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