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 */
88static long double P[] = {
89 8.3319510773868690346226E-4L,
90 4.9000050881978028599627E-1L,
91 1.7500123722550302671919E0L,
92 1.4000100839971580279335E0L,
93};
94static 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 */
103static 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};
138static 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 */
161static 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
189static const long double MAXLOGL = 1.1356523406294143949492E4L;
190static const long double MINLOGL = -1.13994985314888605586758E4L;
191static const long double LOGE2L = 6.9314718055994530941723E-1L;
192static volatile long double z;
193static long double w, W, Wa, Wb, ya, yb, u;
194static const long double huge = 0x1p10000L;
195#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
196static const long double twom10000 = 0x1p-10000L;
197#else
198static volatile long double twom10000 = 0x1p-10000L;
199#endif
200
201static long double reducl( long double );
202static long double powil ( long double, int );
203
204long double
205powl(long double x, long double y)
206{
207/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
208int i, nflg, iyflg, yoddint;
209long e;
210
211if( y == 0.0L )
212	return( 1.0L );
213
214if( x == 1.0L )
215	return( 1.0L );
216
217if( isnan(x) )
218	return( x );
219if( isnan(y) )
220	return( y );
221
222if( y == 1.0L )
223	return( x );
224
225if( !isfinite(y) && x == -1.0L )
226	return( 1.0L );
227
228if( 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	}
239if( 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	}
250if( x >= LDBL_MAX )
251	{
252	if( y > 0.0L )
253		return( INFINITY );
254	return( 0.0L );
255	}
256
257w = floorl(y);
258/* Set iyflg to 1 if y is an integer.  */
259iyflg = 0;
260if( w == y )
261	iyflg = 1;
262
263/* Test for odd integer y.  */
264yoddint = 0;
265if( 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
274if( 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
291nflg = 0;	/* flag = 1 if x<0 raised to integer power */
292if( 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
323if( 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
335if( nflg )
336	x = fabsl(x);
337
338/* separate significand from exponent */
339x = frexpl( x, &i );
340e = i;
341
342/* find significand in antilog table A[] */
343i = 1;
344if( x <= douba(17) )
345	i = 17;
346if( x <= douba(i+8) )
347	i += 8;
348if( x <= douba(i+4) )
349	i += 4;
350if( x <= douba(i+2) )
351	i += 2;
352if( x >= douba(1) )
353	i = -1;
354i += 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 */
364x -= douba(i);
365x -= doubb(i/2);
366x /= 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 */
373z = x*x;
374w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) );
375w = w - ldexpl( z, -1 );   /*  w - 0.5 * z  */
376
377/* Convert to base 2 logarithm:
378 * multiply by log2(e) = 1 + LOG2EA
379 */
380z = LOG2EA * w;
381z += w;
382z += LOG2EA * x;
383z += x;
384
385/* Compute exponent term of the base 2 logarithm. */
386w = -i;
387w = ldexpl( w, -LNXT );	/* divide by NXT */
388w += 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 */
396ya = reducl(y);
397yb = y - ya;
398
399/* (w+z)(ya+yb)
400 * = w*ya + w*yb + z*y
401 */
402F = z * y  +  w * yb;
403Fa = reducl(F);
404Fb = F - Fa;
405
406G = Fa + w * ya;
407Ga = reducl(G);
408Gb = G - Ga;
409
410H = Fb + Gb;
411Ha = reducl(H);
412w = ldexpl( Ga+Ha, LNXT );
413
414/* Test the power of 2 for overflow */
415if( w > MEXP )
416	return (huge * huge);		/* overflow */
417
418if( w < MNEXP )
419	return (twom10000 * twom10000);	/* underflow */
420
421e = w;
422Hb = H - Ha;
423
424if( 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 */
435z = 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 */
440if( e < 0 )
441	i = 0;
442else
443	i = 1;
444i = e/NXT + i;
445e = NXT*i - e;
446w = douba( e );
447z = w * z;      /*    2**-e * ( 1 + (2**Hb-1) )    */
448z = z + w;
449z = ldexpl( z, i );  /* multiply by integer power of 2 */
450
451if( 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
464return( z );
465}
466DEF_STD(powl);
467
468
469/* Find a multiple of 1/NXT that is within 1/NXT of x. */
470static long double
471reducl(long double x)
472{
473long double t;
474
475t = ldexpl( x, LNXT );
476t = floorl( t );
477t = ldexpl( t, -LNXT );
478return(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
519static long double
520powil(long double x, int nn)
521{
522long double ww, y;
523long double s;
524int n, e, sign, asign, lx;
525
526if( 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
536if( nn == 0 )
537	return( 1.0L );
538
539
540if( x < 0.0L )
541	{
542	asign = -1;
543	x = -x;
544	}
545else
546	asign = 0;
547
548
549if( nn < 0 )
550	{
551	sign = -1;
552	n = -nn;
553	}
554else
555	{
556	sign = 1;
557	n = nn;
558	}
559
560/* Overflow detection */
561
562/* Calculate approximate logarithm of answer */
563s = x;
564s = frexpl( s, &lx );
565e = (lx - 1)*n;
566if( (e == 0) || (e > 64) || (e < -64) )
567	{
568	s = (s - 7.0710678118654752e-1L) / (s +  7.0710678118654752e-1L);
569	s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
570	}
571else
572	{
573	s = LOGE2L * e;
574	}
575
576if( s > MAXLOGL )
577	return (huge * huge);		/* overflow */
578
579if( 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 */
585if( s < (-MAXLOGL+2.0L) )
586	{
587	x = 1.0L/x;
588	sign = -sign;
589	}
590
591/* First bit of the power */
592if( n & 1 )
593	y = x;
594
595else
596	{
597	y = 1.0L;
598	asign = 0;
599	}
600
601ww = x;
602n >>= 1;
603while( n )
604	{
605	ww = ww * ww;	/* arg to the 2-to-the-kth power */
606	if( n & 1 )	/* if that bit is set, then include in product */
607		y *= ww;
608	n >>= 1;
609	}
610
611if( asign )
612	y = -y; /* odd power of negative number */
613if( sign < 0 )
614	y = 1.0L/y;
615return(y);
616}
617