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Initial import.

1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
2 *
3 * ***** BEGIN LICENSE BLOCK *****
4 * Version: MPL 1.1/GPL 2.0/LGPL 2.1
5 *
6 * The contents of this file are subject to the Mozilla Public License Version
7 * 1.1 (the "License"); you may not use this file except in compliance with
8 * the License. You may obtain a copy of the License at
9 * http://www.mozilla.org/MPL/
10 *
11 * Software distributed under the License is distributed on an "AS IS" basis,
12 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
13 * for the specific language governing rights and limitations under the
14 * License.
15 *
16 * The Original Code is Mozilla Communicator client code, released
17 * March 31, 1998.
18 *
19 * The Initial Developer of the Original Code is
20 * Sun Microsystems, Inc.
21 * Portions created by the Initial Developer are Copyright (C) 1998
22 * the Initial Developer. All Rights Reserved.
23 *
24 * Contributor(s):
25 *
26 * Alternatively, the contents of this file may be used under the terms of
27 * either of the GNU General Public License Version 2 or later (the "GPL"),
28 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
29 * in which case the provisions of the GPL or the LGPL are applicable instead
30 * of those above. If you wish to allow use of your version of this file only
31 * under the terms of either the GPL or the LGPL, and not to allow others to
32 * use your version of this file under the terms of the MPL, indicate your
33 * decision by deleting the provisions above and replace them with the notice
34 * and other provisions required by the GPL or the LGPL. If you do not delete
35 * the provisions above, a recipient may use your version of this file under
36 * the terms of any one of the MPL, the GPL or the LGPL.
37 *
38 * ***** END LICENSE BLOCK ***** */
39
40 /* @(#)e_pow.c 1.3 95/01/18 */
41 /*
42 * ====================================================
43 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
44 *
45 * Developed at SunSoft, a Sun Microsystems, Inc. business.
46 * Permission to use, copy, modify, and distribute this
47 * software is freely granted, provided that this notice
48 * is preserved.
49 * ====================================================
50 */
51
52 /* __ieee754_pow(x,y) return x**y
53 *
54 * n
55 * Method: Let x = 2 * (1+f)
56 * 1. Compute and return log2(x) in two pieces:
57 * log2(x) = w1 + w2,
58 * where w1 has 53-24 = 29 bit trailing zeros.
59 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
60 * arithmetic, where |y'|<=0.5.
61 * 3. Return x**y = 2**n*exp(y'*log2)
62 *
63 * Special cases:
64 * 1. (anything) ** 0 is 1
65 * 2. (anything) ** 1 is itself
66 * 3. (anything) ** NAN is NAN
67 * 4. NAN ** (anything except 0) is NAN
68 * 5. +-(|x| > 1) ** +INF is +INF
69 * 6. +-(|x| > 1) ** -INF is +0
70 * 7. +-(|x| < 1) ** +INF is +0
71 * 8. +-(|x| < 1) ** -INF is +INF
72 * 9. +-1 ** +-INF is NAN
73 * 10. +0 ** (+anything except 0, NAN) is +0
74 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
75 * 12. +0 ** (-anything except 0, NAN) is +INF
76 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
77 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
78 * 15. +INF ** (+anything except 0,NAN) is +INF
79 * 16. +INF ** (-anything except 0,NAN) is +0
80 * 17. -INF ** (anything) = -0 ** (-anything)
81 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
82 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
83 *
84 * Accuracy:
85 * pow(x,y) returns x**y nearly rounded. In particular
86 * pow(integer,integer)
87 * always returns the correct integer provided it is
88 * representable.
89 *
90 * Constants :
91 * The hexadecimal values are the intended ones for the following
92 * constants. The decimal values may be used, provided that the
93 * compiler will convert from decimal to binary accurately enough
94 * to produce the hexadecimal values shown.
95 */
96
97 #include "fdlibm.h"
98
99 #if defined(_MSC_VER)
100 /* Microsoft Compiler */
101 #pragma warning( disable : 4723 ) /* disables potential divide by 0 warning */
102 #endif
103
104 #ifdef __STDC__
105 static const double
106 #else
107 static double
108 #endif
109 bp[] = {1.0, 1.5,},
110 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
111 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
112 zero = 0.0,
113 one = 1.0,
114 two = 2.0,
115 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
116 really_big = 1.0e300,
117 tiny = 1.0e-300,
118 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
119 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
120 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
121 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
122 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
123 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
124 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
125 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
126 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
127 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
128 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
129 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
130 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
131 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
132 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
133 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
134 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
135 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
136 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
137 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
138 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
139 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
140
141 #ifdef __STDC__
142 double __ieee754_pow(double x, double y)
143 #else
144 double __ieee754_pow(x,y)
145 double x, y;
146 #endif
147 {
148 fd_twoints ux, uy, uz;
149 double y1,t1,p_h,t,z,ax;
150 double z_h,z_l,p_l;
151 double t2,r,s,u,v,w;
152 int i,j,k,yisint,n;
153 int hx,hy,ix,iy;
154 unsigned lx,ly;
155
156 ux.d = x; uy.d = y;
157 hx = __HI(ux); lx = __LO(ux);
158 hy = __HI(uy); ly = __LO(uy);
159 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
160
161 /* y==zero: x**0 = 1 */
162 if((iy|ly)==0) return one;
163
164 /* +-NaN return x+y */
165 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
166 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
167 return x+y;
168
169 /* determine if y is an odd int when x < 0
170 * yisint = 0 ... y is not an integer
171 * yisint = 1 ... y is an odd int
172 * yisint = 2 ... y is an even int
173 */
174 yisint = 0;
175 if(hx<0) {
176 if(iy>=0x43400000) yisint = 2; /* even integer y */
177 else if(iy>=0x3ff00000) {
178 k = (iy>>20)-0x3ff; /* exponent */
179 if(k>20) {
180 j = ly>>(52-k);
181 if((j<<(52-k))==(int)ly) yisint = 2-(j&1);
182 } else if(ly==0) {
183 j = iy>>(20-k);
184 if((j<<(20-k))==iy) yisint = 2-(j&1);
185 }
186 }
187 }
188
189 /* special value of y */
190 if(ly==0) {
191 if (iy==0x7ff00000) { /* y is +-inf */
192 if(((ix-0x3ff00000)|lx)==0)
193 #ifdef _WIN32
194 /* VC++ optimizer reduces y - y to 0 */
195 return y / y;
196 #else
197 return y - y; /* inf**+-1 is NaN */
198 #endif
199 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
200 return (hy>=0)? y: zero;
201 else /* (|x|<1)**-,+inf = inf,0 */
202 return (hy<0)?-y: zero;
203 }
204 if(iy==0x3ff00000) { /* y is +-1 */
205 if(hy<0) return one/x; else return x;
206 }
207 if(hy==0x40000000) return x*x; /* y is 2 */
208 if(hy==0x3fe00000) { /* y is 0.5 */
209 if(hx>=0) /* x >= +0 */
210 return fd_sqrt(x);
211 }
212 }
213
214 ax = fd_fabs(x);
215 /* special value of x */
216 if(lx==0) {
217 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
218 z = ax; /*x is +-0,+-inf,+-1*/
219 if(hy<0) z = one/z; /* z = (1/|x|) */
220 if(hx<0) {
221 if(((ix-0x3ff00000)|yisint)==0) {
222 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
223 } else if(yisint==1) {
224 #ifdef HPUX
225 uz.d = z;
226 __HI(uz) ^= 1<<31; /* some HPUXes cannot negate 0.. */
227 z = uz.d;
228 #else
229 z = -z; /* (x<0)**odd = -(|x|**odd) */
230 #endif
231 }
232 }
233 return z;
234 }
235 }
236
237 /* (x<0)**(non-int) is NaN */
238 if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);
239
240 /* |y| is really_big */
241 if(iy>0x41e00000) { /* if |y| > 2**31 */
242 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
243 if(ix<=0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny;
244 if(ix>=0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny;
245 }
246 /* over/underflow if x is not close to one */
247 if(ix<0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny;
248 if(ix>0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny;
249 /* now |1-x| is tiny <= 2**-20, suffice to compute
250 log(x) by x-x^2/2+x^3/3-x^4/4 */
251 t = x-1; /* t has 20 trailing zeros */
252 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
253 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
254 v = t*ivln2_l-w*ivln2;
255 t1 = u+v;
256 uz.d = t1;
257 __LO(uz) = 0;
258 t1 = uz.d;
259 t2 = v-(t1-u);
260 } else {
261 double s_h,t_h;
262 double s2,s_l,t_l;
263 n = 0;
264 /* take care subnormal number */
265 if(ix<0x00100000)
266 {ax *= two53; n -= 53; uz.d = ax; ix = __HI(uz); }
267 n += ((ix)>>20)-0x3ff;
268 j = ix&0x000fffff;
269 /* determine interval */
270 ix = j|0x3ff00000; /* normalize ix */
271 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
272 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
273 else {k=0;n+=1;ix -= 0x00100000;}
274 uz.d = ax;
275 __HI(uz) = ix;
276 ax = uz.d;
277
278 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
279 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
280 v = one/(ax+bp[k]);
281 s = u*v;
282 s_h = s;
283 uz.d = s_h;
284 __LO(uz) = 0;
285 s_h = uz.d;
286 /* t_h=ax+bp[k] High */
287 t_h = zero;
288 uz.d = t_h;
289 __HI(uz)=((ix>>1)|0x20000000)+0x00080000+(k<<18);
290 t_h = uz.d;
291 t_l = ax - (t_h-bp[k]);
292 s_l = v*((u-s_h*t_h)-s_h*t_l);
293 /* compute log(ax) */
294 s2 = s*s;
295 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
296 r += s_l*(s_h+s);
297 s2 = s_h*s_h;
298 t_h = 3.0+s2+r;
299 uz.d = t_h;
300 __LO(uz) = 0;
301 t_h = uz.d;
302 t_l = r-((t_h-3.0)-s2);
303 /* u+v = s*(1+...) */
304 u = s_h*t_h;
305 v = s_l*t_h+t_l*s;
306 /* 2/(3log2)*(s+...) */
307 p_h = u+v;
308 uz.d = p_h;
309 __LO(uz) = 0;
310 p_h = uz.d;
311 p_l = v-(p_h-u);
312 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
313 z_l = cp_l*p_h+p_l*cp+dp_l[k];
314 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
315 t = (double)n;
316 t1 = (((z_h+z_l)+dp_h[k])+t);
317 uz.d = t1;
318 __LO(uz) = 0;
319 t1 = uz.d;
320 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
321 }
322
323 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
324 if((((hx>>31)+1)|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
325
326 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
327 y1 = y;
328 uy.d = y1;
329 __LO(uy) = 0;
330 y1 = uy.d;
331 p_l = (y-y1)*t1+y*t2;
332 p_h = y1*t1;
333 z = p_l+p_h;
334 uz.d = z;
335 j = __HI(uz);
336 i = __LO(uz);
337
338 if (j>=0x40900000) { /* z >= 1024 */
339 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
340 return s*really_big*really_big; /* overflow */
341 else {
342 if(p_l+ovt>z-p_h) return s*really_big*really_big; /* overflow */
343 }
344 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
345 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
346 return s*tiny*tiny; /* underflow */
347 else {
348 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
349 }
350 }
351 /*
352 * compute 2**(p_h+p_l)
353 */
354 i = j&0x7fffffff;
355 k = (i>>20)-0x3ff;
356 n = 0;
357 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
358 n = j+(0x00100000>>(k+1));
359 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
360 t = zero;
361 uz.d = t;
362 __HI(uz) = (n&~(0x000fffff>>k));
363 t = uz.d;
364 n = ((n&0x000fffff)|0x00100000)>>(20-k);
365 if(j<0) n = -n;
366 p_h -= t;
367 }
368 t = p_l+p_h;
369 uz.d = t;
370 __LO(uz) = 0;
371 t = uz.d;
372 u = t*lg2_h;
373 v = (p_l-(t-p_h))*lg2+t*lg2_l;
374 z = u+v;
375 w = v-(z-u);
376 t = z*z;
377 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
378 r = (z*t1)/(t1-two)-(w+z*w);
379 z = one-(r-z);
380 uz.d = z;
381 j = __HI(uz);
382 j += (n<<20);
383 if((j>>20)<=0) z = fd_scalbn(z,n); /* subnormal output */
384 else { uz.d = z; __HI(uz) += (n<<20); z = uz.d; }
385 return s*z;
386 }

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