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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 | /* Expansions and modifications for 128-bit long double are | |
13 | Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> | |
14 | and are incorporated herein by permission of the author. The author | |
15 | reserves the right to distribute this material elsewhere under different | |
16 | copying permissions. These modifications are distributed here under | |
17 | the following terms: | |
18 | ||
19 | This library is free software; you can redistribute it and/or | |
20 | modify it under the terms of the GNU Lesser General Public | |
21 | License as published by the Free Software Foundation; either | |
22 | version 2.1 of the License, or (at your option) any later version. | |
23 | ||
24 | This library is distributed in the hope that it will be useful, | |
25 | but WITHOUT ANY WARRANTY; without even the implied warranty of | |
26 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
27 | Lesser General Public License for more details. | |
28 | ||
29 | You should have received a copy of the GNU Lesser General Public | |
30 | License along with this library; if not, see | |
31 | <https://www.gnu.org/licenses/>. */ | |
32 | ||
33 | /* __ieee754_powl(x,y) return x**y | |
34 | * | |
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 113-53 = 60 bit trailing zeros. | |
40 | * 2. Perform y*log2(x) = n+y' by simulating muti-precision | |
41 | * arithmetic, where |y'|<=0.5. | |
42 | * 3. Return x**y = 2**n*exp(y'*log2) | |
43 | * | |
44 | * Special cases: | |
45 | * 1. (anything) ** 0 is 1 | |
46 | * 2. (anything) ** 1 is itself | |
47 | * 3. (anything) ** NAN is NAN | |
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 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 | */ | |
66 | ||
67 | #include <math.h> | |
68 | #include <math_private.h> | |
69 | #include <math-underflow.h> | |
70 | ||
71 | static const long double bp[] = { | |
72 | 1.0L, | |
73 | 1.5L, | |
74 | }; | |
75 | ||
76 | /* log_2(1.5) */ | |
77 | static const long double dp_h[] = { | |
78 | 0.0, | |
79 | 5.8496250072115607565592654282227158546448E-1L | |
80 | }; | |
81 | ||
82 | /* Low part of log_2(1.5) */ | |
83 | static const long double dp_l[] = { | |
84 | 0.0, | |
85 | 1.0579781240112554492329533686862998106046E-16L | |
86 | }; | |
87 | ||
88 | static const long double zero = 0.0L, | |
89 | one = 1.0L, | |
90 | two = 2.0L, | |
91 | two113 = 1.0384593717069655257060992658440192E34L, | |
92 | huge = 1.0e300L, | |
93 | tiny = 1.0e-300L; | |
94 | ||
95 | /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) | |
96 | z = (x-1)/(x+1) | |
97 | 1 <= x <= 1.25 | |
98 | Peak relative error 2.3e-37 */ | |
99 | static const long double LN[] = | |
100 | { | |
101 | -3.0779177200290054398792536829702930623200E1L, | |
102 | 6.5135778082209159921251824580292116201640E1L, | |
103 | -4.6312921812152436921591152809994014413540E1L, | |
104 | 1.2510208195629420304615674658258363295208E1L, | |
105 | -9.9266909031921425609179910128531667336670E-1L | |
106 | }; | |
107 | static const long double LD[] = | |
108 | { | |
109 | -5.129862866715009066465422805058933131960E1L, | |
110 | 1.452015077564081884387441590064272782044E2L, | |
111 | -1.524043275549860505277434040464085593165E2L, | |
112 | 7.236063513651544224319663428634139768808E1L, | |
113 | -1.494198912340228235853027849917095580053E1L | |
114 | /* 1.0E0 */ | |
115 | }; | |
116 | ||
117 | /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) | |
118 | 0 <= x <= 0.5 | |
119 | Peak relative error 5.7e-38 */ | |
120 | static const long double PN[] = | |
121 | { | |
122 | 5.081801691915377692446852383385968225675E8L, | |
123 | 9.360895299872484512023336636427675327355E6L, | |
124 | 4.213701282274196030811629773097579432957E4L, | |
125 | 5.201006511142748908655720086041570288182E1L, | |
126 | 9.088368420359444263703202925095675982530E-3L, | |
127 | }; | |
128 | static const long double PD[] = | |
129 | { | |
130 | 3.049081015149226615468111430031590411682E9L, | |
131 | 1.069833887183886839966085436512368982758E8L, | |
132 | 8.259257717868875207333991924545445705394E5L, | |
133 | 1.872583833284143212651746812884298360922E3L, | |
134 | /* 1.0E0 */ | |
135 | }; | |
136 | ||
137 | static const long double | |
138 | /* ln 2 */ | |
139 | lg2 = 6.9314718055994530941723212145817656807550E-1L, | |
140 | lg2_h = 6.9314718055994528622676398299518041312695E-1L, | |
141 | lg2_l = 2.3190468138462996154948554638754786504121E-17L, | |
142 | ovt = 8.0085662595372944372e-0017L, | |
143 | /* 2/(3*log(2)) */ | |
144 | cp = 9.6179669392597560490661645400126142495110E-1L, | |
145 | cp_h = 9.6179669392597555432899980587535537779331E-1L, | |
146 | cp_l = 5.0577616648125906047157785230014751039424E-17L; | |
147 | ||
148 | long double | |
149 | __ieee754_powl (long double x, long double y) | |
150 | { | |
151 | long double z, ax, z_h, z_l, p_h, p_l; | |
152 | long double y1, t1, t2, r, s, sgn, t, u, v, w; | |
153 | long double s2, s_h, s_l, t_h, t_l, ay; | |
154 | int32_t i, j, k, yisint, n; | |
155 | uint32_t ix, iy; | |
156 | int32_t hx, hy, hax; | |
157 | double ohi, xhi, xlo, yhi, ylo; | |
158 | uint32_t lx, ly, lj; | |
159 | ||
160 | ldbl_unpack (x, &xhi, &xlo); | |
161 | EXTRACT_WORDS (hx, lx, xhi); | |
162 | ix = hx & 0x7fffffff; | |
163 | ||
164 | ldbl_unpack (y, &yhi, &ylo); | |
165 | EXTRACT_WORDS (hy, ly, yhi); | |
166 | iy = hy & 0x7fffffff; | |
167 | ||
168 | /* y==zero: x**0 = 1 */ | |
169 | if ((iy | ly) == 0 && !issignaling (x)) | |
170 | return one; | |
171 | ||
172 | /* 1.0**y = 1; -1.0**+-Inf = 1 */ | |
173 | if (x == one && !issignaling (y)) | |
174 | return one; | |
175 | if (x == -1.0L && ((iy - 0x7ff00000) | ly) == 0) | |
176 | return one; | |
177 | ||
178 | /* +-NaN return x+y */ | |
179 | if ((ix >= 0x7ff00000 && ((ix - 0x7ff00000) | lx) != 0) | |
180 | || (iy >= 0x7ff00000 && ((iy - 0x7ff00000) | ly) != 0)) | |
181 | return x + y; | |
182 | ||
183 | /* determine if y is an odd int when x < 0 | |
184 | * yisint = 0 ... y is not an integer | |
185 | * yisint = 1 ... y is an odd int | |
186 | * yisint = 2 ... y is an even int | |
187 | */ | |
188 | yisint = 0; | |
189 | if (hx < 0) | |
190 | { | |
191 | uint32_t low_ye; | |
192 | ||
193 | GET_HIGH_WORD (low_ye, ylo); | |
194 | if ((low_ye & 0x7fffffff) >= 0x43400000) /* Low part >= 2^53 */ | |
195 | yisint = 2; /* even integer y */ | |
196 | else if (iy >= 0x3ff00000) /* 1.0 */ | |
197 | { | |
198 | if (floorl (y) == y) | |
199 | { | |
200 | z = 0.5 * y; | |
201 | if (floorl (z) == z) | |
202 | yisint = 2; | |
203 | else | |
204 | yisint = 1; | |
205 | } | |
206 | } | |
207 | } | |
208 | ||
209 | ax = fabsl (x); | |
210 | ||
211 | /* special value of y */ | |
212 | if (ly == 0) | |
213 | { | |
214 | if (iy == 0x7ff00000) /* y is +-inf */ | |
215 | { | |
216 | if (ax > one) | |
217 | /* (|x|>1)**+-inf = inf,0 */ | |
218 | return (hy >= 0) ? y : zero; | |
219 | else | |
220 | /* (|x|<1)**-,+inf = inf,0 */ | |
221 | return (hy < 0) ? -y : zero; | |
222 | } | |
223 | if (ylo == 0.0) | |
224 | { | |
225 | if (iy == 0x3ff00000) | |
226 | { /* y is +-1 */ | |
227 | if (hy < 0) | |
228 | return one / x; | |
229 | else | |
230 | return x; | |
231 | } | |
232 | if (hy == 0x40000000) | |
233 | return x * x; /* y is 2 */ | |
234 | if (hy == 0x3fe00000) | |
235 | { /* y is 0.5 */ | |
236 | if (hx >= 0) /* x >= +0 */ | |
237 | return sqrtl (x); | |
238 | } | |
239 | } | |
240 | } | |
241 | ||
242 | /* special value of x */ | |
243 | if (lx == 0) | |
244 | { | |
245 | if (ix == 0x7ff00000 || ix == 0 || (ix == 0x3ff00000 && xlo == 0.0)) | |
246 | { | |
247 | z = ax; /*x is +-0,+-inf,+-1 */ | |
248 | if (hy < 0) | |
249 | z = one / z; /* z = (1/|x|) */ | |
250 | if (hx < 0) | |
251 | { | |
252 | if (((ix - 0x3ff00000) | yisint) == 0) | |
253 | { | |
254 | z = (z - z) / (z - z); /* (-1)**non-int is NaN */ | |
255 | } | |
256 | else if (yisint == 1) | |
257 | z = -z; /* (x<0)**odd = -(|x|**odd) */ | |
258 | } | |
259 | return z; | |
260 | } | |
261 | } | |
262 | ||
263 | /* (x<0)**(non-int) is NaN */ | |
264 | if (((((uint32_t) hx >> 31) - 1) | yisint) == 0) | |
265 | return (x - x) / (x - x); | |
266 | ||
267 | /* sgn (sign of result -ve**odd) = -1 else = 1 */ | |
268 | sgn = one; | |
269 | if (((((uint32_t) hx >> 31) - 1) | (yisint - 1)) == 0) | |
270 | sgn = -one; /* (-ve)**(odd int) */ | |
271 | ||
272 | /* |y| is huge. | |
273 | 2^-16495 = 1/2 of smallest representable value. | |
274 | If (1 - 1/131072)^y underflows, y > 1.4986e9 */ | |
275 | if (iy > 0x41d654b0) | |
276 | { | |
277 | /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ | |
278 | if (iy > 0x47d654b0) | |
279 | { | |
280 | if (ix <= 0x3fefffff) | |
281 | return (hy < 0) ? sgn * huge * huge : sgn * tiny * tiny; | |
282 | if (ix >= 0x3ff00000) | |
283 | return (hy > 0) ? sgn * huge * huge : sgn * tiny * tiny; | |
284 | } | |
285 | /* over/underflow if x is not close to one */ | |
286 | if (ix < 0x3fefffff) | |
287 | return (hy < 0) ? sgn * huge * huge : sgn * tiny * tiny; | |
288 | if (ix > 0x3ff00000) | |
289 | return (hy > 0) ? sgn * huge * huge : sgn * tiny * tiny; | |
290 | } | |
291 | ||
292 | ay = y > 0 ? y : -y; | |
293 | if (ay < 0x1p-117) | |
294 | y = y < 0 ? -0x1p-117 : 0x1p-117; | |
295 | ||
296 | n = 0; | |
297 | /* take care subnormal number */ | |
298 | if (ix < 0x00100000) | |
299 | { | |
300 | ax *= two113; | |
301 | n -= 113; | |
302 | ohi = ldbl_high (ax); | |
303 | GET_HIGH_WORD (ix, ohi); | |
304 | } | |
305 | n += ((ix) >> 20) - 0x3ff; | |
306 | j = ix & 0x000fffff; | |
307 | /* determine interval */ | |
308 | ix = j | 0x3ff00000; /* normalize ix */ | |
309 | if (j <= 0x39880) | |
310 | k = 0; /* |x|<sqrt(3/2) */ | |
311 | else if (j < 0xbb670) | |
312 | k = 1; /* |x|<sqrt(3) */ | |
313 | else | |
314 | { | |
315 | k = 0; | |
316 | n += 1; | |
317 | ix -= 0x00100000; | |
318 | } | |
319 | ||
320 | ohi = ldbl_high (ax); | |
321 | GET_HIGH_WORD (hax, ohi); | |
322 | ax = __scalbnl (ax, ((int) ((ix - hax) * 2)) >> 21); | |
323 | ||
324 | /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ | |
325 | u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ | |
326 | v = one / (ax + bp[k]); | |
327 | s = u * v; | |
328 | s_h = ldbl_high (s); | |
329 | ||
330 | /* t_h=ax+bp[k] High */ | |
331 | t_h = ax + bp[k]; | |
332 | t_h = ldbl_high (t_h); | |
333 | t_l = ax - (t_h - bp[k]); | |
334 | s_l = v * ((u - s_h * t_h) - s_h * t_l); | |
335 | /* compute log(ax) */ | |
336 | s2 = s * s; | |
337 | u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4]))); | |
338 | v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2)))); | |
339 | r = s2 * s2 * u / v; | |
340 | r += s_l * (s_h + s); | |
341 | s2 = s_h * s_h; | |
342 | t_h = 3.0 + s2 + r; | |
343 | t_h = ldbl_high (t_h); | |
344 | t_l = r - ((t_h - 3.0) - s2); | |
345 | /* u+v = s*(1+...) */ | |
346 | u = s_h * t_h; | |
347 | v = s_l * t_h + t_l * s; | |
348 | /* 2/(3log2)*(s+...) */ | |
349 | p_h = u + v; | |
350 | p_h = ldbl_high (p_h); | |
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 | t1 = ldbl_high (t1); | |
358 | t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); | |
359 | ||
360 | /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ | |
361 | y1 = ldbl_high (y); | |
362 | p_l = (y - y1) * t1 + y * t2; | |
363 | p_h = y1 * t1; | |
364 | z = p_l + p_h; | |
365 | ohi = ldbl_high (z); | |
366 | EXTRACT_WORDS (j, lj, ohi); | |
367 | if (j >= 0x40d00000) /* z >= 16384 */ | |
368 | { | |
369 | /* if z > 16384 */ | |
370 | if (((j - 0x40d00000) | lj) != 0) | |
371 | return sgn * huge * huge; /* overflow */ | |
372 | else | |
373 | { | |
374 | if (p_l + ovt > z - p_h) | |
375 | return sgn * huge * huge; /* overflow */ | |
376 | } | |
377 | } | |
378 | else if ((j & 0x7fffffff) >= 0x40d01b90) /* z <= -16495 */ | |
379 | { | |
380 | /* z < -16495 */ | |
381 | if (((j - 0xc0d01bc0) | lj) != 0) | |
382 | return sgn * tiny * tiny; /* underflow */ | |
383 | else | |
384 | { | |
385 | if (p_l <= z - p_h) | |
386 | return sgn * tiny * tiny; /* underflow */ | |
387 | } | |
388 | } | |
389 | /* compute 2**(p_h+p_l) */ | |
390 | i = j & 0x7fffffff; | |
391 | k = (i >> 20) - 0x3ff; | |
392 | n = 0; | |
393 | if (i > 0x3fe00000) | |
394 | { /* if |z| > 0.5, set n = [z+0.5] */ | |
395 | n = floorl (z + 0.5L); | |
396 | t = n; | |
397 | p_h -= t; | |
398 | } | |
399 | t = p_l + p_h; | |
400 | t = ldbl_high (t); | |
401 | u = t * lg2_h; | |
402 | v = (p_l - (t - p_h)) * lg2 + t * lg2_l; | |
403 | z = u + v; | |
404 | w = v - (z - u); | |
405 | /* exp(z) */ | |
406 | t = z * z; | |
407 | u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); | |
408 | v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); | |
409 | t1 = z - t * u / v; | |
410 | r = (z * t1) / (t1 - two) - (w + z * w); | |
411 | z = one - (r - z); | |
412 | z = __scalbnl (sgn * z, n); | |
413 | math_check_force_underflow (z); | |
414 | return z; | |
415 | } | |
416 | strong_alias (__ieee754_powl, __powl_finite) |