1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __pow = pow
32
33 /*
34 * pow(x,y) return x**y
35 * n
36 * Method: Let x = 2 * (1+f)
37 * 1. Compute and return log2(x) in two pieces:
38 * log2(x) = w1 + w2,
39 * where w1 has 24 bits trailing zero.
40 * 2. Perform y*log2(x) by simulating muti-precision arithmetic
41 * 3. Return x**y = exp2(y*log(x))
42 *
43 * Special cases:
44 * 1. (anything) ** +-0 is 1
45 * 1'. 1 ** (anything) is 1 (C99; 1 ** +-INF/NAN used to be NAN)
46 * 2. (anything) ** 1 is itself
47 * 3. (anything except 1) ** NAN is NAN ("except 1" is C99)
48 * 4. NAN ** (anything except 0) is NAN
49 * 5. +-(|x| > 1) ** +INF is +INF
50 * 6. +-(|x| > 1) ** -INF is +0
51 * 7. +-(|x| < 1) ** +INF is +0
52 * 8. +-(|x| < 1) ** -INF is +INF
53 * 9. -1 ** +-INF is 1 (C99; -1 ** +-INF used to be NAN)
54 * 10. +0 ** (+anything except 0, NAN) is +0
55 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
56 * 12. +0 ** (-anything except 0, NAN) is +INF
57 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
58 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
59 * 15. +INF ** (+anything except 0,NAN) is +INF
60 * 16. +INF ** (-anything except 0,NAN) is +0
61 * 17. -INF ** (anything) = -0 ** (-anything)
62 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
63 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
64 *
65 * Accuracy:
66 * pow(x,y) returns x**y nearly rounded. In particular
67 * pow(integer,integer)
68 * always returns the correct integer provided it is representable.
69 */
70
71 #include "libm.h"
72 #include "xpg6.h" /* __xpg6 */
73 #define _C99SUSv3_pow _C99SUSv3_pow_treats_Inf_as_an_even_int
74
75 static const double zero = 0.0,
76 one = 1.0,
77 two = 2.0;
78
79 extern const double _TBL_log2_hi[], _TBL_log2_lo[];
80
81 static const double two53 = 9007199254740992.0,
82 A1_hi = 2.8853900432586669921875,
83 A1_lo = 3.8519259825035041963606002e-8,
84 A1 = 2.885390081777926817222541963606002026086e+0000,
85 A2 = 9.617966939207270828380543979852286255862e-0001,
86 A3 = 5.770807680887875964868853124873696201995e-0001,
87 B0_hi = 2.8853900432586669921875,
88 B0_lo = 3.8519259822532793056374320585e-8,
89 B0 = 2.885390081777926814720293056374320585689e+0000,
90 B1 = 9.617966939259755138949202350396200257632e-0001,
91 B2 = 5.770780163585687000782112776448797953382e-0001,
92 B3 = 4.121985488948771523290174512461778354953e-0001,
93 B4 = 3.207590534812432970433641789022666850193e-0001;
94
95 static double
96 log2_x(double x, double *w)
97 {
98 double f, s, z, qn, h, t;
99 int *px = (int *)&x;
100 int *pz = (int *)&z;
101 int i, j, ix, n;
102
103 n = 0;
104 ix = px[HIWORD];
105
106 if (ix >= 0x3fef03f1 && ix < 0x3ff08208) { /* 65/63 > x > 63/65 */
107 double f1, v;
108
109 f = x - one;
110
111 if (((ix - 0x3ff00000) | px[LOWORD]) == 0) {
112 *w = zero;
113 return (zero); /* log2(1)= +0 */
114 }
115
116 qn = one / (two + f);
117 s = f * qn; /* |s|<2**-6 */
118 v = s * s;
119 h = (double)((float)s);
120 f1 = (double)((float)f);
121 t = qn * (((f - two * h) - h * f1) - h * (f - f1));
122 /* s = h+t */
123 f1 = h * B0_lo + s * (v * (B1 + v * (B2 + v * (B3 + v * B4))));
124 t = f1 + t * B0;
125 h *= B0_hi;
126 s = (double)((float)(h + t));
127 *w = t - (s - h);
128 return (s);
129 }
130
131 if (ix < 0x00100000) { /* subnormal x */
132 x *= two53;
133 n = -53;
134 ix = px[HIWORD];
135 }
136
137 /* LARGE N */
138 n += ((ix + 0x1000) >> 20) - 0x3ff;
139 ix = (ix & 0x000fffff) | 0x3ff00000; /* scale x to [1,2] */
140 px[HIWORD] = ix;
141 i = ix + 0x1000;
142 pz[HIWORD] = i & 0xffffe000;
143 pz[LOWORD] = 0;
144 qn = one / (x + z);
145 f = x - z;
146 s = f * qn;
147 h = (double)((float)s);
148 t = qn * ((f - (h + h) * z) - h * f);
149 j = (i >> 13) & 0x7f;
150 f = s * s;
151 t = t * A1 + h * A1_lo;
152 t += (s * f) * (A2 + f * A3);
153 qn = h * A1_hi;
154 s = n + _TBL_log2_hi[j];
155 h = qn + s;
156 t += _TBL_log2_lo[j] - ((h - s) - qn);
157 f = (double)((float)(h + t));
158 *w = t - (f - h);
159 return (f);
160 }
161
162 extern const double _TBL_exp2_hi[], _TBL_exp2_lo[];
163 static const double /* poly app of 2^x-1 on [-1e-10,2^-7+1e-10] */
164 E1 = 6.931471805599453100674958533810346197328e-0001,
165 E2 = 2.402265069587779347846769151717493815979e-0001,
166 E3 = 5.550410866475410512631124892773937864699e-0002,
167 E4 = 9.618143209991026824853712740162451423355e-0003,
168 E5 = 1.333357676549940345096774122231849082991e-0003;
169
170 double
171 pow(double x, double y)
172 {
173 double z, ax;
174 double y1, y2, w1, w2;
175 int sbx, sby, j, k, yisint;
176 int hx, hy, ahx, ahy;
177 unsigned lx, ly;
178 int *pz = (int *)&z;
179
180 hx = ((int *)&x)[HIWORD];
181 lx = ((unsigned *)&x)[LOWORD];
182 hy = ((int *)&y)[HIWORD];
183 ly = ((unsigned *)&y)[LOWORD];
184 ahx = hx & ~0x80000000;
185 ahy = hy & ~0x80000000;
186
187 if ((ahy | ly) == 0) { /* y==zero */
188 if ((ahx | lx) == 0)
189 z = _SVID_libm_err(x, y, 20); /* +-0**+-0 */
190 else if ((ahx | (((lx | -lx) >> 31) & 1)) > 0x7ff00000)
191 z = _SVID_libm_err(x, y, 42); /* NaN**+-0 */
192 else
193 z = one; /* x**+-0 = 1 */
194
195 return (z);
196 } else if (hx == 0x3ff00000 && lx == 0 && (__xpg6 & _C99SUSv3_pow) !=
197 0) {
198 return (one); /* C99: 1**anything = 1 */
199 } else if (ahx > 0x7ff00000 || (ahx == 0x7ff00000 && lx != 0) || ahy >
200 0x7ff00000 || (ahy == 0x7ff00000 && ly != 0)) {
201 return (x * y); /* +-NaN return x*y; + -> * for Cheetah */
202 }
203
204 /* includes Sun: 1**NaN = NaN */
205 sbx = (unsigned)hx >> 31;
206 sby = (unsigned)hy >> 31;
207 ax = fabs(x);
208
209 /*
210 * determine if y is an odd int when x < 0
211 * yisint = 0 ... y is not an integer
212 * yisint = 1 ... y is an odd int
213 * yisint = 2 ... y is an even int
214 */
215 yisint = 0;
216
217 if (sbx) {
218 if (ahy >= 0x43400000) {
219 yisint = 2; /* even integer y */
220 } else if (ahy >= 0x3ff00000) {
221 k = (ahy >> 20) - 0x3ff; /* exponent */
222
223 if (k > 20) {
224 j = ly >> (52 - k);
225
226 if ((j << (52 - k)) == ly)
227 yisint = 2 - (j & 1);
228 } else if (ly == 0) {
229 j = ahy >> (20 - k);
230
231 if ((j << (20 - k)) == ahy)
232 yisint = 2 - (j & 1);
233 }
234 }
235 }
236
237 /* special value of y */
238 if (ly == 0) {
239 if (ahy == 0x7ff00000) { /* y is +-inf */
240 if (((ahx - 0x3ff00000) | lx) == 0) {
241 if ((__xpg6 & _C99SUSv3_pow) != 0)
242 return (one);
243 /* C99: (-1)**+-inf = 1 */
244 else
245 return (y - y);
246
247 /* Sun: (+-1)**+-inf = NaN */
248 } else if (ahx >= 0x3ff00000) {
249 /* (|x|>1)**+,-inf = inf,0 */
250 return (sby == 0 ? y : zero);
251 } else { /* (|x|<1)**-,+inf = inf,0 */
252 return (sby != 0 ? -y : zero);
253 }
254 }
255
256 if (ahy == 0x3ff00000) { /* y is +-1 */
257 if (sby != 0) { /* y is -1 */
258 if (x == zero) /* divided by zero */
259 return (_SVID_libm_err(x, y, 23));
260 else if (ahx < 0x40000 || ((ahx - 0x40000) |
261 lx) == 0) /* overflow */
262 return (_SVID_libm_err(x, y, 21));
263 else
264 return (one / x);
265 } else {
266 return (x);
267 }
268 }
269
270 if (hy == 0x40000000) { /* y is 2 */
271 if (ahx >= 0x5ff00000 && ahx < 0x7ff00000)
272 return (_SVID_libm_err(x, y, 21));
273 /* x*x overflow */
274 else if ((ahx < 0x1e56a09e && (ahx | lx) != 0) ||
275 (ahx == 0x1e56a09e && lx < 0x667f3bcd))
276 return (_SVID_libm_err(x, y, 22));
277 /* x*x underflow */
278 else
279 return (x * x);
280 }
281
282 if (hy == 0x3fe00000) {
283 if (!((ahx | lx) == 0 || ((ahx - 0x7ff00000) | lx) ==
284 0 || sbx == 1))
285 return (sqrt(x)); /* y is 0.5 and x > 0 */
286 }
287 }
288
289 /* special value of x */
290 if (lx == 0) {
291 if (ahx == 0x7ff00000 || ahx == 0 || ahx == 0x3ff00000) {
292 /* x is +-0,+-inf,-1 */
293 z = ax;
294
295 if (sby == 1) {
296 z = one / z; /* z = |x|**y */
297
298 if (ahx == 0)
299 return (_SVID_libm_err(x, y, 23));
300 }
301
302 if (sbx == 1) {
303 if (ahx == 0x3ff00000 && yisint == 0)
304 z = _SVID_libm_err(x, y, 24);
305 /* neg**non-integral is NaN + invalid */
306 else if (yisint == 1)
307 z = -z; /* (x<0)**odd = -(|x|**odd) */
308 }
309
310 return (z);
311 }
312 }
313
314 /* (x<0)**(non-int) is NaN */
315 if (sbx == 1 && yisint == 0)
316 return (_SVID_libm_err(x, y, 24));
317
318 /*
319 * Now ax is finite, y is finite
320 * first compute log2(ax) = w1+w2, with 24 bits w1
321 */
322 w1 = log2_x(ax, &w2);
323
324 /* split up y into y1+y2 and compute (y1+y2)*(w1+w2) */
325 if (((ly & 0x07ffffff) == 0) || ahy >= 0x47e00000 || ahy <=
326 0x38100000) {
327 /* no need to split if y is short or too large or too small */
328 y1 = y * w1;
329 y2 = y * w2;
330 } else {
331 y1 = (double)((float)y);
332 y2 = (y - y1) * w1 + y * w2;
333 y1 *= w1;
334 }
335
336 z = y1 + y2;
337 j = pz[HIWORD];
338
339 if (j >= 0x40900000) { /* z >= 1024 */
340 if (!(j == 0x40900000 && pz[LOWORD] == 0)) { /* z > 1024 */
341 return (_SVID_libm_err(x, y, 21)); /* overflow */
342 } else {
343 w2 = y1 - z;
344 w2 += y2;
345
346 /* rounded to inf */
347 if (w2 >= -8.008566259537296567160e-17)
348 return (_SVID_libm_err(x, y, 21));
349
350 /* overflow */
351 }
352 } else if ((j & ~0x80000000) >= 0x4090cc00) { /* z <= -1075 */
353 if (!(j == 0xc090cc00 && pz[LOWORD] == 0)) { /* z < -1075 */
354 return (_SVID_libm_err(x, y, 22)); /* underflow */
355 } else {
356 w2 = y1 - z;
357 w2 += y2;
358
359 if (w2 <= zero) /* underflow */
360 return (_SVID_libm_err(x, y, 22));
361 }
362 }
363
364 /*
365 * compute 2**(k+f[j]+g)
366 */
367 k = (int)(z * 64.0 + (((hy ^ (ahx - 0x3ff00000)) > 0) ? 0.5 : -0.5));
368 j = k & 63;
369 w1 = y2 - ((double)k * 0.015625 - y1);
370 w2 = _TBL_exp2_hi[j];
371 z = _TBL_exp2_lo[j] + (w2 * w1) * (E1 + w1 * (E2 + w1 * (E3 + w1 * (E4 +
372 w1 * E5))));
373 z += w2;
374 k >>= 6;
375
376 if (k < -1021)
377 z = scalbn(z, k);
378 else /* subnormal output */
379 pz[HIWORD] += k << 20;
380
381 if (sbx == 1 && yisint == 1)
382 z = -z; /* (-ve)**(odd int) */
383
384 return (z);
385 }