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, 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)
41 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
42 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
44 * asin(x) = pi/2 - 2*(s+s*z*R(z))
45 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
46 * For x<=0.98, let pio4_hi = pio2_hi/2, then
48 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
50 * asin(x) = pi/2 - 2*(s+s*z*R(z))
51 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
52 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
55 * if x is NaN, return x itself;
56 * if |x|>1, return NaN with invalid signal.
63 #include <math_private.h>
64 #include <math-underflow.h>
65 #include <libm-alias-finite.h>
67 static const long double
70 pio2_hi
= 0x1.921fb54442d1846ap
+0L, /* pi/2 rounded to nearest to 64
72 pio2_lo
= -0x7.6733ae8fe47c65d8p
-68L, /* pi/2 - pio2_hi rounded to
73 nearest to 64 bits. */
74 pio4_hi
= 0xc.90fdaa22168c235p
-4L, /* pi/4 rounded to nearest to 64
77 /* coefficient for R(x^2) */
79 /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
81 peak relative error 1.9e-21 */
82 pS0
= -1.008714657938491626019651170502036851607E1L
,
83 pS1
= 2.331460313214179572063441834101394865259E1L
,
84 pS2
= -1.863169762159016144159202387315381830227E1L
,
85 pS3
= 5.930399351579141771077475766877674661747E0L
,
86 pS4
= -6.121291917696920296944056882932695185001E-1L,
87 pS5
= 3.776934006243367487161248678019350338383E-3L,
89 qS0
= -6.052287947630949712886794360635592886517E1L
,
90 qS1
= 1.671229145571899593737596543114258558503E2L
,
91 qS2
= -1.707840117062586426144397688315411324388E2L
,
92 qS3
= 7.870295154902110425886636075950077640623E1L
,
93 qS4
= -1.568433562487314651121702982333303458814E1L
;
94 /* 1.000000000000000000000000000000000000000E0 */
97 __ieee754_asinl (long double x
)
99 long double t
, w
, p
, q
, c
, r
, s
;
101 uint32_t se
, i0
, i1
, k
;
103 GET_LDOUBLE_WORDS (se
, i0
, i1
, x
);
105 ix
= (ix
<< 16) | (i0
>> 16);
106 if (ix
>= 0x3fff8000)
108 if (ix
== 0x3fff8000 && ((i0
- 0x80000000) | i1
) == 0)
109 /* asin(1)=+-pi/2 with inexact */
110 return x
* pio2_hi
+ x
* pio2_lo
;
111 return (x
- x
) / (x
- x
); /* asin(|x|>1) is NaN */
113 else if (ix
< 0x3ffe8000)
116 { /* if |x| < 2**-33 */
117 math_check_force_underflow (x
);
119 return x
; /* return x with inexact if x!=0 */
126 t
* (pS1
+ t
* (pS2
+ t
* (pS3
+ t
* (pS4
+ t
* pS5
)))));
127 q
= qS0
+ t
* (qS1
+ t
* (qS2
+ t
* (qS3
+ t
* (qS4
+ t
))));
135 p
= t
* (pS0
+ t
* (pS1
+ t
* (pS2
+ t
* (pS3
+ t
* (pS4
+ t
* pS5
)))));
136 q
= qS0
+ t
* (qS1
+ t
* (qS2
+ t
* (qS3
+ t
* (qS4
+ t
))));
138 if (ix
>= 0x3ffef999)
139 { /* if |x| > 0.975 */
141 t
= pio2_hi
- (2.0 * (s
+ s
* w
) - pio2_lo
);
145 GET_LDOUBLE_WORDS (k
, i0
, i1
, s
);
147 SET_LDOUBLE_WORDS (w
,k
,i0
,i1
);
148 c
= (t
- w
* w
) / (s
+ w
);
150 p
= 2.0 * s
* r
- (pio2_lo
- 2.0 * c
);
151 q
= pio4_hi
- 2.0 * w
;
152 t
= pio4_hi
- (p
- q
);
154 if ((se
& 0x8000) == 0)
159 libm_alias_finite (__ieee754_asinl
, __asinl
)