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 long double |
62 | one = 1.0L, |
63 | pio2_hi = 1.5707963267948966192313216916397514420986L, |
64 | pio2_lo = 4.3359050650618905123985220130216759843812E-35L, |
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 = 5.619049346208901520945464704848780243887E0L, |
71 | rS1 = -4.460504162777731472539175700169871920352E1L, |
72 | rS2 = 1.317669505315409261479577040530751477488E2L, |
73 | rS3 = -1.626532582423661989632442410808596009227E2L, |
74 | rS4 = 3.144806644195158614904369445440583873264E1L, |
75 | rS5 = 9.806674443470740708765165604769099559553E1L, |
76 | rS6 = -5.708468492052010816555762842394927806920E1L, |
77 | rS7 = -1.396540499232262112248553357962639431922E1L, |
78 | rS8 = 1.126243289311910363001762058295832610344E1L, |
79 | rS9 = 4.956179821329901954211277873774472383512E-1L, |
80 | rS10 = -3.313227657082367169241333738391762525780E-1L, |
81 | |
82 | sS0 = -4.645814742084009935700221277307007679325E0L, |
83 | sS1 = 3.879074822457694323970438316317961918430E1L, |
84 | sS2 = -1.221986588013474694623973554726201001066E2L, |
85 | sS3 = 1.658821150347718105012079876756201905822E2L, |
86 | sS4 = -4.804379630977558197953176474426239748977E1L, |
87 | sS5 = -1.004296417397316948114344573811562952793E2L, |
88 | sS6 = 7.530281592861320234941101403870010111138E1L, |
89 | sS7 = 1.270735595411673647119592092304357226607E1L, |
90 | sS8 = -1.815144839646376500705105967064792930282E1L, |
91 | sS9 = -7.821597334910963922204235247786840828217E-2L, |
92 | /* 1.000000000000000000000000000000000000000E0 */ |
93 | |
94 | acosr5625 = 9.7338991014954640492751132535550279812151E-1L, |
95 | pimacosr5625 = 2.1682027434402468335351320579240000860757E0L, |
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 = 2.177690192235413635229046633751390484892E0L, |
102 | P1 = -2.848698225706605746657192566166142909573E1L, |
103 | P2 = 1.040076477655245590871244795403659880304E2L, |
104 | P3 = -1.400087608918906358323551402881238180553E2L, |
105 | P4 = 2.221047917671449176051896400503615543757E1L, |
106 | P5 = 9.643714856395587663736110523917499638702E1L, |
107 | P6 = -5.158406639829833829027457284942389079196E1L, |
108 | P7 = -1.578651828337585944715290382181219741813E1L, |
109 | P8 = 1.093632715903802870546857764647931045906E1L, |
110 | P9 = 5.448925479898460003048760932274085300103E-1L, |
111 | P10 = -3.315886001095605268470690485170092986337E-1L, |
112 | Q0 = -1.958219113487162405143608843774587557016E0L, |
113 | Q1 = 2.614577866876185080678907676023269360520E1L, |
114 | Q2 = -9.990858606464150981009763389881793660938E1L, |
115 | Q3 = 1.443958741356995763628660823395334281596E2L, |
116 | Q4 = -3.206441012484232867657763518369723873129E1L, |
117 | Q5 = -1.048560885341833443564920145642588991492E2L, |
118 | Q6 = 6.745883931909770880159915641984874746358E1L, |
119 | Q7 = 1.806809656342804436118449982647641392951E1L, |
120 | Q8 = -1.770150690652438294290020775359580915464E1L, |
121 | Q9 = -5.659156469628629327045433069052560211164E-1L, |
122 | /* 1.000000000000000000000000000000000000000E0 */ |
123 | |
124 | acosr4375 = 1.1179797320499710475919903296900511518755E0L, |
125 | pimacosr4375 = 2.0236129215398221908706530535894517323217E0L, |
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 = -8.358099012470680544198472400254596543711E2L, |
131 | pS1 = 3.674973957689619490312782828051860366493E3L, |
132 | pS2 = -6.730729094812979665807581609853656623219E3L, |
133 | pS3 = 6.643843795209060298375552684423454077633E3L, |
134 | pS4 = -3.817341990928606692235481812252049415993E3L, |
135 | pS5 = 1.284635388402653715636722822195716476156E3L, |
136 | pS6 = -2.410736125231549204856567737329112037867E2L, |
137 | pS7 = 2.219191969382402856557594215833622156220E1L, |
138 | pS8 = -7.249056260830627156600112195061001036533E-1L, |
139 | pS9 = 1.055923570937755300061509030361395604448E-3L, |
140 | |
141 | qS0 = -5.014859407482408326519083440151745519205E3L, |
142 | qS1 = 2.430653047950480068881028451580393430537E4L, |
143 | qS2 = -4.997904737193653607449250593976069726962E4L, |
144 | qS3 = 5.675712336110456923807959930107347511086E4L, |
145 | qS4 = -3.881523118339661268482937768522572588022E4L, |
146 | qS5 = 1.634202194895541569749717032234510811216E4L, |
147 | qS6 = -4.151452662440709301601820849901296953752E3L, |
148 | qS7 = 5.956050864057192019085175976175695342168E2L, |
149 | qS8 = -4.175375777334867025769346564600396877176E1L; |
150 | /* 1.000000000000000000000000000000000000000E0 */ |
151 | |
152 | long double |
153 | __ieee754_acosl (long double x) |
154 | { |
155 | long double a, z, r, w, p, q, s, t, f2; |
156 | |
157 | if (__glibc_unlikely (isnan (x))) |
158 | return x + x; |
159 | a = __builtin_fabsl (x); |
160 | if (a == 1.0L) |
161 | { |
162 | if (x > 0.0L) |
163 | return 0.0; /* acos(1) = 0 */ |
164 | else |
165 | return (2.0 * pio2_hi) + (2.0 * pio2_lo); /* acos(-1)= pi */ |
166 | } |
167 | else if (a > 1.0L) |
168 | { |
169 | return (x - x) / (x - x); /* acos(|x| > 1) is NaN */ |
170 | } |
171 | if (a < 0.5L) |
172 | { |
173 | if (a < 0x1p-106L) |
174 | return pio2_hi + pio2_lo; |
175 | if (a < 0.4375L) |
176 | { |
177 | /* Arcsine of x. */ |
178 | z = x * x; |
179 | p = (((((((((pS9 * z |
180 | + pS8) * z |
181 | + pS7) * z |
182 | + pS6) * z |
183 | + pS5) * z |
184 | + pS4) * z |
185 | + pS3) * z |
186 | + pS2) * z |
187 | + pS1) * z |
188 | + pS0) * z; |
189 | q = (((((((( z |
190 | + qS8) * z |
191 | + qS7) * z |
192 | + qS6) * z |
193 | + qS5) * z |
194 | + qS4) * z |
195 | + qS3) * z |
196 | + qS2) * z |
197 | + qS1) * z |
198 | + qS0; |
199 | r = x + x * p / q; |
200 | z = pio2_hi - (r - pio2_lo); |
201 | return z; |
202 | } |
203 | /* .4375 <= |x| < .5 */ |
204 | t = a - 0.4375L; |
205 | p = ((((((((((P10 * t |
206 | + P9) * t |
207 | + P8) * t |
208 | + P7) * t |
209 | + P6) * t |
210 | + P5) * t |
211 | + P4) * t |
212 | + P3) * t |
213 | + P2) * t |
214 | + P1) * t |
215 | + P0) * t; |
216 | |
217 | q = (((((((((t |
218 | + Q9) * t |
219 | + Q8) * t |
220 | + Q7) * t |
221 | + Q6) * t |
222 | + Q5) * t |
223 | + Q4) * t |
224 | + Q3) * t |
225 | + Q2) * t |
226 | + Q1) * t |
227 | + Q0; |
228 | r = p / q; |
229 | if (x < 0.0L) |
230 | r = pimacosr4375 - r; |
231 | else |
232 | r = acosr4375 + r; |
233 | return r; |
234 | } |
235 | else if (a < 0.625L) |
236 | { |
237 | t = a - 0.5625L; |
238 | p = ((((((((((rS10 * t |
239 | + rS9) * t |
240 | + rS8) * t |
241 | + rS7) * t |
242 | + rS6) * t |
243 | + rS5) * t |
244 | + rS4) * t |
245 | + rS3) * t |
246 | + rS2) * t |
247 | + rS1) * t |
248 | + rS0) * t; |
249 | |
250 | q = (((((((((t |
251 | + sS9) * t |
252 | + sS8) * t |
253 | + sS7) * t |
254 | + sS6) * t |
255 | + sS5) * t |
256 | + sS4) * t |
257 | + sS3) * t |
258 | + sS2) * t |
259 | + sS1) * t |
260 | + sS0; |
261 | if (x < 0.0L) |
262 | r = pimacosr5625 - p / q; |
263 | else |
264 | r = acosr5625 + p / q; |
265 | return r; |
266 | } |
267 | else |
268 | { /* |x| >= .625 */ |
269 | double shi, slo; |
270 | |
271 | z = (one - a) * 0.5; |
272 | s = sqrtl (z); |
273 | /* Compute an extended precision square root from |
274 | the Newton iteration s -> 0.5 * (s + z / s). |
275 | The change w from s to the improved value is |
276 | w = 0.5 * (s + z / s) - s = (s^2 + z)/2s - s = (z - s^2)/2s. |
277 | Express s = f1 + f2 where f1 * f1 is exactly representable. |
278 | w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s . |
279 | s + w has extended precision. */ |
280 | ldbl_unpack (s, &shi, &slo); |
281 | a = shi; |
282 | f2 = slo; |
283 | w = z - a * a; |
284 | w = w - 2.0 * a * f2; |
285 | w = w - f2 * f2; |
286 | w = w / (2.0 * s); |
287 | /* Arcsine of s. */ |
288 | p = (((((((((pS9 * z |
289 | + pS8) * z |
290 | + pS7) * z |
291 | + pS6) * z |
292 | + pS5) * z |
293 | + pS4) * z |
294 | + pS3) * z |
295 | + pS2) * z |
296 | + pS1) * z |
297 | + pS0) * z; |
298 | q = (((((((( z |
299 | + qS8) * z |
300 | + qS7) * z |
301 | + qS6) * z |
302 | + qS5) * z |
303 | + qS4) * z |
304 | + qS3) * z |
305 | + qS2) * z |
306 | + qS1) * z |
307 | + qS0; |
308 | r = s + (w + s * p / q); |
309 | |
310 | if (x < 0.0L) |
311 | w = pio2_hi + (pio2_lo - r); |
312 | else |
313 | w = r; |
314 | return 2.0 * w; |
315 | } |
316 | } |
317 | libm_alias_finite (__ieee754_acosl, __acosl) |
318 | |