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 <http://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>
70 static const long double bp
[] = {
76 static const long double dp_h
[] = {
78 5.8496250072115607565592654282227158546448E-1L
81 /* Low part of log_2(1.5) */
82 static const long double dp_l
[] = {
84 1.0579781240112554492329533686862998106046E-16L
87 static const long double zero
= 0.0L,
90 two113
= 1.0384593717069655257060992658440192E34L
,
94 /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
97 Peak relative error 2.3e-37 */
98 static const long double LN
[] =
100 -3.0779177200290054398792536829702930623200E1L
,
101 6.5135778082209159921251824580292116201640E1L
,
102 -4.6312921812152436921591152809994014413540E1L
,
103 1.2510208195629420304615674658258363295208E1L
,
104 -9.9266909031921425609179910128531667336670E-1L
106 static const long double LD
[] =
108 -5.129862866715009066465422805058933131960E1L
,
109 1.452015077564081884387441590064272782044E2L
,
110 -1.524043275549860505277434040464085593165E2L
,
111 7.236063513651544224319663428634139768808E1L
,
112 -1.494198912340228235853027849917095580053E1L
116 /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
118 Peak relative error 5.7e-38 */
119 static const long double PN
[] =
121 5.081801691915377692446852383385968225675E8L
,
122 9.360895299872484512023336636427675327355E6L
,
123 4.213701282274196030811629773097579432957E4L
,
124 5.201006511142748908655720086041570288182E1L
,
125 9.088368420359444263703202925095675982530E-3L,
127 static const long double PD
[] =
129 3.049081015149226615468111430031590411682E9L
,
130 1.069833887183886839966085436512368982758E8L
,
131 8.259257717868875207333991924545445705394E5L
,
132 1.872583833284143212651746812884298360922E3L
,
136 static const long double
138 lg2
= 6.9314718055994530941723212145817656807550E-1L,
139 lg2_h
= 6.9314718055994528622676398299518041312695E-1L,
140 lg2_l
= 2.3190468138462996154948554638754786504121E-17L,
141 ovt
= 8.0085662595372944372e-0017L,
143 cp
= 9.6179669392597560490661645400126142495110E-1L,
144 cp_h
= 9.6179669392597555432899980587535537779331E-1L,
145 cp_l
= 5.0577616648125906047157785230014751039424E-17L;
148 __ieee754_powl (long double x
, long double y
)
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
, ay
;
153 int32_t i
, j
, k
, yisint
, n
;
156 double ohi
, xhi
, xlo
, yhi
, ylo
;
159 ldbl_unpack (x
, &xhi
, &xlo
);
160 EXTRACT_WORDS (hx
, lx
, xhi
);
161 ix
= hx
& 0x7fffffff;
163 ldbl_unpack (y
, &yhi
, &ylo
);
164 EXTRACT_WORDS (hy
, ly
, yhi
);
165 iy
= hy
& 0x7fffffff;
167 /* y==zero: x**0 = 1 */
171 /* 1.0**y = 1; -1.0**+-Inf = 1 */
174 if (x
== -1.0L && ((iy
- 0x7ff00000) | ly
) == 0)
177 /* +-NaN return x+y */
178 if ((ix
>= 0x7ff00000 && ((ix
- 0x7ff00000) | lx
) != 0)
179 || (iy
>= 0x7ff00000 && ((iy
- 0x7ff00000) | ly
) != 0))
182 /* determine if y is an odd int when x < 0
183 * yisint = 0 ... y is not an integer
184 * yisint = 1 ... y is an odd int
185 * yisint = 2 ... y is an even int
192 GET_HIGH_WORD (low_ye
, ylo
);
193 if ((low_ye
& 0x7fffffff) >= 0x43400000) /* Low part >= 2^53 */
194 yisint
= 2; /* even integer y */
195 else if (iy
>= 0x3ff00000) /* 1.0 */
197 if (__floorl (y
) == y
)
200 if (__floorl (z
) == z
)
210 /* special value of y */
213 if (iy
== 0x7ff00000) /* y is +-inf */
216 /* (|x|>1)**+-inf = inf,0 */
217 return (hy
>= 0) ? y
: zero
;
219 /* (|x|<1)**-,+inf = inf,0 */
220 return (hy
< 0) ? -y
: zero
;
224 if (iy
== 0x3ff00000)
231 if (hy
== 0x40000000)
232 return x
* x
; /* y is 2 */
233 if (hy
== 0x3fe00000)
235 if (hx
>= 0) /* x >= +0 */
236 return __ieee754_sqrtl (x
);
241 /* special value of x */
244 if (ix
== 0x7ff00000 || ix
== 0 || (ix
== 0x3ff00000 && xlo
== 0.0))
246 z
= ax
; /*x is +-0,+-inf,+-1 */
248 z
= one
/ z
; /* z = (1/|x|) */
251 if (((ix
- 0x3ff00000) | yisint
) == 0)
253 z
= (z
- z
) / (z
- z
); /* (-1)**non-int is NaN */
255 else if (yisint
== 1)
256 z
= -z
; /* (x<0)**odd = -(|x|**odd) */
262 /* (x<0)**(non-int) is NaN */
263 if (((((u_int32_t
) hx
>> 31) - 1) | yisint
) == 0)
264 return (x
- x
) / (x
- x
);
267 2^-16495 = 1/2 of smallest representable value.
268 If (1 - 1/131072)^y underflows, y > 1.4986e9 */
271 /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
274 if (ix
<= 0x3fefffff)
275 return (hy
< 0) ? huge
* huge
: tiny
* tiny
;
276 if (ix
>= 0x3ff00000)
277 return (hy
> 0) ? huge
* huge
: tiny
* tiny
;
279 /* over/underflow if x is not close to one */
281 return (hy
< 0) ? huge
* huge
: tiny
* tiny
;
283 return (hy
> 0) ? huge
* huge
: tiny
* tiny
;
288 y
= y
< 0 ? -0x1p
-117 : 0x1p
-117;
291 /* take care subnormal number */
296 ohi
= ldbl_high (ax
);
297 GET_HIGH_WORD (ix
, ohi
);
299 n
+= ((ix
) >> 20) - 0x3ff;
301 /* determine interval */
302 ix
= j
| 0x3ff00000; /* normalize ix */
304 k
= 0; /* |x|<sqrt(3/2) */
305 else if (j
< 0xbb670)
306 k
= 1; /* |x|<sqrt(3) */
314 ohi
= ldbl_high (ax
);
315 GET_HIGH_WORD (hax
, ohi
);
316 ax
= __scalbnl (ax
, ((int) ((ix
- hax
) * 2)) >> 21);
318 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
319 u
= ax
- bp
[k
]; /* bp[0]=1.0, bp[1]=1.5 */
320 v
= one
/ (ax
+ bp
[k
]);
324 /* t_h=ax+bp[k] High */
326 t_h
= ldbl_high (t_h
);
327 t_l
= ax
- (t_h
- bp
[k
]);
328 s_l
= v
* ((u
- s_h
* t_h
) - s_h
* t_l
);
329 /* compute log(ax) */
331 u
= LN
[0] + s2
* (LN
[1] + s2
* (LN
[2] + s2
* (LN
[3] + s2
* LN
[4])));
332 v
= LD
[0] + s2
* (LD
[1] + s2
* (LD
[2] + s2
* (LD
[3] + s2
* (LD
[4] + s2
))));
334 r
+= s_l
* (s_h
+ s
);
337 t_h
= ldbl_high (t_h
);
338 t_l
= r
- ((t_h
- 3.0) - s2
);
339 /* u+v = s*(1+...) */
341 v
= s_l
* t_h
+ t_l
* s
;
342 /* 2/(3log2)*(s+...) */
344 p_h
= ldbl_high (p_h
);
346 z_h
= cp_h
* p_h
; /* cp_h+cp_l = 2/(3*log2) */
347 z_l
= cp_l
* p_h
+ p_l
* cp
+ dp_l
[k
];
348 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
350 t1
= (((z_h
+ z_l
) + dp_h
[k
]) + t
);
352 t2
= z_l
- (((t1
- t
) - dp_h
[k
]) - z_h
);
354 /* s (sign of result -ve**odd) = -1 else = 1 */
356 if (((((u_int32_t
) hx
>> 31) - 1) | (yisint
- 1)) == 0)
357 s
= -one
; /* (-ve)**(odd int) */
359 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
361 p_l
= (y
- y1
) * t1
+ y
* t2
;
365 EXTRACT_WORDS (j
, lj
, ohi
);
366 if (j
>= 0x40d00000) /* z >= 16384 */
369 if (((j
- 0x40d00000) | lj
) != 0)
370 return s
* huge
* huge
; /* overflow */
373 if (p_l
+ ovt
> z
- p_h
)
374 return s
* huge
* huge
; /* overflow */
377 else if ((j
& 0x7fffffff) >= 0x40d01b90) /* z <= -16495 */
380 if (((j
- 0xc0d01bc0) | lj
) != 0)
381 return s
* tiny
* tiny
; /* underflow */
385 return s
* tiny
* tiny
; /* underflow */
388 /* compute 2**(p_h+p_l) */
390 k
= (i
>> 20) - 0x3ff;
393 { /* if |z| > 0.5, set n = [z+0.5] */
394 n
= __floorl (z
+ 0.5L);
401 v
= (p_l
- (t
- p_h
)) * lg2
+ t
* lg2_l
;
406 u
= PN
[0] + t
* (PN
[1] + t
* (PN
[2] + t
* (PN
[3] + t
* PN
[4])));
407 v
= PD
[0] + t
* (PD
[1] + t
* (PD
[2] + t
* (PD
[3] + t
)));
409 r
= (z
* t1
) / (t1
- two
) - (w
+ z
* w
);
411 z
= __scalbnl (z
, n
);
414 strong_alias (__ieee754_powl
, __powl_finite
)