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 the
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, see
32 <https://www.gnu.org/licenses/>. */
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)
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;
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
49 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
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))
56 * if x is NaN, return x itself;
57 * if |x|>1, return NaN with invalid signal.
64 #include <math-barriers.h>
65 #include <math_private.h>
66 #include <math-underflow.h>
68 static const _Float128
71 pio2_hi
= L(1.5707963267948966192313216916397514420986),
72 pio2_lo
= L(4.3359050650618905123985220130216759843812E-35),
73 pio4_hi
= L(7.8539816339744830961566084581987569936977E-1),
75 /* coefficient for R(x^2) */
77 /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
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),
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 */
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),
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 */
129 asinr5625
= L(5.9740641664535021430381036628424864397707E-1);
134 __ieee754_asinl (_Float128 x
)
136 _Float128 t
, w
, p
, q
, c
, r
, s
;
137 int32_t ix
, sign
, flag
;
138 ieee854_long_double_shape_type u
;
143 ix
= sign
& 0x7fffffff;
144 u
.parts32
.w0
= ix
; /* |x| */
145 if (ix
>= 0x3fff0000) /* |x|>= 1 */
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 */
153 else if (ix
< 0x3ffe0000) /* |x| < 0.5 */
155 if (ix
< 0x3fc60000) /* |x| < 2**-57 */
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 */
165 /* Mark to use pS, qS later on. */
169 else if (ix
< 0x3ffe4000) /* 0.625 */
171 t
= u
.value
- 0.5625;
172 p
= ((((((((((rS10
* t
195 t
= asinr5625
+ p
/ q
;
196 if ((sign
& 0x80000000) == 0)
203 /* 1 > |x| >= 0.625 */
230 if (flag
) /* 2^-57 < |x| < 0.5 */
237 if (ix
>= 0x3ffef333) /* |x| > 0.975 */
240 t
= pio2_hi
- (2.0 * (s
+ s
* w
) - pio2_lo
);
248 c
= (t
- w
* w
) / (s
+ w
);
250 p
= 2.0 * s
* r
- (pio2_lo
- 2.0 * c
);
251 q
= pio4_hi
- 2.0 * w
;
252 t
= pio4_hi
- (p
- q
);
255 if ((sign
& 0x80000000) == 0)
260 strong_alias (__ieee754_asinl
, __asinl_finite
)