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