powl.c raw
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
73 long 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 */
85 static const long double P[] = {
86 8.3319510773868690346226E-4L,
87 4.9000050881978028599627E-1L,
88 1.7500123722550302671919E0L,
89 1.4000100839971580279335E0L,
90 };
91 static 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 */
100 static 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 };
135 static 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 */
158 static 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
184 static const long double MAXLOGL = 1.1356523406294143949492E4L;
185 static const long double MINLOGL = -1.13994985314888605586758E4L;
186 static const long double LOGE2L = 6.9314718055994530941723E-1L;
187 static const long double huge = 0x1p10000L;
188 /* XXX Prevent gcc from erroneously constant folding this. */
189 static const volatile long double twom10000 = 0x1p-10000L;
190
191 static long double reducl(long double);
192 static long double powil(long double, int);
193
194 long 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. */
411 static 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
453 static 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
518 long double powl(long double x, long double y)
519 {
520 return pow(x, y);
521 }
522 #endif
523