2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
9 * ====================================================
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
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.
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.
30 You should have received a copy of the GNU Lesser General Public
31 License along with this library; if not, write to the Free Software
32 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
36 * acos(x) = pi/2 - asin(x)
37 * acos(-x) = pi/2 + asin(x)
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)
45 * acos(x) = 2 asin(sqrt((1-x)/2))
46 * computed with an extended precision square root in the leading term.
48 * acos(x) = pi - 2 asin(sqrt((1-|x|)/2))
51 * if x is NaN, return x itself;
52 * if |x|>1, return NaN with invalid signal.
54 * Functions needed: __ieee754_sqrtl.
58 #include "math_private.h"
61 static const long double
66 pio2_hi
= 1.5707963267948966192313216916397514420986L,
67 pio2_lo
= 4.3359050650618905123985220130216759843812E-35L,
69 /* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
70 -0.0625 <= x <= 0.0625
71 peak relative error 3.3e-35 */
73 rS0
= 5.619049346208901520945464704848780243887E0L
,
74 rS1
= -4.460504162777731472539175700169871920352E1L
,
75 rS2
= 1.317669505315409261479577040530751477488E2L
,
76 rS3
= -1.626532582423661989632442410808596009227E2L
,
77 rS4
= 3.144806644195158614904369445440583873264E1L
,
78 rS5
= 9.806674443470740708765165604769099559553E1L
,
79 rS6
= -5.708468492052010816555762842394927806920E1L
,
80 rS7
= -1.396540499232262112248553357962639431922E1L
,
81 rS8
= 1.126243289311910363001762058295832610344E1L
,
82 rS9
= 4.956179821329901954211277873774472383512E-1L,
83 rS10
= -3.313227657082367169241333738391762525780E-1L,
85 sS0
= -4.645814742084009935700221277307007679325E0L
,
86 sS1
= 3.879074822457694323970438316317961918430E1L
,
87 sS2
= -1.221986588013474694623973554726201001066E2L
,
88 sS3
= 1.658821150347718105012079876756201905822E2L
,
89 sS4
= -4.804379630977558197953176474426239748977E1L
,
90 sS5
= -1.004296417397316948114344573811562952793E2L
,
91 sS6
= 7.530281592861320234941101403870010111138E1L
,
92 sS7
= 1.270735595411673647119592092304357226607E1L
,
93 sS8
= -1.815144839646376500705105967064792930282E1L
,
94 sS9
= -7.821597334910963922204235247786840828217E-2L,
95 /* 1.000000000000000000000000000000000000000E0 */
97 acosr5625
= 9.7338991014954640492751132535550279812151E-1L,
98 pimacosr5625
= 2.1682027434402468335351320579240000860757E0L
,
100 /* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x)
101 -0.0625 <= x <= 0.0625
102 peak relative error 2.1e-35 */
104 P0
= 2.177690192235413635229046633751390484892E0L
,
105 P1
= -2.848698225706605746657192566166142909573E1L
,
106 P2
= 1.040076477655245590871244795403659880304E2L
,
107 P3
= -1.400087608918906358323551402881238180553E2L
,
108 P4
= 2.221047917671449176051896400503615543757E1L
,
109 P5
= 9.643714856395587663736110523917499638702E1L
,
110 P6
= -5.158406639829833829027457284942389079196E1L
,
111 P7
= -1.578651828337585944715290382181219741813E1L
,
112 P8
= 1.093632715903802870546857764647931045906E1L
,
113 P9
= 5.448925479898460003048760932274085300103E-1L,
114 P10
= -3.315886001095605268470690485170092986337E-1L,
115 Q0
= -1.958219113487162405143608843774587557016E0L
,
116 Q1
= 2.614577866876185080678907676023269360520E1L
,
117 Q2
= -9.990858606464150981009763389881793660938E1L
,
118 Q3
= 1.443958741356995763628660823395334281596E2L
,
119 Q4
= -3.206441012484232867657763518369723873129E1L
,
120 Q5
= -1.048560885341833443564920145642588991492E2L
,
121 Q6
= 6.745883931909770880159915641984874746358E1L
,
122 Q7
= 1.806809656342804436118449982647641392951E1L
,
123 Q8
= -1.770150690652438294290020775359580915464E1L
,
124 Q9
= -5.659156469628629327045433069052560211164E-1L,
125 /* 1.000000000000000000000000000000000000000E0 */
127 acosr4375
= 1.1179797320499710475919903296900511518755E0L
,
128 pimacosr4375
= 2.0236129215398221908706530535894517323217E0L
,
130 /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
132 peak relative error 1.9e-35 */
133 pS0
= -8.358099012470680544198472400254596543711E2L
,
134 pS1
= 3.674973957689619490312782828051860366493E3L
,
135 pS2
= -6.730729094812979665807581609853656623219E3L
,
136 pS3
= 6.643843795209060298375552684423454077633E3L
,
137 pS4
= -3.817341990928606692235481812252049415993E3L
,
138 pS5
= 1.284635388402653715636722822195716476156E3L
,
139 pS6
= -2.410736125231549204856567737329112037867E2L
,
140 pS7
= 2.219191969382402856557594215833622156220E1L
,
141 pS8
= -7.249056260830627156600112195061001036533E-1L,
142 pS9
= 1.055923570937755300061509030361395604448E-3L,
144 qS0
= -5.014859407482408326519083440151745519205E3L
,
145 qS1
= 2.430653047950480068881028451580393430537E4L
,
146 qS2
= -4.997904737193653607449250593976069726962E4L
,
147 qS3
= 5.675712336110456923807959930107347511086E4L
,
148 qS4
= -3.881523118339661268482937768522572588022E4L
,
149 qS5
= 1.634202194895541569749717032234510811216E4L
,
150 qS6
= -4.151452662440709301601820849901296953752E3L
,
151 qS7
= 5.956050864057192019085175976175695342168E2L
,
152 qS8
= -4.175375777334867025769346564600396877176E1L
;
153 /* 1.000000000000000000000000000000000000000E0 */
157 __ieee754_acosl (long double x
)
164 long double z
, r
, w
, p
, q
, s
, t
, f2
;
166 ieee854_long_double_shape_type u
;
170 ix
= sign
& 0x7fffffff;
171 u
.parts32
.w0
= ix
; /* |x| */
172 if (ix
>= 0x3fff0000) /* |x| >= 1 */
175 && (u
.parts32
.w1
| u
.parts32
.w2
| u
.parts32
.w3
) == 0)
177 if ((sign
& 0x80000000) == 0)
178 return 0.0; /* acos(1) = 0 */
180 return (2.0 * pio2_hi
) + (2.0 * pio2_lo
); /* acos(-1)= pi */
182 return (x
- x
) / (x
- x
); /* acos(|x| > 1) is NaN */
184 else if (ix
< 0x3ffe0000) /* |x| < 0.5 */
186 if (ix
< 0x3fc60000) /* |x| < 2**-57 */
187 return pio2_hi
+ pio2_lo
;
188 if (ix
< 0x3ffde000) /* |x| < .4375 */
213 z
= pio2_hi
- (r
- pio2_lo
);
216 /* .4375 <= |x| < .5 */
217 t
= u
.value
- 0.4375L;
218 p
= ((((((((((P10
* t
242 if (sign
& 0x80000000)
243 r
= pimacosr4375
- r
;
248 else if (ix
< 0x3ffe4000) /* |x| < 0.625 */
250 t
= u
.value
- 0.5625L;
251 p
= ((((((((((rS10
* t
274 if (sign
& 0x80000000)
275 r
= pimacosr5625
- p
/ q
;
277 r
= acosr5625
+ p
/ q
;
282 z
= (one
- u
.value
) * 0.5;
283 s
= __ieee754_sqrtl (z
);
284 /* Compute an extended precision square root from
285 the Newton iteration s -> 0.5 * (s + z / s).
286 The change w from s to the improved value is
287 w = 0.5 * (s + z / s) - s = (s^2 + z)/2s - s = (z - s^2)/2s.
288 Express s = f1 + f2 where f1 * f1 is exactly representable.
289 w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s .
290 s + w has extended precision. */
295 w
= z
- u
.value
* u
.value
;
296 w
= w
- 2.0 * u
.value
* f2
;
320 r
= s
+ (w
+ s
* p
/ q
);
322 if (sign
& 0x80000000)
323 w
= pio2_hi
+ (pio2_lo
- r
);