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