source: trunk/gcc/libjava/java/lang/e_pow.c

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1
2/* @(#)e_pow.c 5.1 93/09/24 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14/* __ieee754_pow(x,y) return x**y
15 *
16 * n
17 * Method: Let x = 2 * (1+f)
18 * 1. Compute and return log2(x) in two pieces:
19 * log2(x) = w1 + w2,
20 * where w1 has 53-24 = 29 bit trailing zeros.
21 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
22 * arithmetic, where |y'|<=0.5.
23 * 3. Return x**y = 2**n*exp(y'*log2)
24 *
25 * Special cases:
26 * 1. (anything) ** 0 is 1
27 * 2. (anything) ** 1 is itself
28 * 3. (anything) ** NAN is NAN
29 * 4. NAN ** (anything except 0) is NAN
30 * 5. +-(|x| > 1) ** +INF is +INF
31 * 6. +-(|x| > 1) ** -INF is +0
32 * 7. +-(|x| < 1) ** +INF is +0
33 * 8. +-(|x| < 1) ** -INF is +INF
34 * 9. +-1 ** +-INF is NAN
35 * 10. +0 ** (+anything except 0, NAN) is +0
36 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
37 * 12. +0 ** (-anything except 0, NAN) is +INF
38 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
39 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
40 * 15. +INF ** (+anything except 0,NAN) is +INF
41 * 16. +INF ** (-anything except 0,NAN) is +0
42 * 17. -INF ** (anything) = -0 ** (-anything)
43 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
44 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
45 *
46 * Accuracy:
47 * pow(x,y) returns x**y nearly rounded. In particular
48 * pow(integer,integer)
49 * always returns the correct integer provided it is
50 * representable.
51 *
52 * Constants :
53 * The hexadecimal values are the intended ones for the following
54 * constants. The decimal values may be used, provided that the
55 * compiler will convert from decimal to binary accurately enough
56 * to produce the hexadecimal values shown.
57 */
58
59#include "fdlibm.h"
60
61#ifndef _DOUBLE_IS_32BITS
62
63#ifdef __STDC__
64static const double
65#else
66static double
67#endif
68bp[] = {1.0, 1.5,},
69dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
70dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
71zero = 0.0,
72one = 1.0,
73two = 2.0,
74two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
75huge = 1.0e300,
76tiny = 1.0e-300,
77 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
78L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
79L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
80L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
81L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
82L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
83L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
84P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
85P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
86P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
87P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
88P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
89lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
90lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
91lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
92ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
93cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
94cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
95cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
96ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
97ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
98ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
99
100#ifdef __STDC__
101 double __ieee754_pow(double x, double y)
102#else
103 double __ieee754_pow(x,y)
104 double x, y;
105#endif
106{
107 double z,ax,z_h,z_l,p_h,p_l;
108 double y1,t1,t2,r,s,t,u,v,w;
109 int32_t i,j,k,yisint,n;
110 int32_t hx,hy,ix,iy;
111 uint32_t lx,ly;
112
113 EXTRACT_WORDS(hx,lx,x);
114 EXTRACT_WORDS(hy,ly,y);
115 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
116
117 /* y==zero: x**0 = 1 */
118 if((iy|ly)==0) return one;
119
120 /* +-NaN return x+y */
121 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
122 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
123 return x+y;
124
125 /* determine if y is an odd int when x < 0
126 * yisint = 0 ... y is not an integer
127 * yisint = 1 ... y is an odd int
128 * yisint = 2 ... y is an even int
129 */
130 yisint = 0;
131 if(hx<0) {
132 if(iy>=0x43400000) yisint = 2; /* even integer y */
133 else if(iy>=0x3ff00000) {
134 k = (iy>>20)-0x3ff; /* exponent */
135 if(k>20) {
136 j = ly>>(52-k);
137 if((uint32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
138 } else if(ly==0) {
139 j = iy>>(20-k);
140 if((j<<(20-k))==iy) yisint = 2-(j&1);
141 }
142 }
143 }
144
145 /* special value of y */
146 if(ly==0) {
147 if (iy==0x7ff00000) { /* y is +-inf */
148 if(((ix-0x3ff00000)|lx)==0)
149 return y - y; /* inf**+-1 is NaN */
150 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
151 return (hy>=0)? y: zero;
152 else /* (|x|<1)**-,+inf = inf,0 */
153 return (hy<0)?-y: zero;
154 }
155 if(iy==0x3ff00000) { /* y is +-1 */
156 if(hy<0) return one/x; else return x;
157 }
158 if(hy==0x40000000) return x*x; /* y is 2 */
159 if(hy==0x3fe00000) { /* y is 0.5 */
160 if(hx>=0) /* x >= +0 */
161 return __ieee754_sqrt(x);
162 }
163 }
164
165 ax = fabs(x);
166 /* special value of x */
167 if(lx==0) {
168 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
169 z = ax; /*x is +-0,+-inf,+-1*/
170 if(hy<0) z = one/z; /* z = (1/|x|) */
171 if(hx<0) {
172 if(((ix-0x3ff00000)|yisint)==0) {
173 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
174 } else if(yisint==1)
175 z = -z; /* (x<0)**odd = -(|x|**odd) */
176 }
177 return z;
178 }
179 }
180
181 /* (x<0)**(non-int) is NaN */
182 /* CYGNUS LOCAL: This used to be
183 if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);
184 but ANSI C says a right shift of a signed negative quantity is
185 implementation defined. */
186 if(((((uint32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
187
188 /* |y| is huge */
189 if(iy>0x41e00000) { /* if |y| > 2**31 */
190 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
191 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
192 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
193 }
194 /* over/underflow if x is not close to one */
195 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
196 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
197 /* now |1-x| is tiny <= 2**-20, suffice to compute
198 log(x) by x-x^2/2+x^3/3-x^4/4 */
199 t = x-1; /* t has 20 trailing zeros */
200 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
201 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
202 v = t*ivln2_l-w*ivln2;
203 t1 = u+v;
204 SET_LOW_WORD(t1,0);
205 t2 = v-(t1-u);
206 } else {
207 double s2,s_h,s_l,t_h,t_l;
208 n = 0;
209 /* take care subnormal number */
210 if(ix<0x00100000)
211 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
212 n += ((ix)>>20)-0x3ff;
213 j = ix&0x000fffff;
214 /* determine interval */
215 ix = j|0x3ff00000; /* normalize ix */
216 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
217 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
218 else {k=0;n+=1;ix -= 0x00100000;}
219 SET_HIGH_WORD(ax,ix);
220
221 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
222 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
223 v = one/(ax+bp[k]);
224 s = u*v;
225 s_h = s;
226 SET_LOW_WORD(s_h,0);
227 /* t_h=ax+bp[k] High */
228 t_h = zero;
229 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
230 t_l = ax - (t_h-bp[k]);
231 s_l = v*((u-s_h*t_h)-s_h*t_l);
232 /* compute log(ax) */
233 s2 = s*s;
234 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
235 r += s_l*(s_h+s);
236 s2 = s_h*s_h;
237 t_h = 3.0+s2+r;
238 SET_LOW_WORD(t_h,0);
239 t_l = r-((t_h-3.0)-s2);
240 /* u+v = s*(1+...) */
241 u = s_h*t_h;
242 v = s_l*t_h+t_l*s;
243 /* 2/(3log2)*(s+...) */
244 p_h = u+v;
245 SET_LOW_WORD(p_h,0);
246 p_l = v-(p_h-u);
247 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
248 z_l = cp_l*p_h+p_l*cp+dp_l[k];
249 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
250 t = (double)n;
251 t1 = (((z_h+z_l)+dp_h[k])+t);
252 SET_LOW_WORD(t1,0);
253 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
254 }
255
256 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
257 if(((((uint32_t)hx>>31)-1)|(yisint-1))==0)
258 s = -one;/* (-ve)**(odd int) */
259
260 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
261 y1 = y;
262 SET_LOW_WORD(y1,0);
263 p_l = (y-y1)*t1+y*t2;
264 p_h = y1*t1;
265 z = p_l+p_h;
266 EXTRACT_WORDS(j,i,z);
267 if (j>=0x40900000) { /* z >= 1024 */
268 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
269 return s*huge*huge; /* overflow */
270 else {
271 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
272 }
273 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
274 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
275 return s*tiny*tiny; /* underflow */
276 else {
277 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
278 }
279 }
280 /*
281 * compute 2**(p_h+p_l)
282 */
283 i = j&0x7fffffff;
284 k = (i>>20)-0x3ff;
285 n = 0;
286 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
287 n = j+(0x00100000>>(k+1));
288 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
289 t = zero;
290 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
291 n = ((n&0x000fffff)|0x00100000)>>(20-k);
292 if(j<0) n = -n;
293 p_h -= t;
294 }
295 t = p_l+p_h;
296 SET_LOW_WORD(t,0);
297 u = t*lg2_h;
298 v = (p_l-(t-p_h))*lg2+t*lg2_l;
299 z = u+v;
300 w = v-(z-u);
301 t = z*z;
302 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
303 r = (z*t1)/(t1-two)-(w+z*w);
304 z = one-(r-z);
305 GET_HIGH_WORD(j,z);
306 j += (n<<20);
307 if((j>>20)<=0) z = scalbn(z,(int)n); /* subnormal output */
308 else SET_HIGH_WORD(z,j);
309 return s*z;
310}
311
312#endif /* defined(_DOUBLE_IS_32BITS) */
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