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_acosl(x) |
35 | * Method : |
36 | * acos(x) = pi/2 - asin(x) |
37 | * acos(-x) = pi/2 + asin(x) |
38 | * For |x| <= 0.375 |
39 | * acos(x) = pi/2 - asin(x) |
40 | * Between .375 and .5 the approximation is |
41 | * acos(0.4375 + x) = acos(0.4375) + x P(x) / Q(x) |
42 | * Between .5 and .625 the approximation is |
43 | * acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x) |
44 | * For x > 0.625, |
45 | * acos(x) = 2 asin(sqrt((1-x)/2)) |
46 | * computed with an extended precision square root in the leading term. |
47 | * For x < -0.625 |
48 | * acos(x) = pi - 2 asin(sqrt((1-|x|)/2)) |
49 | * |
50 | * Special cases: |
51 | * if x is NaN, return x itself; |
52 | * if |x|>1, return NaN with invalid signal. |
53 | * |
54 | * Functions needed: sqrtl. |
55 | */ |
56 | |
57 | #include <math.h> |
58 | #include <math_private.h> |
59 | #include <libm-alias-finite.h> |
60 | |
61 | static const _Float128 |
62 | one = 1, |
63 | pio2_hi = L(1.5707963267948966192313216916397514420986), |
64 | pio2_lo = L(4.3359050650618905123985220130216759843812E-35), |
65 | |
66 | /* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x) |
67 | -0.0625 <= x <= 0.0625 |
68 | peak relative error 3.3e-35 */ |
69 | |
70 | rS0 = L(5.619049346208901520945464704848780243887E0), |
71 | rS1 = L(-4.460504162777731472539175700169871920352E1), |
72 | rS2 = L(1.317669505315409261479577040530751477488E2), |
73 | rS3 = L(-1.626532582423661989632442410808596009227E2), |
74 | rS4 = L(3.144806644195158614904369445440583873264E1), |
75 | rS5 = L(9.806674443470740708765165604769099559553E1), |
76 | rS6 = L(-5.708468492052010816555762842394927806920E1), |
77 | rS7 = L(-1.396540499232262112248553357962639431922E1), |
78 | rS8 = L(1.126243289311910363001762058295832610344E1), |
79 | rS9 = L(4.956179821329901954211277873774472383512E-1), |
80 | rS10 = L(-3.313227657082367169241333738391762525780E-1), |
81 | |
82 | sS0 = L(-4.645814742084009935700221277307007679325E0), |
83 | sS1 = L(3.879074822457694323970438316317961918430E1), |
84 | sS2 = L(-1.221986588013474694623973554726201001066E2), |
85 | sS3 = L(1.658821150347718105012079876756201905822E2), |
86 | sS4 = L(-4.804379630977558197953176474426239748977E1), |
87 | sS5 = L(-1.004296417397316948114344573811562952793E2), |
88 | sS6 = L(7.530281592861320234941101403870010111138E1), |
89 | sS7 = L(1.270735595411673647119592092304357226607E1), |
90 | sS8 = L(-1.815144839646376500705105967064792930282E1), |
91 | sS9 = L(-7.821597334910963922204235247786840828217E-2), |
92 | /* 1.000000000000000000000000000000000000000E0 */ |
93 | |
94 | acosr5625 = L(9.7338991014954640492751132535550279812151E-1), |
95 | pimacosr5625 = L(2.1682027434402468335351320579240000860757E0), |
96 | |
97 | /* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x) |
98 | -0.0625 <= x <= 0.0625 |
99 | peak relative error 2.1e-35 */ |
100 | |
101 | P0 = L(2.177690192235413635229046633751390484892E0), |
102 | P1 = L(-2.848698225706605746657192566166142909573E1), |
103 | P2 = L(1.040076477655245590871244795403659880304E2), |
104 | P3 = L(-1.400087608918906358323551402881238180553E2), |
105 | P4 = L(2.221047917671449176051896400503615543757E1), |
106 | P5 = L(9.643714856395587663736110523917499638702E1), |
107 | P6 = L(-5.158406639829833829027457284942389079196E1), |
108 | P7 = L(-1.578651828337585944715290382181219741813E1), |
109 | P8 = L(1.093632715903802870546857764647931045906E1), |
110 | P9 = L(5.448925479898460003048760932274085300103E-1), |
111 | P10 = L(-3.315886001095605268470690485170092986337E-1), |
112 | Q0 = L(-1.958219113487162405143608843774587557016E0), |
113 | Q1 = L(2.614577866876185080678907676023269360520E1), |
114 | Q2 = L(-9.990858606464150981009763389881793660938E1), |
115 | Q3 = L(1.443958741356995763628660823395334281596E2), |
116 | Q4 = L(-3.206441012484232867657763518369723873129E1), |
117 | Q5 = L(-1.048560885341833443564920145642588991492E2), |
118 | Q6 = L(6.745883931909770880159915641984874746358E1), |
119 | Q7 = L(1.806809656342804436118449982647641392951E1), |
120 | Q8 = L(-1.770150690652438294290020775359580915464E1), |
121 | Q9 = L(-5.659156469628629327045433069052560211164E-1), |
122 | /* 1.000000000000000000000000000000000000000E0 */ |
123 | |
124 | acosr4375 = L(1.1179797320499710475919903296900511518755E0), |
125 | pimacosr4375 = L(2.0236129215398221908706530535894517323217E0), |
126 | |
127 | /* asin(x) = x + x^3 pS(x^2) / qS(x^2) |
128 | 0 <= x <= 0.5 |
129 | peak relative error 1.9e-35 */ |
130 | pS0 = L(-8.358099012470680544198472400254596543711E2), |
131 | pS1 = L(3.674973957689619490312782828051860366493E3), |
132 | pS2 = L(-6.730729094812979665807581609853656623219E3), |
133 | pS3 = L(6.643843795209060298375552684423454077633E3), |
134 | pS4 = L(-3.817341990928606692235481812252049415993E3), |
135 | pS5 = L(1.284635388402653715636722822195716476156E3), |
136 | pS6 = L(-2.410736125231549204856567737329112037867E2), |
137 | pS7 = L(2.219191969382402856557594215833622156220E1), |
138 | pS8 = L(-7.249056260830627156600112195061001036533E-1), |
139 | pS9 = L(1.055923570937755300061509030361395604448E-3), |
140 | |
141 | qS0 = L(-5.014859407482408326519083440151745519205E3), |
142 | qS1 = L(2.430653047950480068881028451580393430537E4), |
143 | qS2 = L(-4.997904737193653607449250593976069726962E4), |
144 | qS3 = L(5.675712336110456923807959930107347511086E4), |
145 | qS4 = L(-3.881523118339661268482937768522572588022E4), |
146 | qS5 = L(1.634202194895541569749717032234510811216E4), |
147 | qS6 = L(-4.151452662440709301601820849901296953752E3), |
148 | qS7 = L(5.956050864057192019085175976175695342168E2), |
149 | qS8 = L(-4.175375777334867025769346564600396877176E1); |
150 | /* 1.000000000000000000000000000000000000000E0 */ |
151 | |
152 | _Float128 |
153 | __ieee754_acosl (_Float128 x) |
154 | { |
155 | _Float128 z, r, w, p, q, s, t, f2; |
156 | int32_t ix, sign; |
157 | ieee854_long_double_shape_type u; |
158 | |
159 | u.value = x; |
160 | sign = u.parts32.w0; |
161 | ix = sign & 0x7fffffff; |
162 | u.parts32.w0 = ix; /* |x| */ |
163 | if (ix >= 0x3fff0000) /* |x| >= 1 */ |
164 | { |
165 | if (ix == 0x3fff0000 |
166 | && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0) |
167 | { /* |x| == 1 */ |
168 | if ((sign & 0x80000000) == 0) |
169 | return 0.0; /* acos(1) = 0 */ |
170 | else |
171 | return (2.0 * pio2_hi) + (2.0 * pio2_lo); /* acos(-1)= pi */ |
172 | } |
173 | return (x - x) / (x - x); /* acos(|x| > 1) is NaN */ |
174 | } |
175 | else if (ix < 0x3ffe0000) /* |x| < 0.5 */ |
176 | { |
177 | if (ix < 0x3f8e0000) /* |x| < 2**-113 */ |
178 | return pio2_hi + pio2_lo; |
179 | if (ix < 0x3ffde000) /* |x| < .4375 */ |
180 | { |
181 | /* Arcsine of x. */ |
182 | z = x * x; |
183 | p = (((((((((pS9 * z |
184 | + pS8) * z |
185 | + pS7) * z |
186 | + pS6) * z |
187 | + pS5) * z |
188 | + pS4) * z |
189 | + pS3) * z |
190 | + pS2) * z |
191 | + pS1) * z |
192 | + pS0) * z; |
193 | q = (((((((( z |
194 | + qS8) * z |
195 | + qS7) * z |
196 | + qS6) * z |
197 | + qS5) * z |
198 | + qS4) * z |
199 | + qS3) * z |
200 | + qS2) * z |
201 | + qS1) * z |
202 | + qS0; |
203 | r = x + x * p / q; |
204 | z = pio2_hi - (r - pio2_lo); |
205 | return z; |
206 | } |
207 | /* .4375 <= |x| < .5 */ |
208 | t = u.value - L(0.4375); |
209 | p = ((((((((((P10 * t |
210 | + P9) * t |
211 | + P8) * t |
212 | + P7) * t |
213 | + P6) * t |
214 | + P5) * t |
215 | + P4) * t |
216 | + P3) * t |
217 | + P2) * t |
218 | + P1) * t |
219 | + P0) * t; |
220 | |
221 | q = (((((((((t |
222 | + Q9) * t |
223 | + Q8) * t |
224 | + Q7) * t |
225 | + Q6) * t |
226 | + Q5) * t |
227 | + Q4) * t |
228 | + Q3) * t |
229 | + Q2) * t |
230 | + Q1) * t |
231 | + Q0; |
232 | r = p / q; |
233 | if (sign & 0x80000000) |
234 | r = pimacosr4375 - r; |
235 | else |
236 | r = acosr4375 + r; |
237 | return r; |
238 | } |
239 | else if (ix < 0x3ffe4000) /* |x| < 0.625 */ |
240 | { |
241 | t = u.value - L(0.5625); |
242 | p = ((((((((((rS10 * t |
243 | + rS9) * t |
244 | + rS8) * t |
245 | + rS7) * t |
246 | + rS6) * t |
247 | + rS5) * t |
248 | + rS4) * t |
249 | + rS3) * t |
250 | + rS2) * t |
251 | + rS1) * t |
252 | + rS0) * t; |
253 | |
254 | q = (((((((((t |
255 | + sS9) * t |
256 | + sS8) * t |
257 | + sS7) * t |
258 | + sS6) * t |
259 | + sS5) * t |
260 | + sS4) * t |
261 | + sS3) * t |
262 | + sS2) * t |
263 | + sS1) * t |
264 | + sS0; |
265 | if (sign & 0x80000000) |
266 | r = pimacosr5625 - p / q; |
267 | else |
268 | r = acosr5625 + p / q; |
269 | return r; |
270 | } |
271 | else |
272 | { /* |x| >= .625 */ |
273 | z = (one - u.value) * 0.5; |
274 | s = sqrtl (z); |
275 | /* Compute an extended precision square root from |
276 | the Newton iteration s -> 0.5 * (s + z / s). |
277 | The change w from s to the improved value is |
278 | w = 0.5 * (s + z / s) - s = (s^2 + z)/2s - s = (z - s^2)/2s. |
279 | Express s = f1 + f2 where f1 * f1 is exactly representable. |
280 | w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s . |
281 | s + w has extended precision. */ |
282 | u.value = s; |
283 | u.parts32.w2 = 0; |
284 | u.parts32.w3 = 0; |
285 | f2 = s - u.value; |
286 | w = z - u.value * u.value; |
287 | w = w - 2.0 * u.value * f2; |
288 | w = w - f2 * f2; |
289 | w = w / (2.0 * s); |
290 | /* Arcsine of s. */ |
291 | p = (((((((((pS9 * z |
292 | + pS8) * z |
293 | + pS7) * z |
294 | + pS6) * z |
295 | + pS5) * z |
296 | + pS4) * z |
297 | + pS3) * z |
298 | + pS2) * z |
299 | + pS1) * z |
300 | + pS0) * z; |
301 | q = (((((((( z |
302 | + qS8) * z |
303 | + qS7) * z |
304 | + qS6) * z |
305 | + qS5) * z |
306 | + qS4) * z |
307 | + qS3) * z |
308 | + qS2) * z |
309 | + qS1) * z |
310 | + qS0; |
311 | r = s + (w + s * p / q); |
312 | |
313 | if (sign & 0x80000000) |
314 | w = pio2_hi + (pio2_lo - r); |
315 | else |
316 | w = r; |
317 | return 2.0 * w; |
318 | } |
319 | } |
320 | libm_alias_finite (__ieee754_acosl, __acosl) |
321 | |