GCC Code Coverage Report
Directory: ./ Exec Total Coverage
File: lib/libm/src/ld80/e_powl.c Lines: 58 187 31.0 %
Date: 2017-11-13 Branches: 35 128 27.3 %

Line Branch Exec Source
1
/*	$OpenBSD: e_powl.c,v 1.7 2017/01/21 08:29:13 krw Exp $	*/
2
3
/*
4
 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5
 *
6
 * Permission to use, copy, modify, and distribute this software for any
7
 * purpose with or without fee is hereby granted, provided that the above
8
 * copyright notice and this permission notice appear in all copies.
9
 *
10
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17
 */
18
19
/*							powl.c
20
 *
21
 *	Power function, long double precision
22
 *
23
 *
24
 *
25
 * SYNOPSIS:
26
 *
27
 * long double x, y, z, powl();
28
 *
29
 * z = powl( x, y );
30
 *
31
 *
32
 *
33
 * DESCRIPTION:
34
 *
35
 * Computes x raised to the yth power.  Analytically,
36
 *
37
 *      x**y  =  exp( y log(x) ).
38
 *
39
 * Following Cody and Waite, this program uses a lookup table
40
 * of 2**-i/32 and pseudo extended precision arithmetic to
41
 * obtain several extra bits of accuracy in both the logarithm
42
 * and the exponential.
43
 *
44
 *
45
 *
46
 * ACCURACY:
47
 *
48
 * The relative error of pow(x,y) can be estimated
49
 * by   y dl ln(2),   where dl is the absolute error of
50
 * the internally computed base 2 logarithm.  At the ends
51
 * of the approximation interval the logarithm equal 1/32
52
 * and its relative error is about 1 lsb = 1.1e-19.  Hence
53
 * the predicted relative error in the result is 2.3e-21 y .
54
 *
55
 *                      Relative error:
56
 * arithmetic   domain     # trials      peak         rms
57
 *
58
 *    IEEE     +-1000       40000      2.8e-18      3.7e-19
59
 * .001 < x < 1000, with log(x) uniformly distributed.
60
 * -1000 < y < 1000, y uniformly distributed.
61
 *
62
 *    IEEE     0,8700       60000      6.5e-18      1.0e-18
63
 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
64
 *
65
 *
66
 * ERROR MESSAGES:
67
 *
68
 *   message         condition      value returned
69
 * pow overflow     x**y > MAXNUM      INFINITY
70
 * pow underflow   x**y < 1/MAXNUM       0.0
71
 * pow domain      x<0 and y noninteger  0.0
72
 *
73
 */
74
75
#include <float.h>
76
#include <math.h>
77
78
#include "math_private.h"
79
80
/* Table size */
81
#define NXT 32
82
/* log2(Table size) */
83
#define LNXT 5
84
85
/* log(1+x) =  x - .5x^2 + x^3 *  P(z)/Q(z)
86
 * on the domain  2^(-1/32) - 1  <=  x  <=  2^(1/32) - 1
87
 */
88
static long double P[] = {
89
 8.3319510773868690346226E-4L,
90
 4.9000050881978028599627E-1L,
91
 1.7500123722550302671919E0L,
92
 1.4000100839971580279335E0L,
93
};
94
static long double Q[] = {
95
/* 1.0000000000000000000000E0L,*/
96
 5.2500282295834889175431E0L,
97
 8.4000598057587009834666E0L,
98
 4.2000302519914740834728E0L,
99
};
100
/* A[i] = 2^(-i/32), rounded to IEEE long double precision.
101
 * If i is even, A[i] + B[i/2] gives additional accuracy.
102
 */
103
static long double A[33] = {
104
 1.0000000000000000000000E0L,
105
 9.7857206208770013448287E-1L,
106
 9.5760328069857364691013E-1L,
107
 9.3708381705514995065011E-1L,
108
 9.1700404320467123175367E-1L,
109
 8.9735453750155359320742E-1L,
110
 8.7812608018664974155474E-1L,
111
 8.5930964906123895780165E-1L,
112
 8.4089641525371454301892E-1L,
113
 8.2287773907698242225554E-1L,
114
 8.0524516597462715409607E-1L,
115
 7.8799042255394324325455E-1L,
116
 7.7110541270397041179298E-1L,
117
 7.5458221379671136985669E-1L,
118
 7.3841307296974965571198E-1L,
119
 7.2259040348852331001267E-1L,
120
 7.0710678118654752438189E-1L,
121
 6.9195494098191597746178E-1L,
122
 6.7712777346844636413344E-1L,
123
 6.6261832157987064729696E-1L,
124
 6.4841977732550483296079E-1L,
125
 6.3452547859586661129850E-1L,
126
 6.2092890603674202431705E-1L,
127
 6.0762367999023443907803E-1L,
128
 5.9460355750136053334378E-1L,
129
 5.8186242938878875689693E-1L,
130
 5.6939431737834582684856E-1L,
131
 5.5719337129794626814472E-1L,
132
 5.4525386633262882960438E-1L,
133
 5.3357020033841180906486E-1L,
134
 5.2213689121370692017331E-1L,
135
 5.1094857432705833910408E-1L,
136
 5.0000000000000000000000E-1L,
137
};
138
static long double B[17] = {
139
 0.0000000000000000000000E0L,
140
 2.6176170809902549338711E-20L,
141
-1.0126791927256478897086E-20L,
142
 1.3438228172316276937655E-21L,
143
 1.2207982955417546912101E-20L,
144
-6.3084814358060867200133E-21L,
145
 1.3164426894366316434230E-20L,
146
-1.8527916071632873716786E-20L,
147
 1.8950325588932570796551E-20L,
148
 1.5564775779538780478155E-20L,
149
 6.0859793637556860974380E-21L,
150
-2.0208749253662532228949E-20L,
151
 1.4966292219224761844552E-20L,
152
 3.3540909728056476875639E-21L,
153
-8.6987564101742849540743E-22L,
154
-1.2327176863327626135542E-20L,
155
 0.0000000000000000000000E0L,
156
};
157
158
/* 2^x = 1 + x P(x),
159
 * on the interval -1/32 <= x <= 0
160
 */
161
static long double R[] = {
162
 1.5089970579127659901157E-5L,
163
 1.5402715328927013076125E-4L,
164
 1.3333556028915671091390E-3L,
165
 9.6181291046036762031786E-3L,
166
 5.5504108664798463044015E-2L,
167
 2.4022650695910062854352E-1L,
168
 6.9314718055994530931447E-1L,
169
};
170
171
#define douba(k) A[k]
172
#define doubb(k) B[k]
173
#define MEXP (NXT*16384.0L)
174
/* The following if denormal numbers are supported, else -MEXP: */
175
#define MNEXP (-NXT*(16384.0L+64.0L))
176
/* log2(e) - 1 */
177
#define LOG2EA 0.44269504088896340735992L
178
179
#define F W
180
#define Fa Wa
181
#define Fb Wb
182
#define G W
183
#define Ga Wa
184
#define Gb u
185
#define H W
186
#define Ha Wb
187
#define Hb Wb
188
189
static const long double MAXLOGL = 1.1356523406294143949492E4L;
190
static const long double MINLOGL = -1.13994985314888605586758E4L;
191
static const long double LOGE2L = 6.9314718055994530941723E-1L;
192
static volatile long double z;
193
static long double w, W, Wa, Wb, ya, yb, u;
194
static const long double huge = 0x1p10000L;
195
#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
196
static const long double twom10000 = 0x1p-10000L;
197
#else
198
static volatile long double twom10000 = 0x1p-10000L;
199
#endif
200
201
static long double reducl( long double );
202
static long double powil ( long double, int );
203
204
long double
205
powl(long double x, long double y)
206
{
207
/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
208
2072
int i, nflg, iyflg, yoddint;
209
long e;
210
211
1036
if( y == 0.0L )
212
	return( 1.0L );
213
214
1036
if( x == 1.0L )
215
	return( 1.0L );
216
217
1036
if( isnan(x) )
218
	return( x );
219
1036
if( isnan(y) )
220
	return( y );
221
222
1036
if( y == 1.0L )
223
	return( x );
224
225
1036
if( !isfinite(y) && x == -1.0L )
226
	return( 1.0L );
227
228
1036
if( y >= LDBL_MAX )
229
	{
230
	if( x > 1.0L )
231
		return( INFINITY );
232
	if( x > 0.0L && x < 1.0L )
233
		return( 0.0L );
234
	if( x < -1.0L )
235
		return( INFINITY );
236
	if( x > -1.0L && x < 0.0L )
237
		return( 0.0L );
238
	}
239
1036
if( y <= -LDBL_MAX )
240
	{
241
	if( x > 1.0L )
242
		return( 0.0L );
243
	if( x > 0.0L && x < 1.0L )
244
		return( INFINITY );
245
	if( x < -1.0L )
246
		return( 0.0L );
247
	if( x > -1.0L && x < 0.0L )
248
		return( INFINITY );
249
	}
250
1036
if( x >= LDBL_MAX )
251
	{
252
	if( y > 0.0L )
253
		return( INFINITY );
254
	return( 0.0L );
255
	}
256
257
1036
w = floorl(y);
258
/* Set iyflg to 1 if y is an integer.  */
259
iyflg = 0;
260
1036
if( w == y )
261
1036
	iyflg = 1;
262
263
/* Test for odd integer y.  */
264
yoddint = 0;
265
1036
if( iyflg )
266
	{
267
1036
	ya = fabsl(y);
268
1036
	ya = floorl(0.5L * ya);
269
1036
	yb = 0.5L * fabsl(w);
270
1036
	if( ya != yb )
271
444
		yoddint = 1;
272
	}
273
274
1036
if( x <= -LDBL_MAX )
275
	{
276
	if( y > 0.0L )
277
		{
278
		if( yoddint )
279
			return( -INFINITY );
280
		return( INFINITY );
281
		}
282
	if( y < 0.0L )
283
		{
284
		if( yoddint )
285
			return( -0.0L );
286
		return( 0.0 );
287
		}
288
	}
289
290
291
nflg = 0;	/* flag = 1 if x<0 raised to integer power */
292
1036
if( x <= 0.0L )
293
	{
294
	if( x == 0.0L )
295
		{
296
		if( y < 0.0 )
297
			{
298
			if( signbit(x) && yoddint )
299
				return( -INFINITY );
300
			return( INFINITY );
301
			}
302
		if( y > 0.0 )
303
			{
304
			if( signbit(x) && yoddint )
305
				return( -0.0L );
306
			return( 0.0 );
307
			}
308
		if( y == 0.0L )
309
			return( 1.0L );  /*   0**0   */
310
		else
311
			return( 0.0L );  /*   0**y   */
312
		}
313
	else
314
		{
315
		if( iyflg == 0 )
316
			return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
317
		nflg = 1;
318
		}
319
	}
320
321
/* Integer power of an integer.  */
322
323
1036
if( iyflg )
324
	{
325
1036
	i = w;
326
1036
	w = floorl(x);
327

2072
	if( (w == x) && (fabsl(y) < 32768.0) )
328
		{
329
1036
		w = powil( x, (int) y );
330
1036
		return( w );
331
		}
332
	}
333
334
335
if( nflg )
336
	x = fabsl(x);
337
338
/* separate significand from exponent */
339
x = frexpl( x, &i );
340
e = i;
341
342
/* find significand in antilog table A[] */
343
i = 1;
344
if( x <= douba(17) )
345
	i = 17;
346
if( x <= douba(i+8) )
347
	i += 8;
348
if( x <= douba(i+4) )
349
	i += 4;
350
if( x <= douba(i+2) )
351
	i += 2;
352
if( x >= douba(1) )
353
	i = -1;
354
i += 1;
355
356
357
/* Find (x - A[i])/A[i]
358
 * in order to compute log(x/A[i]):
359
 *
360
 * log(x) = log( a x/a ) = log(a) + log(x/a)
361
 *
362
 * log(x/a) = log(1+v),  v = x/a - 1 = (x-a)/a
363
 */
364
x -= douba(i);
365
x -= doubb(i/2);
366
x /= douba(i);
367
368
369
/* rational approximation for log(1+v):
370
 *
371
 * log(1+v)  =  v  -  v**2/2  +  v**3 P(v) / Q(v)
372
 */
373
z = x*x;
374
w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) );
375
w = w - ldexpl( z, -1 );   /*  w - 0.5 * z  */
376
377
/* Convert to base 2 logarithm:
378
 * multiply by log2(e) = 1 + LOG2EA
379
 */
380
z = LOG2EA * w;
381
z += w;
382
z += LOG2EA * x;
383
z += x;
384
385
/* Compute exponent term of the base 2 logarithm. */
386
w = -i;
387
w = ldexpl( w, -LNXT );	/* divide by NXT */
388
w += e;
389
/* Now base 2 log of x is w + z. */
390
391
/* Multiply base 2 log by y, in extended precision. */
392
393
/* separate y into large part ya
394
 * and small part yb less than 1/NXT
395
 */
396
ya = reducl(y);
397
yb = y - ya;
398
399
/* (w+z)(ya+yb)
400
 * = w*ya + w*yb + z*y
401
 */
402
F = z * y  +  w * yb;
403
Fa = reducl(F);
404
Fb = F - Fa;
405
406
G = Fa + w * ya;
407
Ga = reducl(G);
408
Gb = G - Ga;
409
410
H = Fb + Gb;
411
Ha = reducl(H);
412
w = ldexpl( Ga+Ha, LNXT );
413
414
/* Test the power of 2 for overflow */
415
if( w > MEXP )
416
	return (huge * huge);		/* overflow */
417
418
if( w < MNEXP )
419
	return (twom10000 * twom10000);	/* underflow */
420
421
e = w;
422
Hb = H - Ha;
423
424
if( Hb > 0.0L )
425
	{
426
	e += 1;
427
	Hb -= (1.0L/NXT);  /*0.0625L;*/
428
	}
429
430
/* Now the product y * log2(x)  =  Hb + e/NXT.
431
 *
432
 * Compute base 2 exponential of Hb,
433
 * where -0.0625 <= Hb <= 0.
434
 */
435
z = Hb * __polevll( Hb, R, 6 );  /*    z  =  2**Hb - 1    */
436
437
/* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
438
 * Find lookup table entry for the fractional power of 2.
439
 */
440
if( e < 0 )
441
	i = 0;
442
else
443
	i = 1;
444
i = e/NXT + i;
445
e = NXT*i - e;
446
w = douba( e );
447
z = w * z;      /*    2**-e * ( 1 + (2**Hb-1) )    */
448
z = z + w;
449
z = ldexpl( z, i );  /* multiply by integer power of 2 */
450
451
if( nflg )
452
	{
453
/* For negative x,
454
 * find out if the integer exponent
455
 * is odd or even.
456
 */
457
	w = ldexpl( y, -1 );
458
	w = floorl(w);
459
	w = ldexpl( w, 1 );
460
	if( w != y )
461
		z = -z; /* odd exponent */
462
	}
463
464
return( z );
465
1036
}
466
DEF_STD(powl);
467
468
469
/* Find a multiple of 1/NXT that is within 1/NXT of x. */
470
static long double
471
reducl(long double x)
472
{
473
long double t;
474
475
t = ldexpl( x, LNXT );
476
t = floorl( t );
477
t = ldexpl( t, -LNXT );
478
return(t);
479
}
480
481
/*							powil.c
482
 *
483
 *	Real raised to integer power, long double precision
484
 *
485
 *
486
 *
487
 * SYNOPSIS:
488
 *
489
 * long double x, y, powil();
490
 * int n;
491
 *
492
 * y = powil( x, n );
493
 *
494
 *
495
 *
496
 * DESCRIPTION:
497
 *
498
 * Returns argument x raised to the nth power.
499
 * The routine efficiently decomposes n as a sum of powers of
500
 * two. The desired power is a product of two-to-the-kth
501
 * powers of x.  Thus to compute the 32767 power of x requires
502
 * 28 multiplications instead of 32767 multiplications.
503
 *
504
 *
505
 *
506
 * ACCURACY:
507
 *
508
 *
509
 *                      Relative error:
510
 * arithmetic   x domain   n domain  # trials      peak         rms
511
 *    IEEE     .001,1000  -1022,1023  50000       4.3e-17     7.8e-18
512
 *    IEEE        1,2     -1022,1023  20000       3.9e-17     7.6e-18
513
 *    IEEE     .99,1.01     0,8700    10000       3.6e-16     7.2e-17
514
 *
515
 * Returns MAXNUM on overflow, zero on underflow.
516
 *
517
 */
518
519
static long double
520
powil(long double x, int nn)
521
{
522
long double ww, y;
523
long double s;
524
2072
int n, e, sign, asign, lx;
525
526
1036
if( x == 0.0L )
527
	{
528
	if( nn == 0 )
529
		return( 1.0L );
530
	else if( nn < 0 )
531
		return( LDBL_MAX );
532
	else
533
		return( 0.0L );
534
	}
535
536
1036
if( nn == 0 )
537
	return( 1.0L );
538
539
540
1036
if( x < 0.0L )
541
	{
542
	asign = -1;
543
	x = -x;
544
	}
545
else
546
	asign = 0;
547
548
549
1036
if( nn < 0 )
550
	{
551
	sign = -1;
552
1036
	n = -nn;
553
1036
	}
554
else
555
	{
556
	sign = 1;
557
	n = nn;
558
	}
559
560
/* Overflow detection */
561
562
/* Calculate approximate logarithm of answer */
563
s = x;
564
1036
s = frexpl( s, &lx );
565
1036
e = (lx - 1)*n;
566
1036
if( (e == 0) || (e > 64) || (e < -64) )
567
	{
568
148
	s = (s - 7.0710678118654752e-1L) / (s +  7.0710678118654752e-1L);
569
148
	s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
570
148
	}
571
else
572
	{
573
888
	s = LOGE2L * e;
574
	}
575
576
1036
if( s > MAXLOGL )
577
	return (huge * huge);		/* overflow */
578
579
1036
if( s < MINLOGL )
580
	return (twom10000 * twom10000);	/* underflow */
581
/* Handle tiny denormal answer, but with less accuracy
582
 * since roundoff error in 1.0/x will be amplified.
583
 * The precise demarcation should be the gradual underflow threshold.
584
 */
585
1036
if( s < (-MAXLOGL+2.0L) )
586
	{
587
	x = 1.0L/x;
588
	sign = -sign;
589
	}
590
591
/* First bit of the power */
592
1036
if( n & 1 )
593
444
	y = x;
594
595
else
596
	{
597
	y = 1.0L;
598
	asign = 0;
599
	}
600
601
ww = x;
602
1036
n >>= 1;
603
6808
while( n )
604
	{
605
2368
	ww = ww * ww;	/* arg to the 2-to-the-kth power */
606
2368
	if( n & 1 )	/* if that bit is set, then include in product */
607
1924
		y *= ww;
608
2368
	n >>= 1;
609
	}
610
611
1036
if( asign )
612
	y = -y; /* odd power of negative number */
613
1036
if( sign < 0 )
614
1036
	y = 1.0L/y;
615
1036
return(y);
616
1036
}