1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /*
13 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
14 *
15 * Permission to use, copy, modify, and distribute this software for any
16 * purpose with or without fee is hereby granted, provided that the above
17 * copyright notice and this permission notice appear in all copies.
18 *
19 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
20 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
21 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
22 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
23 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
24 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
25 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
26 */
27
28 /* powl(x,y) return x**y
29 *
30 * n
31 * Method: Let x = 2 * (1+f)
32 * 1. Compute and return log2(x) in two pieces:
33 * log2(x) = w1 + w2,
34 * where w1 has 113-53 = 60 bit trailing zeros.
35 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
36 * arithmetic, where |y'|<=0.5.
37 * 3. Return x**y = 2**n*exp(y'*log2)
38 *
39 * Special cases:
40 * 1. (anything) ** 0 is 1
41 * 2. (anything) ** 1 is itself
42 * 3. (anything) ** NAN is NAN
43 * 4. NAN ** (anything except 0) is NAN
44 * 5. +-(|x| > 1) ** +INF is +INF
45 * 6. +-(|x| > 1) ** -INF is +0
46 * 7. +-(|x| < 1) ** +INF is +0
47 * 8. +-(|x| < 1) ** -INF is +INF
48 * 9. +-1 ** +-INF is NAN
49 * 10. +0 ** (+anything except 0, NAN) is +0
50 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
51 * 12. +0 ** (-anything except 0, NAN) is +INF
52 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
53 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
54 * 15. +INF ** (+anything except 0,NAN) is +INF
55 * 16. +INF ** (-anything except 0,NAN) is +0
56 * 17. -INF ** (anything) = -0 ** (-anything)
57 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
58 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
59 *
60 */
61
62
63
64 static const long double bp[] = {
65 1.0L,
66 1.5L,
67 };
68
69 /* log_2(1.5) */
70 static const long double dp_h[] = {
71 0.0L,
72 5.8496250072115607565592654282227158546448E-1L
73 };
74
75 /* Low part of log_2(1.5) */
76 static const long double dp_l[] = {
77 0.0L,
78 1.0579781240112554492329533686862998106046E-16L
79 };
80
81 static const long double zero = 0.0L,
82 one = 1.0L,
83 two = 2.0L,
84 two113 = 1.0384593717069655257060992658440192E34L,
85 huge = 1.0e3000L,
86 tiny = 1.0e-3000L;
87
88 /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
89 z = (x-1)/(x+1)
90 1 <= x <= 1.25
91 Peak relative error 2.3e-37 */
92 static const long double LN[] =
93 {
94 -3.0779177200290054398792536829702930623200E1L,
95 6.5135778082209159921251824580292116201640E1L,
96 -4.6312921812152436921591152809994014413540E1L,
97 1.2510208195629420304615674658258363295208E1L,
98 -9.9266909031921425609179910128531667336670E-1L
99 };
100 static const long double LD[] =
101 {
102 -5.129862866715009066465422805058933131960E1L,
103 1.452015077564081884387441590064272782044E2L,
104 -1.524043275549860505277434040464085593165E2L,
105 7.236063513651544224319663428634139768808E1L,
106 -1.494198912340228235853027849917095580053E1L
107 /* 1.0E0 */
108 };
109
110 /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
111 0 <= x <= 0.5
112 Peak relative error 5.7e-38 */
113 static const long double PN[] =
114 {
115 5.081801691915377692446852383385968225675E8L,
116 9.360895299872484512023336636427675327355E6L,
117 4.213701282274196030811629773097579432957E4L,
118 5.201006511142748908655720086041570288182E1L,
119 9.088368420359444263703202925095675982530E-3L,
120 };
121 static const long double PD[] =
122 {
123 3.049081015149226615468111430031590411682E9L,
124 1.069833887183886839966085436512368982758E8L,
125 8.259257717868875207333991924545445705394E5L,
126 1.872583833284143212651746812884298360922E3L,
127 /* 1.0E0 */
128 };
129
130 static const long double
131 /* ln 2 */
132 lg2 = 6.9314718055994530941723212145817656807550E-1L,
133 lg2_h = 6.9314718055994528622676398299518041312695E-1L,
134 lg2_l = 2.3190468138462996154948554638754786504121E-17L,
135 ovt = 1.38926478993351112510650782611372039878722890e-34L,
136 // ovt = 8.0085662595372944372e-0017L,
137 /* 2/(3*log(2)) */
138 cp = 9.6179669392597560490661645400126142495110E-1L,
139 cp_h = 9.6179669392597555432899980587535537779331E-1L,
140 cp_l = 5.0577616648125906047157785230014751039424E-17L;
141
142 long double
powl(long double x,long double y)143 powl(long double x, long double y)
144 {
145 long double z, ax, z_h, z_l, p_h, p_l;
146 long double yy1, t1, t2, r, s, t, u, v, w;
147 long double s2, s_h, s_l, t_h, t_l;
148 int32_t i, j, k, yisint, n;
149 u_int32_t ix, iy;
150 int32_t hx, hy;
151 ieee_quad_shape_type o, p, q;
152
153 p.value = x;
154 hx = p.parts32.mswhi;
155 ix = hx & 0x7fffffff;
156
157 q.value = y;
158 hy = q.parts32.mswhi;
159 iy = hy & 0x7fffffff;
160
161
162 /* y==zero: x**0 = 1 */
163 if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) {
164 if (issignalingl(x))
165 return x + y;
166 return one;
167 }
168
169 /* 1.0**y = 1; -1.0**+-Inf = 1 */
170 if (x == one) {
171 if (issignalingl(y))
172 return x + y;
173 return one;
174 }
175 if (x == -1.0L && iy == 0x7fff0000
176 && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
177 return one;
178
179 /* +-NaN return x+y */
180 if ((ix > 0x7fff0000)
181 || ((ix == 0x7fff0000)
182 && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0))
183 || (iy > 0x7fff0000)
184 || ((iy == 0x7fff0000)
185 && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0)))
186 return x + y;
187
188 /* determine if y is an odd int when x < 0
189 * yisint = 0 ... y is not an integer
190 * yisint = 1 ... y is an odd int
191 * yisint = 2 ... y is an even int
192 */
193 yisint = 0;
194 if (hx < 0)
195 {
196 if (iy >= 0x40700000) /* 2^113 */
197 yisint = 2; /* even integer y */
198 else if (iy >= 0x3fff0000) /* 1.0 */
199 {
200 if (floorl (y) == y)
201 {
202 z = 0.5L * y;
203 if (floorl (z) == z)
204 yisint = 2;
205 else
206 yisint = 1;
207 }
208 }
209 }
210
211 ax = fabsl (x);
212 /* special value of x */
213 if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0)
214 {
215 if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000)
216 {
217 z = ax; /*x is +-inf,+-1 */
218 if (hy < 0)
219 {
220 if (ix == 0 && iy != 0x7fff0000) {
221 return __math_divzerol(hx < 0 && yisint == 1);
222 }
223 z = one / z; /* z = (1/|x|) */
224 }
225 if (hx < 0)
226 {
227 if (((ix - 0x3fff0000) | yisint) == 0)
228 {
229 z = __math_invalidl(z); /* (-1)**non-int is NaN */
230 }
231 else if (yisint == 1)
232 z = -z; /* (x<0)**odd = -(|x|**odd) */
233 }
234 return z;
235 }
236 }
237
238 /* special value of y */
239 if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
240 {
241 if (iy == 0x7fff0000) /* y is +-inf */
242 {
243 if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi |
244 p.parts32.lswlo) == 0)
245 return __math_invalidl(y); /* +-1**inf is NaN */
246 else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */
247 return (hy >= 0) ? y : zero;
248 else /* (|x|<1)**-,+inf = inf,0 */
249 return (hy < 0) ? -y : zero;
250 }
251 if (iy == 0x3fff0000)
252 { /* y is +-1 */
253 if (hy < 0)
254 return check_oflowl(one / x);
255 else
256 return x;
257 }
258 if (hy == 0x40000000)
259 return check_uflowl(check_oflowl(x * x)); /* y is 2 */
260 if (hy == 0x3ffe0000)
261 { /* y is 0.5 */
262 if (hx >= 0) /* x >= +0 */
263 return sqrtl (x);
264 }
265 }
266
267 /* (x<0)**(non-int) is NaN */
268 if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
269 return __math_invalidl(x);
270
271 /* |y| is huge.
272 2^-16495 = 1/2 of smallest representable value.
273 If (1 - 1/131072)^y underflows, y > 1.4986e9 */
274 if (iy > 0x401d654b)
275 {
276 int neg = (hx < 0) & yisint;
277 /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
278 if (iy > 0x407d654b)
279 {
280 if (ix <= 0x3ffeffff)
281 return (hy < 0) ? __math_oflowl(neg) : __math_uflowl(neg);
282 if (ix >= 0x3fff0000)
283 return (hy > 0) ? __math_oflowl(neg) : __math_uflowl(neg);
284 }
285 /* over/underflow if x is not close to one */
286 if (ix < 0x3ffeffff)
287 return (hy < 0) ? __math_oflowl(neg) : __math_uflowl(neg);
288 if (ix > 0x3fff0000)
289 return (hy > 0) ? __math_oflowl(neg) : __math_uflowl(neg);
290 }
291
292 n = 0;
293 /* take care subnormal number */
294 if (ix < 0x00010000)
295 {
296 ax *= two113;
297 n -= 113;
298 o.value = ax;
299 ix = o.parts32.mswhi;
300 }
301 n += ((ix) >> 16) - 0x3fff;
302 j = ix & 0x0000ffff;
303 /* determine interval */
304 ix = j | 0x3fff0000; /* normalize ix */
305 if (j <= 0x3988)
306 k = 0; /* |x|<sqrt(3/2) */
307 else if (j < 0xbb67)
308 k = 1; /* |x|<sqrt(3) */
309 else
310 {
311 k = 0;
312 n += 1;
313 ix -= 0x00010000;
314 }
315
316 o.value = ax;
317 o.parts32.mswhi = ix;
318 ax = o.value;
319
320 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
321 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
322 v = one / (ax + bp[k]);
323 s = u * v;
324 s_h = s;
325
326 o.value = s_h;
327 o.parts32.lswlo = 0;
328 o.parts32.lswhi &= 0xf8000000;
329 s_h = o.value;
330 /* t_h=ax+bp[k] High */
331 t_h = ax + bp[k];
332 o.value = t_h;
333 o.parts32.lswlo = 0;
334 o.parts32.lswhi &= 0xf8000000;
335 t_h = o.value;
336 t_l = ax - (t_h - bp[k]);
337 s_l = v * ((u - s_h * t_h) - s_h * t_l);
338 /* compute log(ax) */
339 s2 = s * s;
340 u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
341 v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
342 r = s2 * s2 * u / v;
343 r += s_l * (s_h + s);
344 s2 = s_h * s_h;
345 t_h = 3.0L + s2 + r;
346 o.value = t_h;
347 o.parts32.lswlo = 0;
348 o.parts32.lswhi &= 0xf8000000;
349 t_h = o.value;
350 t_l = r - ((t_h - 3.0L) - s2);
351 /* u+v = s*(1+...) */
352 u = s_h * t_h;
353 v = s_l * t_h + t_l * s;
354 /* 2/(3log2)*(s+...) */
355 p_h = u + v;
356 o.value = p_h;
357 o.parts32.lswlo = 0;
358 o.parts32.lswhi &= 0xf8000000;
359 p_h = o.value;
360 p_l = v - (p_h - u);
361 z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
362 z_l = cp_l * p_h + p_l * cp + dp_l[k];
363 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
364 t = (long double) n;
365 t1 = (((z_h + z_l) + dp_h[k]) + t);
366 o.value = t1;
367 o.parts32.lswlo = 0;
368 o.parts32.lswhi &= 0xf8000000;
369 t1 = o.value;
370 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
371
372 /* s (sign of result -ve**odd) = -1 else = 1 */
373 s = one;
374 if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
375 s = -one; /* (-ve)**(odd int) */
376
377 /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
378 yy1 = y;
379 o.value = yy1;
380 o.parts32.lswlo = 0;
381 o.parts32.lswhi &= 0xf8000000;
382 yy1 = o.value;
383 p_l = (y - yy1) * t1 + y * t2;
384 p_h = yy1 * t1;
385 z = p_l + p_h;
386 o.value = z;
387 j = o.parts32.mswhi;
388 if (j >= 0x400d0000) /* z >= 16384 */
389 {
390 /* if z > 16384 */
391 if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi |
392 o.parts32.lswlo) != 0)
393 return __math_oflowl(s<0); /* overflow */
394 else
395 {
396 if (p_l + ovt > z - p_h)
397 return __math_oflowl(s<0); /* overflow */
398 }
399 }
400 else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */
401 {
402 /* z < -16495 */
403 if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi |
404 o.parts32.lswlo)
405 != 0)
406 return __math_uflowl(s<0); /* underflow */
407 else
408 {
409 if (p_l <= z - p_h)
410 return __math_uflowl(s<0); /* underflow */
411 }
412 }
413 /* compute 2**(p_h+p_l) */
414 i = j & 0x7fffffff;
415 k = (i >> 16) - 0x3fff;
416 n = 0;
417 if (i > 0x3ffe0000)
418 { /* if |z| > 0.5, set n = [z+0.5] */
419 n = floorl (z + 0.5L);
420 t = n;
421 p_h -= t;
422 }
423 t = p_l + p_h;
424 o.value = t;
425 o.parts32.lswlo = 0;
426 o.parts32.lswhi &= 0xf8000000;
427 t = o.value;
428 u = t * lg2_h;
429 v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
430 z = u + v;
431 w = v - (z - u);
432 /* exp(z) */
433 t = z * z;
434 u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
435 v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
436 t1 = z - t * u / v;
437 r = (z * t1) / (t1 - two) - (w + z * w);
438 z = one - (r - z);
439 o.value = z;
440 j = o.parts32.mswhi;
441 j += (n << 16);
442 if ((j >> 16) <= 0)
443 z = scalbnl (z, n); /* subnormal output */
444 else
445 {
446 o.parts32.mswhi = j;
447 z = o.value;
448 }
449 return s * z;
450 }
451
452 #if defined(_HAVE_ALIAS_ATTRIBUTE)
453 #ifndef __clang__
454 #pragma GCC diagnostic ignored "-Wmissing-attributes"
455 #endif
456 __strong_reference(powl, _powl);
457 #endif
458
