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
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
|
/* Implementation of the degree trignometric functions COSD, SIND, TAND.
Copyright (C) 2020 Free Software Foundation, Inc.
Contributed by Steven G. Kargl <kargl@gcc.gnu.org>
This file is part of the GNU Fortran runtime library (libgfortran).
Libgfortran is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.
Libgfortran is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
Under Section 7 of GPL version 3, you are granted additional
permissions described in the GCC Runtime Library Exception, version
3.1, as published by the Free Software Foundation.
You should have received a copy of the GNU General Public License and
a copy of the GCC Runtime Library Exception along with this program;
see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
<http://www.gnu.org/licenses/>. */
#include "libgfortran.h"
#include <math.h>
/*
For real x, let {x}_P or x_P be the closest representible number in the
floating point representation which uses P binary bits of fractional
precision (with IEEE rounding semantics).
Similarly, let f_P(x) be shorthand for {f(x)}_P.
Let ulp_P(x) be the unit of least precision for x: in other words the
maximal value of |a_P - b_P| where a_P <= x <= b_P and a_P != b_P.
Let x ~= y <-> | x - y | < ulp_P(x - y).
Let deg(x) be the value of x radians in degrees.
Values for each precision P were selected as follows.
COSD_SMALL = 2**{-N} such that for all x <= COSD_SMALL:
* cos(deg(x)) ~= 1, or equivalently:
| 1 - cos(deg(x)) | < ulp_P(1).
Unfortunately for SIND (and therefore TAND) a similar relation is only
possible for REAL(4) and REAL(8). With REAL(10) and REAL(16), enough
precision is available such that sin_P(x) != x_P for some x less than any
value. (There are values where this equality holds, but the distance has
inflection points.)
For REAL(4) and REAL(8), we can select SIND_SMALL such that:
* sin(deg(x)) ~= deg(x), or equivalently:
| deg(x) - sin(deg(x)) | < ulp_P(deg(x)).
*/
/* Build _gfortran_sind_r4, _gfortran_cosd_r4, and _gfortran_tand_r4 */
#define FTYPE GFC_REAL_4
#define SIND sind_r4
#define COSD cosd_r4
#define TAND tand_r4
#define SUFFIX(x) x ## f
#define TINY 0x1.p-100f /* ~= 7.889e-31 */
#define COSD_SMALL 0x1.p-7f /* = 7.8125e-3 */
#define SIND_SMALL 0x1.p-5f /* = 3.125e-2 */
#define COSD30 8.66025388e-01f
#define PIO180H 1.74560547e-02f /* high 12 bits. */
#define PIO180L -2.76216747e-06f /* Next 24 bits. */
#include "trigd_lib.inc"
#undef FTYPE
#undef TINY
#undef COSD_SMALL
#undef SIND_SMALL
#undef COSD30
#undef PIO180H
#undef PIO180L
#undef SIND
#undef COSD
#undef TAND
#undef SUFFIX
/* Build _gfortran_sind_r8, _gfortran_cosd_r8, and _gfortran_tand_r8. */
#define FTYPE GFC_REAL_8
#define SIND sind_r8
#define COSD cosd_r8
#define TAND tand_r8
#define SUFFIX(x) x
#define TINY 0x1.p-1000 /* ~= 9.33e-302 (min exp -1074) */
#define COSD_SMALL 0x1.p-21 /* ~= 4.768e-7 */
#define SIND_SMALL 0x1.p-19 /* ~= 9.537e-7 */
#define COSD30 8.6602540378443860e-01
#define PIO180H 1.7453283071517944e-02 /* high 21 bits. */
#define PIO180L 9.4484253514332993e-09 /* Next 53 bits. */
#include "trigd_lib.inc"
#undef FTYPE
#undef TINY
#undef COSD_SMALL
#undef SIND_SMALL
#undef COSD30
#undef PIO180H
#undef PIO180L
#undef SIND
#undef COSD
#undef TAND
#undef SUFFIX
/* Build _gfortran_sind_r10, _gfortran_cosd_r10, and _gfortran_tand_r10. */
#ifdef HAVE_GFC_REAL_10
#define FTYPE GFC_REAL_10
#define SIND sind_r10
#define COSD cosd_r10
#define TAND tand_r10
#define SUFFIX(x) x ## l /* L */
#define TINY 0x1.p-16400L /* ~= 1.28e-4937 (min exp -16494) */
#define COSD_SMALL 0x1.p-26L /* ~= 1.490e-8 */
#undef SIND_SMALL /* not precise */
#define COSD30 8.66025403784438646787e-01L
#define PIO180H 1.74532925229868851602e-02L /* high 32 bits */
#define PIO180L -3.04358939097084072823e-12L /* Next 64 bits */
#include "trigd_lib.inc"
#undef FTYPE
#undef TINY
#undef COSD_SMALL
#undef SIND_SMALL
#undef COSD30
#undef PIO180H
#undef PIO180L
#undef SIND
#undef COSD
#undef TAND
#undef SUFFIX
#endif /* HAVE_GFC_REAL_10 */
/* Build _gfortran_sind_r16, _gfortran_cosd_r16, and _gfortran_tand_r16. */
#ifdef HAVE_GFC_REAL_16
#define FTYPE GFC_REAL_16
#define SIND sind_r16
#define COSD cosd_r16
#define TAND tand_r16
#ifdef GFC_REAL_16_IS_FLOAT128 /* libquadmath. */
#define SUFFIX(x) x ## q
#else
#define SUFFIX(x) x ## l
#endif /* GFC_REAL_16_IS_FLOAT128 */
#define TINY SUFFIX(0x1.p-16400) /* ~= 1.28e-4937 */
#define COSD_SMALL SUFFIX(0x1.p-51) /* ~= 4.441e-16 */
#undef SIND_SMALL /* not precise */
#define COSD30 SUFFIX(8.66025403784438646763723170752936183e-01)
#define PIO180H SUFFIX(1.74532925199433197605003442731685936e-02)
#define PIO180L SUFFIX(-2.39912634365882824665106671063098954e-17)
#include "trigd_lib.inc"
#undef FTYPE
#undef COSD_SMALL
#undef SIND_SMALL
#undef COSD30
#undef PIO180H
#undef PIO180L
#undef PIO180
#undef D2R
#undef CPYSGN
#undef FABS
#undef FMOD
#undef SIN
#undef COS
#undef TAN
#undef SIND
#undef COSD
#undef TAND
#undef SUFFIX
#endif /* HAVE_GFC_REAL_16 */
|
