xref: /freebsd/lib/msun/ld80/e_powl.c (revision c697fb7f) 
1 /*-
2  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
3  *
4  * Permission to use, copy, modify, and distribute this software for any
5  * purpose with or without fee is hereby granted, provided that the above
6  * copyright notice and this permission notice appear in all copies.
7  *
8  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
9  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
10  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
11  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
12  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
13  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
14  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
15  */
16 
17 #include <sys/cdefs.h>
18 __FBSDID("$FreeBSD$");
19 
20 #include <math.h>
21 
22 #include "math_private.h"
23 
24 /*
25  * Polynomial evaluator:
26  *  P[0] x^n  +  P[1] x^(n-1)  +  ...  +  P[n]
27  */
28 static inline long double
29 __polevll(long double x, long double *PP, int n)
30 {
31 	long double y;
32 	long double *P;
33 
34 	P = PP;
35 	y = *P++;
36 	do {
37 		y = y * x + *P++;
38 	} while (--n);
39 
40 	return (y);
41 }
42 
43 /*
44  * Polynomial evaluator:
45  *  x^n  +  P[0] x^(n-1)  +  P[1] x^(n-2)  +  ...  +  P[n]
46  */
47 static inline long double
48 __p1evll(long double x, long double *PP, int n)
49 {
50 	long double y;
51 	long double *P;
52 
53 	P = PP;
54 	n -= 1;
55 	y = x + *P++;
56 	do {
57 		y = y * x + *P++;
58 	} while (--n);
59 
60 	return (y);
61 }
62 
63 /*							powl.c
64  *
65  *	Power function, long double precision
66  *
67  *
68  *
69  * SYNOPSIS:
70  *
71  * long double x, y, z, powl();
72  *
73  * z = powl( x, y );
74  *
75  *
76  *
77  * DESCRIPTION:
78  *
79  * Computes x raised to the yth power.  Analytically,
80  *
81  *      x**y  =  exp( y log(x) ).
82  *
83  * Following Cody and Waite, this program uses a lookup table
84  * of 2**-i/32 and pseudo extended precision arithmetic to
85  * obtain several extra bits of accuracy in both the logarithm
86  * and the exponential.
87  *
88  *
89  *
90  * ACCURACY:
91  *
92  * The relative error of pow(x,y) can be estimated
93  * by   y dl ln(2),   where dl is the absolute error of
94  * the internally computed base 2 logarithm.  At the ends
95  * of the approximation interval the logarithm equal 1/32
96  * and its relative error is about 1 lsb = 1.1e-19.  Hence
97  * the predicted relative error in the result is 2.3e-21 y .
98  *
99  *                      Relative error:
100  * arithmetic   domain     # trials      peak         rms
101  *
102  *    IEEE     +-1000       40000      2.8e-18      3.7e-19
103  * .001 < x < 1000, with log(x) uniformly distributed.
104  * -1000 < y < 1000, y uniformly distributed.
105  *
106  *    IEEE     0,8700       60000      6.5e-18      1.0e-18
107  * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
108  *
109  *
110  * ERROR MESSAGES:
111  *
112  *   message         condition      value returned
113  * pow overflow     x**y > MAXNUM      INFINITY
114  * pow underflow   x**y < 1/MAXNUM       0.0
115  * pow domain      x<0 and y noninteger  0.0
116  *
117  */
118 
119 #include <sys/cdefs.h>
120 __FBSDID("$FreeBSD$");
121 
122 #include <float.h>
123 #include <math.h>
124 
125 #include "math_private.h"
126 
127 /* Table size */
128 #define NXT 32
129 /* log2(Table size) */
130 #define LNXT 5
131 
132 /* log(1+x) =  x - .5x^2 + x^3 *  P(z)/Q(z)
133  * on the domain  2^(-1/32) - 1  <=  x  <=  2^(1/32) - 1
134  */
135 static long double P[] = {
136  8.3319510773868690346226E-4L,
137  4.9000050881978028599627E-1L,
138  1.7500123722550302671919E0L,
139  1.4000100839971580279335E0L,
140 };
141 static long double Q[] = {
142 /* 1.0000000000000000000000E0L,*/
143  5.2500282295834889175431E0L,
144  8.4000598057587009834666E0L,
145  4.2000302519914740834728E0L,
146 };
147 /* A[i] = 2^(-i/32), rounded to IEEE long double precision.
148  * If i is even, A[i] + B[i/2] gives additional accuracy.
149  */
150 static long double A[33] = {
151  1.0000000000000000000000E0L,
152  9.7857206208770013448287E-1L,
153  9.5760328069857364691013E-1L,
154  9.3708381705514995065011E-1L,
155  9.1700404320467123175367E-1L,
156  8.9735453750155359320742E-1L,
157  8.7812608018664974155474E-1L,
158  8.5930964906123895780165E-1L,
159  8.4089641525371454301892E-1L,
160  8.2287773907698242225554E-1L,
161  8.0524516597462715409607E-1L,
162  7.8799042255394324325455E-1L,
163  7.7110541270397041179298E-1L,
164  7.5458221379671136985669E-1L,
165  7.3841307296974965571198E-1L,
166  7.2259040348852331001267E-1L,
167  7.0710678118654752438189E-1L,
168  6.9195494098191597746178E-1L,
169  6.7712777346844636413344E-1L,
170  6.6261832157987064729696E-1L,
171  6.4841977732550483296079E-1L,
172  6.3452547859586661129850E-1L,
173  6.2092890603674202431705E-1L,
174  6.0762367999023443907803E-1L,
175  5.9460355750136053334378E-1L,
176  5.8186242938878875689693E-1L,
177  5.6939431737834582684856E-1L,
178  5.5719337129794626814472E-1L,
179  5.4525386633262882960438E-1L,
180  5.3357020033841180906486E-1L,
181  5.2213689121370692017331E-1L,
182  5.1094857432705833910408E-1L,
183  5.0000000000000000000000E-1L,
184 };
185 static long double B[17] = {
186  0.0000000000000000000000E0L,
187  2.6176170809902549338711E-20L,
188 -1.0126791927256478897086E-20L,
189  1.3438228172316276937655E-21L,
190  1.2207982955417546912101E-20L,
191 -6.3084814358060867200133E-21L,
192  1.3164426894366316434230E-20L,
193 -1.8527916071632873716786E-20L,
194  1.8950325588932570796551E-20L,
195  1.5564775779538780478155E-20L,
196  6.0859793637556860974380E-21L,
197 -2.0208749253662532228949E-20L,
198  1.4966292219224761844552E-20L,
199  3.3540909728056476875639E-21L,
200 -8.6987564101742849540743E-22L,
201 -1.2327176863327626135542E-20L,
202  0.0000000000000000000000E0L,
203 };
204 
205 /* 2^x = 1 + x P(x),
206  * on the interval -1/32 <= x <= 0
207  */
208 static long double R[] = {
209  1.5089970579127659901157E-5L,
210  1.5402715328927013076125E-4L,
211  1.3333556028915671091390E-3L,
212  9.6181291046036762031786E-3L,
213  5.5504108664798463044015E-2L,
214  2.4022650695910062854352E-1L,
215  6.9314718055994530931447E-1L,
216 };
217 
218 #define douba(k) A[k]
219 #define doubb(k) B[k]
220 #define MEXP (NXT*16384.0L)
221 /* The following if denormal numbers are supported, else -MEXP: */
222 #define MNEXP (-NXT*(16384.0L+64.0L))
223 /* log2(e) - 1 */
224 #define LOG2EA 0.44269504088896340735992L
225 
226 #define F W
227 #define Fa Wa
228 #define Fb Wb
229 #define G W
230 #define Ga Wa
231 #define Gb u
232 #define H W
233 #define Ha Wb
234 #define Hb Wb
235 
236 static const long double MAXLOGL = 1.1356523406294143949492E4L;
237 static const long double MINLOGL = -1.13994985314888605586758E4L;
238 static const long double LOGE2L = 6.9314718055994530941723E-1L;
239 static volatile long double z;
240 static long double w, W, Wa, Wb, ya, yb, u;
241 static const long double huge = 0x1p10000L;
242 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */
243 static const long double twom10000 = 0x1p-10000L;
244 #else
245 static volatile long double twom10000 = 0x1p-10000L;
246 #endif
247 
248 static long double reducl( long double );
249 static long double powil ( long double, int );
250 
251 long double
252 powl(long double x, long double y)
253 {
254 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
255 int i, nflg, iyflg, yoddint;
256 long e;
257 
258 if( y == 0.0L )
259 	return( 1.0L );
260 
261 if( x == 1.0L )
262 	return( 1.0L );
263 
264 if( isnan(x) )
265 	return ( nan_mix(x, y) );
266 if( isnan(y) )
267 	return ( nan_mix(x, y) );
268 
269 if( y == 1.0L )
270 	return( x );
271 
272 if( !isfinite(y) && x == -1.0L )
273 	return( 1.0L );
274 
275 if( y >= LDBL_MAX )
276 	{
277 	if( x > 1.0L )
278 		return( INFINITY );
279 	if( x > 0.0L && x < 1.0L )
280 		return( 0.0L );
281 	if( x < -1.0L )
282 		return( INFINITY );
283 	if( x > -1.0L && x < 0.0L )
284 		return( 0.0L );
285 	}
286 if( y <= -LDBL_MAX )
287 	{
288 	if( x > 1.0L )
289 		return( 0.0L );
290 	if( x > 0.0L && x < 1.0L )
291 		return( INFINITY );
292 	if( x < -1.0L )
293 		return( 0.0L );
294 	if( x > -1.0L && x < 0.0L )
295 		return( INFINITY );
296 	}
297 if( x >= LDBL_MAX )
298 	{
299 	if( y > 0.0L )
300 		return( INFINITY );
301 	return( 0.0L );
302 	}
303 
304 w = floorl(y);
305 /* Set iyflg to 1 if y is an integer.  */
306 iyflg = 0;
307 if( w == y )
308 	iyflg = 1;
309 
310 /* Test for odd integer y.  */
311 yoddint = 0;
312 if( iyflg )
313 	{
314 	ya = fabsl(y);
315 	ya = floorl(0.5L * ya);
316 	yb = 0.5L * fabsl(w);
317 	if( ya != yb )
318 		yoddint = 1;
319 	}
320 
321 if( x <= -LDBL_MAX )
322 	{
323 	if( y > 0.0L )
324 		{
325 		if( yoddint )
326 			return( -INFINITY );
327 		return( INFINITY );
328 		}
329 	if( y < 0.0L )
330 		{
331 		if( yoddint )
332 			return( -0.0L );
333 		return( 0.0 );
334 		}
335 	}
336 
337 
338 nflg = 0;	/* flag = 1 if x<0 raised to integer power */
339 if( x <= 0.0L )
340 	{
341 	if( x == 0.0L )
342 		{
343 		if( y < 0.0 )
344 			{
345 			if( signbit(x) && yoddint )
346 				return( -INFINITY );
347 			return( INFINITY );
348 			}
349 		if( y > 0.0 )
350 			{
351 			if( signbit(x) && yoddint )
352 				return( -0.0L );
353 			return( 0.0 );
354 			}
355 		if( y == 0.0L )
356 			return( 1.0L );  /*   0**0   */
357 		else
358 			return( 0.0L );  /*   0**y   */
359 		}
360 	else
361 		{
362 		if( iyflg == 0 )
363 			return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
364 		nflg = 1;
365 		}
366 	}
367 
368 /* Integer power of an integer.  */
369 
370 if( iyflg )
371 	{
372 	i = w;
373 	w = floorl(x);
374 	if( (w == x) && (fabsl(y) < 32768.0) )
375 		{
376 		w = powil( x, (int) y );
377 		return( w );
378 		}
379 	}
380 
381 
382 if( nflg )
383 	x = fabsl(x);
384 
385 /* separate significand from exponent */
386 x = frexpl( x, &i );
387 e = i;
388 
389 /* find significand in antilog table A[] */
390 i = 1;
391 if( x <= douba(17) )
392 	i = 17;
393 if( x <= douba(i+8) )
394 	i += 8;
395 if( x <= douba(i+4) )
396 	i += 4;
397 if( x <= douba(i+2) )
398 	i += 2;
399 if( x >= douba(1) )
400 	i = -1;
401 i += 1;
402 
403 
404 /* Find (x - A[i])/A[i]
405  * in order to compute log(x/A[i]):
406  *
407  * log(x) = log( a x/a ) = log(a) + log(x/a)
408  *
409  * log(x/a) = log(1+v),  v = x/a - 1 = (x-a)/a
410  */
411 x -= douba(i);
412 x -= doubb(i/2);
413 x /= douba(i);
414 
415 
416 /* rational approximation for log(1+v):
417  *
418  * log(1+v)  =  v  -  v**2/2  +  v**3 P(v) / Q(v)
419  */
420 z = x*x;
421 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) );
422 w = w - ldexpl( z, -1 );   /*  w - 0.5 * z  */
423 
424 /* Convert to base 2 logarithm:
425  * multiply by log2(e) = 1 + LOG2EA
426  */
427 z = LOG2EA * w;
428 z += w;
429 z += LOG2EA * x;
430 z += x;
431 
432 /* Compute exponent term of the base 2 logarithm. */
433 w = -i;
434 w = ldexpl( w, -LNXT );	/* divide by NXT */
435 w += e;
436 /* Now base 2 log of x is w + z. */
437 
438 /* Multiply base 2 log by y, in extended precision. */
439 
440 /* separate y into large part ya
441  * and small part yb less than 1/NXT
442  */
443 ya = reducl(y);
444 yb = y - ya;
445 
446 /* (w+z)(ya+yb)
447  * = w*ya + w*yb + z*y
448  */
449 F = z * y  +  w * yb;
450 Fa = reducl(F);
451 Fb = F - Fa;
452 
453 G = Fa + w * ya;
454 Ga = reducl(G);
455 Gb = G - Ga;
456 
457 H = Fb + Gb;
458 Ha = reducl(H);
459 w = ldexpl( Ga+Ha, LNXT );
460 
461 /* Test the power of 2 for overflow */
462 if( w > MEXP )
463 	return (huge * huge);		/* overflow */
464 
465 if( w < MNEXP )
466 	return (twom10000 * twom10000);	/* underflow */
467 
468 e = w;
469 Hb = H - Ha;
470 
471 if( Hb > 0.0L )
472 	{
473 	e += 1;
474 	Hb -= (1.0L/NXT);  /*0.0625L;*/
475 	}
476 
477 /* Now the product y * log2(x)  =  Hb + e/NXT.
478  *
479  * Compute base 2 exponential of Hb,
480  * where -0.0625 <= Hb <= 0.
481  */
482 z = Hb * __polevll( Hb, R, 6 );  /*    z  =  2**Hb - 1    */
483 
484 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
485  * Find lookup table entry for the fractional power of 2.
486  */
487 if( e < 0 )
488 	i = 0;
489 else
490 	i = 1;
491 i = e/NXT + i;
492 e = NXT*i - e;
493 w = douba( e );
494 z = w * z;      /*    2**-e * ( 1 + (2**Hb-1) )    */
495 z = z + w;
496 z = ldexpl( z, i );  /* multiply by integer power of 2 */
497 
498 if( nflg )
499 	{
500 /* For negative x,
501  * find out if the integer exponent
502  * is odd or even.
503  */
504 	w = ldexpl( y, -1 );
505 	w = floorl(w);
506 	w = ldexpl( w, 1 );
507 	if( w != y )
508 		z = -z; /* odd exponent */
509 	}
510 
511 return( z );
512 }
513 
514 
515 /* Find a multiple of 1/NXT that is within 1/NXT of x. */
516 static inline long double
517 reducl(long double x)
518 {
519 long double t;
520 
521 t = ldexpl( x, LNXT );
522 t = floorl( t );
523 t = ldexpl( t, -LNXT );
524 return(t);
525 }
526 
527 /*							powil.c
528  *
529  *	Real raised to integer power, long double precision
530  *
531  *
532  *
533  * SYNOPSIS:
534  *
535  * long double x, y, powil();
536  * int n;
537  *
538  * y = powil( x, n );
539  *
540  *
541  *
542  * DESCRIPTION:
543  *
544  * Returns argument x raised to the nth power.
545  * The routine efficiently decomposes n as a sum of powers of
546  * two. The desired power is a product of two-to-the-kth
547  * powers of x.  Thus to compute the 32767 power of x requires
548  * 28 multiplications instead of 32767 multiplications.
549  *
550  *
551  *
552  * ACCURACY:
553  *
554  *
555  *                      Relative error:
556  * arithmetic   x domain   n domain  # trials      peak         rms
557  *    IEEE     .001,1000  -1022,1023  50000       4.3e-17     7.8e-18
558  *    IEEE        1,2     -1022,1023  20000       3.9e-17     7.6e-18
559  *    IEEE     .99,1.01     0,8700    10000       3.6e-16     7.2e-17
560  *
561  * Returns MAXNUM on overflow, zero on underflow.
562  *
563  */
564 
565 static long double
566 powil(long double x, int nn)
567 {
568 long double ww, y;
569 long double s;
570 int n, e, sign, asign, lx;
571 
572 if( x == 0.0L )
573 	{
574 	if( nn == 0 )
575 		return( 1.0L );
576 	else if( nn < 0 )
577 		return( LDBL_MAX );
578 	else
579 		return( 0.0L );
580 	}
581 
582 if( nn == 0 )
583 	return( 1.0L );
584 
585 
586 if( x < 0.0L )
587 	{
588 	asign = -1;
589 	x = -x;
590 	}
591 else
592 	asign = 0;
593 
594 
595 if( nn < 0 )
596 	{
597 	sign = -1;
598 	n = -nn;
599 	}
600 else
601 	{
602 	sign = 1;
603 	n = nn;
604 	}
605 
606 /* Overflow detection */
607 
608 /* Calculate approximate logarithm of answer */
609 s = x;
610 s = frexpl( s, &lx );
611 e = (lx - 1)*n;
612 if( (e == 0) || (e > 64) || (e < -64) )
613 	{
614 	s = (s - 7.0710678118654752e-1L) / (s +  7.0710678118654752e-1L);
615 	s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
616 	}
617 else
618 	{
619 	s = LOGE2L * e;
620 	}
621 
622 if( s > MAXLOGL )
623 	return (huge * huge);		/* overflow */
624 
625 if( s < MINLOGL )
626 	return (twom10000 * twom10000);	/* underflow */
627 /* Handle tiny denormal answer, but with less accuracy
628  * since roundoff error in 1.0/x will be amplified.
629  * The precise demarcation should be the gradual underflow threshold.
630  */
631 if( s < (-MAXLOGL+2.0L) )
632 	{
633 	x = 1.0L/x;
634 	sign = -sign;
635 	}
636 
637 /* First bit of the power */
638 if( n & 1 )
639 	y = x;
640 
641 else
642 	{
643 	y = 1.0L;
644 	asign = 0;
645 	}
646 
647 ww = x;
648 n >>= 1;
649 while( n )
650 	{
651 	ww = ww * ww;	/* arg to the 2-to-the-kth power */
652 	if( n & 1 )	/* if that bit is set, then include in product */
653 		y *= ww;
654 	n >>= 1;
655 	}
656 
657 if( asign )
658 	y = -y; /* odd power of negative number */
659 if( sign < 0 )
660 	y = 1.0L/y;
661 return(y);
662 }
663