596009227E2), |
109 | rS4 = L(-3.144806644195158614904369445440583873264E1), |
110 | rS5 = L(-9.806674443470740708765165604769099559553E1), |
111 | rS6 = L(5.708468492052010816555762842394927806920E1), |
112 | rS7 = L(1.396540499232262112248553357962639431922E1), |
113 | rS8 = L(-1.126243289311910363001762058295832610344E1), |
114 | rS9 = L(-4.956179821329901954211277873774472383512E-1), |
115 | rS10 = L(3.313227657082367169241333738391762525780E-1), |
116 | |
117 | sS0 = L(-4.645814742084009935700221277307007679325E0), |
118 | sS1 = L(3.879074822457694323970438316317961918430E1), |
119 | sS2 = L(-1.221986588013474694623973554726201001066E2), |
120 | sS3 = L(1.658821150347718105012079876756201905822E2), |
121 | sS4 = L(-4.804379630977558197953176474426239748977E1), |
122 | sS5 = L(-1.004296417397316948114344573811562952793E2), |
123 | sS6 = L(7.530281592861320234941101403870010111138E1), |
124 | sS7 = L(1.270735595411673647119592092304357226607E1), |
125 | sS8 = L(-1.815144839646376500705105967064792930282E1), |
126 | sS9 = L(-7.821597334910963922204235247786840828217E-2), |
127 | /* 1.000000000000000000000000000000000000000E0 */ |
128 | |
129 | asinr5625 = L(5.9740641664535021430381036628424864397707E-1); |
130 | |
131 | |
132 | |
133 | _Float128 |
134 | __ieee754_asinl (_Float128 x) |
135 | { |
136 | _Float128 t, w, p, q, c, r, s; |
137 | int32_t ix, sign, flag; |
138 | ieee854_long_double_shape_type u; |
139 | |
140 | flag = 0; |
141 | u.value = x; |
142 | sign = u.parts32.w0; |
143 | ix = sign & 0x7fffffff; |
144 | u.parts32.w0 = ix; /* |x| */ |
145 | if (ix >= 0x3fff0000) /* |x|>= 1 */ |
146 | { |
147 | if (ix == 0x3fff0000 |
148 | && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0) |
149 | /* asin(1)=+-pi/2 with inexact */ |
150 | return x * pio2_hi + x * pio2_lo; |
151 | return (x - x) / (x - x); /* asin(|x|>1) is NaN */ |
152 | } |
153 | else if (ix < 0x3ffe0000) /* |x| < 0.5 */ |
154 | { |
155 | if (ix < 0x3fc60000) /* |x| < 2**-57 */ |
156 | { |
157 | math_check_force_underflow (x); |
158 | _Float128 force_inexact = huge + x; |
159 | math_force_eval (force_inexact); |
160 | return x; /* return x with inexact if x!=0 */ |
161 | } |
162 | else |
163 | { |
164 | t = x * x; |
165 | /* Mark to use pS, qS later on. */ |
166 | flag = 1; |
167 | } |
168 | } |
169 | else if (ix < 0x3ffe4000) /* 0.625 */ |
170 | { |
171 | t = u.value - 0.5625; |
172 | p = ((((((((((rS10 * t |
173 | + rS9) * t |
174 | + rS8) * t |
175 | + rS7) * t |
176 | + rS6) * t |
177 | + rS5) * t |
178 | + rS4) * t |
179 | + rS3) * t |
180 | + rS2) * t |
181 | + rS1) * t |
182 | + rS0) * t; |
183 | |
184 | q = ((((((((( t |
185 | + sS9) * t |
186 | + sS8) * t |
187 | + sS7) * t |
188 | + sS6) * t |
189 | + sS5) * t |
190 | + sS4) * t |
191 | + sS3) * t |
192 | + sS2) * t |
193 | + sS1) * t |
194 | + sS0; |
195 | t = asinr5625 + p / q; |
196 | if ((sign & 0x80000000) == 0) |
197 | return t; |
198 | else |
199 | return -t; |
200 | } |
201 | else |
202 | { |
203 | /* 1 > |x| >= 0.625 */ |
204 | w = one - u.value; |
205 | t = w * 0.5; |
206 | } |
207 | |
208 | p = (((((((((pS9 * t |
209 | + pS8) * t |
210 | + pS7) * t |
211 | + pS6) * t |
212 | + pS5) * t |
213 | + pS4) * t |
214 | + pS3) * t |
215 | + pS2) * t |
216 | + pS1) * t |
217 | + pS0) * t; |
218 | |
219 | q = (((((((( t |
220 | + qS8) * t |
221 | + qS7) * t |
222 | + qS6) * t |
223 | + qS5) * t |
224 | + qS4) * t |
225 | + qS3) * t |
226 | + qS2) * t |
227 | + qS1) * t |
228 | + qS0; |
229 | |
230 | if (flag) /* 2^-57 < |x| < 0.5 */ |
231 | { |
232 | w = p / q; |
233 | return x + x * w; |
234 | } |
235 | |
236 | s = sqrtl (t); |
237 | if (ix >= 0x3ffef333) /* |x| > 0.975 */ |
238 | { |
239 | w = p / q; |
240 | t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); |
241 | } |
242 | else |
243 | { |
244 | u.value = s; |
245 | u.parts32.w3 = 0; |
246 | u.parts32.w2 = 0; |
247 | w = u.value; |
248 | c = (t - w * w) / (s + w); |
249 | r = p / q; |
250 | p = 2.0 * s * r - (pio2_lo - 2.0 * c); |
251 | q = pio4_hi - 2.0 * w; |
252 | t = pio4_hi - (p - q); |
253 | } |
254 | |
255 | if ((sign & 0x80000000) == 0) |
256 | return t; |
257 | else |
258 | return -t; |
259 | } |
260 | strong_alias (__ieee754_asinl, __asinl_finite) |
261 | |