1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */
2 /*
3 * Copyright (c) 2008 Stephen L. Moshier <[email protected]>
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
powl(long double x,long double y)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
powl(long double x,long double y)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 (y == 0.0)
216 return 1.0; /* x**0 = 1, even if x is nan */
217 if (y == 1.0)
218 return x;
219 /* if y*log2(x) < log2(LDBL_TRUE_MIN)-1 then x^y uflows to 0
220 if y*log2(x) > -log2(LDBL_TRUE_MIN)+1 > LDBL_MAX_EXP then x^y oflows
221 if |x|!=1 then |log2(x)| > |log(x)| > LDBL_EPSILON/2 so
222 x^y oflows/uflows if |y|*LDBL_EPSILON/2 > -log2(LDBL_TRUE_MIN)+1 */
223 if (fabsl(y) > 2*(-LDBL_MIN_EXP+LDBL_MANT_DIG+1)/LDBL_EPSILON) {
224 /* y is not an odd int */
225 if (x == -1.0)
226 return 1.0;
227 if (y == INFINITY) {
228 if (x > 1.0 || x < -1.0)
229 return INFINITY;
230 return 0.0;
231 }
232 if (y == -INFINITY) {
233 if (x > 1.0 || x < -1.0)
234 return 0.0;
235 return INFINITY;
236 }
237 if ((x > 1.0 || x < -1.0) == (y > 0))
238 return huge * huge;
239 return twom10000 * twom10000;
240 }
241 if (x == INFINITY) {
242 if (y > 0.0)
243 return INFINITY;
244 return 0.0;
245 }
246
247 w = floorl(y);
248
249 /* Set iyflg to 1 if y is an integer. */
250 iyflg = 0;
251 if (w == y)
252 iyflg = 1;
253
254 /* Test for odd integer y. */
255 yoddint = 0;
256 if (iyflg) {
257 ya = fabsl(y);
258 ya = floorl(0.5 * ya);
259 yb = 0.5 * fabsl(w);
260 if( ya != yb )
261 yoddint = 1;
262 }
263
264 if (x == -INFINITY) {
265 if (y > 0.0) {
266 if (yoddint)
267 return -INFINITY;
268 return INFINITY;
269 }
270 if (y < 0.0) {
271 if (yoddint)
272 return -0.0;
273 return 0.0;
274 }
275 }
276 nflg = 0; /* (x<0)**(odd int) */
277 if (x <= 0.0) {
278 if (x == 0.0) {
279 if (y < 0.0) {
280 if (signbit(x) && yoddint)
281 /* (-0.0)**(-odd int) = -inf, divbyzero */
282 return -1.0/0.0;
283 /* (+-0.0)**(negative) = inf, divbyzero */
284 return 1.0/0.0;
285 }
286 if (signbit(x) && yoddint)
287 return -0.0;
288 return 0.0;
289 }
290 if (iyflg == 0)
291 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
292 /* (x<0)**(integer) */
293 if (yoddint)
294 nflg = 1; /* negate result */
295 x = -x;
296 }
297 /* (+integer)**(integer) */
298 if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
299 w = powil(x, (int)y);
300 return nflg ? -w : w;
301 }
302
303 /* separate significand from exponent */
304 x = frexpl(x, &i);
305 e = i;
306
307 /* find significand in antilog table A[] */
308 i = 1;
309 if (x <= A[17])
310 i = 17;
311 if (x <= A[i+8])
312 i += 8;
313 if (x <= A[i+4])
314 i += 4;
315 if (x <= A[i+2])
316 i += 2;
317 if (x >= A[1])
318 i = -1;
319 i += 1;
320
321 /* Find (x - A[i])/A[i]
322 * in order to compute log(x/A[i]):
323 *
324 * log(x) = log( a x/a ) = log(a) + log(x/a)
325 *
326 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
327 */
328 x -= A[i];
329 x -= B[i/2];
330 x /= A[i];
331
332 /* rational approximation for log(1+v):
333 *
334 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
335 */
336 z = x*x;
337 w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
338 w = w - 0.5*z;
339
340 /* Convert to base 2 logarithm:
341 * multiply by log2(e) = 1 + LOG2EA
342 */
343 z = LOG2EA * w;
344 z += w;
345 z += LOG2EA * x;
346 z += x;
347
348 /* Compute exponent term of the base 2 logarithm. */
349 w = -i;
350 w /= NXT;
351 w += e;
352 /* Now base 2 log of x is w + z. */
353
354 /* Multiply base 2 log by y, in extended precision. */
355
356 /* separate y into large part ya
357 * and small part yb less than 1/NXT
358 */
359 ya = reducl(y);
360 yb = y - ya;
361
362 /* (w+z)(ya+yb)
363 * = w*ya + w*yb + z*y
364 */
365 F = z * y + w * yb;
366 Fa = reducl(F);
367 Fb = F - Fa;
368
369 G = Fa + w * ya;
370 Ga = reducl(G);
371 Gb = G - Ga;
372
373 H = Fb + Gb;
374 Ha = reducl(H);
375 w = (Ga + Ha) * NXT;
376
377 /* Test the power of 2 for overflow */
378 if (w > MEXP)
379 return huge * huge; /* overflow */
380 if (w < MNEXP)
381 return twom10000 * twom10000; /* underflow */
382
383 e = w;
384 Hb = H - Ha;
385
386 if (Hb > 0.0) {
387 e += 1;
388 Hb -= 1.0/NXT; /*0.0625L;*/
389 }
390
391 /* Now the product y * log2(x) = Hb + e/NXT.
392 *
393 * Compute base 2 exponential of Hb,
394 * where -0.0625 <= Hb <= 0.
395 */
396 z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */
397
398 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
399 * Find lookup table entry for the fractional power of 2.
400 */
401 if (e < 0)
402 i = 0;
403 else
404 i = 1;
405 i = e/NXT + i;
406 e = NXT*i - e;
407 w = A[e];
408 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
409 z = z + w;
410 z = scalbnl(z, i); /* multiply by integer power of 2 */
411
412 if (nflg)
413 z = -z;
414 return z;
415 }
416
417
418 /* Find a multiple of 1/NXT that is within 1/NXT of x. */
reducl(long double x)419 static long double reducl(long double x)
420 {
421 long double t;
422
423 t = x * NXT;
424 t = floorl(t);
425 t = t / NXT;
426 return t;
427 }
428
429 /*
430 * Positive real raised to integer power, long double precision
431 *
432 *
433 * SYNOPSIS:
434 *
435 * long double x, y, powil();
436 * int n;
437 *
438 * y = powil( x, n );
439 *
440 *
441 * DESCRIPTION:
442 *
443 * Returns argument x>0 raised to the nth power.
444 * The routine efficiently decomposes n as a sum of powers of
445 * two. The desired power is a product of two-to-the-kth
446 * powers of x. Thus to compute the 32767 power of x requires
447 * 28 multiplications instead of 32767 multiplications.
448 *
449 *
450 * ACCURACY:
451 *
452 * Relative error:
453 * arithmetic x domain n domain # trials peak rms
454 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
455 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
456 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
457 *
458 * Returns MAXNUM on overflow, zero on underflow.
459 */
460
powil(long double x,int nn)461 static long double powil(long double x, int nn)
462 {
463 long double ww, y;
464 long double s;
465 int n, e, sign, lx;
466
467 if (nn == 0)
468 return 1.0;
469
470 if (nn < 0) {
471 sign = -1;
472 n = -nn;
473 } else {
474 sign = 1;
475 n = nn;
476 }
477
478 /* Overflow detection */
479
480 /* Calculate approximate logarithm of answer */
481 s = x;
482 s = frexpl( s, &lx);
483 e = (lx - 1)*n;
484 if ((e == 0) || (e > 64) || (e < -64)) {
485 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
486 s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
487 } else {
488 s = LOGE2L * e;
489 }
490
491 if (s > MAXLOGL)
492 return huge * huge; /* overflow */
493
494 if (s < MINLOGL)
495 return twom10000 * twom10000; /* underflow */
496 /* Handle tiny denormal answer, but with less accuracy
497 * since roundoff error in 1.0/x will be amplified.
498 * The precise demarcation should be the gradual underflow threshold.
499 */
500 if (s < -MAXLOGL+2.0) {
501 x = 1.0/x;
502 sign = -sign;
503 }
504
505 /* First bit of the power */
506 if (n & 1)
507 y = x;
508 else
509 y = 1.0;
510
511 ww = x;
512 n >>= 1;
513 while (n) {
514 ww = ww * ww; /* arg to the 2-to-the-kth power */
515 if (n & 1) /* if that bit is set, then include in product */
516 y *= ww;
517 n >>= 1;
518 }
519
520 if (sign < 0)
521 y = 1.0/y;
522 return y;
523 }
524 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
525 // TODO: broken implementation to make things compile
powl(long double x,long double y)526 long double powl(long double x, long double y)
527 {
528 return pow(x, y);
529 }
530 #endif
531