Line data Source code
1 : /* @(#)e_pow.c 1.5 04/04/22 SMI */
2 : /*
3 : * ====================================================
4 : * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
5 : *
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 : //#include <sys/cdefs.h>
13 : //__FBSDID("$FreeBSD$");
14 :
15 : /* __ieee754_pow(x,y) return x**y
16 : *
17 : * n
18 : * Method: Let x = 2 * (1+f)
19 : * 1. Compute and return log2(x) in two pieces:
20 : * log2(x) = w1 + w2,
21 : * where w1 has 53-24 = 29 bit trailing zeros.
22 : * 2. Perform y*log2(x) = n+y' by simulating multi-precision
23 : * arithmetic, where |y'|<=0.5.
24 : * 3. Return x**y = 2**n*exp(y'*log2)
25 : *
26 : * Special cases:
27 : * 1. (anything) ** 0 is 1
28 : * 2. (anything) ** 1 is itself
29 : * 3. (anything) ** NAN is NAN except 1 ** NAN = 1
30 : * 4. NAN ** (anything except 0) is NAN
31 : * 5. +-(|x| > 1) ** +INF is +INF
32 : * 6. +-(|x| > 1) ** -INF is +0
33 : * 7. +-(|x| < 1) ** +INF is +0
34 : * 8. +-(|x| < 1) ** -INF is +INF
35 : * 9. +-1 ** +-INF is 1
36 : * 10. +0 ** (+anything except 0, NAN) is +0
37 : * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
38 : * 12. +0 ** (-anything except 0, NAN) is +INF
39 : * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
40 : * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
41 : * 15. +INF ** (+anything except 0,NAN) is +INF
42 : * 16. +INF ** (-anything except 0,NAN) is +0
43 : * 17. -INF ** (anything) = -0 ** (-anything)
44 : * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
45 : * 19. (-anything except 0 and inf) ** (non-integer) is NAN
46 : *
47 : * Accuracy:
48 : * pow(x,y) returns x**y nearly rounded. In particular
49 : * pow(integer,integer)
50 : * always returns the correct integer provided it is
51 : * representable.
52 : *
53 : * Constants :
54 : * The hexadecimal values are the intended ones for the following
55 : * constants. The decimal values may be used, provided that the
56 : * compiler will convert from decimal to binary accurately enough
57 : * to produce the hexadecimal values shown.
58 : */
59 :
60 : #include "math_private.h"
61 :
62 : static const double
63 : bp[] = {1.0, 1.5,},
64 : dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
65 : dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
66 : zero = 0.0,
67 : one = 1.0,
68 : two = 2.0,
69 : two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
70 : huge = 1.0e300,
71 : tiny = 1.0e-300,
72 : /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
73 : L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
74 : L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
75 : L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
76 : L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
77 : L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
78 : L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
79 : P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
80 : P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
81 : P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
82 : P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
83 : P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
84 : lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
85 : lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
86 : lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
87 : ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
88 : cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
89 : cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
90 : cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
91 : ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
92 : ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
93 : ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
94 :
95 : double
96 0 : __ieee754_pow(double x, double y)
97 : {
98 : double z,ax,z_h,z_l,p_h,p_l;
99 : double y1,t1,t2,r,s,t,u,v,w;
100 : int32_t i,j,k,yisint,n;
101 : int32_t hx,hy,ix,iy;
102 : u_int32_t lx,ly;
103 :
104 0 : EXTRACT_WORDS(hx,lx,x);
105 0 : EXTRACT_WORDS(hy,ly,y);
106 0 : ix = hx&0x7fffffff; iy = hy&0x7fffffff;
107 :
108 : /* y==zero: x**0 = 1 */
109 0 : if((iy|ly)==0) return one;
110 :
111 : /* x==1: 1**y = 1, even if y is NaN */
112 0 : if (hx==0x3ff00000 && lx == 0) return one;
113 :
114 : /* y!=zero: result is NaN if either arg is NaN */
115 0 : if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
116 0 : iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
117 0 : return (x+0.0)+(y+0.0);
118 :
119 : /* determine if y is an odd int when x < 0
120 : * yisint = 0 ... y is not an integer
121 : * yisint = 1 ... y is an odd int
122 : * yisint = 2 ... y is an even int
123 : */
124 0 : yisint = 0;
125 0 : if(hx<0) {
126 0 : if(iy>=0x43400000) yisint = 2; /* even integer y */
127 0 : else if(iy>=0x3ff00000) {
128 0 : k = (iy>>20)-0x3ff; /* exponent */
129 0 : if(k>20) {
130 0 : j = ly>>(52-k);
131 0 : if((j<<(52-k))==ly) yisint = 2-(j&1);
132 0 : } else if(ly==0) {
133 0 : j = iy>>(20-k);
134 0 : if((j<<(20-k))==iy) yisint = 2-(j&1);
135 : }
136 : }
137 : }
138 :
139 : /* special value of y */
140 0 : if(ly==0) {
141 0 : if (iy==0x7ff00000) { /* y is +-inf */
142 0 : if(((ix-0x3ff00000)|lx)==0)
143 0 : return one; /* (-1)**+-inf is 1 */
144 0 : else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
145 0 : return (hy>=0)? y: zero;
146 : else /* (|x|<1)**-,+inf = inf,0 */
147 0 : return (hy<0)?-y: zero;
148 : }
149 0 : if(iy==0x3ff00000) { /* y is +-1 */
150 0 : if(hy<0) return one/x; else return x;
151 : }
152 0 : if(hy==0x40000000) return x*x; /* y is 2 */
153 0 : if(hy==0x3fe00000) { /* y is 0.5 */
154 0 : if(hx>=0) /* x >= +0 */
155 0 : return sqrt(x);
156 : }
157 : }
158 :
159 0 : ax = fabs(x);
160 : /* special value of x */
161 0 : if(lx==0) {
162 0 : if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
163 0 : z = ax; /*x is +-0,+-inf,+-1*/
164 0 : if(hy<0) z = one/z; /* z = (1/|x|) */
165 0 : if(hx<0) {
166 0 : if(((ix-0x3ff00000)|yisint)==0) {
167 0 : z = (z-z)/(z-z); /* (-1)**non-int is NaN */
168 0 : } else if(yisint==1)
169 0 : z = -z; /* (x<0)**odd = -(|x|**odd) */
170 : }
171 0 : return z;
172 : }
173 : }
174 :
175 : /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
176 : n = (hx>>31)+1;
177 : but ANSI C says a right shift of a signed negative quantity is
178 : implementation defined. */
179 0 : n = ((u_int32_t)hx>>31)-1;
180 :
181 : /* (x<0)**(non-int) is NaN */
182 0 : if((n|yisint)==0) return (x-x)/(x-x);
183 :
184 0 : s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
185 0 : if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
186 :
187 : /* |y| is huge */
188 0 : if(iy>0x41e00000) { /* if |y| > 2**31 */
189 0 : if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
190 0 : if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
191 0 : if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
192 : }
193 : /* over/underflow if x is not close to one */
194 0 : if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
195 0 : if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
196 : /* now |1-x| is tiny <= 2**-20, suffice to compute
197 : log(x) by x-x^2/2+x^3/3-x^4/4 */
198 0 : t = ax-one; /* t has 20 trailing zeros */
199 0 : w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
200 0 : u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
201 0 : v = t*ivln2_l-w*ivln2;
202 0 : t1 = u+v;
203 0 : SET_LOW_WORD(t1,0);
204 0 : t2 = v-(t1-u);
205 : } else {
206 : double ss,s2,s_h,s_l,t_h,t_l;
207 0 : n = 0;
208 : /* take care subnormal number */
209 0 : if(ix<0x00100000)
210 0 : {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
211 0 : n += ((ix)>>20)-0x3ff;
212 0 : j = ix&0x000fffff;
213 : /* determine interval */
214 0 : ix = j|0x3ff00000; /* normalize ix */
215 0 : if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
216 0 : else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
217 0 : else {k=0;n+=1;ix -= 0x00100000;}
218 0 : SET_HIGH_WORD(ax,ix);
219 :
220 : /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
221 0 : u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
222 0 : v = one/(ax+bp[k]);
223 0 : ss = u*v;
224 0 : s_h = ss;
225 0 : SET_LOW_WORD(s_h,0);
226 : /* t_h=ax+bp[k] High */
227 0 : t_h = zero;
228 0 : SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
229 0 : t_l = ax - (t_h-bp[k]);
230 0 : s_l = v*((u-s_h*t_h)-s_h*t_l);
231 : /* compute log(ax) */
232 0 : s2 = ss*ss;
233 0 : r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
234 0 : r += s_l*(s_h+ss);
235 0 : s2 = s_h*s_h;
236 0 : t_h = 3.0+s2+r;
237 0 : SET_LOW_WORD(t_h,0);
238 0 : t_l = r-((t_h-3.0)-s2);
239 : /* u+v = ss*(1+...) */
240 0 : u = s_h*t_h;
241 0 : v = s_l*t_h+t_l*ss;
242 : /* 2/(3log2)*(ss+...) */
243 0 : p_h = u+v;
244 0 : SET_LOW_WORD(p_h,0);
245 0 : p_l = v-(p_h-u);
246 0 : z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
247 0 : z_l = cp_l*p_h+p_l*cp+dp_l[k];
248 : /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
249 0 : t = (double)n;
250 0 : t1 = (((z_h+z_l)+dp_h[k])+t);
251 0 : SET_LOW_WORD(t1,0);
252 0 : t2 = z_l-(((t1-t)-dp_h[k])-z_h);
253 : }
254 :
255 : /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
256 0 : y1 = y;
257 0 : SET_LOW_WORD(y1,0);
258 0 : p_l = (y-y1)*t1+y*t2;
259 0 : p_h = y1*t1;
260 0 : z = p_l+p_h;
261 0 : EXTRACT_WORDS(j,i,z);
262 0 : if (j>=0x40900000) { /* z >= 1024 */
263 0 : if(((j-0x40900000)|i)!=0) /* if z > 1024 */
264 0 : return s*huge*huge; /* overflow */
265 : else {
266 0 : if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
267 : }
268 0 : } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
269 0 : if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
270 0 : return s*tiny*tiny; /* underflow */
271 : else {
272 0 : if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
273 : }
274 : }
275 : /*
276 : * compute 2**(p_h+p_l)
277 : */
278 0 : i = j&0x7fffffff;
279 0 : k = (i>>20)-0x3ff;
280 0 : n = 0;
281 0 : if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
282 0 : n = j+(0x00100000>>(k+1));
283 0 : k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
284 0 : t = zero;
285 0 : SET_HIGH_WORD(t,n&~(0x000fffff>>k));
286 0 : n = ((n&0x000fffff)|0x00100000)>>(20-k);
287 0 : if(j<0) n = -n;
288 0 : p_h -= t;
289 : }
290 0 : t = p_l+p_h;
291 0 : SET_LOW_WORD(t,0);
292 0 : u = t*lg2_h;
293 0 : v = (p_l-(t-p_h))*lg2+t*lg2_l;
294 0 : z = u+v;
295 0 : w = v-(z-u);
296 0 : t = z*z;
297 0 : t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
298 0 : r = (z*t1)/(t1-two)-(w+z*w);
299 0 : z = one-(r-z);
300 0 : GET_HIGH_WORD(j,z);
301 0 : j += (n<<20);
302 0 : if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
303 0 : else SET_HIGH_WORD(z,j);
304 0 : return s*z;
305 : }
|
