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 <sys/cdefs.h>
18 #include <math.h>
19
20 #include "math_private.h"
21
22 /*
23 * Polynomial evaluator:
24 * P[0] x^n + P[1] x^(n-1) + ... + P[n]
25 */
26 static inline long double
__polevll(long double x,long double * PP,int n)27 __polevll(long double x, long double *PP, int n)
28 {
29 long double y;
30 long double *P;
31
32 P = PP;
33 y = *P++;
34 do {
35 y = y * x + *P++;
36 } while (--n);
37
38 return (y);
39 }
40
41 /*
42 * Polynomial evaluator:
43 * x^n + P[0] x^(n-1) + P[1] x^(n-2) + ... + P[n]
44 */
45 static inline long double
__p1evll(long double x,long double * PP,int n)46 __p1evll(long double x, long double *PP, int n)
47 {
48 long double y;
49 long double *P;
50
51 P = PP;
52 n -= 1;
53 y = x + *P++;
54 do {
55 y = y * x + *P++;
56 } while (--n);
57
58 return (y);
59 }
60
61 /* powl.c
62 *
63 * Power function, long double precision
64 *
65 *
66 *
67 * SYNOPSIS:
68 *
69 * long double x, y, z, powl();
70 *
71 * z = powl( x, y );
72 *
73 *
74 *
75 * DESCRIPTION:
76 *
77 * Computes x raised to the yth power. Analytically,
78 *
79 * x**y = exp( y log(x) ).
80 *
81 * Following Cody and Waite, this program uses a lookup table
82 * of 2**-i/32 and pseudo extended precision arithmetic to
83 * obtain several extra bits of accuracy in both the logarithm
84 * and the exponential.
85 *
86 *
87 *
88 * ACCURACY:
89 *
90 * The relative error of pow(x,y) can be estimated
91 * by y dl ln(2), where dl is the absolute error of
92 * the internally computed base 2 logarithm. At the ends
93 * of the approximation interval the logarithm equal 1/32
94 * and its relative error is about 1 lsb = 1.1e-19. Hence
95 * the predicted relative error in the result is 2.3e-21 y .
96 *
97 * Relative error:
98 * arithmetic domain # trials peak rms
99 *
100 * IEEE +-1000 40000 2.8e-18 3.7e-19
101 * .001 < x < 1000, with log(x) uniformly distributed.
102 * -1000 < y < 1000, y uniformly distributed.
103 *
104 * IEEE 0,8700 60000 6.5e-18 1.0e-18
105 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
106 *
107 *
108 * ERROR MESSAGES:
109 *
110 * message condition value returned
111 * pow overflow x**y > MAXNUM INFINITY
112 * pow underflow x**y < 1/MAXNUM 0.0
113 * pow domain x<0 and y noninteger 0.0
114 *
115 */
116
117 #include <sys/cdefs.h>
118 #include <float.h>
119 #include <math.h>
120
121 #include "math_private.h"
122
123 /* Table size */
124 #define NXT 32
125 /* log2(Table size) */
126 #define LNXT 5
127
128 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z)
129 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1
130 */
131 static long double P[] = {
132 8.3319510773868690346226E-4L,
133 4.9000050881978028599627E-1L,
134 1.7500123722550302671919E0L,
135 1.4000100839971580279335E0L,
136 };
137 static long double Q[] = {
138 /* 1.0000000000000000000000E0L,*/
139 5.2500282295834889175431E0L,
140 8.4000598057587009834666E0L,
141 4.2000302519914740834728E0L,
142 };
143 /* A[i] = 2^(-i/32), rounded to IEEE long double precision.
144 * If i is even, A[i] + B[i/2] gives additional accuracy.
145 */
146 static long double A[33] = {
147 1.0000000000000000000000E0L,
148 9.7857206208770013448287E-1L,
149 9.5760328069857364691013E-1L,
150 9.3708381705514995065011E-1L,
151 9.1700404320467123175367E-1L,
152 8.9735453750155359320742E-1L,
153 8.7812608018664974155474E-1L,
154 8.5930964906123895780165E-1L,
155 8.4089641525371454301892E-1L,
156 8.2287773907698242225554E-1L,
157 8.0524516597462715409607E-1L,
158 7.8799042255394324325455E-1L,
159 7.7110541270397041179298E-1L,
160 7.5458221379671136985669E-1L,
161 7.3841307296974965571198E-1L,
162 7.2259040348852331001267E-1L,
163 7.0710678118654752438189E-1L,
164 6.9195494098191597746178E-1L,
165 6.7712777346844636413344E-1L,
166 6.6261832157987064729696E-1L,
167 6.4841977732550483296079E-1L,
168 6.3452547859586661129850E-1L,
169 6.2092890603674202431705E-1L,
170 6.0762367999023443907803E-1L,
171 5.9460355750136053334378E-1L,
172 5.8186242938878875689693E-1L,
173 5.6939431737834582684856E-1L,
174 5.5719337129794626814472E-1L,
175 5.4525386633262882960438E-1L,
176 5.3357020033841180906486E-1L,
177 5.2213689121370692017331E-1L,
178 5.1094857432705833910408E-1L,
179 5.0000000000000000000000E-1L,
180 };
181 static long double B[17] = {
182 0.0000000000000000000000E0L,
183 2.6176170809902549338711E-20L,
184 -1.0126791927256478897086E-20L,
185 1.3438228172316276937655E-21L,
186 1.2207982955417546912101E-20L,
187 -6.3084814358060867200133E-21L,
188 1.3164426894366316434230E-20L,
189 -1.8527916071632873716786E-20L,
190 1.8950325588932570796551E-20L,
191 1.5564775779538780478155E-20L,
192 6.0859793637556860974380E-21L,
193 -2.0208749253662532228949E-20L,
194 1.4966292219224761844552E-20L,
195 3.3540909728056476875639E-21L,
196 -8.6987564101742849540743E-22L,
197 -1.2327176863327626135542E-20L,
198 0.0000000000000000000000E0L,
199 };
200
201 /* 2^x = 1 + x P(x),
202 * on the interval -1/32 <= x <= 0
203 */
204 static long double R[] = {
205 1.5089970579127659901157E-5L,
206 1.5402715328927013076125E-4L,
207 1.3333556028915671091390E-3L,
208 9.6181291046036762031786E-3L,
209 5.5504108664798463044015E-2L,
210 2.4022650695910062854352E-1L,
211 6.9314718055994530931447E-1L,
212 };
213
214 #define douba(k) A[k]
215 #define doubb(k) B[k]
216 #define MEXP (NXT*16384.0L)
217 /* The following if denormal numbers are supported, else -MEXP: */
218 #define MNEXP (-NXT*(16384.0L+64.0L))
219 /* log2(e) - 1 */
220 #define LOG2EA 0.44269504088896340735992L
221
222 #define F W
223 #define Fa Wa
224 #define Fb Wb
225 #define G W
226 #define Ga Wa
227 #define Gb u
228 #define H W
229 #define Ha Wb
230 #define Hb Wb
231
232 static const long double MAXLOGL = 1.1356523406294143949492E4L;
233 static const long double MINLOGL = -1.13994985314888605586758E4L;
234 static const long double LOGE2L = 6.9314718055994530941723E-1L;
235 static volatile long double z;
236 static long double w, W, Wa, Wb, ya, yb, u;
237 static const long double huge = 0x1p10000L;
238 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */
239 static const long double twom10000 = 0x1p-10000L;
240 #else
241 static volatile long double twom10000 = 0x1p-10000L;
242 #endif
243
244 static long double reducl( long double );
245 static long double powil ( long double, int );
246
247 long double
powl(long double x,long double y)248 powl(long double x, long double y)
249 {
250 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
251 int i, nflg, iyflg, yoddint;
252 long e;
253
254 if( y == 0.0L )
255 return( 1.0L );
256
257 if( x == 1.0L )
258 return( 1.0L );
259
260 if( isnan(x) )
261 return ( nan_mix(x, y) );
262 if( isnan(y) )
263 return ( nan_mix(x, y) );
264
265 if( y == 1.0L )
266 return( x );
267
268 if( !isfinite(y) && x == -1.0L )
269 return( 1.0L );
270
271 if( y >= LDBL_MAX )
272 {
273 if( x > 1.0L )
274 return( INFINITY );
275 if( x > 0.0L && x < 1.0L )
276 return( 0.0L );
277 if( x < -1.0L )
278 return( INFINITY );
279 if( x > -1.0L && x < 0.0L )
280 return( 0.0L );
281 }
282 if( y <= -LDBL_MAX )
283 {
284 if( x > 1.0L )
285 return( 0.0L );
286 if( x > 0.0L && x < 1.0L )
287 return( INFINITY );
288 if( x < -1.0L )
289 return( 0.0L );
290 if( x > -1.0L && x < 0.0L )
291 return( INFINITY );
292 }
293 if( x >= LDBL_MAX )
294 {
295 if( y > 0.0L )
296 return( INFINITY );
297 return( 0.0L );
298 }
299
300 w = floorl(y);
301 /* Set iyflg to 1 if y is an integer. */
302 iyflg = 0;
303 if( w == y )
304 iyflg = 1;
305
306 /* Test for odd integer y. */
307 yoddint = 0;
308 if( iyflg )
309 {
310 ya = fabsl(y);
311 ya = floorl(0.5L * ya);
312 yb = 0.5L * fabsl(w);
313 if( ya != yb )
314 yoddint = 1;
315 }
316
317 if( x <= -LDBL_MAX )
318 {
319 if( y > 0.0L )
320 {
321 if( yoddint )
322 return( -INFINITY );
323 return( INFINITY );
324 }
325 if( y < 0.0L )
326 {
327 if( yoddint )
328 return( -0.0L );
329 return( 0.0 );
330 }
331 }
332
333
334 nflg = 0; /* flag = 1 if x<0 raised to integer power */
335 if( x <= 0.0L )
336 {
337 if( x == 0.0L )
338 {
339 if( y < 0.0 )
340 {
341 if( signbit(x) && yoddint )
342 return( -INFINITY );
343 return( INFINITY );
344 }
345 if( y > 0.0 )
346 {
347 if( signbit(x) && yoddint )
348 return( -0.0L );
349 return( 0.0 );
350 }
351 if( y == 0.0L )
352 return( 1.0L ); /* 0**0 */
353 else
354 return( 0.0L ); /* 0**y */
355 }
356 else
357 {
358 if( iyflg == 0 )
359 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
360 nflg = 1;
361 }
362 }
363
364 /* Integer power of an integer. */
365
366 if( iyflg )
367 {
368 i = w;
369 w = floorl(x);
370 if( (w == x) && (fabsl(y) < 32768.0) )
371 {
372 w = powil( x, (int) y );
373 return( w );
374 }
375 }
376
377
378 if( nflg )
379 x = fabsl(x);
380
381 /* separate significand from exponent */
382 x = frexpl( x, &i );
383 e = i;
384
385 /* find significand in antilog table A[] */
386 i = 1;
387 if( x <= douba(17) )
388 i = 17;
389 if( x <= douba(i+8) )
390 i += 8;
391 if( x <= douba(i+4) )
392 i += 4;
393 if( x <= douba(i+2) )
394 i += 2;
395 if( x >= douba(1) )
396 i = -1;
397 i += 1;
398
399
400 /* Find (x - A[i])/A[i]
401 * in order to compute log(x/A[i]):
402 *
403 * log(x) = log( a x/a ) = log(a) + log(x/a)
404 *
405 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
406 */
407 x -= douba(i);
408 x -= doubb(i/2);
409 x /= douba(i);
410
411
412 /* rational approximation for log(1+v):
413 *
414 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
415 */
416 z = x*x;
417 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) );
418 w = w - ldexpl( z, -1 ); /* w - 0.5 * z */
419
420 /* Convert to base 2 logarithm:
421 * multiply by log2(e) = 1 + LOG2EA
422 */
423 z = LOG2EA * w;
424 z += w;
425 z += LOG2EA * x;
426 z += x;
427
428 /* Compute exponent term of the base 2 logarithm. */
429 w = -i;
430 w = ldexpl( w, -LNXT ); /* divide by NXT */
431 w += e;
432 /* Now base 2 log of x is w + z. */
433
434 /* Multiply base 2 log by y, in extended precision. */
435
436 /* separate y into large part ya
437 * and small part yb less than 1/NXT
438 */
439 ya = reducl(y);
440 yb = y - ya;
441
442 /* (w+z)(ya+yb)
443 * = w*ya + w*yb + z*y
444 */
445 F = z * y + w * yb;
446 Fa = reducl(F);
447 Fb = F - Fa;
448
449 G = Fa + w * ya;
450 Ga = reducl(G);
451 Gb = G - Ga;
452
453 H = Fb + Gb;
454 Ha = reducl(H);
455 w = ldexpl( Ga+Ha, LNXT );
456
457 /* Test the power of 2 for overflow */
458 if( w > MEXP )
459 return (huge * huge); /* overflow */
460
461 if( w < MNEXP )
462 return (twom10000 * twom10000); /* underflow */
463
464 e = w;
465 Hb = H - Ha;
466
467 if( Hb > 0.0L )
468 {
469 e += 1;
470 Hb -= (1.0L/NXT); /*0.0625L;*/
471 }
472
473 /* Now the product y * log2(x) = Hb + e/NXT.
474 *
475 * Compute base 2 exponential of Hb,
476 * where -0.0625 <= Hb <= 0.
477 */
478 z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */
479
480 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
481 * Find lookup table entry for the fractional power of 2.
482 */
483 if( e < 0 )
484 i = 0;
485 else
486 i = 1;
487 i = e/NXT + i;
488 e = NXT*i - e;
489 w = douba( e );
490 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
491 z = z + w;
492 z = ldexpl( z, i ); /* multiply by integer power of 2 */
493
494 if( nflg )
495 {
496 /* For negative x,
497 * find out if the integer exponent
498 * is odd or even.
499 */
500 w = ldexpl( y, -1 );
501 w = floorl(w);
502 w = ldexpl( w, 1 );
503 if( w != y )
504 z = -z; /* odd exponent */
505 }
506
507 return( z );
508 }
509
510
511 /* Find a multiple of 1/NXT that is within 1/NXT of x. */
512 static inline long double
reducl(long double x)513 reducl(long double x)
514 {
515 long double t;
516
517 t = ldexpl( x, LNXT );
518 t = floorl( t );
519 t = ldexpl( t, -LNXT );
520 return(t);
521 }
522
523 /* powil.c
524 *
525 * Real raised to integer power, long double precision
526 *
527 *
528 *
529 * SYNOPSIS:
530 *
531 * long double x, y, powil();
532 * int n;
533 *
534 * y = powil( x, n );
535 *
536 *
537 *
538 * DESCRIPTION:
539 *
540 * Returns argument x raised to the nth power.
541 * The routine efficiently decomposes n as a sum of powers of
542 * two. The desired power is a product of two-to-the-kth
543 * powers of x. Thus to compute the 32767 power of x requires
544 * 28 multiplications instead of 32767 multiplications.
545 *
546 *
547 *
548 * ACCURACY:
549 *
550 *
551 * Relative error:
552 * arithmetic x domain n domain # trials peak rms
553 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
554 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
555 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
556 *
557 * Returns MAXNUM on overflow, zero on underflow.
558 *
559 */
560
561 static long double
powil(long double x,int nn)562 powil(long double x, int nn)
563 {
564 long double ww, y;
565 long double s;
566 int n, e, sign, asign, lx;
567
568 if( x == 0.0L )
569 {
570 if( nn == 0 )
571 return( 1.0L );
572 else if( nn < 0 )
573 return( LDBL_MAX );
574 else
575 return( 0.0L );
576 }
577
578 if( nn == 0 )
579 return( 1.0L );
580
581
582 if( x < 0.0L )
583 {
584 asign = -1;
585 x = -x;
586 }
587 else
588 asign = 0;
589
590
591 if( nn < 0 )
592 {
593 sign = -1;
594 n = -nn;
595 }
596 else
597 {
598 sign = 1;
599 n = nn;
600 }
601
602 /* Overflow detection */
603
604 /* Calculate approximate logarithm of answer */
605 s = x;
606 s = frexpl( s, &lx );
607 e = (lx - 1)*n;
608 if( (e == 0) || (e > 64) || (e < -64) )
609 {
610 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
611 s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
612 }
613 else
614 {
615 s = LOGE2L * e;
616 }
617
618 if( s > MAXLOGL )
619 return (huge * huge); /* overflow */
620
621 if( s < MINLOGL )
622 return (twom10000 * twom10000); /* underflow */
623 /* Handle tiny denormal answer, but with less accuracy
624 * since roundoff error in 1.0/x will be amplified.
625 * The precise demarcation should be the gradual underflow threshold.
626 */
627 if( s < (-MAXLOGL+2.0L) )
628 {
629 x = 1.0L/x;
630 sign = -sign;
631 }
632
633 /* First bit of the power */
634 if( n & 1 )
635 y = x;
636
637 else
638 {
639 y = 1.0L;
640 asign = 0;
641 }
642
643 ww = x;
644 n >>= 1;
645 while( n )
646 {
647 ww = ww * ww; /* arg to the 2-to-the-kth power */
648 if( n & 1 ) /* if that bit is set, then include in product */
649 y *= ww;
650 n >>= 1;
651 }
652
653 if( asign )
654 y = -y; /* odd power of negative number */
655 if( sign < 0 )
656 y = 1.0L/y;
657 return(y);
658 }
659