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