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
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
|
/*
* IBM Accurate Mathematical Library
* Copyright (c) International Business Machines Corp., 2001
*
* 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 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 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, write to the Free Software
* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
*/
/****************************************************************/
/* MODULE_NAME: sincos32.c */
/* */
/* FUNCTIONS: ss32 */
/* cc32 */
/* c32 */
/* sin32 */
/* cos32 */
/* mpsin */
/* mpcos */
/* mpranred */
/* mpsin1 */
/* mpcos1 */
/* */
/* FILES NEEDED: endian.h mpa.h sincos32.h */
/* mpa.c */
/* */
/* Multi Precision sin() and cos() function with p=32 for sin()*/
/* cos() arcsin() and arccos() routines */
/* In addition mpranred() routine performs range reduction of */
/* a double number x into multi precision number y, */
/* such that y=x-n*pi/2, abs(y)<pi/4, n=0,+-1,+-2,.... */
/****************************************************************/
#include "endian.h"
#include "mpa.h"
#include "sincos32.h"
#include "math_private.h"
/****************************************************************/
/* Compute Multi-Precision sin() function for given p. Receive */
/* Multi Precision number x and result stored at y */
/****************************************************************/
static void ss32(mp_no *x, mp_no *y, int p) {
int i;
double a;
#if 0
double b;
static const mp_no mpone = {1,{1.0,1.0}};
#endif
mp_no mpt1,x2,gor,sum ,mpk={1,{1.0}};
#if 0
mp_no mpt2;
#endif
for (i=1;i<=p;i++) mpk.d[i]=0;
__mul(x,x,&x2,p);
__cpy(&oofac27,&gor,p);
__cpy(&gor,&sum,p);
for (a=27.0;a>1.0;a-=2.0) {
mpk.d[1]=a*(a-1.0);
__mul(&gor,&mpk,&mpt1,p);
__cpy(&mpt1,&gor,p);
__mul(&x2,&sum,&mpt1,p);
__sub(&gor,&mpt1,&sum,p);
}
__mul(x,&sum,y,p);
}
/**********************************************************************/
/* Compute Multi-Precision cos() function for given p. Receive Multi */
/* Precision number x and result stored at y */
/**********************************************************************/
static void cc32(mp_no *x, mp_no *y, int p) {
int i;
double a;
#if 0
double b;
static const mp_no mpone = {1,{1.0,1.0}};
#endif
mp_no mpt1,x2,gor,sum ,mpk={1,{1.0}};
#if 0
mp_no mpt2;
#endif
for (i=1;i<=p;i++) mpk.d[i]=0;
__mul(x,x,&x2,p);
mpk.d[1]=27.0;
__mul(&oofac27,&mpk,&gor,p);
__cpy(&gor,&sum,p);
for (a=26.0;a>2.0;a-=2.0) {
mpk.d[1]=a*(a-1.0);
__mul(&gor,&mpk,&mpt1,p);
__cpy(&mpt1,&gor,p);
__mul(&x2,&sum,&mpt1,p);
__sub(&gor,&mpt1,&sum,p);
}
__mul(&x2,&sum,y,p);
}
/***************************************************************************/
/* c32() computes both sin(x), cos(x) as Multi precision numbers */
/***************************************************************************/
void __c32(mp_no *x, mp_no *y, mp_no *z, int p) {
static const mp_no mpt={1,{1.0,2.0}}, one={1,{1.0,1.0}};
mp_no u,t,t1,t2,c,s;
int i;
__cpy(x,&u,p);
u.e=u.e-1;
cc32(&u,&c,p);
ss32(&u,&s,p);
for (i=0;i<24;i++) {
__mul(&c,&s,&t,p);
__sub(&s,&t,&t1,p);
__add(&t1,&t1,&s,p);
__sub(&mpt,&c,&t1,p);
__mul(&t1,&c,&t2,p);
__add(&t2,&t2,&c,p);
}
__sub(&one,&c,y,p);
__cpy(&s,z,p);
}
/************************************************************************/
/*Routine receive double x and two double results of sin(x) and return */
/*result which is more accurate */
/*Computing sin(x) with multi precision routine c32 */
/************************************************************************/
double __sin32(double x, double res, double res1) {
int p;
mp_no a,b,c;
p=32;
__dbl_mp(res,&a,p);
__dbl_mp(0.5*(res1-res),&b,p);
__add(&a,&b,&c,p);
if (x>0.8)
{ __sub(&hp,&c,&a,p);
__c32(&a,&b,&c,p);
}
else __c32(&c,&a,&b,p); /* b=sin(0.5*(res+res1)) */
__dbl_mp(x,&c,p); /* c = x */
__sub(&b,&c,&a,p);
/* if a>0 return min(res,res1), otherwise return max(res,res1) */
if (a.d[0]>0) return (res<res1)?res:res1;
else return (res>res1)?res:res1;
}
/************************************************************************/
/*Routine receive double x and two double results of cos(x) and return */
/*result which is more accurate */
/*Computing cos(x) with multi precision routine c32 */
/************************************************************************/
double __cos32(double x, double res, double res1) {
int p;
mp_no a,b,c;
p=32;
__dbl_mp(res,&a,p);
__dbl_mp(0.5*(res1-res),&b,p);
__add(&a,&b,&c,p);
if (x>2.4)
{ __sub(&pi,&c,&a,p);
__c32(&a,&b,&c,p);
b.d[0]=-b.d[0];
}
else if (x>0.8)
{ __sub(&hp,&c,&a,p);
__c32(&a,&c,&b,p);
}
else __c32(&c,&b,&a,p); /* b=cos(0.5*(res+res1)) */
__dbl_mp(x,&c,p); /* c = x */
__sub(&b,&c,&a,p);
/* if a>0 return max(res,res1), otherwise return min(res,res1) */
if (a.d[0]>0) return (res>res1)?res:res1;
else return (res<res1)?res:res1;
}
/*******************************************************************/
/*Compute sin(x+dx) as Multi Precision number and return result as */
/* double */
/*******************************************************************/
double __mpsin(double x, double dx) {
int p;
double y;
mp_no a,b,c;
p=32;
__dbl_mp(x,&a,p);
__dbl_mp(dx,&b,p);
__add(&a,&b,&c,p);
if (x>0.8) { __sub(&hp,&c,&a,p); __c32(&a,&b,&c,p); }
else __c32(&c,&a,&b,p); /* b = sin(x+dx) */
__mp_dbl(&b,&y,p);
return y;
}
/*******************************************************************/
/* Compute cos()of double-length number (x+dx) as Multi Precision */
/* number and return result as double */
/*******************************************************************/
double __mpcos(double x, double dx) {
int p;
double y;
mp_no a,b,c;
p=32;
__dbl_mp(x,&a,p);
__dbl_mp(dx,&b,p);
__add(&a,&b,&c,p);
if (x>0.8)
{ __sub(&hp,&c,&b,p);
__c32(&b,&a,&c,p);
}
else __c32(&c,&a,&b,p); /* a = cos(x+dx) */
__mp_dbl(&a,&y,p);
return y;
}
/******************************************************************/
/* mpranred() performs range reduction of a double number x into */
/* multi precision number y, such that y=x-n*pi/2, abs(y)<pi/4, */
/* n=0,+-1,+-2,.... */
/* Return int which indicates in which quarter of circle x is */
/******************************************************************/
int __mpranred(double x, mp_no *y, int p)
{
number v;
double t,xn;
int i,k,n;
static const mp_no one = {1,{1.0,1.0}};
mp_no a,b,c;
if (ABS(x) < 2.8e14) {
t = (x*hpinv.d + toint.d);
xn = t - toint.d;
v.d = t;
n =v.i[LOW_HALF]&3;
__dbl_mp(xn,&a,p);
__mul(&a,&hp,&b,p);
__dbl_mp(x,&c,p);
__sub(&c,&b,y,p);
return n;
}
else { /* if x is very big more precision required */
__dbl_mp(x,&a,p);
a.d[0]=1.0;
k = a.e-5;
if (k < 0) k=0;
b.e = -k;
b.d[0] = 1.0;
for (i=0;i<p;i++) b.d[i+1] = toverp[i+k];
__mul(&a,&b,&c,p);
t = c.d[c.e];
for (i=1;i<=p-c.e;i++) c.d[i]=c.d[i+c.e];
for (i=p+1-c.e;i<=p;i++) c.d[i]=0;
c.e=0;
if (c.d[1] >= 8388608.0)
{ t +=1.0;
__sub(&c,&one,&b,p);
__mul(&b,&hp,y,p);
}
else __mul(&c,&hp,y,p);
n = (int) t;
if (x < 0) { y->d[0] = - y->d[0]; n = -n; }
return (n&3);
}
}
/*******************************************************************/
/* Multi-Precision sin() function subroutine, for p=32. It is */
/* based on the routines mpranred() and c32(). */
/*******************************************************************/
double __mpsin1(double x)
{
int p;
int n;
mp_no u,s,c;
double y;
p=32;
n=__mpranred(x,&u,p); /* n is 0, 1, 2 or 3 */
__c32(&u,&c,&s,p);
switch (n) { /* in which quarter of unit circle y is*/
case 0:
__mp_dbl(&s,&y,p);
return y;
break;
case 2:
__mp_dbl(&s,&y,p);
return -y;
break;
case 1:
__mp_dbl(&c,&y,p);
return y;
break;
case 3:
__mp_dbl(&c,&y,p);
return -y;
break;
}
return 0; /* unreachable, to make the compiler happy */
}
/*****************************************************************/
/* Multi-Precision cos() function subroutine, for p=32. It is */
/* based on the routines mpranred() and c32(). */
/*****************************************************************/
double __mpcos1(double x)
{
int p;
int n;
mp_no u,s,c;
double y;
p=32;
n=__mpranred(x,&u,p); /* n is 0, 1, 2 or 3 */
__c32(&u,&c,&s,p);
switch (n) { /* in what quarter of unit circle y is*/
case 0:
__mp_dbl(&c,&y,p);
return y;
break;
case 2:
__mp_dbl(&c,&y,p);
return -y;
break;
case 1:
__mp_dbl(&s,&y,p);
return -y;
break;
case 3:
__mp_dbl(&s,&y,p);
return y;
break;
}
return 0; /* unreachable, to make the compiler happy */
}
/******************************************************************/
|
