about summary refs log tree commit diff
path: root/sysdeps/ieee754/dbl-64/s_sin.c
blob: 13c08227201e304807c6b1515f27375d2a2c9af2 (plain) (blame)
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
/*
 * IBM Accurate Mathematical Library
 * written by International Business Machines Corp.
 * Copyright (C) 2001-2021 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 <https://www.gnu.org/licenses/>.
 */
/****************************************************************************/
/*                                                                          */
/* MODULE_NAME:usncs.c                                                      */
/*                                                                          */
/* FUNCTIONS: usin                                                          */
/*            ucos                                                          */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h  usncs.h                     */
/*		 branred.c sincos.tbl					    */
/*                                                                          */
/* An ultimate sin and cos routine. Given an IEEE double machine number x   */
/* it computes sin(x) or cos(x) with ~0.55 ULP.				    */
/* Assumption: Machine arithmetic operations are performed in               */
/* round to nearest mode of IEEE 754 standard.                              */
/*                                                                          */
/****************************************************************************/


#include <errno.h>
#include <float.h>
#include "endian.h"
#include "mydefs.h"
#include "usncs.h"
#include "MathLib.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <libm-alias-double.h>
#include <fenv.h>

/* Helper macros to compute sin of the input values.  */
#define POLYNOMIAL2(xx) ((((s5 * (xx) + s4) * (xx) + s3) * (xx) + s2) * (xx))

#define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1)

/* The computed polynomial is a variation of the Taylor series expansion for
   sin(a):

   a - a^3/3! + a^5/5! - a^7/7! + a^9/9! + (1 - a^2) * da / 2

   The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
   on.  The result is returned to LHS.  */
#define TAYLOR_SIN(xx, a, da) \
({									      \
  double t = ((POLYNOMIAL (xx)  * (a) - 0.5 * (da))  * (xx) + (da));	      \
  double res = (a) + t;							      \
  res;									      \
})

#define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
({									      \
  int4 k = u.i[LOW_HALF] << 2;						      \
  sn = __sincostab.x[k];						      \
  ssn = __sincostab.x[k + 1];						      \
  cs = __sincostab.x[k + 2];						      \
  ccs = __sincostab.x[k + 3];						      \
})

#ifndef SECTION
# define SECTION
#endif

extern const union
{
  int4 i[880];
  double x[440];
} __sincostab attribute_hidden;

static const double
  sn3 = -1.66666666666664880952546298448555E-01,
  sn5 = 8.33333214285722277379541354343671E-03,
  cs2 = 4.99999999999999999999950396842453E-01,
  cs4 = -4.16666666666664434524222570944589E-02,
  cs6 = 1.38888874007937613028114285595617E-03;

int __branred (double x, double *a, double *aa);

/* Given a number partitioned into X and DX, this function computes the cosine
   of the number by combining the sin and cos of X (as computed by a variation
   of the Taylor series) with the values looked up from the sin/cos table to
   get the result.  */
static __always_inline double
do_cos (double x, double dx)
{
  mynumber u;

  if (x < 0)
    dx = -dx;

  u.x = big + fabs (x);
  x = fabs (x) - (u.x - big) + dx;

  double xx, s, sn, ssn, c, cs, ccs, cor;
  xx = x * x;
  s = x + x * xx * (sn3 + xx * sn5);
  c = xx * (cs2 + xx * (cs4 + xx * cs6));
  SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
  cor = (ccs - s * ssn - cs * c) - sn * s;
  return cs + cor;
}

/* Given a number partitioned into X and DX, this function computes the sine of
   the number by combining the sin and cos of X (as computed by a variation of
   the Taylor series) with the values looked up from the sin/cos table to get
   the result.  */
static __always_inline double
do_sin (double x, double dx)
{
  double xold = x;
  /* Max ULP is 0.501 if |x| < 0.126, otherwise ULP is 0.518.  */
  if (fabs (x) < 0.126)
    return TAYLOR_SIN (x * x, x, dx);

  mynumber u;

  if (x <= 0)
    dx = -dx;
  u.x = big + fabs (x);
  x = fabs (x) - (u.x - big);

  double xx, s, sn, ssn, c, cs, ccs, cor;
  xx = x * x;
  s = x + (dx + x * xx * (sn3 + xx * sn5));
  c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
  SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
  cor = (ssn + s * ccs - sn * c) + cs * s;
  return copysign (sn + cor, xold);
}

/* Reduce range of x to within PI/2 with abs (x) < 105414350.  The high part
   is written to *a, the low part to *da.  Range reduction is accurate to 136
   bits so that when x is large and *a very close to zero, all 53 bits of *a
   are correct.  */
static __always_inline int4
reduce_sincos (double x, double *a, double *da)
{
  mynumber v;

  double t = (x * hpinv + toint);
  double xn = t - toint;
  v.x = t;
  double y = (x - xn * mp1) - xn * mp2;
  int4 n = v.i[LOW_HALF] & 3;

  double b, db, t1, t2;
  t1 = xn * pp3;
  t2 = y - t1;
  db = (y - t2) - t1;

  t1 = xn * pp4;
  b = t2 - t1;
  db += (t2 - b) - t1;

  *a = b;
  *da = db;
  return n;
}

/* Compute sin or cos (A + DA) for the given quadrant N.  */
static __always_inline double
do_sincos (double a, double da, int4 n)
{
  double retval;

  if (n & 1)
    /* Max ULP is 0.513.  */
    retval = do_cos (a, da);
  else
    /* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518.  */
    retval = do_sin (a, da);

  return (n & 2) ? -retval : retval;
}


/*******************************************************************/
/* An ultimate sin routine. Given an IEEE double machine number x  */
/* it computes the rounded value of sin(x).			   */
/*******************************************************************/
#ifndef IN_SINCOS
double
SECTION
__sin (double x)
{
  double t, a, da;
  mynumber u;
  int4 k, m, n;
  double retval = 0;

  SET_RESTORE_ROUND_53BIT (FE_TONEAREST);

  u.x = x;
  m = u.i[HIGH_HALF];
  k = 0x7fffffff & m;		/* no sign           */
  if (k < 0x3e500000)		/* if x->0 =>sin(x)=x */
    {
      math_check_force_underflow (x);
      retval = x;
    }
/*--------------------------- 2^-26<|x|< 0.855469---------------------- */
  else if (k < 0x3feb6000)
    {
      /* Max ULP is 0.548.  */
      retval = do_sin (x, 0);
    }				/*   else  if (k < 0x3feb6000)    */

/*----------------------- 0.855469  <|x|<2.426265  ----------------------*/
  else if (k < 0x400368fd)
    {
      t = hp0 - fabs (x);
      /* Max ULP is 0.51.  */
      retval = copysign (do_cos (t, hp1), x);
    }				/*   else  if (k < 0x400368fd)    */

/*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
  else if (k < 0x419921FB)
    {
      n = reduce_sincos (x, &a, &da);
      retval = do_sincos (a, da, n);
    }				/*   else  if (k <  0x419921FB )    */

/* --------------------105414350 <|x| <2^1024------------------------------*/
  else if (k < 0x7ff00000)
    {
      n = __branred (x, &a, &da);
      retval = do_sincos (a, da, n);
    }
/*--------------------- |x| > 2^1024 ----------------------------------*/
  else
    {
      if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
	__set_errno (EDOM);
      retval = x / x;
    }

  return retval;
}


/*******************************************************************/
/* An ultimate cos routine. Given an IEEE double machine number x  */
/* it computes the rounded value of cos(x).			   */
/*******************************************************************/

double
SECTION
__cos (double x)
{
  double y, a, da;
  mynumber u;
  int4 k, m, n;

  double retval = 0;

  SET_RESTORE_ROUND_53BIT (FE_TONEAREST);

  u.x = x;
  m = u.i[HIGH_HALF];
  k = 0x7fffffff & m;

  /* |x|<2^-27 => cos(x)=1 */
  if (k < 0x3e400000)
    retval = 1.0;

  else if (k < 0x3feb6000)
    {				/* 2^-27 < |x| < 0.855469 */
      /* Max ULP is 0.51.  */
      retval = do_cos (x, 0);
    }				/*   else  if (k < 0x3feb6000)    */

  else if (k < 0x400368fd)
    { /* 0.855469  <|x|<2.426265  */ ;
      y = hp0 - fabs (x);
      a = y + hp1;
      da = (y - a) + hp1;
      /* Max ULP is 0.501 if xx < 0.01588 or 0.518 otherwise.
	 Range reduction uses 106 bits here which is sufficient.  */
      retval = do_sin (a, da);
    }				/*   else  if (k < 0x400368fd)    */

  else if (k < 0x419921FB)
    {				/* 2.426265<|x|< 105414350 */
      n = reduce_sincos (x, &a, &da);
      retval = do_sincos (a, da, n + 1);
    }				/*   else  if (k <  0x419921FB )    */

  /* 105414350 <|x| <2^1024 */
  else if (k < 0x7ff00000)
    {
      n = __branred (x, &a, &da);
      retval = do_sincos (a, da, n + 1);
    }

  else
    {
      if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
	__set_errno (EDOM);
      retval = x / x;		/* |x| > 2^1024 */
    }

  return retval;
}

#ifndef __cos
libm_alias_double (__cos, cos)
#endif
#ifndef __sin
libm_alias_double (__sin, sin)
#endif

#endif