1 /*-
2 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
3 *
4 * Permission to use, copy, modify, and distribute this software for any
5 * purpose with or without fee is hereby granted, provided that the above
6 * copyright notice and this permission notice appear in all copies.
7 *
8 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
9 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
10 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
11 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
12 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
13 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
14 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
15 */
16
17 #include <math.h>
18
19 #include "math_private.h"
20
21 /*
22 * Polynomial evaluator:
23 * P[0] x^n + P[1] x^(n-1) + ... + P[n]
24 */
25 static inline long double
__polevll(long double x,const long double * PP,int n)26 __polevll(long double x, const long double *PP, int n)
27 {
28 long double y;
29 const long double *P;
30
31 P = PP;
32 y = *P++;
33 do {
34 y = y * x + *P++;
35 } while (--n);
36
37 return (y);
38 }
39
40 /*
41 * Polynomial evaluator:
42 * x^n + P[0] x^(n-1) + P[1] x^(n-2) + ... + P[n]
43 */
44 static inline long double
__p1evll(long double x,const long double * PP,int n)45 __p1evll(long double x, const long double *PP, int n)
46 {
47 long double y;
48 const long double *P;
49
50 P = PP;
51 n -= 1;
52 y = x + *P++;
53 do {
54 y = y * x + *P++;
55 } while (--n);
56
57 return (y);
58 }
59
60 /* powl.c
61 *
62 * Power function, long double precision
63 *
64 *
65 *
66 * SYNOPSIS:
67 *
68 * long double x, y, z, powl();
69 *
70 * z = powl( x, y );
71 *
72 *
73 *
74 * DESCRIPTION:
75 *
76 * Computes x raised to the yth power. Analytically,
77 *
78 * x**y = exp( y log(x) ).
79 *
80 * Following Cody and Waite, this program uses a lookup table
81 * of 2**-i/32 and pseudo extended precision arithmetic to
82 * obtain several extra bits of accuracy in both the logarithm
83 * and the exponential.
84 *
85 *
86 *
87 * ACCURACY:
88 *
89 * The relative error of pow(x,y) can be estimated
90 * by y dl ln(2), where dl is the absolute error of
91 * the internally computed base 2 logarithm. At the ends
92 * of the approximation interval the logarithm equal 1/32
93 * and its relative error is about 1 lsb = 1.1e-19. Hence
94 * the predicted relative error in the result is 2.3e-21 y .
95 *
96 * Relative error:
97 * arithmetic domain # trials peak rms
98 *
99 * IEEE +-1000 40000 2.8e-18 3.7e-19
100 * .001 < x < 1000, with log(x) uniformly distributed.
101 * -1000 < y < 1000, y uniformly distributed.
102 *
103 * IEEE 0,8700 60000 6.5e-18 1.0e-18
104 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
105 *
106 *
107 * ERROR MESSAGES:
108 *
109 * message condition value returned
110 * pow overflow x**y > MAXNUM INFINITY
111 * pow underflow x**y < 1/MAXNUM 0.0
112 * pow domain x<0 and y noninteger 0.0
113 *
114 */
115
116 #include <float.h>
117 #include <math.h>
118
119 #include "math_private.h"
120
121 /* Table size */
122 #define NXT 32
123 /* log2(Table size) */
124 #define LNXT 5
125
126 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z)
127 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1
128 */
129 static const long double P[] = {
130 8.3319510773868690346226E-4L,
131 4.9000050881978028599627E-1L,
132 1.7500123722550302671919E0L,
133 1.4000100839971580279335E0L,
134 };
135 static const long double Q[] = {
136 /* 1.0000000000000000000000E0L,*/
137 5.2500282295834889175431E0L,
138 8.4000598057587009834666E0L,
139 4.2000302519914740834728E0L,
140 };
141 /* A[i] = 2^(-i/32), rounded to IEEE long double precision.
142 * If i is even, A[i] + B[i/2] gives additional accuracy.
143 */
144 static const long double A[33] = {
145 1.0000000000000000000000E0L,
146 9.7857206208770013448287E-1L,
147 9.5760328069857364691013E-1L,
148 9.3708381705514995065011E-1L,
149 9.1700404320467123175367E-1L,
150 8.9735453750155359320742E-1L,
151 8.7812608018664974155474E-1L,
152 8.5930964906123895780165E-1L,
153 8.4089641525371454301892E-1L,
154 8.2287773907698242225554E-1L,
155 8.0524516597462715409607E-1L,
156 7.8799042255394324325455E-1L,
157 7.7110541270397041179298E-1L,
158 7.5458221379671136985669E-1L,
159 7.3841307296974965571198E-1L,
160 7.2259040348852331001267E-1L,
161 7.0710678118654752438189E-1L,
162 6.9195494098191597746178E-1L,
163 6.7712777346844636413344E-1L,
164 6.6261832157987064729696E-1L,
165 6.4841977732550483296079E-1L,
166 6.3452547859586661129850E-1L,
167 6.2092890603674202431705E-1L,
168 6.0762367999023443907803E-1L,
169 5.9460355750136053334378E-1L,
170 5.8186242938878875689693E-1L,
171 5.6939431737834582684856E-1L,
172 5.5719337129794626814472E-1L,
173 5.4525386633262882960438E-1L,
174 5.3357020033841180906486E-1L,
175 5.2213689121370692017331E-1L,
176 5.1094857432705833910408E-1L,
177 5.0000000000000000000000E-1L,
178 };
179 static const long double B[17] = {
180 0.0000000000000000000000E0L,
181 2.6176170809902549338711E-20L,
182 -1.0126791927256478897086E-20L,
183 1.3438228172316276937655E-21L,
184 1.2207982955417546912101E-20L,
185 -6.3084814358060867200133E-21L,
186 1.3164426894366316434230E-20L,
187 -1.8527916071632873716786E-20L,
188 1.8950325588932570796551E-20L,
189 1.5564775779538780478155E-20L,
190 6.0859793637556860974380E-21L,
191 -2.0208749253662532228949E-20L,
192 1.4966292219224761844552E-20L,
193 3.3540909728056476875639E-21L,
194 -8.6987564101742849540743E-22L,
195 -1.2327176863327626135542E-20L,
196 0.0000000000000000000000E0L,
197 };
198
199 /* 2^x = 1 + x P(x),
200 * on the interval -1/32 <= x <= 0
201 */
202 static const long double R[] = {
203 1.5089970579127659901157E-5L,
204 1.5402715328927013076125E-4L,
205 1.3333556028915671091390E-3L,
206 9.6181291046036762031786E-3L,
207 5.5504108664798463044015E-2L,
208 2.4022650695910062854352E-1L,
209 6.9314718055994530931447E-1L,
210 };
211
212 #define douba(k) A[k]
213 #define doubb(k) B[k]
214 #define MEXP (NXT*16384.0L)
215 /* The following if denormal numbers are supported, else -MEXP: */
216 #define MNEXP (-NXT*(16384.0L+64.0L))
217 /* log2(e) - 1 */
218 #define LOG2EA 0.44269504088896340735992L
219
220 #define F W
221 #define Fa Wa
222 #define Fb Wb
223 #define G W
224 #define Ga Wa
225 #define Gb u
226 #define H W
227 #define Ha Wb
228 #define Hb Wb
229
230 static const long double MAXLOGL = 1.1356523406294143949492E4L;
231 static const long double MINLOGL = -1.13994985314888605586758E4L;
232 static const long double LOGE2L = 6.9314718055994530941723E-1L;
233 static _Thread_local volatile long double z;
234 static _Thread_local long double w, W, Wa, Wb, ya, yb, u;
235 static const long double huge = 0x1p10000L;
236 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */
237 static const long double twom10000 = 0x1p-10000L;
238 #else
239 static _Thread_local volatile long double twom10000 = 0x1p-10000L;
240 #endif
241
242 static long double reducl( long double );
243 static long double powil ( long double, int );
244
245 long double
powl(long double x,long double y)246 powl(long double x, long double y)
247 {
248 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
249 int i, nflg, iyflg, yoddint;
250 long e;
251
252 if( y == 0.0L )
253 return( 1.0L );
254
255 if( x == 1.0L )
256 return( 1.0L );
257
258 if( isnan(x) )
259 return ( nan_mix(x, y) );
260 if( isnan(y) )
261 return ( nan_mix(x, y) );
262
263 if( y == 1.0L )
264 return( x );
265
266 if( !isfinite(y) && x == -1.0L )
267 return( 1.0L );
268
269 if( y >= LDBL_MAX )
270 {
271 if( x > 1.0L )
272 return( INFINITY );
273 if( x > 0.0L && x < 1.0L )
274 return( 0.0L );
275 if( x < -1.0L )
276 return( INFINITY );
277 if( x > -1.0L && x < 0.0L )
278 return( 0.0L );
279 }
280 if( y <= -LDBL_MAX )
281 {
282 if( x > 1.0L )
283 return( 0.0L );
284 if( x > 0.0L && x < 1.0L )
285 return( INFINITY );
286 if( x < -1.0L )
287 return( 0.0L );
288 if( x > -1.0L && x < 0.0L )
289 return( INFINITY );
290 }
291 if( x >= LDBL_MAX )
292 {
293 if( y > 0.0L )
294 return( INFINITY );
295 return( 0.0L );
296 }
297
298 w = floorl(y);
299 /* Set iyflg to 1 if y is an integer. */
300 iyflg = 0;
301 if( w == y )
302 iyflg = 1;
303
304 /* Test for odd integer y. */
305 yoddint = 0;
306 if( iyflg )
307 {
308 ya = fabsl(y);
309 ya = floorl(0.5L * ya);
310 yb = 0.5L * fabsl(w);
311 if( ya != yb )
312 yoddint = 1;
313 }
314
315 if( x <= -LDBL_MAX )
316 {
317 if( y > 0.0L )
318 {
319 if( yoddint )
320 return( -INFINITY );
321 return( INFINITY );
322 }
323 if( y < 0.0L )
324 {
325 if( yoddint )
326 return( -0.0L );
327 return( 0.0 );
328 }
329 }
330
331
332 nflg = 0; /* flag = 1 if x<0 raised to integer power */
333 if( x <= 0.0L )
334 {
335 if( x == 0.0L )
336 {
337 if( y < 0.0 )
338 {
339 if( signbit(x) && yoddint )
340 return( -INFINITY );
341 return( INFINITY );
342 }
343 if( y > 0.0 )
344 {
345 if( signbit(x) && yoddint )
346 return( -0.0L );
347 return( 0.0 );
348 }
349 if( y == 0.0L )
350 return( 1.0L ); /* 0**0 */
351 else
352 return( 0.0L ); /* 0**y */
353 }
354 else
355 {
356 if( iyflg == 0 )
357 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
358 nflg = 1;
359 }
360 }
361
362 /* Integer power of an integer. */
363
364 if( iyflg )
365 {
366 i = w;
367 w = floorl(x);
368 if( (w == x) && (fabsl(y) < 32768.0) )
369 {
370 w = powil( x, (int) y );
371 return( w );
372 }
373 }
374
375
376 if( nflg )
377 x = fabsl(x);
378
379 /* separate significand from exponent */
380 x = frexpl( x, &i );
381 e = i;
382
383 /* find significand in antilog table A[] */
384 i = 1;
385 if( x <= douba(17) )
386 i = 17;
387 if( x <= douba(i+8) )
388 i += 8;
389 if( x <= douba(i+4) )
390 i += 4;
391 if( x <= douba(i+2) )
392 i += 2;
393 if( x >= douba(1) )
394 i = -1;
395 i += 1;
396
397
398 /* Find (x - A[i])/A[i]
399 * in order to compute log(x/A[i]):
400 *
401 * log(x) = log( a x/a ) = log(a) + log(x/a)
402 *
403 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
404 */
405 x -= douba(i);
406 x -= doubb(i/2);
407 x /= douba(i);
408
409
410 /* rational approximation for log(1+v):
411 *
412 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
413 */
414 z = x*x;
415 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) );
416 w = w - ldexpl( z, -1 ); /* w - 0.5 * z */
417
418 /* Convert to base 2 logarithm:
419 * multiply by log2(e) = 1 + LOG2EA
420 */
421 z = LOG2EA * w;
422 z += w;
423 z += LOG2EA * x;
424 z += x;
425
426 /* Compute exponent term of the base 2 logarithm. */
427 w = -i;
428 w = ldexpl( w, -LNXT ); /* divide by NXT */
429 w += e;
430 /* Now base 2 log of x is w + z. */
431
432 /* Multiply base 2 log by y, in extended precision. */
433
434 /* separate y into large part ya
435 * and small part yb less than 1/NXT
436 */
437 ya = reducl(y);
438 yb = y - ya;
439
440 /* (w+z)(ya+yb)
441 * = w*ya + w*yb + z*y
442 */
443 F = z * y + w * yb;
444 Fa = reducl(F);
445 Fb = F - Fa;
446
447 G = Fa + w * ya;
448 Ga = reducl(G);
449 Gb = G - Ga;
450
451 H = Fb + Gb;
452 Ha = reducl(H);
453 w = ldexpl( Ga+Ha, LNXT );
454
455 /* Test the power of 2 for overflow */
456 if( w > MEXP )
457 return (huge * huge); /* overflow */
458
459 if( w < MNEXP )
460 return (twom10000 * twom10000); /* underflow */
461
462 e = w;
463 Hb = H - Ha;
464
465 if( Hb > 0.0L )
466 {
467 e += 1;
468 Hb -= (1.0L/NXT); /*0.0625L;*/
469 }
470
471 /* Now the product y * log2(x) = Hb + e/NXT.
472 *
473 * Compute base 2 exponential of Hb,
474 * where -0.0625 <= Hb <= 0.
475 */
476 z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */
477
478 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
479 * Find lookup table entry for the fractional power of 2.
480 */
481 if( e < 0 )
482 i = 0;
483 else
484 i = 1;
485 i = e/NXT + i;
486 e = NXT*i - e;
487 w = douba( e );
488 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
489 z = z + w;
490 z = ldexpl( z, i ); /* multiply by integer power of 2 */
491
492 if( nflg )
493 {
494 /* For negative x,
495 * find out if the integer exponent
496 * is odd or even.
497 */
498 w = ldexpl( y, -1 );
499 w = floorl(w);
500 w = ldexpl( w, 1 );
501 if( w != y )
502 z = -z; /* odd exponent */
503 }
504
505 return( z );
506 }
507
508
509 /* Find a multiple of 1/NXT that is within 1/NXT of x. */
510 static inline long double
reducl(long double x)511 reducl(long double x)
512 {
513 long double t;
514
515 t = ldexpl( x, LNXT );
516 t = floorl( t );
517 t = ldexpl( t, -LNXT );
518 return(t);
519 }
520
521 /* powil.c
522 *
523 * Real raised to integer power, long double precision
524 *
525 *
526 *
527 * SYNOPSIS:
528 *
529 * long double x, y, powil();
530 * int n;
531 *
532 * y = powil( x, n );
533 *
534 *
535 *
536 * DESCRIPTION:
537 *
538 * Returns argument x raised to the nth power.
539 * The routine efficiently decomposes n as a sum of powers of
540 * two. The desired power is a product of two-to-the-kth
541 * powers of x. Thus to compute the 32767 power of x requires
542 * 28 multiplications instead of 32767 multiplications.
543 *
544 *
545 *
546 * ACCURACY:
547 *
548 *
549 * Relative error:
550 * arithmetic x domain n domain # trials peak rms
551 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
552 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
553 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
554 *
555 * Returns MAXNUM on overflow, zero on underflow.
556 *
557 */
558
559 static long double
powil(long double x,int nn)560 powil(long double x, int nn)
561 {
562 long double ww, y;
563 long double s;
564 int n, e, sign, asign, lx;
565
566 if( x == 0.0L )
567 {
568 if( nn == 0 )
569 return( 1.0L );
570 else if( nn < 0 )
571 return( LDBL_MAX );
572 else
573 return( 0.0L );
574 }
575
576 if( nn == 0 )
577 return( 1.0L );
578
579
580 if( x < 0.0L )
581 {
582 asign = -1;
583 x = -x;
584 }
585 else
586 asign = 0;
587
588
589 if( nn < 0 )
590 {
591 sign = -1;
592 n = -nn;
593 }
594 else
595 {
596 sign = 1;
597 n = nn;
598 }
599
600 /* Overflow detection */
601
602 /* Calculate approximate logarithm of answer */
603 s = x;
604 s = frexpl( s, &lx );
605 e = (lx - 1)*n;
606 if( (e == 0) || (e > 64) || (e < -64) )
607 {
608 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
609 s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
610 }
611 else
612 {
613 s = LOGE2L * e;
614 }
615
616 if( s > MAXLOGL )
617 return (huge * huge); /* overflow */
618
619 if( s < MINLOGL )
620 return (twom10000 * twom10000); /* underflow */
621 /* Handle tiny denormal answer, but with less accuracy
622 * since roundoff error in 1.0/x will be amplified.
623 * The precise demarcation should be the gradual underflow threshold.
624 */
625 if( s < (-MAXLOGL+2.0L) )
626 {
627 x = 1.0L/x;
628 sign = -sign;
629 }
630
631 /* First bit of the power */
632 if( n & 1 )
633 y = x;
634
635 else
636 {
637 y = 1.0L;
638 asign = 0;
639 }
640
641 ww = x;
642 n >>= 1;
643 while( n )
644 {
645 ww = ww * ww; /* arg to the 2-to-the-kth power */
646 if( n & 1 ) /* if that bit is set, then include in product */
647 y *= ww;
648 n >>= 1;
649 }
650
651 if( asign )
652 y = -y; /* odd power of negative number */
653 if( sign < 0 )
654 y = 1.0L/y;
655 return(y);
656 }
657