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
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
|
/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2013 Free Software Foundation, Inc.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
* the Free Software Foundation; either version 2.1 of the License, or
* (at your option) any later version.
*
* This program 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 Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this program; if not, see <http://www.gnu.org/licenses/>.
*/
/************************************************************************/
/* MODULE_NAME: atnat.c */
/* */
/* FUNCTIONS: uatan */
/* atanMp */
/* signArctan */
/* */
/* */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat.h */
/* mpatan.c mpatan2.c mpsqrt.c */
/* uatan.tbl */
/* */
/* An ultimate atan() routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of atan(x). */
/* */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
/************************************************************************/
#include <dla.h>
#include "mpa.h"
#include "MathLib.h"
#include "uatan.tbl"
#include "atnat.h"
#include <math.h>
void __mpatan (mp_no *, mp_no *, int); /* see definition in mpatan.c */
static double atanMp (double, const int[]);
/* Fix the sign of y and return */
static double
__signArctan (double x, double y)
{
return __copysign (y, x);
}
/* An ultimate atan() routine. Given an IEEE double machine number x, */
/* routine computes the correctly rounded (to nearest) value of atan(x). */
double
atan (double x)
{
double cor, s1, ss1, s2, ss2, t1, t2, t3, t7, t8, t9, t10, u, u2, u3,
v, vv, w, ww, y, yy, z, zz;
#ifndef DLA_FMS
double t4, t5, t6;
#endif
int i, ux, dx;
static const int pr[M] = { 6, 8, 10, 32 };
number num;
num.d = x;
ux = num.i[HIGH_HALF];
dx = num.i[LOW_HALF];
/* x=NaN */
if (((ux & 0x7ff00000) == 0x7ff00000)
&& (((ux & 0x000fffff) | dx) != 0x00000000))
return x + x;
/* Regular values of x, including denormals +-0 and +-INF */
u = (x < 0) ? -x : x;
if (u < C)
{
if (u < B)
{
if (u < A)
return x;
else
{ /* A <= u < B */
v = x * x;
yy = d11.d + v * d13.d;
yy = d9.d + v * yy;
yy = d7.d + v * yy;
yy = d5.d + v * yy;
yy = d3.d + v * yy;
yy *= x * v;
if ((y = x + (yy - U1 * x)) == x + (yy + U1 * x))
return y;
EMULV (x, x, v, vv, t1, t2, t3, t4, t5); /* v+vv=x^2 */
s1 = f17.d + v * f19.d;
s1 = f15.d + v * s1;
s1 = f13.d + v * s1;
s1 = f11.d + v * s1;
s1 *= v;
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (x, 0, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7,
t8);
ADD2 (x, 0, s2, ss2, s1, ss1, t1, t2);
if ((y = s1 + (ss1 - U5 * s1)) == s1 + (ss1 + U5 * s1))
return y;
return atanMp (x, pr);
}
}
else
{ /* B <= u < C */
i = (TWO52 + TWO8 * u) - TWO52;
i -= 16;
z = u - cij[i][0].d;
yy = cij[i][5].d + z * cij[i][6].d;
yy = cij[i][4].d + z * yy;
yy = cij[i][3].d + z * yy;
yy = cij[i][2].d + z * yy;
yy *= z;
t1 = cij[i][1].d;
if (i < 112)
{
if (i < 48)
u2 = U21; /* u < 1/4 */
else
u2 = U22;
} /* 1/4 <= u < 1/2 */
else
{
if (i < 176)
u2 = U23; /* 1/2 <= u < 3/4 */
else
u2 = U24;
} /* 3/4 <= u <= 1 */
if ((y = t1 + (yy - u2 * t1)) == t1 + (yy + u2 * t1))
return __signArctan (x, y);
z = u - hij[i][0].d;
s1 = hij[i][14].d + z * hij[i][15].d;
s1 = hij[i][13].d + z * s1;
s1 = hij[i][12].d + z * s1;
s1 = hij[i][11].d + z * s1;
s1 *= z;
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
if ((y = s2 + (ss2 - U6 * s2)) == s2 + (ss2 + U6 * s2))
return __signArctan (x, y);
return atanMp (x, pr);
}
}
else
{
if (u < D)
{ /* C <= u < D */
w = 1 / u;
EMULV (w, u, t1, t2, t3, t4, t5, t6, t7);
ww = w * ((1 - t1) - t2);
i = (TWO52 + TWO8 * w) - TWO52;
i -= 16;
z = (w - cij[i][0].d) + ww;
yy = cij[i][5].d + z * cij[i][6].d;
yy = cij[i][4].d + z * yy;
yy = cij[i][3].d + z * yy;
yy = cij[i][2].d + z * yy;
yy = HPI1 - z * yy;
t1 = HPI - cij[i][1].d;
if (i < 112)
u3 = U31; /* w < 1/2 */
else
u3 = U32; /* w >= 1/2 */
if ((y = t1 + (yy - u3)) == t1 + (yy + u3))
return __signArctan (x, y);
DIV2 (1 , 0, u, 0, w, ww, t1, t2, t3, t4, t5, t6, t7, t8, t9,
t10);
t1 = w - hij[i][0].d;
EADD (t1, ww, z, zz);
s1 = hij[i][14].d + z * hij[i][15].d;
s1 = hij[i][13].d + z * s1;
s1 = hij[i][12].d + z * s1;
s1 = hij[i][11].d + z * s1;
s1 *= z;
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
SUB2 (HPI, HPI1, s2, ss2, s1, ss1, t1, t2);
if ((y = s1 + (ss1 - U7)) == s1 + (ss1 + U7))
return __signArctan (x, y);
return atanMp (x, pr);
}
else
{
if (u < E)
{ /* D <= u < E */
w = 1 / u;
v = w * w;
EMULV (w, u, t1, t2, t3, t4, t5, t6, t7);
yy = d11.d + v * d13.d;
yy = d9.d + v * yy;
yy = d7.d + v * yy;
yy = d5.d + v * yy;
yy = d3.d + v * yy;
yy *= w * v;
ww = w * ((1 - t1) - t2);
ESUB (HPI, w, t3, cor);
yy = ((HPI1 + cor) - ww) - yy;
if ((y = t3 + (yy - U4)) == t3 + (yy + U4))
return __signArctan (x, y);
DIV2 (1 , 0, u, 0, w, ww, t1, t2, t3, t4, t5, t6, t7, t8,
t9, t10);
MUL2 (w, ww, w, ww, v, vv, t1, t2, t3, t4, t5, t6, t7, t8);
s1 = f17.d + v * f19.d;
s1 = f15.d + v * s1;
s1 = f13.d + v * s1;
s1 = f11.d + v * s1;
s1 *= v;
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (w, ww, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (w, ww, s2, ss2, s1, ss1, t1, t2);
SUB2 (HPI, HPI1, s1, ss1, s2, ss2, t1, t2);
if ((y = s2 + (ss2 - U8)) == s2 + (ss2 + U8))
return __signArctan (x, y);
return atanMp (x, pr);
}
else
{
/* u >= E */
if (x > 0)
return HPI;
else
return MHPI;
}
}
}
}
/* Final stages. Compute atan(x) by multiple precision arithmetic */
static double
atanMp (double x, const int pr[])
{
mp_no mpx, mpy, mpy2, mperr, mpt1, mpy1;
double y1, y2;
int i, p;
for (i = 0; i < M; i++)
{
p = pr[i];
__dbl_mp (x, &mpx, p);
__mpatan (&mpx, &mpy, p);
__dbl_mp (u9[i].d, &mpt1, p);
__mul (&mpy, &mpt1, &mperr, p);
__add (&mpy, &mperr, &mpy1, p);
__sub (&mpy, &mperr, &mpy2, p);
__mp_dbl (&mpy1, &y1, p);
__mp_dbl (&mpy2, &y2, p);
if (y1 == y2)
return y1;
}
return y1; /*if impossible to do exact computing */
}
#ifdef NO_LONG_DOUBLE
weak_alias (atan, atanl)
#endif
|
