1 | /* |
2 | * IBM Accurate Mathematical Library |
3 | * written by International Business Machines Corp. |
4 | * Copyright (C) 2001-2018 Free Software Foundation, Inc. |
5 | * |
6 | * This program is free software; you can redistribute it and/or modify |
7 | * it under the terms of the GNU Lesser General Public License as published by |
8 | * the Free Software Foundation; either version 2.1 of the License, or |
9 | * (at your option) any later version. |
10 | * |
11 | * This program is distributed in the hope that it will be useful, |
12 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
13 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
14 | * GNU Lesser General Public License for more details. |
15 | * |
16 | * You should have received a copy of the GNU Lesser General Public License |
17 | * along with this program; if not, see <http://www.gnu.org/licenses/>. |
18 | */ |
19 | /***************************************************************************/ |
20 | /* MODULE_NAME: upow.c */ |
21 | /* */ |
22 | /* FUNCTIONS: upow */ |
23 | /* power1 */ |
24 | /* my_log2 */ |
25 | /* log1 */ |
26 | /* checkint */ |
27 | /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */ |
28 | /* halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c */ |
29 | /* uexp.c upow.c */ |
30 | /* root.tbl uexp.tbl upow.tbl */ |
31 | /* An ultimate power routine. Given two IEEE double machine numbers y,x */ |
32 | /* it computes the correctly rounded (to nearest) value of x^y. */ |
33 | /* Assumption: Machine arithmetic operations are performed in */ |
34 | /* round to nearest mode of IEEE 754 standard. */ |
35 | /* */ |
36 | /***************************************************************************/ |
37 | #include <math.h> |
38 | #include "endian.h" |
39 | #include "upow.h" |
40 | #include <dla.h> |
41 | #include "mydefs.h" |
42 | #include "MathLib.h" |
43 | #include "upow.tbl" |
44 | #include <math_private.h> |
45 | #include <fenv.h> |
46 | |
47 | #ifndef SECTION |
48 | # define SECTION |
49 | #endif |
50 | |
51 | static const double huge = 1.0e300, tiny = 1.0e-300; |
52 | |
53 | double __exp1 (double x, double xx, double error); |
54 | static double log1 (double x, double *delta, double *error); |
55 | static double my_log2 (double x, double *delta, double *error); |
56 | double __slowpow (double x, double y, double z); |
57 | static double power1 (double x, double y); |
58 | static int checkint (double x); |
59 | |
60 | /* An ultimate power routine. Given two IEEE double machine numbers y, x it |
61 | computes the correctly rounded (to nearest) value of X^y. */ |
62 | double |
63 | SECTION |
64 | __ieee754_pow (double x, double y) |
65 | { |
66 | double z, a, aa, error, t, a1, a2, y1, y2; |
67 | mynumber u, v; |
68 | int k; |
69 | int4 qx, qy; |
70 | v.x = y; |
71 | u.x = x; |
72 | if (v.i[LOW_HALF] == 0) |
73 | { /* of y */ |
74 | qx = u.i[HIGH_HALF] & 0x7fffffff; |
75 | /* Is x a NaN? */ |
76 | if ((((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000)) |
77 | && (y != 0 || issignaling (x))) |
78 | return x + x; |
79 | if (y == 1.0) |
80 | return x; |
81 | if (y == 2.0) |
82 | return x * x; |
83 | if (y == -1.0) |
84 | return 1.0 / x; |
85 | if (y == 0) |
86 | return 1.0; |
87 | } |
88 | /* else */ |
89 | if (((u.i[HIGH_HALF] > 0 && u.i[HIGH_HALF] < 0x7ff00000) || /* x>0 and not x->0 */ |
90 | (u.i[HIGH_HALF] == 0 && u.i[LOW_HALF] != 0)) && |
91 | /* 2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */ |
92 | (v.i[HIGH_HALF] & 0x7fffffff) < 0x4ff00000) |
93 | { /* if y<-1 or y>1 */ |
94 | double retval; |
95 | |
96 | { |
97 | SET_RESTORE_ROUND (FE_TONEAREST); |
98 | |
99 | /* Avoid internal underflow for tiny y. The exact value of y does |
100 | not matter if |y| <= 2**-64. */ |
101 | if (fabs (y) < 0x1p-64) |
102 | y = y < 0 ? -0x1p-64 : 0x1p-64; |
103 | z = log1 (x, &aa, &error); /* x^y =e^(y log (X)) */ |
104 | t = y * CN; |
105 | y1 = t - (t - y); |
106 | y2 = y - y1; |
107 | t = z * CN; |
108 | a1 = t - (t - z); |
109 | a2 = (z - a1) + aa; |
110 | a = y1 * a1; |
111 | aa = y2 * a1 + y * a2; |
112 | a1 = a + aa; |
113 | a2 = (a - a1) + aa; |
114 | error = error * fabs (y); |
115 | t = __exp1 (a1, a2, 1.9e16 * error); /* return -10 or 0 if wasn't computed exactly */ |
116 | retval = (t > 0) ? t : power1 (x, y); |
117 | } |
118 | |
119 | if (isinf (retval)) |
120 | retval = huge * huge; |
121 | else if (retval == 0) |
122 | retval = tiny * tiny; |
123 | else |
124 | math_check_force_underflow_nonneg (retval); |
125 | return retval; |
126 | } |
127 | |
128 | if (x == 0) |
129 | { |
130 | if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0) |
131 | || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000) /* NaN */ |
132 | return y + y; |
133 | if (fabs (y) > 1.0e20) |
134 | return (y > 0) ? 0 : 1.0 / 0.0; |
135 | k = checkint (y); |
136 | if (k == -1) |
137 | return y < 0 ? 1.0 / x : x; |
138 | else |
139 | return y < 0 ? 1.0 / 0.0 : 0.0; /* return 0 */ |
140 | } |
141 | |
142 | qx = u.i[HIGH_HALF] & 0x7fffffff; /* no sign */ |
143 | qy = v.i[HIGH_HALF] & 0x7fffffff; /* no sign */ |
144 | |
145 | if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0)) /* NaN */ |
146 | return x + y; |
147 | if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) /* NaN */ |
148 | return x == 1.0 && !issignaling (y) ? 1.0 : y + y; |
149 | |
150 | /* if x<0 */ |
151 | if (u.i[HIGH_HALF] < 0) |
152 | { |
153 | k = checkint (y); |
154 | if (k == 0) |
155 | { |
156 | if (qy == 0x7ff00000) |
157 | { |
158 | if (x == -1.0) |
159 | return 1.0; |
160 | else if (x > -1.0) |
161 | return v.i[HIGH_HALF] < 0 ? INF.x : 0.0; |
162 | else |
163 | return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x; |
164 | } |
165 | else if (qx == 0x7ff00000) |
166 | return y < 0 ? 0.0 : INF.x; |
167 | return (x - x) / (x - x); /* y not integer and x<0 */ |
168 | } |
169 | else if (qx == 0x7ff00000) |
170 | { |
171 | if (k < 0) |
172 | return y < 0 ? nZERO.x : nINF.x; |
173 | else |
174 | return y < 0 ? 0.0 : INF.x; |
175 | } |
176 | /* if y even or odd */ |
177 | if (k == 1) |
178 | return __ieee754_pow (-x, y); |
179 | else |
180 | { |
181 | double retval; |
182 | { |
183 | SET_RESTORE_ROUND (FE_TONEAREST); |
184 | retval = -__ieee754_pow (-x, y); |
185 | } |
186 | if (isinf (retval)) |
187 | retval = -huge * huge; |
188 | else if (retval == 0) |
189 | retval = -tiny * tiny; |
190 | return retval; |
191 | } |
192 | } |
193 | /* x>0 */ |
194 | |
195 | if (qx == 0x7ff00000) /* x= 2^-0x3ff */ |
196 | return y > 0 ? x : 0; |
197 | |
198 | if (qy > 0x45f00000 && qy < 0x7ff00000) |
199 | { |
200 | if (x == 1.0) |
201 | return 1.0; |
202 | if (y > 0) |
203 | return (x > 1.0) ? huge * huge : tiny * tiny; |
204 | if (y < 0) |
205 | return (x < 1.0) ? huge * huge : tiny * tiny; |
206 | } |
207 | |
208 | if (x == 1.0) |
209 | return 1.0; |
210 | if (y > 0) |
211 | return (x > 1.0) ? INF.x : 0; |
212 | if (y < 0) |
213 | return (x < 1.0) ? INF.x : 0; |
214 | return 0; /* unreachable, to make the compiler happy */ |
215 | } |
216 | |
217 | #ifndef __ieee754_pow |
218 | strong_alias (__ieee754_pow, __pow_finite) |
219 | #endif |
220 | |
221 | /* Compute x^y using more accurate but more slow log routine. */ |
222 | static double |
223 | SECTION |
224 | power1 (double x, double y) |
225 | { |
226 | double z, a, aa, error, t, a1, a2, y1, y2; |
227 | z = my_log2 (x, &aa, &error); |
228 | t = y * CN; |
229 | y1 = t - (t - y); |
230 | y2 = y - y1; |
231 | t = z * CN; |
232 | a1 = t - (t - z); |
233 | a2 = z - a1; |
234 | a = y * z; |
235 | aa = ((y1 * a1 - a) + y1 * a2 + y2 * a1) + y2 * a2 + aa * y; |
236 | a1 = a + aa; |
237 | a2 = (a - a1) + aa; |
238 | error = error * fabs (y); |
239 | t = __exp1 (a1, a2, 1.9e16 * error); |
240 | return (t >= 0) ? t : __slowpow (x, y, z); |
241 | } |
242 | |
243 | /* Compute log(x) (x is left argument). The result is the returned double + the |
244 | parameter DELTA. The result is bounded by ERROR. */ |
245 | static double |
246 | SECTION |
247 | log1 (double x, double *delta, double *error) |
248 | { |
249 | unsigned int i, j; |
250 | int m; |
251 | double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0; |
252 | mynumber u, v; |
253 | #ifdef BIG_ENDI |
254 | mynumber /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */ |
255 | #else |
256 | # ifdef LITTLE_ENDI |
257 | mynumber /**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */ |
258 | # endif |
259 | #endif |
260 | |
261 | u.x = x; |
262 | m = u.i[HIGH_HALF]; |
263 | *error = 0; |
264 | *delta = 0; |
265 | if (m < 0x00100000) /* 1<x<2^-1007 */ |
266 | { |
267 | x = x * t52.x; |
268 | add = -52.0; |
269 | u.x = x; |
270 | m = u.i[HIGH_HALF]; |
271 | } |
272 | |
273 | if ((m & 0x000fffff) < 0x0006a09e) |
274 | { |
275 | u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000; |
276 | two52.i[LOW_HALF] = (m >> 20); |
277 | } |
278 | else |
279 | { |
280 | u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000; |
281 | two52.i[LOW_HALF] = (m >> 20) + 1; |
282 | } |
283 | |
284 | v.x = u.x + bigu.x; |
285 | uu = v.x - bigu.x; |
286 | i = (v.i[LOW_HALF] & 0x000003ff) << 2; |
287 | if (two52.i[LOW_HALF] == 1023) /* nx = 0 */ |
288 | { |
289 | if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */ |
290 | { |
291 | t = x - 1.0; |
292 | t1 = (t + 5.0e6) - 5.0e6; |
293 | t2 = t - t1; |
294 | e1 = t - 0.5 * t1 * t1; |
295 | e2 = (t * t * t * (r3 + t * (r4 + t * (r5 + t * (r6 + t |
296 | * (r7 + t * r8))))) |
297 | - 0.5 * t2 * (t + t1)); |
298 | res = e1 + e2; |
299 | *error = 1.0e-21 * fabs (t); |
300 | *delta = (e1 - res) + e2; |
301 | return res; |
302 | } /* |x-1| < 1.5*2**-10 */ |
303 | else |
304 | { |
305 | v.x = u.x * (ui.x[i] + ui.x[i + 1]) + bigv.x; |
306 | vv = v.x - bigv.x; |
307 | j = v.i[LOW_HALF] & 0x0007ffff; |
308 | j = j + j + j; |
309 | eps = u.x - uu * vv; |
310 | e1 = eps * ui.x[i]; |
311 | e2 = eps * (ui.x[i + 1] + vj.x[j] * (ui.x[i] + ui.x[i + 1])); |
312 | e = e1 + e2; |
313 | e2 = ((e1 - e) + e2); |
314 | t = ui.x[i + 2] + vj.x[j + 1]; |
315 | t1 = t + e; |
316 | t2 = ((((t - t1) + e) + (ui.x[i + 3] + vj.x[j + 2])) + e2 + e * e |
317 | * (p2 + e * (p3 + e * p4))); |
318 | res = t1 + t2; |
319 | *error = 1.0e-24; |
320 | *delta = (t1 - res) + t2; |
321 | return res; |
322 | } |
323 | } /* nx = 0 */ |
324 | else /* nx != 0 */ |
325 | { |
326 | eps = u.x - uu; |
327 | nx = (two52.x - two52e.x) + add; |
328 | e1 = eps * ui.x[i]; |
329 | e2 = eps * ui.x[i + 1]; |
330 | e = e1 + e2; |
331 | e2 = (e1 - e) + e2; |
332 | t = nx * ln2a.x + ui.x[i + 2]; |
333 | t1 = t + e; |
334 | t2 = ((((t - t1) + e) + nx * ln2b.x + ui.x[i + 3] + e2) + e * e |
335 | * (q2 + e * (q3 + e * (q4 + e * (q5 + e * q6))))); |
336 | res = t1 + t2; |
337 | *error = 1.0e-21; |
338 | *delta = (t1 - res) + t2; |
339 | return res; |
340 | } /* nx != 0 */ |
341 | } |
342 | |
343 | /* Slower but more accurate routine of log. The returned result is double + |
344 | DELTA. The result is bounded by ERROR. */ |
345 | static double |
346 | SECTION |
347 | my_log2 (double x, double *delta, double *error) |
348 | { |
349 | unsigned int i, j; |
350 | int m; |
351 | double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0; |
352 | double ou1, ou2, lu1, lu2, ov, lv1, lv2, a, a1, a2; |
353 | double y, yy, z, zz, j1, j2, j7, j8; |
354 | #ifndef DLA_FMS |
355 | double j3, j4, j5, j6; |
356 | #endif |
357 | mynumber u, v; |
358 | #ifdef BIG_ENDI |
359 | mynumber /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */ |
360 | #else |
361 | # ifdef LITTLE_ENDI |
362 | mynumber /**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */ |
363 | # endif |
364 | #endif |
365 | |
366 | u.x = x; |
367 | m = u.i[HIGH_HALF]; |
368 | *error = 0; |
369 | *delta = 0; |
370 | add = 0; |
371 | if (m < 0x00100000) |
372 | { /* x < 2^-1022 */ |
373 | x = x * t52.x; |
374 | add = -52.0; |
375 | u.x = x; |
376 | m = u.i[HIGH_HALF]; |
377 | } |
378 | |
379 | if ((m & 0x000fffff) < 0x0006a09e) |
380 | { |
381 | u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000; |
382 | two52.i[LOW_HALF] = (m >> 20); |
383 | } |
384 | else |
385 | { |
386 | u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000; |
387 | two52.i[LOW_HALF] = (m >> 20) + 1; |
388 | } |
389 | |
390 | v.x = u.x + bigu.x; |
391 | uu = v.x - bigu.x; |
392 | i = (v.i[LOW_HALF] & 0x000003ff) << 2; |
393 | /*------------------------------------- |x-1| < 2**-11------------------------------- */ |
394 | if ((two52.i[LOW_HALF] == 1023) && (i == 1200)) |
395 | { |
396 | t = x - 1.0; |
397 | EMULV (t, s3, y, yy, j1, j2, j3, j4, j5); |
398 | ADD2 (-0.5, 0, y, yy, z, zz, j1, j2); |
399 | MUL2 (t, 0, z, zz, y, yy, j1, j2, j3, j4, j5, j6, j7, j8); |
400 | MUL2 (t, 0, y, yy, z, zz, j1, j2, j3, j4, j5, j6, j7, j8); |
401 | |
402 | e1 = t + z; |
403 | e2 = ((((t - e1) + z) + zz) + t * t * t |
404 | * (ss3 + t * (s4 + t * (s5 + t * (s6 + t * (s7 + t * s8)))))); |
405 | res = e1 + e2; |
406 | *error = 1.0e-25 * fabs (t); |
407 | *delta = (e1 - res) + e2; |
408 | return res; |
409 | } |
410 | /*----------------------------- |x-1| > 2**-11 -------------------------- */ |
411 | else |
412 | { /*Computing log(x) according to log table */ |
413 | nx = (two52.x - two52e.x) + add; |
414 | ou1 = ui.x[i]; |
415 | ou2 = ui.x[i + 1]; |
416 | lu1 = ui.x[i + 2]; |
417 | lu2 = ui.x[i + 3]; |
418 | v.x = u.x * (ou1 + ou2) + bigv.x; |
419 | vv = v.x - bigv.x; |
420 | j = v.i[LOW_HALF] & 0x0007ffff; |
421 | j = j + j + j; |
422 | eps = u.x - uu * vv; |
423 | ov = vj.x[j]; |
424 | lv1 = vj.x[j + 1]; |
425 | lv2 = vj.x[j + 2]; |
426 | a = (ou1 + ou2) * (1.0 + ov); |
427 | a1 = (a + 1.0e10) - 1.0e10; |
428 | a2 = a * (1.0 - a1 * uu * vv); |
429 | e1 = eps * a1; |
430 | e2 = eps * a2; |
431 | e = e1 + e2; |
432 | e2 = (e1 - e) + e2; |
433 | t = nx * ln2a.x + lu1 + lv1; |
434 | t1 = t + e; |
435 | t2 = ((((t - t1) + e) + (lu2 + lv2 + nx * ln2b.x + e2)) + e * e |
436 | * (p2 + e * (p3 + e * p4))); |
437 | res = t1 + t2; |
438 | *error = 1.0e-27; |
439 | *delta = (t1 - res) + t2; |
440 | return res; |
441 | } |
442 | } |
443 | |
444 | /* This function receives a double x and checks if it is an integer. If not, |
445 | it returns 0, else it returns 1 if even or -1 if odd. */ |
446 | static int |
447 | SECTION |
448 | checkint (double x) |
449 | { |
450 | union |
451 | { |
452 | int4 i[2]; |
453 | double x; |
454 | } u; |
455 | int k; |
456 | unsigned int m, n; |
457 | u.x = x; |
458 | m = u.i[HIGH_HALF] & 0x7fffffff; /* no sign */ |
459 | if (m >= 0x7ff00000) |
460 | return 0; /* x is +/-inf or NaN */ |
461 | if (m >= 0x43400000) |
462 | return 1; /* |x| >= 2**53 */ |
463 | if (m < 0x40000000) |
464 | return 0; /* |x| < 2, can not be 0 or 1 */ |
465 | n = u.i[LOW_HALF]; |
466 | k = (m >> 20) - 1023; /* 1 <= k <= 52 */ |
467 | if (k == 52) |
468 | return (n & 1) ? -1 : 1; /* odd or even */ |
469 | if (k > 20) |
470 | { |
471 | if (n << (k - 20) != 0) |
472 | return 0; /* if not integer */ |
473 | return (n << (k - 21) != 0) ? -1 : 1; |
474 | } |
475 | if (n) |
476 | return 0; /*if not integer */ |
477 | if (k == 20) |
478 | return (m & 1) ? -1 : 1; |
479 | if (m << (k + 12) != 0) |
480 | return 0; |
481 | return (m << (k + 11) != 0) ? -1 : 1; |
482 | } |
483 | |