aboutsummaryrefslogtreecommitdiff
path: root/sysdeps/ieee754/dbl-64/e_exp.c
blob: 6a7122f5857fb27b0e3a076407ba104d14cd5a7c (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
/* EXP function - Compute double precision exponential */
/*
 * IBM Accurate Mathematical Library
 * written by International Business Machines Corp.
 * Copyright (C) 2001-2017 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:uexp.c                                                     */
/*                                                                         */
/*  FUNCTION:uexp                                                          */
/*           exp1                                                          */
/*                                                                         */
/* FILES NEEDED:dla.h endian.h mpa.h mydefs.h uexp.h                       */
/*              mpa.c mpexp.x                                              */
/*                                                                         */
/* An ultimate exp routine. Given an IEEE double machine number x          */
/* it computes the correctly rounded (to nearest) value of e^x             */
/* Assumption: Machine arithmetic operations are performed in              */
/* round to nearest mode of IEEE 754 standard.                             */
/*                                                                         */
/***************************************************************************/

/*  IBM exp(x) replaced by following exp(x) in 2017. IBM exp1(x,xx) remains.  */
/* exp(x)
   Hybrid algorithm of Peter Tang's Table driven method (for large
   arguments) and an accurate table (for small arguments).
   Written by K.C. Ng, November 1988.
   Revised by Patrick McGehearty, Nov 2017 to use j/64 instead of j/32
   Method (large arguments):
	1. Argument Reduction: given the input x, find r and integer k
	   and j such that
	             x = (k+j/64)*(ln2) + r,  |r| <= (1/128)*ln2

	2. exp(x) = 2^k * (2^(j/64) + 2^(j/64)*expm1(r))
	   a. expm1(r) is approximated by a polynomial:
	      expm1(r) ~ r + t1*r^2 + t2*r^3 + ... + t5*r^6
	      Here t1 = 1/2 exactly.
	   b. 2^(j/64) is represented to twice double precision
	      as TBL[2j]+TBL[2j+1].

   Note: If divide were fast enough, we could use another approximation
	 in 2.a:
	      expm1(r) ~ (2r)/(2-R), R = r - r^2*(t1 + t2*r^2)
	      (for the same t1 and t2 as above)

   Special cases:
	exp(INF) is INF, exp(NaN) is NaN;
	exp(-INF)=  0;
	for finite argument, only exp(0)=1 is exact.

   Accuracy:
	According to an error analysis, the error is always less than
	an ulp (unit in the last place).  The largest errors observed
	are less than 0.55 ulp for normal results and less than 0.75 ulp
	for subnormal results.

   Misc. info.
	For IEEE double
		if x >  7.09782712893383973096e+02 then exp(x) overflow
		if x < -7.45133219101941108420e+02 then exp(x) underflow.  */

#include <math.h>
#include <math-svid-compat.h>
#include <math_private.h>
#include <errno.h>
#include "endian.h"
#include "uexp.h"
#include "uexp.tbl"
#include "mydefs.h"
#include "MathLib.h"
#include <fenv.h>
#include <float.h>

extern double __ieee754_exp (double);

#include "eexp.tbl"

static const double
  half = 0.5,
  one = 1.0;


double
__ieee754_exp (double x_arg)
{
  double z, t;
  double retval;
  int hx, ix, k, j, m;
  int fe_val;
  union
  {
    int i_part[2];
    double x;
  } xx;
  union
  {
    int y_part[2];
    double y;
  } yy;
  xx.x = x_arg;

  ix = xx.i_part[HIGH_HALF];
  hx = ix & ~0x80000000;

  if (hx < 0x3ff0a2b2)
    {				/* |x| < 3/2 ln 2 */
      if (hx < 0x3f862e42)
	{			/* |x| < 1/64 ln 2 */
	  if (hx < 0x3ed00000)
	    {			/* |x| < 2^-18 */
	      if (hx < 0x3e300000)
		{
		  retval = one + xx.x;
		  return retval;
		}
	      retval = one + xx.x * (one + half * xx.x);
	      return retval;
	    }
	  /* Use FE_TONEAREST rounding mode for computing yy.y.
	     Avoid set/reset of rounding mode if in FE_TONEAREST mode.  */
	  fe_val = get_rounding_mode ();
	  if (fe_val == FE_TONEAREST)
	    {
	      t = xx.x * xx.x;
	      yy.y = xx.x + (t * (half + xx.x * t2)
			     + (t * t) * (t3 + xx.x * t4 + t * t5));
	      retval = one + yy.y;
	    }
	  else
	    {
	      libc_fesetround (FE_TONEAREST);
	      t = xx.x * xx.x;
	      yy.y = xx.x + (t * (half + xx.x * t2)
			     + (t * t) * (t3 + xx.x * t4 + t * t5));
	      retval = one + yy.y;
	      libc_fesetround (fe_val);
	    }
	  return retval;
	}

      /* Find the multiple of 2^-6 nearest x.  */
      k = hx >> 20;
      j = (0x00100000 | (hx & 0x000fffff)) >> (0x40c - k);
      j = (j - 1) & ~1;
      if (ix < 0)
	j += 134;
      /* Use FE_TONEAREST rounding mode for computing yy.y.
	 Avoid set/reset of rounding mode if in FE_TONEAREST mode.  */
      fe_val = get_rounding_mode ();
      if (fe_val == FE_TONEAREST)
	{
	  z = xx.x - TBL2[j];
	  t = z * z;
	  yy.y = z + (t * (half + (z * t2))
		      + (t * t) * (t3 + z * t4 + t * t5));
	  retval = TBL2[j + 1] + TBL2[j + 1] * yy.y;
	}
      else
	{
	  libc_fesetround (FE_TONEAREST);
	  z = xx.x - TBL2[j];
	  t = z * z;
	  yy.y = z + (t * (half + (z * t2))
		      + (t * t) * (t3 + z * t4 + t * t5));
	  retval = TBL2[j + 1] + TBL2[j + 1] * yy.y;
	  libc_fesetround (fe_val);
	}
      return retval;
    }

  if (hx >= 0x40862e42)
    {				/* x is large, infinite, or nan.  */
      if (hx >= 0x7ff00000)
	{
	  if (ix == 0xfff00000 && xx.i_part[LOW_HALF] == 0)
	    return zero;	/* exp(-inf) = 0.  */
	  return (xx.x * xx.x);	/* exp(nan/inf) is nan or inf.  */
	}
      if (xx.x > threshold1)
	{			/* Set overflow error condition.  */
	  retval = hhuge * hhuge;
	  return retval;
	}
      if (-xx.x > threshold2)
	{			/* Set underflow error condition.  */
	  double force_underflow = tiny * tiny;
	  math_force_eval (force_underflow);
	  retval = force_underflow;
	  return retval;
	}
    }

  /* Use FE_TONEAREST rounding mode for computing yy.y.
     Avoid set/reset of rounding mode if already in FE_TONEAREST mode.  */
  fe_val = get_rounding_mode ();
  if (fe_val == FE_TONEAREST)
    {
      t = invln2_64 * xx.x;
      if (ix < 0)
	t -= half;
      else
	t += half;
      k = (int) t;
      j = (k & 0x3f) << 1;
      m = k >> 6;
      z = (xx.x - k * ln2_64hi) - k * ln2_64lo;

      /* z is now in primary range.  */
      t = z * z;
      yy.y = z + (t * (half + z * t2) + (t * t) * (t3 + z * t4 + t * t5));
      yy.y = TBL[j] + (TBL[j + 1] + TBL[j] * yy.y);
    }
  else
    {
      libc_fesetround (FE_TONEAREST);
      t = invln2_64 * xx.x;
      if (ix < 0)
	t -= half;
      else
	t += half;
      k = (int) t;
      j = (k & 0x3f) << 1;
      m = k >> 6;
      z = (xx.x - k * ln2_64hi) - k * ln2_64lo;

      /* z is now in primary range.  */
      t = z * z;
      yy.y = z + (t * (half + z * t2) + (t * t) * (t3 + z * t4 + t * t5));
      yy.y = TBL[j] + (TBL[j + 1] + TBL[j] * yy.y);
      libc_fesetround (fe_val);
    }

  if (m < -1021)
    {
      yy.y_part[HIGH_HALF] += (m + 54) << 20;
      retval = twom54 * yy.y;
      if (retval < DBL_MIN)
	{
	  double force_underflow = tiny * tiny;
	  math_force_eval (force_underflow);
	}
      return retval;
    }
  yy.y_part[HIGH_HALF] += m << 20;
  return yy.y;
}
#ifndef __ieee754_exp
strong_alias (__ieee754_exp, __exp_finite)
#endif

#ifndef SECTION
# define SECTION
#endif

/* Compute e^(x+xx).  The routine also receives bound of error of previous
   calculation.  If after computing exp the error exceeds the allowed bounds,
   the routine returns a non-positive number.  Otherwise it returns the
   computed result, which is always positive.  */
double
SECTION
__exp1 (double x, double xx, double error)
{
  double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
  mynumber junk1, junk2, binexp = {{0, 0}};
  int4 i, j, m, n, ex;

  junk1.x = x;
  m = junk1.i[HIGH_HALF];
  n = m & hugeint;		/* no sign */

  if (n > smallint && n < bigint)
    {
      y = x * log2e.x + three51.x;
      bexp = y - three51.x;	/*  multiply the result by 2**bexp        */

      junk1.x = y;

      eps = bexp * ln_two2.x;	/* x = bexp*ln(2) + t - eps               */
      t = x - bexp * ln_two1.x;

      y = t + three33.x;
      base = y - three33.x;	/* t rounded to a multiple of 2**-18      */
      junk2.x = y;
      del = (t - base) + (xx - eps);	/*  x = bexp*ln(2) + base + del      */
      eps = del + del * del * (p3.x * del + p2.x);

      binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 1023) << 20;

      i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
      j = (junk2.i[LOW_HALF] & 511) << 1;

      al = coar.x[i] * fine.x[j];
      bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
	     + coar.x[i + 1] * fine.x[j + 1]);

      rem = (bet + bet * eps) + al * eps;
      res = al + rem;
      cor = (al - res) + rem;
      if (res == (res + cor * (1.0 + error + err_1)))
	return res * binexp.x;
      else
	return -10.0;
    }

  if (n <= smallint)
    return 1.0;			/*  if x->0 e^x=1 */

  if (n >= badint)
    {
      if (n > infint)
	return (zero / zero);	/* x is NaN,  return invalid */
      if (n < infint)
	return ((x > 0) ? (hhuge * hhuge) : (tiny * tiny));
      /* x is finite,  cause either overflow or underflow  */
      if (junk1.i[LOW_HALF] != 0)
	return (zero / zero);	/*  x is NaN  */
      return ((x > 0) ? inf.x : zero);	/* |x| = inf;  return either inf or 0 */
    }

  y = x * log2e.x + three51.x;
  bexp = y - three51.x;
  junk1.x = y;
  eps = bexp * ln_two2.x;
  t = x - bexp * ln_two1.x;
  y = t + three33.x;
  base = y - three33.x;
  junk2.x = y;
  del = (t - base) + (xx - eps);
  eps = del + del * del * (p3.x * del + p2.x);
  i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
  j = (junk2.i[LOW_HALF] & 511) << 1;
  al = coar.x[i] * fine.x[j];
  bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
	 + coar.x[i + 1] * fine.x[j + 1]);
  rem = (bet + bet * eps) + al * eps;
  res = al + rem;
  cor = (al - res) + rem;
  if (m >> 31)
    {
      ex = junk1.i[LOW_HALF];
      if (res < 1.0)
	{
	  res += res;
	  cor += cor;
	  ex -= 1;
	}
      if (ex >= -1022)
	{
	  binexp.i[HIGH_HALF] = (1023 + ex) << 20;
	  if (res == (res + cor * (1.0 + error + err_1)))
	    return res * binexp.x;
	  else
	    return -10.0;
	}
      ex = -(1022 + ex);
      binexp.i[HIGH_HALF] = (1023 - ex) << 20;
      res *= binexp.x;
      cor *= binexp.x;
      eps = 1.00000000001 + (error + err_1) * binexp.x;
      t = 1.0 + res;
      y = ((1.0 - t) + res) + cor;
      res = t + y;
      cor = (t - res) + y;
      if (res == (res + eps * cor))
	{
	  binexp.i[HIGH_HALF] = 0x00100000;
	  return (res - 1.0) * binexp.x;
	}
      else
	return -10.0;
    }
  else
    {
      binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 767) << 20;
      if (res == (res + cor * (1.0 + error + err_1)))
	return res * binexp.x * t256.x;
      else
	return -10.0;
    }
}