1 /* $OpenBSD: e_powl.c,v 1.5 2013/11/12 20:35:19 martynas 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
76
77 /* Table size */
78 #define NXT 32
79 /* log2(Table size) */
80 #define LNXT 5
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 douba(k) A[k]
169 #define doubb(k) B[k]
170 #define MEXP (NXT*16384.0L)
171 /* The following if denormal numbers are supported, else -MEXP: */
172 #define MNEXP (-NXT*(16384.0L+64.0L))
173 /* log2(e) - 1 */
174 #define LOG2EA 0.44269504088896340735992L
175
176 static const long double MAXLOGL = 1.1356523406294143949492E4L;
177 static const long double MINLOGL = -1.13994985314888605586758E4L;
178 static const long double LOGE2L = 6.9314718055994530941723E-1L;
179 static const long double huge = 0x1p10000L;
180 static const long double twom10000 = 0x1p-10000L;
181
182 static long double reducl( long double );
183 static long double powil ( long double, int );
184
185 long double
powl(long double x,long double y)186 powl(long double x, long double y)
187 {
188 long double w, ya, yb, z;
189 long double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb;
190 int i, nflg, iyflg, yoddint;
191 long e;
192
193 if( y == 0.0L ) {
194 if (issignalingl(x))
195 return x + y;
196 return( 1.0L );
197 }
198
199 if( x == 1.0L ) {
200 if (issignalingl(y))
201 return x + y;
202 return( 1.0L );
203 }
204
205 w = floorl(y);
206 /* Set iyflg to 1 if y is an integer. */
207 iyflg = (w == y);
208
209 /* flag = 1 if x is negative */
210 nflg = signbit(x);
211
212 /* Test for odd integer y and negative x (including negative zero) */
213 yoddint = 0;
214 if( iyflg && nflg )
215 {
216 ya = ldexpl(y, -1);
217 yoddint = (ya != floorl(ya));
218 }
219
220 if( x == 0.0L) {
221 if( y < 0 )
222 return __math_divzerol(yoddint);
223 }
224
225 if( isnanl(x) )
226 return( x );
227 if( isnanl(y) )
228 return( y );
229
230 if( y == 1.0L )
231 return( x );
232
233 if( !isfinite(y) ) {
234 if (x == -1.0L)
235 return( 1.0L );
236 if (y < 0) {
237 if (fabsl(x) < 1)
238 return( (long double)INFINITY );
239 return( 0.0L );
240 } else {
241 if (fabsl(x) < 1)
242 return( 0.0L );
243 return( (long double)INFINITY );
244 }
245 }
246
247 /* y >= 2**80, infinity (x > 1) or zero (x < 1) */
248 if( y >= 0x1p80L)
249 {
250 if( x > 1.0L )
251 return __math_oflowl(0);
252 if( x > 0.0L && x < 1.0L )
253 return( 0.0L );
254 if( x < -1.0L )
255 return __math_oflowl(1);
256 if( x > -1.0L && x < 0.0L )
257 return( -0.0L );
258 }
259 /* y <= -2**80, zero (x > 1) or infinity (x < 1) */
260 if( y <= -0x1p80L)
261 {
262 if( x > 1.0L )
263 return __math_uflowl(0);
264 if( x > 0.0L && x < 1.0L )
265 return( (long double)INFINITY );
266 if( x < -1.0L )
267 return( 0.0L );
268 if( x > -1.0L && x < 0.0L )
269 return( (long double)INFINITY );
270 }
271 if( x >= LDBL_MAX )
272 {
273 if( y > 0.0L )
274 return( (long double)INFINITY );
275 return( 0.0L );
276 }
277
278
279
280 if( x <= -LDBL_MAX )
281 {
282 if( y > 0.0L )
283 {
284 if( yoddint )
285 return( -(long double)INFINITY );
286 return( (long double)INFINITY );
287 }
288 if( y < 0.0L )
289 {
290 if( yoddint )
291 return( -0.0L );
292 return( 0.0L );
293 }
294 }
295
296
297 if( x <= 0.0L )
298 {
299 if( x == 0.0L )
300 {
301 if( y < 0.0L )
302 {
303 if( yoddint )
304 return( -(long double)INFINITY );
305 return( (long double)INFINITY );
306 }
307 if( y > 0.0L )
308 {
309 if( yoddint )
310 return( -0.0L );
311 return( 0.0L );
312 }
313 if( y == 0.0L )
314 return( 1.0L ); /* 0**0 */
315 else
316 return( 0.0L ); /* 0**y */
317 }
318 else
319 {
320 if( iyflg == 0 )
321 return __math_invalidl(x); /* (x<0)**(non-int) is NaN */
322 }
323 }
324
325 /* Integer power of an integer. */
326
327 if( iyflg )
328 {
329 w = floorl(x);
330 if( (w == x) && (fabsl(y) < 32768.0L) )
331 {
332 w = powil( x, (int) y );
333 return( w );
334 }
335 }
336
337
338 if( nflg )
339 x = fabsl(x);
340
341 /* separate significand from exponent */
342 x = frexpl( x, &i );
343 e = i;
344
345 /* find significand in antilog table A[] */
346 i = 1;
347 if( x <= douba(17) )
348 i = 17;
349 if( x <= douba(i+8) )
350 i += 8;
351 if( x <= douba(i+4) )
352 i += 4;
353 if( x <= douba(i+2) )
354 i += 2;
355 if( x >= douba(1) )
356 i = -1;
357 i += 1;
358
359
360 /* Find (x - A[i])/A[i]
361 * in order to compute log(x/A[i]):
362 *
363 * log(x) = log( a x/a ) = log(a) + log(x/a)
364 *
365 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
366 */
367 x -= douba(i);
368 x -= doubb(i/2);
369 x /= douba(i);
370
371
372 /* rational approximation for log(1+v):
373 *
374 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
375 */
376 z = x*x;
377 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) );
378 w = w - ldexpl( z, -1 ); /* w - 0.5 * z */
379
380 /* Convert to base 2 logarithm:
381 * multiply by log2(e) = 1 + LOG2EA
382 */
383 z = LOG2EA * w;
384 z += w;
385 z += LOG2EA * x;
386 z += x;
387
388 /* Compute exponent term of the base 2 logarithm. */
389 w = -i;
390 w = ldexpl( w, -LNXT ); /* divide by NXT */
391 w += e;
392 /* Now base 2 log of x is w + z. */
393
394 /* Multiply base 2 log by y, in extended precision. */
395
396 /* separate y into large part ya
397 * and small part yb less than 1/NXT
398 */
399 ya = reducl(y);
400 yb = y - ya;
401
402 /* (w+z)(ya+yb)
403 * = w*ya + w*yb + z*y
404 */
405 F = z * y + w * yb;
406 Fa = reducl(F);
407 Fb = F - Fa;
408
409 G = Fa + w * ya;
410 Ga = reducl(G);
411 Gb = G - Ga;
412
413 H = Fb + Gb;
414 Ha = reducl(H);
415 w = ldexpl( Ga+Ha, LNXT );
416
417 /* Test the power of 2 for overflow */
418 if( w > MEXP )
419 return __math_oflowl(yoddint); /* overflow */
420
421 if( w < MNEXP )
422 return __math_uflowl(yoddint); /* underflow */
423
424 e = w;
425 Hb = H - Ha;
426
427 if( Hb > 0.0L )
428 {
429 e += 1;
430 Hb -= (1.0L/NXT); /*0.0625L;*/
431 }
432
433 /* Now the product y * log2(x) = Hb + e/NXT.
434 *
435 * Compute base 2 exponential of Hb,
436 * where -0.0625 <= Hb <= 0.
437 */
438 z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */
439
440 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
441 * Find lookup table entry for the fractional power of 2.
442 */
443 if( e < 0 )
444 i = 0;
445 else
446 i = 1;
447 i = e/NXT + i;
448 e = NXT*i - e;
449 w = douba( e );
450 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
451 z = z + w;
452 z = ldexpl( z, i ); /* multiply by integer power of 2 */
453
454 if( nflg )
455 {
456 /* For negative x,
457 * find out if the integer exponent
458 * is odd or even.
459 */
460 w = ldexpl( y, -1 );
461 w = floorl(w);
462 w = ldexpl( w, 1 );
463 if( w != y )
464 z = -z; /* odd exponent */
465 }
466
467 return( z );
468 }
469
470
471 /* Find a multiple of 1/NXT that is within 1/NXT of x. */
472 static long double
reducl(long double x)473 reducl(long double x)
474 {
475 long double t = x;
476
477 if (t < LDBL_MAX / NXT) {
478 t = ldexpl( t, LNXT );
479 t = floorl( t );
480 t = ldexpl( t, -LNXT );
481 }
482 return(t);
483 }
484
485 /* powil.c
486 *
487 * Real raised to integer power, long double precision
488 *
489 *
490 *
491 * SYNOPSIS:
492 *
493 * long double x, y, powil();
494 * int n;
495 *
496 * y = powil( x, n );
497 *
498 *
499 *
500 * DESCRIPTION:
501 *
502 * Returns argument x raised to the nth power.
503 * The routine efficiently decomposes n as a sum of powers of
504 * two. The desired power is a product of two-to-the-kth
505 * powers of x. Thus to compute the 32767 power of x requires
506 * 28 multiplications instead of 32767 multiplications.
507 *
508 *
509 *
510 * ACCURACY:
511 *
512 *
513 * Relative error:
514 * arithmetic x domain n domain # trials peak rms
515 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
516 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
517 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
518 *
519 * Returns MAXNUM on overflow, zero on underflow.
520 *
521 */
522
523 static long double
powil(long double x,int nn)524 powil(long double x, int nn)
525 {
526 long double ww, y;
527 long double s;
528 int n, e, sign, asign, lx;
529
530 if( x == 0.0L )
531 {
532 if( nn == 0 )
533 return( 1.0L );
534 else if( nn < 0 )
535 return( LDBL_MAX );
536 else
537 return( 0.0L );
538 }
539
540 if( nn == 0 )
541 return( 1.0L );
542
543
544 if( x < 0.0L )
545 {
546 asign = -1;
547 x = -x;
548 }
549 else
550 asign = 0;
551
552
553 if( nn < 0 )
554 {
555 sign = -1;
556 n = -nn;
557 }
558 else
559 {
560 sign = 1;
561 n = nn;
562 }
563
564 /* Overflow detection */
565
566 /* Calculate approximate logarithm of answer */
567 s = x;
568 s = frexpl( s, &lx );
569 e = (lx - 1)*n;
570 if( (e == 0) || (e > 64) || (e < -64) )
571 {
572 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
573 s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
574 }
575 else
576 {
577 s = LOGE2L * e;
578 }
579
580 if( s > MAXLOGL )
581 return __math_oflowl(asign < 0 && (nn & 1)); /* overflow */
582
583 if( s < MINLOGL )
584 return __math_uflowl(asign < 0 && (nn & 1)); /* underflow */
585
586 /* Handle tiny denormal answer, but with less accuracy
587 * since roundoff error in 1.0/x will be amplified.
588 * The precise demarcation should be the gradual underflow threshold.
589 */
590 if( s < (-MAXLOGL+2.0L) )
591 {
592 x = 1.0L/x;
593 sign = -sign;
594 }
595
596 /* First bit of the power */
597 if( n & 1 )
598 y = x;
599
600 else
601 {
602 y = 1.0L;
603 asign = 0;
604 }
605
606 ww = x;
607 n >>= 1;
608 while( n )
609 {
610 ww = ww * ww; /* arg to the 2-to-the-kth power */
611 if( n & 1 ) /* if that bit is set, then include in product */
612 y *= ww;
613 n >>= 1;
614 }
615
616 if( asign )
617 y = -y; /* odd power of negative number */
618 if( sign < 0 )
619 y = 1.0L/y;
620 return(y);
621 }
622
623 #if defined(_HAVE_ALIAS_ATTRIBUTE)
624 #ifndef __clang__
625 #pragma GCC diagnostic ignored "-Wmissing-attributes"
626 #endif
627 __strong_reference(powl, _powl);
628 #endif
629
