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 multi-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 #include <sys/cdefs.h>
63 __FBSDID("$FreeBSD$");
64
65 #include <float.h>
66 #include <math.h>
67
68 #include "math_private.h"
69
70 static const long double bp[] = {
71 1.0L,
72 1.5L,
73 };
74
75 /* log_2(1.5) */
76 static const long double dp_h[] = {
77 0.0,
78 5.8496250072115607565592654282227158546448E-1L
79 };
80
81 /* Low part of log_2(1.5) */
82 static const long double dp_l[] = {
83 0.0,
84 1.0579781240112554492329533686862998106046E-16L
85 };
86
87 static const long double zero = 0.0L,
88 one = 1.0L,
89 two = 2.0L,
90 two113 = 1.0384593717069655257060992658440192E34L,
91 huge = 1.0e3000L,
92 tiny = 1.0e-3000L;
93
94 /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
95 z = (x-1)/(x+1)
96 1 <= x <= 1.25
97 Peak relative error 2.3e-37 */
98 static const long double LN[] =
99 {
100 -3.0779177200290054398792536829702930623200E1L,
101 6.5135778082209159921251824580292116201640E1L,
102 -4.6312921812152436921591152809994014413540E1L,
103 1.2510208195629420304615674658258363295208E1L,
104 -9.9266909031921425609179910128531667336670E-1L
105 };
106 static const long double LD[] =
107 {
108 -5.129862866715009066465422805058933131960E1L,
109 1.452015077564081884387441590064272782044E2L,
110 -1.524043275549860505277434040464085593165E2L,
111 7.236063513651544224319663428634139768808E1L,
112 -1.494198912340228235853027849917095580053E1L
113 /* 1.0E0 */
114 };
115
116 /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
117 0 <= x <= 0.5
118 Peak relative error 5.7e-38 */
119 static const long double PN[] =
120 {
121 5.081801691915377692446852383385968225675E8L,
122 9.360895299872484512023336636427675327355E6L,
123 4.213701282274196030811629773097579432957E4L,
124 5.201006511142748908655720086041570288182E1L,
125 9.088368420359444263703202925095675982530E-3L,
126 };
127 static const long double PD[] =
128 {
129 3.049081015149226615468111430031590411682E9L,
130 1.069833887183886839966085436512368982758E8L,
131 8.259257717868875207333991924545445705394E5L,
132 1.872583833284143212651746812884298360922E3L,
133 /* 1.0E0 */
134 };
135
136 static const long double
137 /* ln 2 */
138 lg2 = 6.9314718055994530941723212145817656807550E-1L,
139 lg2_h = 6.9314718055994528622676398299518041312695E-1L,
140 lg2_l = 2.3190468138462996154948554638754786504121E-17L,
141 ovt = 8.0085662595372944372e-0017L,
142 /* 2/(3*log(2)) */
143 cp = 9.6179669392597560490661645400126142495110E-1L,
144 cp_h = 9.6179669392597555432899980587535537779331E-1L,
145 cp_l = 5.0577616648125906047157785230014751039424E-17L;
146
147 long double
powl(long double x,long double y)148 powl(long double x, long double y)
149 {
150 long double z, ax, z_h, z_l, p_h, p_l;
151 long double yy1, t1, t2, r, s, t, u, v, w;
152 long double s2, s_h, s_l, t_h, t_l;
153 int32_t i, j, k, yisint, n;
154 u_int32_t ix, iy;
155 int32_t hx, hy;
156 ieee_quad_shape_type o, p, q;
157
158 p.value = x;
159 hx = p.parts32.mswhi;
160 ix = hx & 0x7fffffff;
161
162 q.value = y;
163 hy = q.parts32.mswhi;
164 iy = hy & 0x7fffffff;
165
166
167 /* y==zero: x**0 = 1 */
168 if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
169 return one;
170
171 /* 1.0**y = 1; -1.0**+-Inf = 1 */
172 if (x == one)
173 return one;
174 if (x == -1.0L && iy == 0x7fff0000
175 && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
176 return one;
177
178 /* +-NaN return x+y */
179 if ((ix > 0x7fff0000)
180 || ((ix == 0x7fff0000)
181 && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0))
182 || (iy > 0x7fff0000)
183 || ((iy == 0x7fff0000)
184 && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0)))
185 return nan_mix(x, y);
186
187 /* determine if y is an odd int when x < 0
188 * yisint = 0 ... y is not an integer
189 * yisint = 1 ... y is an odd int
190 * yisint = 2 ... y is an even int
191 */
192 yisint = 0;
193 if (hx < 0)
194 {
195 if (iy >= 0x40700000) /* 2^113 */
196 yisint = 2; /* even integer y */
197 else if (iy >= 0x3fff0000) /* 1.0 */
198 {
199 if (floorl (y) == y)
200 {
201 z = 0.5 * y;
202 if (floorl (z) == z)
203 yisint = 2;
204 else
205 yisint = 1;
206 }
207 }
208 }
209
210 /* special value of y */
211 if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
212 {
213 if (iy == 0x7fff0000) /* y is +-inf */
214 {
215 if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi |
216 p.parts32.lswlo) == 0)
217 return y - y; /* +-1**inf is NaN */
218 else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */
219 return (hy >= 0) ? y : zero;
220 else /* (|x|<1)**-,+inf = inf,0 */
221 return (hy < 0) ? -y : zero;
222 }
223 if (iy == 0x3fff0000)
224 { /* y is +-1 */
225 if (hy < 0)
226 return one / x;
227 else
228 return x;
229 }
230 if (hy == 0x40000000)
231 return x * x; /* y is 2 */
232 if (hy == 0x3ffe0000)
233 { /* y is 0.5 */
234 if (hx >= 0) /* x >= +0 */
235 return sqrtl (x);
236 }
237 }
238
239 ax = fabsl (x);
240 /* special value of x */
241 if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0)
242 {
243 if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000)
244 {
245 z = ax; /*x is +-0,+-inf,+-1 */
246 if (hy < 0)
247 z = one / z; /* z = (1/|x|) */
248 if (hx < 0)
249 {
250 if (((ix - 0x3fff0000) | yisint) == 0)
251 {
252 z = (z - z) / (z - z); /* (-1)**non-int is NaN */
253 }
254 else if (yisint == 1)
255 z = -z; /* (x<0)**odd = -(|x|**odd) */
256 }
257 return z;
258 }
259 }
260
261 /* (x<0)**(non-int) is NaN */
262 if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
263 return (x - x) / (x - x);
264
265 /* |y| is huge.
266 2^-16495 = 1/2 of smallest representable value.
267 If (1 - 1/131072)^y underflows, y > 1.4986e9 */
268 if (iy > 0x401d654b)
269 {
270 /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
271 if (iy > 0x407d654b)
272 {
273 if (ix <= 0x3ffeffff)
274 return (hy < 0) ? huge * huge : tiny * tiny;
275 if (ix >= 0x3fff0000)
276 return (hy > 0) ? huge * huge : tiny * tiny;
277 }
278 /* over/underflow if x is not close to one */
279 if (ix < 0x3ffeffff)
280 return (hy < 0) ? huge * huge : tiny * tiny;
281 if (ix > 0x3fff0000)
282 return (hy > 0) ? huge * huge : tiny * tiny;
283 }
284
285 n = 0;
286 /* take care subnormal number */
287 if (ix < 0x00010000)
288 {
289 ax *= two113;
290 n -= 113;
291 o.value = ax;
292 ix = o.parts32.mswhi;
293 }
294 n += ((ix) >> 16) - 0x3fff;
295 j = ix & 0x0000ffff;
296 /* determine interval */
297 ix = j | 0x3fff0000; /* normalize ix */
298 if (j <= 0x3988)
299 k = 0; /* |x|<sqrt(3/2) */
300 else if (j < 0xbb67)
301 k = 1; /* |x|<sqrt(3) */
302 else
303 {
304 k = 0;
305 n += 1;
306 ix -= 0x00010000;
307 }
308
309 o.value = ax;
310 o.parts32.mswhi = ix;
311 ax = o.value;
312
313 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
314 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
315 v = one / (ax + bp[k]);
316 s = u * v;
317 s_h = s;
318
319 o.value = s_h;
320 o.parts32.lswlo = 0;
321 o.parts32.lswhi &= 0xf8000000;
322 s_h = o.value;
323 /* t_h=ax+bp[k] High */
324 t_h = ax + bp[k];
325 o.value = t_h;
326 o.parts32.lswlo = 0;
327 o.parts32.lswhi &= 0xf8000000;
328 t_h = o.value;
329 t_l = ax - (t_h - bp[k]);
330 s_l = v * ((u - s_h * t_h) - s_h * t_l);
331 /* compute log(ax) */
332 s2 = s * s;
333 u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
334 v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
335 r = s2 * s2 * u / v;
336 r += s_l * (s_h + s);
337 s2 = s_h * s_h;
338 t_h = 3.0 + s2 + r;
339 o.value = t_h;
340 o.parts32.lswlo = 0;
341 o.parts32.lswhi &= 0xf8000000;
342 t_h = o.value;
343 t_l = r - ((t_h - 3.0) - s2);
344 /* u+v = s*(1+...) */
345 u = s_h * t_h;
346 v = s_l * t_h + t_l * s;
347 /* 2/(3log2)*(s+...) */
348 p_h = u + v;
349 o.value = p_h;
350 o.parts32.lswlo = 0;
351 o.parts32.lswhi &= 0xf8000000;
352 p_h = o.value;
353 p_l = v - (p_h - u);
354 z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
355 z_l = cp_l * p_h + p_l * cp + dp_l[k];
356 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
357 t = (long double) n;
358 t1 = (((z_h + z_l) + dp_h[k]) + t);
359 o.value = t1;
360 o.parts32.lswlo = 0;
361 o.parts32.lswhi &= 0xf8000000;
362 t1 = o.value;
363 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
364
365 /* s (sign of result -ve**odd) = -1 else = 1 */
366 s = one;
367 if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
368 s = -one; /* (-ve)**(odd int) */
369
370 /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
371 yy1 = y;
372 o.value = yy1;
373 o.parts32.lswlo = 0;
374 o.parts32.lswhi &= 0xf8000000;
375 yy1 = o.value;
376 p_l = (y - yy1) * t1 + y * t2;
377 p_h = yy1 * t1;
378 z = p_l + p_h;
379 o.value = z;
380 j = o.parts32.mswhi;
381 if (j >= 0x400d0000) /* z >= 16384 */
382 {
383 /* if z > 16384 */
384 if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi |
385 o.parts32.lswlo) != 0)
386 return s * huge * huge; /* overflow */
387 else
388 {
389 if (p_l + ovt > z - p_h)
390 return s * huge * huge; /* overflow */
391 }
392 }
393 else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */
394 {
395 /* z < -16495 */
396 if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi |
397 o.parts32.lswlo)
398 != 0)
399 return s * tiny * tiny; /* underflow */
400 else
401 {
402 if (p_l <= z - p_h)
403 return s * tiny * tiny; /* underflow */
404 }
405 }
406 /* compute 2**(p_h+p_l) */
407 i = j & 0x7fffffff;
408 k = (i >> 16) - 0x3fff;
409 n = 0;
410 if (i > 0x3ffe0000)
411 { /* if |z| > 0.5, set n = [z+0.5] */
412 n = floorl (z + 0.5L);
413 t = n;
414 p_h -= t;
415 }
416 t = p_l + p_h;
417 o.value = t;
418 o.parts32.lswlo = 0;
419 o.parts32.lswhi &= 0xf8000000;
420 t = o.value;
421 u = t * lg2_h;
422 v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
423 z = u + v;
424 w = v - (z - u);
425 /* exp(z) */
426 t = z * z;
427 u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
428 v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
429 t1 = z - t * u / v;
430 r = (z * t1) / (t1 - two) - (w + z * w);
431 z = one - (r - z);
432 o.value = z;
433 j = o.parts32.mswhi;
434 j += (n << 16);
435 if ((j >> 16) <= 0)
436 z = scalbnl (z, n); /* subnormal output */
437 else
438 {
439 o.parts32.mswhi = j;
440 z = o.value;
441 }
442 return s * z;
443 }
444