1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
|
/* Correctly-rounded arc-tangent of binary32 value.
Copyright (c) 2022-2024 Alexei Sibidanov.
The original version of this file was copied from the CORE-MATH
project (file src/binary32/atan/atanf.c, revision a8066a5).
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/
#include <math.h>
#include <stdint.h>
#include <libm-alias-float.h>
#include "math_config.h"
float
__atanf (float x)
{
const double pi2 = 0x1.921fb54442d18p+0;
uint32_t t = asuint (x);
int e = (t >> 23) & 0xff;
bool gt = e >= 127;
uint32_t ta = t & 0x7fffffff;
if (__glibc_unlikely (ta >= 0x4c700518u)) /* |x| > 0x1.e00a3p+25 */
{
if (ta > 0x7f800000u)
return x + x; /* nan */
return copysign (pi2, (double) x);
}
if (__glibc_unlikely (e < 127 - 13))
{
if (__glibc_unlikely (e < 127 - 25))
{
if (!(t << 1))
return x;
return fmaf (-x, fabsf (x), x);
}
return fmaf (-0x1.5555555555555p-2f * x, x * x, x);
}
/* now |x| >= 0x1p-13 */
double z = x;
if (gt)
z = 1 / z; /* gt is non-zero for |x| >= 1 */
double z2 = z * z;
double z4 = z2 * z2;
double z8 = z4 * z4;
/* polynomials generated using rminimax
(https://gitlab.inria.fr/sfilip/rminimax) with the following command:
./ratapprox --function="atan(x)" --dom=[0.000122070,1]
--num=[x,x^3,x^5,x^7,x^9,x^11,x^13] --den=[1,x^2,x^4,x^6,x^8,x^10,x^12]
--output=atanf.sollya --log (see output atanf.sollya) The coefficient
cd[0] was slightly reduced from the original value 0x1.51eccde075d67p-2 to
avoid an exceptional case for |x| = 0x1.1ad646p-4 and rounding to nearest.
*/
static const double cn[] =
{
0x1.51eccde075d67p-2, 0x1.a76bb5637f2f2p-1, 0x1.81e0eed20de88p-1,
0x1.376c8ca67d11dp-2, 0x1.aec7b69202ac6p-5, 0x1.9561899acc73ep-9,
0x1.bf9fa5b67e6p-16
};
static const double cd[] =
{
0x1.51eccde075d66p-2, 0x1.dfbdd7b392d28p-1, 0x1p+0,
0x1.fd22bf0e89b54p-2, 0x1.d91ff8b576282p-4, 0x1.653ea99fc9bbp-7,
0x1.1e7fcc202340ap-12
};
double cn0 = cn[0] + z2 * cn[1];
double cn2 = cn[2] + z2 * cn[3];
double cn4 = cn[4] + z2 * cn[5];
double cn6 = cn[6];
cn0 += z4 * cn2;
cn4 += z4 * cn6;
cn0 += z8 * cn4;
cn0 *= z;
double cd0 = cd[0] + z2 * cd[1];
double cd2 = cd[2] + z2 * cd[3];
double cd4 = cd[4] + z2 * cd[5];
double cd6 = cd[6];
cd0 += z4 * cd2;
cd4 += z4 * cd6;
cd0 += z8 * cd4;
double r = cn0 / cd0;
if (!gt)
return r; /* for |x| < 1, (float) r is correctly rounded */
#define PI_OVER2_H 0x1.9p0
#define PI_OVER2_L 0x1.0fdaa22168c23p-7
/* now r approximates atan(1/x), we use atan(x) + atan(1/x) = sign(x)*pi/2,
where PI_OVER2_H + PI_OVER2_L approximates pi/2.
With sign(z)*L + (-r + sign(z)*H), it fails for x=0x1.98c252p+12 and
rounding upward.
With sign(z)*PI - r, where PI is a double approximation of pi to nearest,
it fails for x=0x1.ddf9f6p+0 and rounding upward. */
r = copysign (PI_OVER2_L, z) - r + copysign (PI_OVER2_H, z);
return r;
}
libm_alias_float (__atan, atan)
|
