summaryrefslogtreecommitdiff
path: root/sysdeps/ieee754/dbl-64/e_pow.c
blob: 2dd66d87b7420fa269445e3a476da503dc53499e (plain)
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
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
/*
 * IBM Accurate Mathematical Library
 * written by International Business Machines Corp.
 * Copyright (C) 2001-2014 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: upow.c                                                    */
/*                                                                         */
/*  FUNCTIONS: upow                                                        */
/*             power1                                                      */
/*             my_log2                                                     */
/*             log1                                                        */
/*             checkint                                                    */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h                             */
/*               halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c       */
/*                          uexp.c  upow.c				   */
/*               root.tbl uexp.tbl upow.tbl                                */
/* An ultimate power routine. Given two IEEE double machine numbers y,x    */
/* it computes the correctly rounded (to nearest) value of x^y.            */
/* Assumption: Machine arithmetic operations are performed in              */
/* round to nearest mode of IEEE 754 standard.                             */
/*                                                                         */
/***************************************************************************/
#include <math.h>
#include "endian.h"
#include "upow.h"
#include <dla.h>
#include "mydefs.h"
#include "MathLib.h"
#include "upow.tbl"
#include <math_private.h>
#include <fenv.h>

#ifndef SECTION
# define SECTION
#endif

static const double huge = 1.0e300, tiny = 1.0e-300;

double __exp1 (double x, double xx, double error);
static double log1 (double x, double *delta, double *error);
static double my_log2 (double x, double *delta, double *error);
double __slowpow (double x, double y, double z);
static double power1 (double x, double y);
static int checkint (double x);

/* An ultimate power routine. Given two IEEE double machine numbers y, x it
   computes the correctly rounded (to nearest) value of X^y.  */
double
SECTION
__ieee754_pow (double x, double y)
{
  double z, a, aa, error, t, a1, a2, y1, y2;
  mynumber u, v;
  int k;
  int4 qx, qy;
  v.x = y;
  u.x = x;
  if (v.i[LOW_HALF] == 0)
    {				/* of y */
      qx = u.i[HIGH_HALF] & 0x7fffffff;
      /* Is x a NaN?  */
      if (((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000))
	return x;
      if (y == 1.0)
	return x;
      if (y == 2.0)
	return x * x;
      if (y == -1.0)
	return 1.0 / x;
      if (y == 0)
	return 1.0;
    }
  /* else */
  if (((u.i[HIGH_HALF] > 0 && u.i[HIGH_HALF] < 0x7ff00000) ||	/* x>0 and not x->0 */
       (u.i[HIGH_HALF] == 0 && u.i[LOW_HALF] != 0)) &&
      /*   2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */
      (v.i[HIGH_HALF] & 0x7fffffff) < 0x4ff00000)
    {				/* if y<-1 or y>1   */
      double retval;

      {
	SET_RESTORE_ROUND (FE_TONEAREST);

	/* Avoid internal underflow for tiny y.  The exact value of y does
	   not matter if |y| <= 2**-64.  */
	if (ABS (y) < 0x1p-64)
	  y = y < 0 ? -0x1p-64 : 0x1p-64;
	z = log1 (x, &aa, &error);	/* x^y  =e^(y log (X)) */
	t = y * CN;
	y1 = t - (t - y);
	y2 = y - y1;
	t = z * CN;
	a1 = t - (t - z);
	a2 = (z - a1) + aa;
	a = y1 * a1;
	aa = y2 * a1 + y * a2;
	a1 = a + aa;
	a2 = (a - a1) + aa;
	error = error * ABS (y);
	t = __exp1 (a1, a2, 1.9e16 * error);	/* return -10 or 0 if wasn't computed exactly */
	retval = (t > 0) ? t : power1 (x, y);
      }

      if (__isinf (retval))
	retval = huge * huge;
      else if (retval == 0)
	retval = tiny * tiny;
      return retval;
    }

  if (x == 0)
    {
      if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
	  || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000)	/* NaN */
	return y;
      if (ABS (y) > 1.0e20)
	return (y > 0) ? 0 : 1.0 / 0.0;
      k = checkint (y);
      if (k == -1)
	return y < 0 ? 1.0 / x : x;
      else
	return y < 0 ? 1.0 / 0.0 : 0.0;	/* return 0 */
    }

  qx = u.i[HIGH_HALF] & 0x7fffffff;	/*   no sign   */
  qy = v.i[HIGH_HALF] & 0x7fffffff;	/*   no sign   */

  if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0))	/* NaN */
    return x;
  if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0))	/* NaN */
    return x == 1.0 ? 1.0 : y;

  /* if x<0 */
  if (u.i[HIGH_HALF] < 0)
    {
      k = checkint (y);
      if (k == 0)
	{
	  if (qy == 0x7ff00000)
	    {
	      if (x == -1.0)
		return 1.0;
	      else if (x > -1.0)
		return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
	      else
		return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x;
	    }
	  else if (qx == 0x7ff00000)
	    return y < 0 ? 0.0 : INF.x;
	  return (x - x) / (x - x);	/* y not integer and x<0 */
	}
      else if (qx == 0x7ff00000)
	{
	  if (k < 0)
	    return y < 0 ? nZERO.x : nINF.x;
	  else
	    return y < 0 ? 0.0 : INF.x;
	}
      /* if y even or odd */
      if (k == 1)
	return __ieee754_pow (-x, y);
      else
	{
	  double retval;
	  {
	    SET_RESTORE_ROUND (FE_TONEAREST);
	    retval = -__ieee754_pow (-x, y);
	  }
	  if (__isinf (retval))
	    retval = -huge * huge;
	  else if (retval == 0)
	    retval = -tiny * tiny;
	  return retval;
	}
    }
  /* x>0 */

  if (qx == 0x7ff00000)		/* x= 2^-0x3ff */
    return y > 0 ? x : 0;

  if (qy > 0x45f00000 && qy < 0x7ff00000)
    {
      if (x == 1.0)
	return 1.0;
      if (y > 0)
	return (x > 1.0) ? huge * huge : tiny * tiny;
      if (y < 0)
	return (x < 1.0) ? huge * huge : tiny * tiny;
    }

  if (x == 1.0)
    return 1.0;
  if (y > 0)
    return (x > 1.0) ? INF.x : 0;
  if (y < 0)
    return (x < 1.0) ? INF.x : 0;
  return 0;			/* unreachable, to make the compiler happy */
}

#ifndef __ieee754_pow
strong_alias (__ieee754_pow, __pow_finite)
#endif

/* Compute x^y using more accurate but more slow log routine.  */
static double
SECTION
power1 (double x, double y)
{
  double z, a, aa, error, t, a1, a2, y1, y2;
  z = my_log2 (x, &aa, &error);
  t = y * CN;
  y1 = t - (t - y);
  y2 = y - y1;
  t = z * CN;
  a1 = t - (t - z);
  a2 = z - a1;
  a = y * z;
  aa = ((y1 * a1 - a) + y1 * a2 + y2 * a1) + y2 * a2 + aa * y;
  a1 = a + aa;
  a2 = (a - a1) + aa;
  error = error * ABS (y);
  t = __exp1 (a1, a2, 1.9e16 * error);
  return (t >= 0) ? t : __slowpow (x, y, z);
}

/* Compute log(x) (x is left argument). The result is the returned double + the
   parameter DELTA.  The result is bounded by ERROR.  */
static double
SECTION
log1 (double x, double *delta, double *error)
{
  int i, j, m;
  double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
  mynumber u, v;
#ifdef BIG_ENDI
  mynumber /**/ two52 = {{0x43300000, 0x00000000}};	/* 2**52  */
#else
# ifdef LITTLE_ENDI
  mynumber /**/ two52 = {{0x00000000, 0x43300000}};	/* 2**52  */
# endif
#endif

  u.x = x;
  m = u.i[HIGH_HALF];
  *error = 0;
  *delta = 0;
  if (m < 0x00100000)		/*  1<x<2^-1007 */
    {
      x = x * t52.x;
      add = -52.0;
      u.x = x;
      m = u.i[HIGH_HALF];
    }

  if ((m & 0x000fffff) < 0x0006a09e)
    {
      u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
      two52.i[LOW_HALF] = (m >> 20);
    }
  else
    {
      u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
      two52.i[LOW_HALF] = (m >> 20) + 1;
    }

  v.x = u.x + bigu.x;
  uu = v.x - bigu.x;
  i = (v.i[LOW_HALF] & 0x000003ff) << 2;
  if (two52.i[LOW_HALF] == 1023)	/* nx = 0              */
    {
      if (i > 1192 && i < 1208)	/* |x-1| < 1.5*2**-10  */
	{
	  t = x - 1.0;
	  t1 = (t + 5.0e6) - 5.0e6;
	  t2 = t - t1;
	  e1 = t - 0.5 * t1 * t1;
	  e2 = (t * t * t * (r3 + t * (r4 + t * (r5 + t * (r6 + t
							   * (r7 + t * r8)))))
		- 0.5 * t2 * (t + t1));
	  res = e1 + e2;
	  *error = 1.0e-21 * ABS (t);
	  *delta = (e1 - res) + e2;
	  return res;
	}			/* |x-1| < 1.5*2**-10  */
      else
	{
	  v.x = u.x * (ui.x[i] + ui.x[i + 1]) + bigv.x;
	  vv = v.x - bigv.x;
	  j = v.i[LOW_HALF] & 0x0007ffff;
	  j = j + j + j;
	  eps = u.x - uu * vv;
	  e1 = eps * ui.x[i];
	  e2 = eps * (ui.x[i + 1] + vj.x[j] * (ui.x[i] + ui.x[i + 1]));
	  e = e1 + e2;
	  e2 = ((e1 - e) + e2);
	  t = ui.x[i + 2] + vj.x[j + 1];
	  t1 = t + e;
	  t2 = ((((t - t1) + e) + (ui.x[i + 3] + vj.x[j + 2])) + e2 + e * e
		* (p2 + e * (p3 + e * p4)));
	  res = t1 + t2;
	  *error = 1.0e-24;
	  *delta = (t1 - res) + t2;
	  return res;
	}
    }				/* nx = 0 */
  else				/* nx != 0   */
    {
      eps = u.x - uu;
      nx = (two52.x - two52e.x) + add;
      e1 = eps * ui.x[i];
      e2 = eps * ui.x[i + 1];
      e = e1 + e2;
      e2 = (e1 - e) + e2;
      t = nx * ln2a.x + ui.x[i + 2];
      t1 = t + e;
      t2 = ((((t - t1) + e) + nx * ln2b.x + ui.x[i + 3] + e2) + e * e
	    * (q2 + e * (q3 + e * (q4 + e * (q5 + e * q6)))));
      res = t1 + t2;
      *error = 1.0e-21;
      *delta = (t1 - res) + t2;
      return res;
    }				/* nx != 0   */
}

/* Slower but more accurate routine of log.  The returned result is double +
   DELTA.  The result is bounded by ERROR.  */
static double
SECTION
my_log2 (double x, double *delta, double *error)
{
  int i, j, m;
  double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
  double ou1, ou2, lu1, lu2, ov, lv1, lv2, a, a1, a2;
  double y, yy, z, zz, j1, j2, j7, j8;
#ifndef DLA_FMS
  double j3, j4, j5, j6;
#endif
  mynumber u, v;
#ifdef BIG_ENDI
  mynumber /**/ two52 = {{0x43300000, 0x00000000}};	/* 2**52  */
#else
# ifdef LITTLE_ENDI
  mynumber /**/ two52 = {{0x00000000, 0x43300000}};	/* 2**52  */
# endif
#endif

  u.x = x;
  m = u.i[HIGH_HALF];
  *error = 0;
  *delta = 0;
  add = 0;
  if (m < 0x00100000)
    {				/* x < 2^-1022 */
      x = x * t52.x;
      add = -52.0;
      u.x = x;
      m = u.i[HIGH_HALF];
    }

  if ((m & 0x000fffff) < 0x0006a09e)
    {
      u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
      two52.i[LOW_HALF] = (m >> 20);
    }
  else
    {
      u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
      two52.i[LOW_HALF] = (m >> 20) + 1;
    }

  v.x = u.x + bigu.x;
  uu = v.x - bigu.x;
  i = (v.i[LOW_HALF] & 0x000003ff) << 2;
  /*------------------------------------- |x-1| < 2**-11-------------------------------  */
  if ((two52.i[LOW_HALF] == 1023) && (i == 1200))
    {
      t = x - 1.0;
      EMULV (t, s3, y, yy, j1, j2, j3, j4, j5);
      ADD2 (-0.5, 0, y, yy, z, zz, j1, j2);
      MUL2 (t, 0, z, zz, y, yy, j1, j2, j3, j4, j5, j6, j7, j8);
      MUL2 (t, 0, y, yy, z, zz, j1, j2, j3, j4, j5, j6, j7, j8);

      e1 = t + z;
      e2 = ((((t - e1) + z) + zz) + t * t * t
	    * (ss3 + t * (s4 + t * (s5 + t * (s6 + t * (s7 + t * s8))))));
      res = e1 + e2;
      *error = 1.0e-25 * ABS (t);
      *delta = (e1 - res) + e2;
      return res;
    }
  /*----------------------------- |x-1| > 2**-11  --------------------------  */
  else
    {				/*Computing log(x) according to log table                        */
      nx = (two52.x - two52e.x) + add;
      ou1 = ui.x[i];
      ou2 = ui.x[i + 1];
      lu1 = ui.x[i + 2];
      lu2 = ui.x[i + 3];
      v.x = u.x * (ou1 + ou2) + bigv.x;
      vv = v.x - bigv.x;
      j = v.i[LOW_HALF] & 0x0007ffff;
      j = j + j + j;
      eps = u.x - uu * vv;
      ov = vj.x[j];
      lv1 = vj.x[j + 1];
      lv2 = vj.x[j + 2];
      a = (ou1 + ou2) * (1.0 + ov);
      a1 = (a + 1.0e10) - 1.0e10;
      a2 = a * (1.0 - a1 * uu * vv);
      e1 = eps * a1;
      e2 = eps * a2;
      e = e1 + e2;
      e2 = (e1 - e) + e2;
      t = nx * ln2a.x + lu1 + lv1;
      t1 = t + e;
      t2 = ((((t - t1) + e) + (lu2 + lv2 + nx * ln2b.x + e2)) + e * e
	    * (p2 + e * (p3 + e * p4)));
      res = t1 + t2;
      *error = 1.0e-27;
      *delta = (t1 - res) + t2;
      return res;
    }
}

/* This function receives a double x and checks if it is an integer.  If not,
   it returns 0, else it returns 1 if even or -1 if odd.  */
static int
SECTION
checkint (double x)
{
  union
  {
    int4 i[2];
    double x;
  } u;
  int k, m, n;
  u.x = x;
  m = u.i[HIGH_HALF] & 0x7fffffff;	/* no sign */
  if (m >= 0x7ff00000)
    return 0;			/*  x is +/-inf or NaN  */
  if (m >= 0x43400000)
    return 1;			/*  |x| >= 2**53   */
  if (m < 0x40000000)
    return 0;			/* |x| < 2,  can not be 0 or 1  */
  n = u.i[LOW_HALF];
  k = (m >> 20) - 1023;		/*  1 <= k <= 52   */
  if (k == 52)
    return (n & 1) ? -1 : 1;	/* odd or even */
  if (k > 20)
    {
      if (n << (k - 20))
	return 0;		/* if not integer */
      return (n << (k - 21)) ? -1 : 1;
    }
  if (n)
    return 0;			/*if  not integer */
  if (k == 20)
    return (m & 1) ? -1 : 1;
  if (m << (k + 12))
    return 0;
  return (m << (k + 11)) ? -1 : 1;
}