1/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */
2/*
3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
4 *
5 * Permission to use, copy, modify, and distribute this software for any
6 * purpose with or without fee is hereby granted, provided that the above
7 * copyright notice and this permission notice appear in all copies.
8 *
9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
16 */
17/*                                                      powl.c
18 *
19 *      Power function, long double precision
20 *
21 *
22 * SYNOPSIS:
23 *
24 * long double x, y, z, powl();
25 *
26 * z = powl( x, y );
27 *
28 *
29 * DESCRIPTION:
30 *
31 * Computes x raised to the yth power.  Analytically,
32 *
33 *      x**y  =  exp( y log(x) ).
34 *
35 * Following Cody and Waite, this program uses a lookup table
36 * of 2**-i/32 and pseudo extended precision arithmetic to
37 * obtain several extra bits of accuracy in both the logarithm
38 * and the exponential.
39 *
40 *
41 * ACCURACY:
42 *
43 * The relative error of pow(x,y) can be estimated
44 * by   y dl ln(2),   where dl is the absolute error of
45 * the internally computed base 2 logarithm.  At the ends
46 * of the approximation interval the logarithm equal 1/32
47 * and its relative error is about 1 lsb = 1.1e-19.  Hence
48 * the predicted relative error in the result is 2.3e-21 y .
49 *
50 *                      Relative error:
51 * arithmetic   domain     # trials      peak         rms
52 *
53 *    IEEE     +-1000       40000      2.8e-18      3.7e-19
54 * .001 < x < 1000, with log(x) uniformly distributed.
55 * -1000 < y < 1000, y uniformly distributed.
56 *
57 *    IEEE     0,8700       60000      6.5e-18      1.0e-18
58 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
59 *
60 *
61 * ERROR MESSAGES:
62 *
63 *   message         condition      value returned
64 * pow overflow     x**y > MAXNUM      INFINITY
65 * pow underflow   x**y < 1/MAXNUM       0.0
66 * pow domain      x<0 and y noninteger  0.0
67 *
68 */
69
70#include "libm.h"
71
72#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
73long double powl(long double x, long double y)
74{
75	return pow(x, y);
76}
77#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
78
79/* Table size */
80#define NXT 32
81
82/* log(1+x) =  x - .5x^2 + x^3 *  P(z)/Q(z)
83 * on the domain  2^(-1/32) - 1  <=  x  <=  2^(1/32) - 1
84 */
85static const long double P[] = {
86 8.3319510773868690346226E-4L,
87 4.9000050881978028599627E-1L,
88 1.7500123722550302671919E0L,
89 1.4000100839971580279335E0L,
90};
91static const long double Q[] = {
92/* 1.0000000000000000000000E0L,*/
93 5.2500282295834889175431E0L,
94 8.4000598057587009834666E0L,
95 4.2000302519914740834728E0L,
96};
97/* A[i] = 2^(-i/32), rounded to IEEE long double precision.
98 * If i is even, A[i] + B[i/2] gives additional accuracy.
99 */
100static const long double A[33] = {
101 1.0000000000000000000000E0L,
102 9.7857206208770013448287E-1L,
103 9.5760328069857364691013E-1L,
104 9.3708381705514995065011E-1L,
105 9.1700404320467123175367E-1L,
106 8.9735453750155359320742E-1L,
107 8.7812608018664974155474E-1L,
108 8.5930964906123895780165E-1L,
109 8.4089641525371454301892E-1L,
110 8.2287773907698242225554E-1L,
111 8.0524516597462715409607E-1L,
112 7.8799042255394324325455E-1L,
113 7.7110541270397041179298E-1L,
114 7.5458221379671136985669E-1L,
115 7.3841307296974965571198E-1L,
116 7.2259040348852331001267E-1L,
117 7.0710678118654752438189E-1L,
118 6.9195494098191597746178E-1L,
119 6.7712777346844636413344E-1L,
120 6.6261832157987064729696E-1L,
121 6.4841977732550483296079E-1L,
122 6.3452547859586661129850E-1L,
123 6.2092890603674202431705E-1L,
124 6.0762367999023443907803E-1L,
125 5.9460355750136053334378E-1L,
126 5.8186242938878875689693E-1L,
127 5.6939431737834582684856E-1L,
128 5.5719337129794626814472E-1L,
129 5.4525386633262882960438E-1L,
130 5.3357020033841180906486E-1L,
131 5.2213689121370692017331E-1L,
132 5.1094857432705833910408E-1L,
133 5.0000000000000000000000E-1L,
134};
135static const long double B[17] = {
136 0.0000000000000000000000E0L,
137 2.6176170809902549338711E-20L,
138-1.0126791927256478897086E-20L,
139 1.3438228172316276937655E-21L,
140 1.2207982955417546912101E-20L,
141-6.3084814358060867200133E-21L,
142 1.3164426894366316434230E-20L,
143-1.8527916071632873716786E-20L,
144 1.8950325588932570796551E-20L,
145 1.5564775779538780478155E-20L,
146 6.0859793637556860974380E-21L,
147-2.0208749253662532228949E-20L,
148 1.4966292219224761844552E-20L,
149 3.3540909728056476875639E-21L,
150-8.6987564101742849540743E-22L,
151-1.2327176863327626135542E-20L,
152 0.0000000000000000000000E0L,
153};
154
155/* 2^x = 1 + x P(x),
156 * on the interval -1/32 <= x <= 0
157 */
158static const long double R[] = {
159 1.5089970579127659901157E-5L,
160 1.5402715328927013076125E-4L,
161 1.3333556028915671091390E-3L,
162 9.6181291046036762031786E-3L,
163 5.5504108664798463044015E-2L,
164 2.4022650695910062854352E-1L,
165 6.9314718055994530931447E-1L,
166};
167
168#define MEXP (NXT*16384.0L)
169/* The following if denormal numbers are supported, else -MEXP: */
170#define MNEXP (-NXT*(16384.0L+64.0L))
171/* log2(e) - 1 */
172#define LOG2EA 0.44269504088896340735992L
173
174#define F W
175#define Fa Wa
176#define Fb Wb
177#define G W
178#define Ga Wa
179#define Gb u
180#define H W
181#define Ha Wb
182#define Hb Wb
183
184static const long double MAXLOGL = 1.1356523406294143949492E4L;
185static const long double MINLOGL = -1.13994985314888605586758E4L;
186static const long double LOGE2L = 6.9314718055994530941723E-1L;
187static const long double huge = 0x1p10000L;
188/* XXX Prevent gcc from erroneously constant folding this. */
189static const volatile long double twom10000 = 0x1p-10000L;
190
191static long double reducl(long double);
192static long double powil(long double, int);
193
194long double powl(long double x, long double y)
195{
196	/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
197	int i, nflg, iyflg, yoddint;
198	long e;
199	volatile long double z=0;
200	long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
201
202	/* make sure no invalid exception is raised by nan comparision */
203	if (isnan(x)) {
204		if (!isnan(y) && y == 0.0)
205			return 1.0;
206		return x;
207	}
208	if (isnan(y)) {
209		if (x == 1.0)
210			return 1.0;
211		return y;
212	}
213	if (x == 1.0)
214		return 1.0; /* 1**y = 1, even if y is nan */
215	if (x == -1.0 && !isfinite(y))
216		return 1.0; /* -1**inf = 1 */
217	if (y == 0.0)
218		return 1.0; /* x**0 = 1, even if x is nan */
219	if (y == 1.0)
220		return x;
221	if (y >= LDBL_MAX) {
222		if (x > 1.0 || x < -1.0)
223			return INFINITY;
224		if (x != 0.0)
225			return 0.0;
226	}
227	if (y <= -LDBL_MAX) {
228		if (x > 1.0 || x < -1.0)
229			return 0.0;
230		if (x != 0.0 || y == -INFINITY)
231			return INFINITY;
232	}
233	if (x >= LDBL_MAX) {
234		if (y > 0.0)
235			return INFINITY;
236		return 0.0;
237	}
238
239	w = floorl(y);
240
241	/* Set iyflg to 1 if y is an integer. */
242	iyflg = 0;
243	if (w == y)
244		iyflg = 1;
245
246	/* Test for odd integer y. */
247	yoddint = 0;
248	if (iyflg) {
249		ya = fabsl(y);
250		ya = floorl(0.5 * ya);
251		yb = 0.5 * fabsl(w);
252		if( ya != yb )
253			yoddint = 1;
254	}
255
256	if (x <= -LDBL_MAX) {
257		if (y > 0.0) {
258			if (yoddint)
259				return -INFINITY;
260			return INFINITY;
261		}
262		if (y < 0.0) {
263			if (yoddint)
264				return -0.0;
265			return 0.0;
266		}
267	}
268	nflg = 0; /* (x<0)**(odd int) */
269	if (x <= 0.0) {
270		if (x == 0.0) {
271			if (y < 0.0) {
272				if (signbit(x) && yoddint)
273					/* (-0.0)**(-odd int) = -inf, divbyzero */
274					return -1.0/0.0;
275				/* (+-0.0)**(negative) = inf, divbyzero */
276				return 1.0/0.0;
277			}
278			if (signbit(x) && yoddint)
279				return -0.0;
280			return 0.0;
281		}
282		if (iyflg == 0)
283			return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
284		/* (x<0)**(integer) */
285		if (yoddint)
286			nflg = 1; /* negate result */
287		x = -x;
288	}
289	/* (+integer)**(integer)  */
290	if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
291		w = powil(x, (int)y);
292		return nflg ? -w : w;
293	}
294
295	/* separate significand from exponent */
296	x = frexpl(x, &i);
297	e = i;
298
299	/* find significand in antilog table A[] */
300	i = 1;
301	if (x <= A[17])
302		i = 17;
303	if (x <= A[i+8])
304		i += 8;
305	if (x <= A[i+4])
306		i += 4;
307	if (x <= A[i+2])
308		i += 2;
309	if (x >= A[1])
310		i = -1;
311	i += 1;
312
313	/* Find (x - A[i])/A[i]
314	 * in order to compute log(x/A[i]):
315	 *
316	 * log(x) = log( a x/a ) = log(a) + log(x/a)
317	 *
318	 * log(x/a) = log(1+v),  v = x/a - 1 = (x-a)/a
319	 */
320	x -= A[i];
321	x -= B[i/2];
322	x /= A[i];
323
324	/* rational approximation for log(1+v):
325	 *
326	 * log(1+v)  =  v  -  v**2/2  +  v**3 P(v) / Q(v)
327	 */
328	z = x*x;
329	w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
330	w = w - 0.5*z;
331
332	/* Convert to base 2 logarithm:
333	 * multiply by log2(e) = 1 + LOG2EA
334	 */
335	z = LOG2EA * w;
336	z += w;
337	z += LOG2EA * x;
338	z += x;
339
340	/* Compute exponent term of the base 2 logarithm. */
341	w = -i;
342	w /= NXT;
343	w += e;
344	/* Now base 2 log of x is w + z. */
345
346	/* Multiply base 2 log by y, in extended precision. */
347
348	/* separate y into large part ya
349	 * and small part yb less than 1/NXT
350	 */
351	ya = reducl(y);
352	yb = y - ya;
353
354	/* (w+z)(ya+yb)
355	 * = w*ya + w*yb + z*y
356	 */
357	F = z * y  +  w * yb;
358	Fa = reducl(F);
359	Fb = F - Fa;
360
361	G = Fa + w * ya;
362	Ga = reducl(G);
363	Gb = G - Ga;
364
365	H = Fb + Gb;
366	Ha = reducl(H);
367	w = (Ga + Ha) * NXT;
368
369	/* Test the power of 2 for overflow */
370	if (w > MEXP)
371		return huge * huge;  /* overflow */
372	if (w < MNEXP)
373		return twom10000 * twom10000;  /* underflow */
374
375	e = w;
376	Hb = H - Ha;
377
378	if (Hb > 0.0) {
379		e += 1;
380		Hb -= 1.0/NXT;  /*0.0625L;*/
381	}
382
383	/* Now the product y * log2(x)  =  Hb + e/NXT.
384	 *
385	 * Compute base 2 exponential of Hb,
386	 * where -0.0625 <= Hb <= 0.
387	 */
388	z = Hb * __polevll(Hb, R, 6);  /*  z = 2**Hb - 1  */
389
390	/* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
391	 * Find lookup table entry for the fractional power of 2.
392	 */
393	if (e < 0)
394		i = 0;
395	else
396		i = 1;
397	i = e/NXT + i;
398	e = NXT*i - e;
399	w = A[e];
400	z = w * z;  /*  2**-e * ( 1 + (2**Hb-1) )  */
401	z = z + w;
402	z = scalbnl(z, i);  /* multiply by integer power of 2 */
403
404	if (nflg)
405		z = -z;
406	return z;
407}
408
409
410/* Find a multiple of 1/NXT that is within 1/NXT of x. */
411static long double reducl(long double x)
412{
413	long double t;
414
415	t = x * NXT;
416	t = floorl(t);
417	t = t / NXT;
418	return t;
419}
420
421/*
422 *      Positive real raised to integer power, long double precision
423 *
424 *
425 * SYNOPSIS:
426 *
427 * long double x, y, powil();
428 * int n;
429 *
430 * y = powil( x, n );
431 *
432 *
433 * DESCRIPTION:
434 *
435 * Returns argument x>0 raised to the nth power.
436 * The routine efficiently decomposes n as a sum of powers of
437 * two. The desired power is a product of two-to-the-kth
438 * powers of x.  Thus to compute the 32767 power of x requires
439 * 28 multiplications instead of 32767 multiplications.
440 *
441 *
442 * ACCURACY:
443 *
444 *                      Relative error:
445 * arithmetic   x domain   n domain  # trials      peak         rms
446 *    IEEE     .001,1000  -1022,1023  50000       4.3e-17     7.8e-18
447 *    IEEE        1,2     -1022,1023  20000       3.9e-17     7.6e-18
448 *    IEEE     .99,1.01     0,8700    10000       3.6e-16     7.2e-17
449 *
450 * Returns MAXNUM on overflow, zero on underflow.
451 */
452
453static long double powil(long double x, int nn)
454{
455	long double ww, y;
456	long double s;
457	int n, e, sign, lx;
458
459	if (nn == 0)
460		return 1.0;
461
462	if (nn < 0) {
463		sign = -1;
464		n = -nn;
465	} else {
466		sign = 1;
467		n = nn;
468	}
469
470	/* Overflow detection */
471
472	/* Calculate approximate logarithm of answer */
473	s = x;
474	s = frexpl( s, &lx);
475	e = (lx - 1)*n;
476	if ((e == 0) || (e > 64) || (e < -64)) {
477		s = (s - 7.0710678118654752e-1L) / (s +  7.0710678118654752e-1L);
478		s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
479	} else {
480		s = LOGE2L * e;
481	}
482
483	if (s > MAXLOGL)
484		return huge * huge;  /* overflow */
485
486	if (s < MINLOGL)
487		return twom10000 * twom10000;  /* underflow */
488	/* Handle tiny denormal answer, but with less accuracy
489	 * since roundoff error in 1.0/x will be amplified.
490	 * The precise demarcation should be the gradual underflow threshold.
491	 */
492	if (s < -MAXLOGL+2.0) {
493		x = 1.0/x;
494		sign = -sign;
495	}
496
497	/* First bit of the power */
498	if (n & 1)
499		y = x;
500	else
501		y = 1.0;
502
503	ww = x;
504	n >>= 1;
505	while (n) {
506		ww = ww * ww;   /* arg to the 2-to-the-kth power */
507		if (n & 1)     /* if that bit is set, then include in product */
508			y *= ww;
509		n >>= 1;
510	}
511
512	if (sign < 0)
513		y = 1.0/y;
514	return y;
515}
516#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
517// TODO: broken implementation to make things compile
518long double powl(long double x, long double y)
519{
520	return pow(x, y);
521}
522#endif
523