]>
git.ipfire.org Git - thirdparty/glibc.git/blob - sysdeps/ieee754/ldbl-128ibm/e_powl.c
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
9 * ====================================================
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
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.
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.
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/>. */
33 /* __ieee754_powl(x,y) return x**y
36 * Method: Let x = 2 * (1+f)
37 * 1. Compute and return log2(x) in two pieces:
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)
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
68 #include <math_private.h>
69 #include <math-underflow.h>
70 #include <libm-alias-finite.h>
72 static const long double bp
[] = {
78 static const long double dp_h
[] = {
80 5.8496250072115607565592654282227158546448E-1L
83 /* Low part of log_2(1.5) */
84 static const long double dp_l
[] = {
86 1.0579781240112554492329533686862998106046E-16L
89 static const long double zero
= 0.0L,
92 two113
= 1.0384593717069655257060992658440192E34L
,
96 /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
99 Peak relative error 2.3e-37 */
100 static const long double LN
[] =
102 -3.0779177200290054398792536829702930623200E1L
,
103 6.5135778082209159921251824580292116201640E1L
,
104 -4.6312921812152436921591152809994014413540E1L
,
105 1.2510208195629420304615674658258363295208E1L
,
106 -9.9266909031921425609179910128531667336670E-1L
108 static const long double LD
[] =
110 -5.129862866715009066465422805058933131960E1L
,
111 1.452015077564081884387441590064272782044E2L
,
112 -1.524043275549860505277434040464085593165E2L
,
113 7.236063513651544224319663428634139768808E1L
,
114 -1.494198912340228235853027849917095580053E1L
118 /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
120 Peak relative error 5.7e-38 */
121 static const long double PN
[] =
123 5.081801691915377692446852383385968225675E8L
,
124 9.360895299872484512023336636427675327355E6L
,
125 4.213701282274196030811629773097579432957E4L
,
126 5.201006511142748908655720086041570288182E1L
,
127 9.088368420359444263703202925095675982530E-3L,
129 static const long double PD
[] =
131 3.049081015149226615468111430031590411682E9L
,
132 1.069833887183886839966085436512368982758E8L
,
133 8.259257717868875207333991924545445705394E5L
,
134 1.872583833284143212651746812884298360922E3L
,
138 static const long double
140 lg2
= 6.9314718055994530941723212145817656807550E-1L,
141 lg2_h
= 6.9314718055994528622676398299518041312695E-1L,
142 lg2_l
= 2.3190468138462996154948554638754786504121E-17L,
143 ovt
= 8.0085662595372944372e-0017L,
145 cp
= 9.6179669392597560490661645400126142495110E-1L,
146 cp_h
= 9.6179669392597555432899980587535537779331E-1L,
147 cp_l
= 5.0577616648125906047157785230014751039424E-17L;
150 __ieee754_powl (long double x
, long double y
)
152 long double z
, ax
, z_h
, z_l
, p_h
, p_l
;
153 long double y1
, t1
, t2
, r
, s
, sgn
, t
, u
, v
, w
;
154 long double s2
, s_h
, s_l
, t_h
, t_l
, ay
;
155 int32_t i
, j
, k
, yisint
, n
;
158 double ohi
, xhi
, xlo
, yhi
, ylo
;
161 ldbl_unpack (x
, &xhi
, &xlo
);
162 EXTRACT_WORDS (hx
, lx
, xhi
);
163 ix
= hx
& 0x7fffffff;
165 ldbl_unpack (y
, &yhi
, &ylo
);
166 EXTRACT_WORDS (hy
, ly
, yhi
);
167 iy
= hy
& 0x7fffffff;
169 /* y==zero: x**0 = 1 */
170 if ((iy
| ly
) == 0 && !issignaling (x
))
173 /* 1.0**y = 1; -1.0**+-Inf = 1 */
174 if (x
== one
&& !issignaling (y
))
176 if (x
== -1.0L && ((iy
- 0x7ff00000) | ly
) == 0)
179 /* +-NaN return x+y */
180 if ((ix
>= 0x7ff00000 && ((ix
- 0x7ff00000) | lx
) != 0)
181 || (iy
>= 0x7ff00000 && ((iy
- 0x7ff00000) | ly
) != 0))
184 /* determine if y is an odd int when x < 0
185 * yisint = 0 ... y is not an integer
186 * yisint = 1 ... y is an odd int
187 * yisint = 2 ... y is an even int
194 GET_HIGH_WORD (low_ye
, ylo
);
195 if ((low_ye
& 0x7fffffff) >= 0x43400000) /* Low part >= 2^53 */
196 yisint
= 2; /* even integer y */
197 else if (iy
>= 0x3ff00000) /* 1.0 */
212 /* special value of y */
215 if (iy
== 0x7ff00000) /* y is +-inf */
218 /* (|x|>1)**+-inf = inf,0 */
219 return (hy
>= 0) ? y
: zero
;
221 /* (|x|<1)**-,+inf = inf,0 */
222 return (hy
< 0) ? -y
: zero
;
226 if (iy
== 0x3ff00000)
233 if (hy
== 0x40000000)
234 return x
* x
; /* y is 2 */
235 if (hy
== 0x3fe00000)
237 if (hx
>= 0) /* x >= +0 */
243 /* special value of x */
246 if (ix
== 0x7ff00000 || ix
== 0 || (ix
== 0x3ff00000 && xlo
== 0.0))
248 z
= ax
; /*x is +-0,+-inf,+-1 */
250 z
= one
/ z
; /* z = (1/|x|) */
253 if (((ix
- 0x3ff00000) | yisint
) == 0)
255 z
= (z
- z
) / (z
- z
); /* (-1)**non-int is NaN */
257 else if (yisint
== 1)
258 z
= -z
; /* (x<0)**odd = -(|x|**odd) */
264 /* (x<0)**(non-int) is NaN */
265 if (((((uint32_t) hx
>> 31) - 1) | yisint
) == 0)
266 return (x
- x
) / (x
- x
);
268 /* sgn (sign of result -ve**odd) = -1 else = 1 */
270 if (((((uint32_t) hx
>> 31) - 1) | (yisint
- 1)) == 0)
271 sgn
= -one
; /* (-ve)**(odd int) */
274 2^-16495 = 1/2 of smallest representable value.
275 If (1 - 1/131072)^y underflows, y > 1.4986e9 */
278 /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
281 if (ix
<= 0x3fefffff)
282 return (hy
< 0) ? sgn
* huge
* huge
: sgn
* tiny
* tiny
;
283 if (ix
>= 0x3ff00000)
284 return (hy
> 0) ? sgn
* huge
* huge
: sgn
* tiny
* tiny
;
286 /* over/underflow if x is not close to one */
288 return (hy
< 0) ? sgn
* huge
* huge
: sgn
* tiny
* tiny
;
290 return (hy
> 0) ? sgn
* huge
* huge
: sgn
* tiny
* tiny
;
295 y
= y
< 0 ? -0x1p
-117 : 0x1p
-117;
298 /* take care subnormal number */
303 ohi
= ldbl_high (ax
);
304 GET_HIGH_WORD (ix
, ohi
);
306 n
+= ((ix
) >> 20) - 0x3ff;
308 /* determine interval */
309 ix
= j
| 0x3ff00000; /* normalize ix */
311 k
= 0; /* |x|<sqrt(3/2) */
312 else if (j
< 0xbb670)
313 k
= 1; /* |x|<sqrt(3) */
321 ohi
= ldbl_high (ax
);
322 GET_HIGH_WORD (hax
, ohi
);
323 ax
= __scalbnl (ax
, ((int) ((ix
- hax
) * 2)) >> 21);
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
]);
331 /* t_h=ax+bp[k] High */
333 t_h
= ldbl_high (t_h
);
334 t_l
= ax
- (t_h
- bp
[k
]);
335 s_l
= v
* ((u
- s_h
* t_h
) - s_h
* t_l
);
336 /* compute log(ax) */
338 u
= LN
[0] + s2
* (LN
[1] + s2
* (LN
[2] + s2
* (LN
[3] + s2
* LN
[4])));
339 v
= LD
[0] + s2
* (LD
[1] + s2
* (LD
[2] + s2
* (LD
[3] + s2
* (LD
[4] + s2
))));
341 r
+= s_l
* (s_h
+ s
);
344 t_h
= ldbl_high (t_h
);
345 t_l
= r
- ((t_h
- 3.0) - s2
);
346 /* u+v = s*(1+...) */
348 v
= s_l
* t_h
+ t_l
* s
;
349 /* 2/(3log2)*(s+...) */
351 p_h
= ldbl_high (p_h
);
353 z_h
= cp_h
* p_h
; /* cp_h+cp_l = 2/(3*log2) */
354 z_l
= cp_l
* p_h
+ p_l
* cp
+ dp_l
[k
];
355 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
357 t1
= (((z_h
+ z_l
) + dp_h
[k
]) + t
);
359 t2
= z_l
- (((t1
- t
) - dp_h
[k
]) - z_h
);
361 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
363 p_l
= (y
- y1
) * t1
+ y
* t2
;
367 EXTRACT_WORDS (j
, lj
, ohi
);
368 if (j
>= 0x40d00000) /* z >= 16384 */
371 if (((j
- 0x40d00000) | lj
) != 0)
372 return sgn
* huge
* huge
; /* overflow */
375 if (p_l
+ ovt
> z
- p_h
)
376 return sgn
* huge
* huge
; /* overflow */
379 else if ((j
& 0x7fffffff) >= 0x40d01b90) /* z <= -16495 */
382 if (((j
- 0xc0d01bc0) | lj
) != 0)
383 return sgn
* tiny
* tiny
; /* underflow */
387 return sgn
* tiny
* tiny
; /* underflow */
390 /* compute 2**(p_h+p_l) */
392 k
= (i
>> 20) - 0x3ff;
395 { /* if |z| > 0.5, set n = [z+0.5] */
396 n
= floorl (z
+ 0.5L);
403 v
= (p_l
- (t
- p_h
)) * lg2
+ t
* lg2_l
;
408 u
= PN
[0] + t
* (PN
[1] + t
* (PN
[2] + t
* (PN
[3] + t
* PN
[4])));
409 v
= PD
[0] + t
* (PD
[1] + t
* (PD
[2] + t
* (PD
[3] + t
)));
411 r
= (z
* t1
) / (t1
- two
) - (w
+ z
* w
);
413 z
= __scalbnl (sgn
* z
, n
);
414 math_check_force_underflow (z
);
417 libm_alias_finite (__ieee754_powl
, __powl
)