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. */
189#ifdef __wasilibc_unmodified_upstream // WASI doesn't need old GCC workarounds
190static const volatile long double twom10000 = 0x1p-10000L;
191#else
192static const long double twom10000 = 0x1p-10000L;
193#endif
194
195static long double reducl(long double);
196static long double powil(long double, int);
197
198long double powl(long double x, long double y)
199{
200 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
201 int i, nflg, iyflg, yoddint;
202 long e;
203 volatile long double z=0;
204 long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
205
206 /* make sure no invalid exception is raised by nan comparision */
207 if (isnan(x)) {
208 if (!isnan(y) && y == 0.0)
209 return 1.0;
210 return x;
211 }
212 if (isnan(y)) {
213 if (x == 1.0)
214 return 1.0;
215 return y;
216 }
217 if (x == 1.0)
218 return 1.0; /* 1**y = 1, even if y is nan */
219 if (y == 0.0)
220 return 1.0; /* x**0 = 1, even if x is nan */
221 if (y == 1.0)
222 return x;
223 /* if y*log2(x) < log2(LDBL_TRUE_MIN)-1 then x^y uflows to 0
224 if y*log2(x) > -log2(LDBL_TRUE_MIN)+1 > LDBL_MAX_EXP then x^y oflows
225 if |x|!=1 then |log2(x)| > |log(x)| > LDBL_EPSILON/2 so
226 x^y oflows/uflows if |y|*LDBL_EPSILON/2 > -log2(LDBL_TRUE_MIN)+1 */
227 if (fabsl(y) > 2*(-LDBL_MIN_EXP+LDBL_MANT_DIG+1)/LDBL_EPSILON) {
228 /* y is not an odd int */
229 if (x == -1.0)
230 return 1.0;
231 if (y == INFINITY) {
232 if (x > 1.0 || x < -1.0)
233 return INFINITY;
234 return 0.0;
235 }
236 if (y == -INFINITY) {
237 if (x > 1.0 || x < -1.0)
238 return 0.0;
239 return INFINITY;
240 }
241 if ((x > 1.0 || x < -1.0) == (y > 0))
242 return huge * huge;
243 return twom10000 * twom10000;
244 }
245 if (x == INFINITY) {
246 if (y > 0.0)
247 return INFINITY;
248 return 0.0;
249 }
250
251 w = floorl(y);
252
253 /* Set iyflg to 1 if y is an integer. */
254 iyflg = 0;
255 if (w == y)
256 iyflg = 1;
257
258 /* Test for odd integer y. */
259 yoddint = 0;
260 if (iyflg) {
261 ya = fabsl(y);
262 ya = floorl(0.5 * ya);
263 yb = 0.5 * fabsl(w);
264 if( ya != yb )
265 yoddint = 1;
266 }
267
268 if (x == -INFINITY) {
269 if (y > 0.0) {
270 if (yoddint)
271 return -INFINITY;
272 return INFINITY;
273 }
274 if (y < 0.0) {
275 if (yoddint)
276 return -0.0;
277 return 0.0;
278 }
279 }
280 nflg = 0; /* (x<0)**(odd int) */
281 if (x <= 0.0) {
282 if (x == 0.0) {
283 if (y < 0.0) {
284 if (signbit(x) && yoddint)
285 /* (-0.0)**(-odd int) = -inf, divbyzero */
286 return -1.0/0.0;
287 /* (+-0.0)**(negative) = inf, divbyzero */
288 return 1.0/0.0;
289 }
290 if (signbit(x) && yoddint)
291 return -0.0;
292 return 0.0;
293 }
294 if (iyflg == 0)
295 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
296 /* (x<0)**(integer) */
297 if (yoddint)
298 nflg = 1; /* negate result */
299 x = -x;
300 }
301 /* (+integer)**(integer) */
302 if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
303 w = powil(x, (int)y);
304 return nflg ? -w : w;
305 }
306
307 /* separate significand from exponent */
308 x = frexpl(x, &i);
309 e = i;
310
311 /* find significand in antilog table A[] */
312 i = 1;
313 if (x <= A[17])
314 i = 17;
315 if (x <= A[i+8])
316 i += 8;
317 if (x <= A[i+4])
318 i += 4;
319 if (x <= A[i+2])
320 i += 2;
321 if (x >= A[1])
322 i = -1;
323 i += 1;
324
325 /* Find (x - A[i])/A[i]
326 * in order to compute log(x/A[i]):
327 *
328 * log(x) = log( a x/a ) = log(a) + log(x/a)
329 *
330 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
331 */
332 x -= A[i];
333 x -= B[i/2];
334 x /= A[i];
335
336 /* rational approximation for log(1+v):
337 *
338 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
339 */
340 z = x*x;
341 w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
342 w = w - 0.5*z;
343
344 /* Convert to base 2 logarithm:
345 * multiply by log2(e) = 1 + LOG2EA
346 */
347 z = LOG2EA * w;
348 z += w;
349 z += LOG2EA * x;
350 z += x;
351
352 /* Compute exponent term of the base 2 logarithm. */
353 w = -i;
354 w /= NXT;
355 w += e;
356 /* Now base 2 log of x is w + z. */
357
358 /* Multiply base 2 log by y, in extended precision. */
359
360 /* separate y into large part ya
361 * and small part yb less than 1/NXT
362 */
363 ya = reducl(y);
364 yb = y - ya;
365
366 /* (w+z)(ya+yb)
367 * = w*ya + w*yb + z*y
368 */
369 F = z * y + w * yb;
370 Fa = reducl(F);
371 Fb = F - Fa;
372
373 G = Fa + w * ya;
374 Ga = reducl(G);
375 Gb = G - Ga;
376
377 H = Fb + Gb;
378 Ha = reducl(H);
379 w = (Ga + Ha) * NXT;
380
381 /* Test the power of 2 for overflow */
382 if (w > MEXP)
383 return huge * huge; /* overflow */
384 if (w < MNEXP)
385 return twom10000 * twom10000; /* underflow */
386
387 e = w;
388 Hb = H - Ha;
389
390 if (Hb > 0.0) {
391 e += 1;
392 Hb -= 1.0/NXT; /*0.0625L;*/
393 }
394
395 /* Now the product y * log2(x) = Hb + e/NXT.
396 *
397 * Compute base 2 exponential of Hb,
398 * where -0.0625 <= Hb <= 0.
399 */
400 z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */
401
402 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
403 * Find lookup table entry for the fractional power of 2.
404 */
405 if (e < 0)
406 i = 0;
407 else
408 i = 1;
409 i = e/NXT + i;
410 e = NXT*i - e;
411 w = A[e];
412 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
413 z = z + w;
414 z = scalbnl(z, i); /* multiply by integer power of 2 */
415
416 if (nflg)
417 z = -z;
418 return z;
419}
420
421
422/* Find a multiple of 1/NXT that is within 1/NXT of x. */
423static long double reducl(long double x)
424{
425 long double t;
426
427 t = x * NXT;
428 t = floorl(t);
429 t = t / NXT;
430 return t;
431}
432
433/*
434 * Positive real raised to integer power, long double precision
435 *
436 *
437 * SYNOPSIS:
438 *
439 * long double x, y, powil();
440 * int n;
441 *
442 * y = powil( x, n );
443 *
444 *
445 * DESCRIPTION:
446 *
447 * Returns argument x>0 raised to the nth power.
448 * The routine efficiently decomposes n as a sum of powers of
449 * two. The desired power is a product of two-to-the-kth
450 * powers of x. Thus to compute the 32767 power of x requires
451 * 28 multiplications instead of 32767 multiplications.
452 *
453 *
454 * ACCURACY:
455 *
456 * Relative error:
457 * arithmetic x domain n domain # trials peak rms
458 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
459 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
460 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
461 *
462 * Returns MAXNUM on overflow, zero on underflow.
463 */
464
465static long double powil(long double x, int nn)
466{
467 long double ww, y;
468 long double s;
469 int n, e, sign, lx;
470
471 if (nn == 0)
472 return 1.0;
473
474 if (nn < 0) {
475 sign = -1;
476 n = -nn;
477 } else {
478 sign = 1;
479 n = nn;
480 }
481
482 /* Overflow detection */
483
484 /* Calculate approximate logarithm of answer */
485 s = x;
486 s = frexpl( s, &lx);
487 e = (lx - 1)*n;
488 if ((e == 0) || (e > 64) || (e < -64)) {
489 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
490 s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
491 } else {
492 s = LOGE2L * e;
493 }
494
495 if (s > MAXLOGL)
496 return huge * huge; /* overflow */
497
498 if (s < MINLOGL)
499 return twom10000 * twom10000; /* underflow */
500 /* Handle tiny denormal answer, but with less accuracy
501 * since roundoff error in 1.0/x will be amplified.
502 * The precise demarcation should be the gradual underflow threshold.
503 */
504 if (s < -MAXLOGL+2.0) {
505 x = 1.0/x;
506 sign = -sign;
507 }
508
509 /* First bit of the power */
510 if (n & 1)
511 y = x;
512 else
513 y = 1.0;
514
515 ww = x;
516 n >>= 1;
517 while (n) {
518 ww = ww * ww; /* arg to the 2-to-the-kth power */
519 if (n & 1) /* if that bit is set, then include in product */
520 y *= ww;
521 n >>= 1;
522 }
523
524 if (sign < 0)
525 y = 1.0/y;
526 return y;
527}
528#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
529// TODO: broken implementation to make things compile
530long double powl(long double x, long double y)
531{
532 return pow(x, y);
533}
534#endif