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1f5649f8 UD |
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 | ||
cc7375ce RM |
12 | /* Expansions and modifications for 128-bit long double are |
13 | Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> | |
0ac5ae23 | 14 | and are incorporated herein by permission of the author. The author |
9cd2726c | 15 | reserves the right to distribute this material elsewhere under different |
0ac5ae23 | 16 | copying permissions. These modifications are distributed here under |
9cd2726c | 17 | the following terms: |
cc7375ce RM |
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, write to the Free Software | |
31 | Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ | |
1f5649f8 UD |
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 | ||
70 | static const long double bp[] = { | |
71 | 1.0L, | |
72 | 1.5L, | |
73 | }; | |
74 | ||
75 | /* log_2(1.5) */ | |
76 | static const long double dp_h[] = { | |
77 | 0.0, | |
78 | 5.8496250072115607565592654282227158546448E-1L | |
79 | }; | |
80 | ||
81 | /* Low part of log_2(1.5) */ | |
82 | static const long double dp_l[] = { | |
83 | 0.0, | |
84 | 1.0579781240112554492329533686862998106046E-16L | |
85 | }; | |
86 | ||
87 | static const long double zero = 0.0L, | |
88 | one = 1.0L, | |
89 | two = 2.0L, | |
90 | two113 = 1.0384593717069655257060992658440192E34L, | |
91 | huge = 1.0e3000L, | |
92 | tiny = 1.0e-3000L; | |
93 | ||
94 | /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) | |
95 | z = (x-1)/(x+1) | |
96 | 1 <= x <= 1.25 | |
97 | Peak relative error 2.3e-37 */ | |
98 | static const long double LN[] = | |
99 | { | |
100 | -3.0779177200290054398792536829702930623200E1L, | |
101 | 6.5135778082209159921251824580292116201640E1L, | |
102 | -4.6312921812152436921591152809994014413540E1L, | |
103 | 1.2510208195629420304615674658258363295208E1L, | |
104 | -9.9266909031921425609179910128531667336670E-1L | |
105 | }; | |
106 | static const long double LD[] = | |
107 | { | |
108 | -5.129862866715009066465422805058933131960E1L, | |
109 | 1.452015077564081884387441590064272782044E2L, | |
110 | -1.524043275549860505277434040464085593165E2L, | |
111 | 7.236063513651544224319663428634139768808E1L, | |
112 | -1.494198912340228235853027849917095580053E1L | |
113 | /* 1.0E0 */ | |
114 | }; | |
115 | ||
116 | /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) | |
117 | 0 <= x <= 0.5 | |
118 | Peak relative error 5.7e-38 */ | |
119 | static const long double PN[] = | |
120 | { | |
121 | 5.081801691915377692446852383385968225675E8L, | |
122 | 9.360895299872484512023336636427675327355E6L, | |
123 | 4.213701282274196030811629773097579432957E4L, | |
124 | 5.201006511142748908655720086041570288182E1L, | |
125 | 9.088368420359444263703202925095675982530E-3L, | |
126 | }; | |
127 | static const long double PD[] = | |
128 | { | |
129 | 3.049081015149226615468111430031590411682E9L, | |
130 | 1.069833887183886839966085436512368982758E8L, | |
131 | 8.259257717868875207333991924545445705394E5L, | |
132 | 1.872583833284143212651746812884298360922E3L, | |
133 | /* 1.0E0 */ | |
134 | }; | |
135 | ||
136 | static const long double | |
137 | /* ln 2 */ | |
138 | lg2 = 6.9314718055994530941723212145817656807550E-1L, | |
139 | lg2_h = 6.9314718055994528622676398299518041312695E-1L, | |
140 | lg2_l = 2.3190468138462996154948554638754786504121E-17L, | |
141 | ovt = 8.0085662595372944372e-0017L, | |
142 | /* 2/(3*log(2)) */ | |
143 | cp = 9.6179669392597560490661645400126142495110E-1L, | |
144 | cp_h = 9.6179669392597555432899980587535537779331E-1L, | |
145 | cp_l = 5.0577616648125906047157785230014751039424E-17L; | |
146 | ||
1f5649f8 UD |
147 | long double |
148 | __ieee754_powl (long double x, long double y) | |
1f5649f8 UD |
149 | { |
150 | long double z, ax, z_h, z_l, p_h, p_l; | |
151 | long double y1, t1, t2, r, s, t, u, v, w; | |
152 | long double s2, s_h, s_l, t_h, t_l; | |
153 | int32_t i, j, k, yisint, n; | |
154 | u_int32_t ix, iy; | |
155 | int32_t hx, hy; | |
156 | ieee854_long_double_shape_type o, p, q; | |
157 | ||
158 | p.value = x; | |
159 | hx = p.parts32.w0; | |
160 | ix = hx & 0x7fffffff; | |
161 | ||
162 | q.value = y; | |
163 | hy = q.parts32.w0; | |
164 | iy = hy & 0x7fffffff; | |
165 | ||
166 | ||
167 | /* y==zero: x**0 = 1 */ | |
168 | if ((iy | q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0) | |
169 | return one; | |
170 | ||
52e1b618 UD |
171 | /* 1.0**y = 1; -1.0**+-Inf = 1 */ |
172 | if (x == one) | |
173 | return one; | |
174 | if (x == -1.0L && iy == 0x7fff0000 | |
175 | && (q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0) | |
176 | return one; | |
177 | ||
1f5649f8 UD |
178 | /* +-NaN return x+y */ |
179 | if ((ix > 0x7fff0000) | |
180 | || ((ix == 0x7fff0000) | |
181 | && ((p.parts32.w1 | p.parts32.w2 | p.parts32.w3) != 0)) | |
182 | || (iy > 0x7fff0000) | |
183 | || ((iy == 0x7fff0000) | |
184 | && ((q.parts32.w1 | q.parts32.w2 | q.parts32.w3) != 0))) | |
185 | return x + y; | |
186 | ||
187 | /* determine if y is an odd int when x < 0 | |
188 | * yisint = 0 ... y is not an integer | |
189 | * yisint = 1 ... y is an odd int | |
190 | * yisint = 2 ... y is an even int | |
191 | */ | |
192 | yisint = 0; | |
193 | if (hx < 0) | |
194 | { | |
195 | if (iy >= 0x40700000) /* 2^113 */ | |
196 | yisint = 2; /* even integer y */ | |
197 | else if (iy >= 0x3fff0000) /* 1.0 */ | |
198 | { | |
199 | if (__floorl (y) == y) | |
200 | { | |
201 | z = 0.5 * y; | |
202 | if (__floorl (z) == z) | |
203 | yisint = 2; | |
204 | else | |
205 | yisint = 1; | |
206 | } | |
207 | } | |
208 | } | |
209 | ||
210 | /* special value of y */ | |
211 | if ((q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0) | |
212 | { | |
213 | if (iy == 0x7fff0000) /* y is +-inf */ | |
214 | { | |
215 | if (((ix - 0x3fff0000) | p.parts32.w1 | p.parts32.w2 | p.parts32.w3) | |
216 | == 0) | |
f964490f | 217 | return y - y; /* +-1**inf is NaN */ |
1f5649f8 UD |
218 | else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */ |
219 | return (hy >= 0) ? y : zero; | |
220 | else /* (|x|<1)**-,+inf = inf,0 */ | |
221 | return (hy < 0) ? -y : zero; | |
222 | } | |
223 | if (iy == 0x3fff0000) | |
224 | { /* y is +-1 */ | |
225 | if (hy < 0) | |
226 | return one / x; | |
227 | else | |
228 | return x; | |
229 | } | |
230 | if (hy == 0x40000000) | |
231 | return x * x; /* y is 2 */ | |
232 | if (hy == 0x3ffe0000) | |
233 | { /* y is 0.5 */ | |
234 | if (hx >= 0) /* x >= +0 */ | |
235 | return __ieee754_sqrtl (x); | |
236 | } | |
237 | } | |
238 | ||
239 | ax = fabsl (x); | |
240 | /* special value of x */ | |
241 | if ((p.parts32.w1 | p.parts32.w2 | p.parts32.w3) == 0) | |
242 | { | |
243 | if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000) | |
244 | { | |
245 | z = ax; /*x is +-0,+-inf,+-1 */ | |
246 | if (hy < 0) | |
247 | z = one / z; /* z = (1/|x|) */ | |
248 | if (hx < 0) | |
249 | { | |
250 | if (((ix - 0x3fff0000) | yisint) == 0) | |
251 | { | |
252 | z = (z - z) / (z - z); /* (-1)**non-int is NaN */ | |
253 | } | |
254 | else if (yisint == 1) | |
255 | z = -z; /* (x<0)**odd = -(|x|**odd) */ | |
256 | } | |
257 | return z; | |
258 | } | |
259 | } | |
260 | ||
261 | /* (x<0)**(non-int) is NaN */ | |
262 | if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0) | |
263 | return (x - x) / (x - x); | |
264 | ||
265 | /* |y| is huge. | |
266 | 2^-16495 = 1/2 of smallest representable value. | |
267 | If (1 - 1/131072)^y underflows, y > 1.4986e9 */ | |
268 | if (iy > 0x401d654b) | |
269 | { | |
270 | /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ | |
271 | if (iy > 0x407d654b) | |
272 | { | |
273 | if (ix <= 0x3ffeffff) | |
274 | return (hy < 0) ? huge * huge : tiny * tiny; | |
275 | if (ix >= 0x3fff0000) | |
276 | return (hy > 0) ? huge * huge : tiny * tiny; | |
277 | } | |
278 | /* over/underflow if x is not close to one */ | |
279 | if (ix < 0x3ffeffff) | |
280 | return (hy < 0) ? huge * huge : tiny * tiny; | |
281 | if (ix > 0x3fff0000) | |
282 | return (hy > 0) ? huge * huge : tiny * tiny; | |
283 | } | |
284 | ||
285 | n = 0; | |
286 | /* take care subnormal number */ | |
287 | if (ix < 0x00010000) | |
288 | { | |
289 | ax *= two113; | |
290 | n -= 113; | |
291 | o.value = ax; | |
292 | ix = o.parts32.w0; | |
293 | } | |
294 | n += ((ix) >> 16) - 0x3fff; | |
295 | j = ix & 0x0000ffff; | |
296 | /* determine interval */ | |
297 | ix = j | 0x3fff0000; /* normalize ix */ | |
298 | if (j <= 0x3988) | |
299 | k = 0; /* |x|<sqrt(3/2) */ | |
300 | else if (j < 0xbb67) | |
301 | k = 1; /* |x|<sqrt(3) */ | |
302 | else | |
303 | { | |
304 | k = 0; | |
305 | n += 1; | |
306 | ix -= 0x00010000; | |
307 | } | |
308 | ||
309 | o.value = ax; | |
310 | o.parts32.w0 = ix; | |
311 | ax = o.value; | |
312 | ||
313 | /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ | |
314 | u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ | |
315 | v = one / (ax + bp[k]); | |
316 | s = u * v; | |
317 | s_h = s; | |
318 | ||
319 | o.value = s_h; | |
320 | o.parts32.w3 = 0; | |
321 | o.parts32.w2 &= 0xf8000000; | |
322 | s_h = o.value; | |
323 | /* t_h=ax+bp[k] High */ | |
324 | t_h = ax + bp[k]; | |
325 | o.value = t_h; | |
326 | o.parts32.w3 = 0; | |
327 | o.parts32.w2 &= 0xf8000000; | |
328 | t_h = o.value; | |
329 | t_l = ax - (t_h - bp[k]); | |
330 | s_l = v * ((u - s_h * t_h) - s_h * t_l); | |
331 | /* compute log(ax) */ | |
332 | s2 = s * s; | |
333 | u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4]))); | |
334 | v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2)))); | |
335 | r = s2 * s2 * u / v; | |
336 | r += s_l * (s_h + s); | |
337 | s2 = s_h * s_h; | |
338 | t_h = 3.0 + s2 + r; | |
339 | o.value = t_h; | |
340 | o.parts32.w3 = 0; | |
341 | o.parts32.w2 &= 0xf8000000; | |
342 | t_h = o.value; | |
343 | t_l = r - ((t_h - 3.0) - s2); | |
344 | /* u+v = s*(1+...) */ | |
345 | u = s_h * t_h; | |
346 | v = s_l * t_h + t_l * s; | |
347 | /* 2/(3log2)*(s+...) */ | |
348 | p_h = u + v; | |
349 | o.value = p_h; | |
350 | o.parts32.w3 = 0; | |
351 | o.parts32.w2 &= 0xf8000000; | |
352 | p_h = o.value; | |
353 | p_l = v - (p_h - u); | |
354 | z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */ | |
355 | z_l = cp_l * p_h + p_l * cp + dp_l[k]; | |
356 | /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ | |
357 | t = (long double) n; | |
358 | t1 = (((z_h + z_l) + dp_h[k]) + t); | |
359 | o.value = t1; | |
360 | o.parts32.w3 = 0; | |
361 | o.parts32.w2 &= 0xf8000000; | |
362 | t1 = o.value; | |
363 | t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); | |
364 | ||
365 | /* s (sign of result -ve**odd) = -1 else = 1 */ | |
366 | s = one; | |
367 | if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0) | |
368 | s = -one; /* (-ve)**(odd int) */ | |
369 | ||
370 | /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ | |
371 | y1 = y; | |
372 | o.value = y1; | |
373 | o.parts32.w3 = 0; | |
374 | o.parts32.w2 &= 0xf8000000; | |
375 | y1 = o.value; | |
376 | p_l = (y - y1) * t1 + y * t2; | |
377 | p_h = y1 * t1; | |
378 | z = p_l + p_h; | |
379 | o.value = z; | |
380 | j = o.parts32.w0; | |
381 | if (j >= 0x400d0000) /* z >= 16384 */ | |
382 | { | |
383 | /* if z > 16384 */ | |
384 | if (((j - 0x400d0000) | o.parts32.w1 | o.parts32.w2 | o.parts32.w3) != 0) | |
385 | return s * huge * huge; /* overflow */ | |
386 | else | |
387 | { | |
388 | if (p_l + ovt > z - p_h) | |
389 | return s * huge * huge; /* overflow */ | |
390 | } | |
391 | } | |
392 | else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */ | |
393 | { | |
394 | /* z < -16495 */ | |
395 | if (((j - 0xc00d01bc) | o.parts32.w1 | o.parts32.w2 | o.parts32.w3) | |
396 | != 0) | |
397 | return s * tiny * tiny; /* underflow */ | |
398 | else | |
399 | { | |
400 | if (p_l <= z - p_h) | |
401 | return s * tiny * tiny; /* underflow */ | |
402 | } | |
403 | } | |
404 | /* compute 2**(p_h+p_l) */ | |
405 | i = j & 0x7fffffff; | |
406 | k = (i >> 16) - 0x3fff; | |
407 | n = 0; | |
408 | if (i > 0x3ffe0000) | |
409 | { /* if |z| > 0.5, set n = [z+0.5] */ | |
410 | n = __floorl (z + 0.5L); | |
411 | t = n; | |
412 | p_h -= t; | |
413 | } | |
414 | t = p_l + p_h; | |
415 | o.value = t; | |
416 | o.parts32.w3 = 0; | |
417 | o.parts32.w2 &= 0xf8000000; | |
418 | t = o.value; | |
419 | u = t * lg2_h; | |
420 | v = (p_l - (t - p_h)) * lg2 + t * lg2_l; | |
421 | z = u + v; | |
422 | w = v - (z - u); | |
423 | /* exp(z) */ | |
424 | t = z * z; | |
425 | u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); | |
426 | v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); | |
427 | t1 = z - t * u / v; | |
428 | r = (z * t1) / (t1 - two) - (w + z * w); | |
429 | z = one - (r - z); | |
430 | o.value = z; | |
431 | j = o.parts32.w0; | |
432 | j += (n << 16); | |
433 | if ((j >> 16) <= 0) | |
434 | z = __scalbnl (z, n); /* subnormal output */ | |
435 | else | |
436 | { | |
437 | o.parts32.w0 = j; | |
438 | z = o.value; | |
439 | } | |
440 | return s * z; | |
441 | } | |
0ac5ae23 | 442 | strong_alias (__ieee754_powl, __powl_finite) |