1 /*	$OpenBSD: e_powl.c,v 1.5 2013/11/12 20:35:19 martynas 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 int i, nflg, iyflg, yoddint;
209 long e;
210 
211 if( y == 0.0L )
212 	return( 1.0L );
213 
214 if( x == 1.0L )
215 	return( 1.0L );
216 
217 if( isnan(x) )
218 	return( x );
219 if( isnan(y) )
220 	return( y );
221 
222 if( y == 1.0L )
223 	return( x );
224 
225 if( !isfinite(y) && x == -1.0L )
226 	return( 1.0L );
227 
228 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 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 if( x >= LDBL_MAX )
251 	{
252 	if( y > 0.0L )
253 		return( INFINITY );
254 	return( 0.0L );
255 	}
256 
257 w = floorl(y);
258 /* Set iyflg to 1 if y is an integer.  */
259 iyflg = 0;
260 if( w == y )
261 	iyflg = 1;
262 
263 /* Test for odd integer y.  */
264 yoddint = 0;
265 if( iyflg )
266 	{
267 	ya = fabsl(y);
268 	ya = floorl(0.5L * ya);
269 	yb = 0.5L * fabsl(w);
270 	if( ya != yb )
271 		yoddint = 1;
272 	}
273 
274 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 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 if( iyflg )
324 	{
325 	i = w;
326 	w = floorl(x);
327 	if( (w == x) && (fabsl(y) < 32768.0) )
328 		{
329 		w = powil( x, (int) y );
330 		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 }
466 
467 
468 /* Find a multiple of 1/NXT that is within 1/NXT of x. */
469 static long double
470 reducl(long double x)
471 {
472 long double t;
473 
474 t = ldexpl( x, LNXT );
475 t = floorl( t );
476 t = ldexpl( t, -LNXT );
477 return(t);
478 }
479 
480 /*							powil.c
481  *
482  *	Real raised to integer power, long double precision
483  *
484  *
485  *
486  * SYNOPSIS:
487  *
488  * long double x, y, powil();
489  * int n;
490  *
491  * y = powil( x, n );
492  *
493  *
494  *
495  * DESCRIPTION:
496  *
497  * Returns argument x raised to the nth power.
498  * The routine efficiently decomposes n as a sum of powers of
499  * two. The desired power is a product of two-to-the-kth
500  * powers of x.  Thus to compute the 32767 power of x requires
501  * 28 multiplications instead of 32767 multiplications.
502  *
503  *
504  *
505  * ACCURACY:
506  *
507  *
508  *                      Relative error:
509  * arithmetic   x domain   n domain  # trials      peak         rms
510  *    IEEE     .001,1000  -1022,1023  50000       4.3e-17     7.8e-18
511  *    IEEE        1,2     -1022,1023  20000       3.9e-17     7.6e-18
512  *    IEEE     .99,1.01     0,8700    10000       3.6e-16     7.2e-17
513  *
514  * Returns MAXNUM on overflow, zero on underflow.
515  *
516  */
517 
518 static long double
519 powil(long double x, int nn)
520 {
521 long double ww, y;
522 long double s;
523 int n, e, sign, asign, lx;
524 
525 if( x == 0.0L )
526 	{
527 	if( nn == 0 )
528 		return( 1.0L );
529 	else if( nn < 0 )
530 		return( LDBL_MAX );
531 	else
532 		return( 0.0L );
533 	}
534 
535 if( nn == 0 )
536 	return( 1.0L );
537 
538 
539 if( x < 0.0L )
540 	{
541 	asign = -1;
542 	x = -x;
543 	}
544 else
545 	asign = 0;
546 
547 
548 if( nn < 0 )
549 	{
550 	sign = -1;
551 	n = -nn;
552 	}
553 else
554 	{
555 	sign = 1;
556 	n = nn;
557 	}
558 
559 /* Overflow detection */
560 
561 /* Calculate approximate logarithm of answer */
562 s = x;
563 s = frexpl( s, &lx );
564 e = (lx - 1)*n;
565 if( (e == 0) || (e > 64) || (e < -64) )
566 	{
567 	s = (s - 7.0710678118654752e-1L) / (s +  7.0710678118654752e-1L);
568 	s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
569 	}
570 else
571 	{
572 	s = LOGE2L * e;
573 	}
574 
575 if( s > MAXLOGL )
576 	return (huge * huge);		/* overflow */
577 
578 if( s < MINLOGL )
579 	return (twom10000 * twom10000);	/* underflow */
580 /* Handle tiny denormal answer, but with less accuracy
581  * since roundoff error in 1.0/x will be amplified.
582  * The precise demarcation should be the gradual underflow threshold.
583  */
584 if( s < (-MAXLOGL+2.0L) )
585 	{
586 	x = 1.0L/x;
587 	sign = -sign;
588 	}
589 
590 /* First bit of the power */
591 if( n & 1 )
592 	y = x;
593 
594 else
595 	{
596 	y = 1.0L;
597 	asign = 0;
598 	}
599 
600 ww = x;
601 n >>= 1;
602 while( n )
603 	{
604 	ww = ww * ww;	/* arg to the 2-to-the-kth power */
605 	if( n & 1 )	/* if that bit is set, then include in product */
606 		y *= ww;
607 	n >>= 1;
608 	}
609 
610 if( asign )
611 	y = -y; /* odd power of negative number */
612 if( sign < 0 )
613 	y = 1.0L/y;
614 return(y);
615 }
616