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)