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-barriers.h> |
69 | #include <math_private.h> |
70 | #include <libm-alias-finite.h> |
71 | |
72 | static const _Float128 bp[] = { |
73 | 1, |
74 | L(1.5), |
75 | }; |
76 | |
77 | /* log_2(1.5) */ |
78 | static const _Float128 dp_h[] = { |
79 | 0.0, |
80 | L(5.8496250072115607565592654282227158546448E-1) |
81 | }; |
82 | |
83 | /* Low part of log_2(1.5) */ |
84 | static const _Float128 dp_l[] = { |
85 | 0.0, |
86 | L(1.0579781240112554492329533686862998106046E-16) |
87 | }; |
88 | |
89 | static const _Float128 zero = 0, |
90 | one = 1, |
91 | two = 2, |
92 | two113 = L(1.0384593717069655257060992658440192E34), |
93 | huge = L(1.0e3000), |
94 | tiny = L(1.0e-3000); |
95 | |
96 | /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) |
97 | z = (x-1)/(x+1) |
98 | 1 <= x <= 1.25 |
99 | Peak relative error 2.3e-37 */ |
100 | static const _Float128 LN[] = |
101 | { |
102 | L(-3.0779177200290054398792536829702930623200E1), |
103 | L(6.5135778082209159921251824580292116201640E1), |
104 | L(-4.6312921812152436921591152809994014413540E1), |
105 | L(1.2510208195629420304615674658258363295208E1), |
106 | L(-9.9266909031921425609179910128531667336670E-1) |
107 | }; |
108 | static const _Float128 LD[] = |
109 | { |
110 | L(-5.129862866715009066465422805058933131960E1), |
111 | L(1.452015077564081884387441590064272782044E2), |
112 | L(-1.524043275549860505277434040464085593165E2), |
113 | L(7.236063513651544224319663428634139768808E1), |
114 | L(-1.494198912340228235853027849917095580053E1) |
115 | /* 1.0E0 */ |
116 | }; |
117 | |
118 | /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) |
119 | 0 <= x <= 0.5 |
120 | Peak relative error 5.7e-38 */ |
121 | static const _Float128 PN[] = |
122 | { |
123 | L(5.081801691915377692446852383385968225675E8), |
124 | L(9.360895299872484512023336636427675327355E6), |
125 | L(4.213701282274196030811629773097579432957E4), |
126 | L(5.201006511142748908655720086041570288182E1), |
127 | L(9.088368420359444263703202925095675982530E-3), |
128 | }; |
129 | static const _Float128 PD[] = |
130 | { |
131 | L(3.049081015149226615468111430031590411682E9), |
132 | L(1.069833887183886839966085436512368982758E8), |
133 | L(8.259257717868875207333991924545445705394E5), |
134 | L(1.872583833284143212651746812884298360922E3), |
135 | /* 1.0E0 */ |
136 | }; |
137 | |
138 | static const _Float128 |
139 | /* ln 2 */ |
140 | lg2 = L(6.9314718055994530941723212145817656807550E-1), |
141 | lg2_h = L(6.9314718055994528622676398299518041312695E-1), |
142 | lg2_l = L(2.3190468138462996154948554638754786504121E-17), |
143 | ovt = L(8.0085662595372944372e-0017), |
144 | /* 2/(3*log(2)) */ |
145 | cp = L(9.6179669392597560490661645400126142495110E-1), |
146 | cp_h = L(9.6179669392597555432899980587535537779331E-1), |
147 | cp_l = L(5.0577616648125906047157785230014751039424E-17); |
148 | |
149 | _Float128 |
150 | __ieee754_powl (_Float128 x, _Float128 y) |
151 | { |
152 | _Float128 z, ax, z_h, z_l, p_h, p_l; |
153 | _Float128 y1, t1, t2, r, s, sgn, t, u, v, w; |
154 | _Float128 s2, s_h, s_l, t_h, t_l, ay; |
155 | int32_t i, j, k, yisint, n; |
156 | uint32_t ix, iy; |
157 | int32_t hx, hy; |
158 | ieee854_long_double_shape_type o, p, q; |
159 | |
160 | p.value = x; |
161 | hx = p.parts32.w0; |
162 | ix = hx & 0x7fffffff; |
163 | |
164 | q.value = y; |
165 | hy = q.parts32.w0; |
166 | iy = hy & 0x7fffffff; |
167 | |
168 | |
169 | /* y==zero: x**0 = 1 */ |
170 | if ((iy | q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0 |
171 | && !issignaling (x)) |
172 | return one; |
173 | |
174 | /* 1.0**y = 1; -1.0**+-Inf = 1 */ |
175 | if (x == one && !issignaling (y)) |
176 | return one; |
177 | if (x == -1 && iy == 0x7fff0000 |
178 | && (q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0) |
179 | return one; |
180 | |
181 | /* +-NaN return x+y */ |
182 | if ((ix > 0x7fff0000) |
183 | || ((ix == 0x7fff0000) |
184 | && ((p.parts32.w1 | p.parts32.w2 | p.parts32.w3) != 0)) |
185 | || (iy > 0x7fff0000) |
186 | || ((iy == 0x7fff0000) |
187 | && ((q.parts32.w1 | q.parts32.w2 | q.parts32.w3) != 0))) |
188 | return x + y; |
189 | |
190 | /* determine if y is an odd int when x < 0 |
191 | * yisint = 0 ... y is not an integer |
192 | * yisint = 1 ... y is an odd int |
193 | * yisint = 2 ... y is an even int |
194 | */ |
195 | yisint = 0; |
196 | if (hx < 0) |
197 | { |
198 | if (iy >= 0x40700000) /* 2^113 */ |
199 | yisint = 2; /* even integer y */ |
200 | else if (iy >= 0x3fff0000) /* 1.0 */ |
201 | { |
202 | if (floorl (y) == y) |
203 | { |
204 | z = 0.5 * y; |
205 | if (floorl (z) == z) |
206 | yisint = 2; |
207 | else |
208 | yisint = 1; |
209 | } |
210 | } |
211 | } |
212 | |
213 | /* special value of y */ |
214 | if ((q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0) |
215 | { |
216 | if (iy == 0x7fff0000) /* y is +-inf */ |
217 | { |
218 | if (((ix - 0x3fff0000) | p.parts32.w1 | p.parts32.w2 | p.parts32.w3) |
219 | == 0) |
220 | return y - y; /* +-1**inf is NaN */ |
221 | else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */ |
222 | return (hy >= 0) ? y : zero; |
223 | else /* (|x|<1)**-,+inf = inf,0 */ |
224 | return (hy < 0) ? -y : zero; |
225 | } |
226 | if (iy == 0x3fff0000) |
227 | { /* y is +-1 */ |
228 | if (hy < 0) |
229 | return one / x; |
230 | else |
231 | return x; |
232 | } |
233 | if (hy == 0x40000000) |
234 | return x * x; /* y is 2 */ |
235 | if (hy == 0x3ffe0000) |
236 | { /* y is 0.5 */ |
237 | if (hx >= 0) /* x >= +0 */ |
238 | return sqrtl (x); |
239 | } |
240 | } |
241 | |
242 | ax = fabsl (x); |
243 | /* special value of x */ |
244 | if ((p.parts32.w1 | p.parts32.w2 | p.parts32.w3) == 0) |
245 | { |
246 | if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000) |
247 | { |
248 | z = ax; /*x is +-0,+-inf,+-1 */ |
249 | if (hy < 0) |
250 | z = one / z; /* z = (1/|x|) */ |
251 | if (hx < 0) |
252 | { |
253 | if (((ix - 0x3fff0000) | yisint) == 0) |
254 | { |
255 | z = (z - z) / (z - z); /* (-1)**non-int is NaN */ |
256 | } |
257 | else if (yisint == 1) |
258 | z = -z; /* (x<0)**odd = -(|x|**odd) */ |
259 | } |
260 | return z; |
261 | } |
262 | } |
263 | |
264 | /* (x<0)**(non-int) is NaN */ |
265 | if (((((uint32_t) hx >> 31) - 1) | yisint) == 0) |
266 | return (x - x) / (x - x); |
267 | |
268 | /* sgn (sign of result -ve**odd) = -1 else = 1 */ |
269 | sgn = one; |
270 | if (((((uint32_t) hx >> 31) - 1) | (yisint - 1)) == 0) |
271 | sgn = -one; /* (-ve)**(odd int) */ |
272 | |
273 | /* |y| is huge. |
274 | 2^-16495 = 1/2 of smallest representable value. |
275 | If (1 - 1/131072)^y underflows, y > 1.4986e9 */ |
276 | if (iy > 0x401d654b) |
277 | { |
278 | /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ |
279 | if (iy > 0x407d654b) |
280 | { |
281 | if (ix <= 0x3ffeffff) |
282 | return (hy < 0) ? huge * huge : tiny * tiny; |
283 | if (ix >= 0x3fff0000) |
284 | return (hy > 0) ? huge * huge : tiny * tiny; |
285 | } |
286 | /* over/underflow if x is not close to one */ |
287 | if (ix < 0x3ffeffff) |
288 | return (hy < 0) ? sgn * huge * huge : sgn * tiny * tiny; |
289 | if (ix > 0x3fff0000) |
290 | return (hy > 0) ? sgn * huge * huge : sgn * tiny * tiny; |
291 | } |
292 | |
293 | ay = y > 0 ? y : -y; |
294 | if (ay < 0x1p-128) |
295 | y = y < 0 ? -0x1p-128 : 0x1p-128; |
296 | |
297 | n = 0; |
298 | /* take care subnormal number */ |
299 | if (ix < 0x00010000) |
300 | { |
301 | ax *= two113; |
302 | n -= 113; |
303 | o.value = ax; |
304 | ix = o.parts32.w0; |
305 | } |
306 | n += ((ix) >> 16) - 0x3fff; |
307 | j = ix & 0x0000ffff; |
308 | /* determine interval */ |
309 | ix = j | 0x3fff0000; /* normalize ix */ |
310 | if (j <= 0x3988) |
311 | k = 0; /* |x|<sqrt(3/2) */ |
312 | else if (j < 0xbb67) |
313 | k = 1; /* |x|<sqrt(3) */ |
314 | else |
315 | { |
316 | k = 0; |
317 | n += 1; |
318 | ix -= 0x00010000; |
319 | } |
320 | |
321 | o.value = ax; |
322 | o.parts32.w0 = ix; |
323 | ax = o.value; |
324 | |
325 | /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
326 | u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ |
327 | v = one / (ax + bp[k]); |
328 | s = u * v; |
329 | s_h = s; |
330 | |
331 | o.value = s_h; |
332 | o.parts32.w3 = 0; |
333 | o.parts32.w2 &= 0xf8000000; |
334 | s_h = o.value; |
335 | /* t_h=ax+bp[k] High */ |
336 | t_h = ax + bp[k]; |
337 | o.value = t_h; |
338 | o.parts32.w3 = 0; |
339 | o.parts32.w2 &= 0xf8000000; |
340 | t_h = o.value; |
341 | t_l = ax - (t_h - bp[k]); |
342 | s_l = v * ((u - s_h * t_h) - s_h * t_l); |
343 | /* compute log(ax) */ |
344 | s2 = s * s; |
345 | u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4]))); |
346 | v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2)))); |
347 | r = s2 * s2 * u / v; |
348 | r += s_l * (s_h + s); |
349 | s2 = s_h * s_h; |
350 | t_h = 3.0 + s2 + r; |
351 | o.value = t_h; |
352 | o.parts32.w3 = 0; |
353 | o.parts32.w2 &= 0xf8000000; |
354 | t_h = o.value; |
355 | t_l = r - ((t_h - 3.0) - s2); |
356 | /* u+v = s*(1+...) */ |
357 | u = s_h * t_h; |
358 | v = s_l * t_h + t_l * s; |
359 | /* 2/(3log2)*(s+...) */ |
360 | p_h = u + v; |
361 | o.value = p_h; |
362 | o.parts32.w3 = 0; |
363 | o.parts32.w2 &= 0xf8000000; |
364 | p_h = o.value; |
365 | p_l = v - (p_h - u); |
366 | z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */ |
367 | z_l = cp_l * p_h + p_l * cp + dp_l[k]; |
368 | /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
369 | t = (_Float128) n; |
370 | t1 = (((z_h + z_l) + dp_h[k]) + t); |
371 | o.value = t1; |
372 | o.parts32.w3 = 0; |
373 | o.parts32.w2 &= 0xf8000000; |
374 | t1 = o.value; |
375 | t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); |
376 | |
377 | /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ |
378 | y1 = y; |
379 | o.value = y1; |
380 | o.parts32.w3 = 0; |
381 | o.parts32.w2 &= 0xf8000000; |
382 | y1 = o.value; |
383 | p_l = (y - y1) * t1 + y * t2; |
384 | p_h = y1 * t1; |
385 | z = p_l + p_h; |
386 | o.value = z; |
387 | j = o.parts32.w0; |
388 | if (j >= 0x400d0000) /* z >= 16384 */ |
389 | { |
390 | /* if z > 16384 */ |
391 | if (((j - 0x400d0000) | o.parts32.w1 | o.parts32.w2 | o.parts32.w3) != 0) |
392 | return sgn * huge * huge; /* overflow */ |
393 | else |
394 | { |
395 | if (p_l + ovt > z - p_h) |
396 | return sgn * huge * huge; /* overflow */ |
397 | } |
398 | } |
399 | else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */ |
400 | { |
401 | /* z < -16495 */ |
402 | if (((j - 0xc00d01bc) | o.parts32.w1 | o.parts32.w2 | o.parts32.w3) |
403 | != 0) |
404 | return sgn * tiny * tiny; /* underflow */ |
405 | else |
406 | { |
407 | if (p_l <= z - p_h) |
408 | return sgn * tiny * tiny; /* underflow */ |
409 | } |
410 | } |
411 | /* compute 2**(p_h+p_l) */ |
412 | i = j & 0x7fffffff; |
413 | k = (i >> 16) - 0x3fff; |
414 | n = 0; |
415 | if (i > 0x3ffe0000) |
416 | { /* if |z| > 0.5, set n = [z+0.5] */ |
417 | n = floorl (z + L(0.5)); |
418 | t = n; |
419 | p_h -= t; |
420 | } |
421 | t = p_l + p_h; |
422 | o.value = t; |
423 | o.parts32.w3 = 0; |
424 | o.parts32.w2 &= 0xf8000000; |
425 | t = o.value; |
426 | u = t * lg2_h; |
427 | v = (p_l - (t - p_h)) * lg2 + t * lg2_l; |
428 | z = u + v; |
429 | w = v - (z - u); |
430 | /* exp(z) */ |
431 | t = z * z; |
432 | u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); |
433 | v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); |
434 | t1 = z - t * u / v; |
435 | r = (z * t1) / (t1 - two) - (w + z * w); |
436 | z = one - (r - z); |
437 | o.value = z; |
438 | j = o.parts32.w0; |
439 | j += (n << 16); |
440 | if ((j >> 16) <= 0) |
441 | { |
442 | z = __scalbnl (z, n); /* subnormal output */ |
443 | _Float128 force_underflow = z * z; |
444 | math_force_eval (force_underflow); |
445 | } |
446 | else |
447 | { |
448 | o.parts32.w0 = j; |
449 | z = o.value; |
450 | } |
451 | return sgn * z; |
452 | } |
453 | libm_alias_finite (__ieee754_powl, __powl) |
454 | |