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 "cdefs-compat.h"
13 //__FBSDID("$FreeBSD: src/lib/msun/src/e_pow.c,v 1.14 2011/10/21 06:26:07 das Exp $");
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 muti-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
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 NAN
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 <float.h>
61 #include <openlibm_math.h>
62
63 #include "math_private.h"
64
65 static const double
66 bp[] = {1.0, 1.5,},
67 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
68 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
69 zero = 0.0,
70 one = 1.0,
71 two = 2.0,
72 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
73 huge = 1.0e300,
74 tiny = 1.0e-300,
75 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
76 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
77 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
78 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
79 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
80 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
81 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
82 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
83 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
84 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
85 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
86 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
87 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
88 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
89 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
90 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
91 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
92 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
93 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
94 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
95 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
96 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
97
98 OLM_DLLEXPORT double
__ieee754_pow(double x,double y)99 __ieee754_pow(double x, double y)
100 {
101 double z,ax,z_h,z_l,p_h,p_l;
102 double y1,t1,t2,r,s,t,u,v,w;
103 int32_t i,j,k,yisint,n;
104 int32_t hx,hy,ix,iy;
105 u_int32_t lx,ly;
106
107 EXTRACT_WORDS(hx,lx,x);
108 EXTRACT_WORDS(hy,ly,y);
109 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
110
111 /* y==zero: x**0 = 1 */
112 if((iy|ly)==0) return one;
113
114 /* x==1: 1**y = 1, even if y is NaN */
115 if (hx==0x3ff00000 && lx == 0) return one;
116
117 /* y!=zero: result is NaN if either arg is NaN */
118 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
119 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
120 return (x+0.0)+(y+0.0);
121
122 /* determine if y is an odd int when x < 0
123 * yisint = 0 ... y is not an integer
124 * yisint = 1 ... y is an odd int
125 * yisint = 2 ... y is an even int
126 */
127 yisint = 0;
128 if(hx<0) {
129 if(iy>=0x43400000) yisint = 2; /* even integer y */
130 else if(iy>=0x3ff00000) {
131 k = (iy>>20)-0x3ff; /* exponent */
132 if(k>20) {
133 j = ly>>(52-k);
134 if((j<<(52-k))==ly) yisint = 2-(j&1);
135 } else if(ly==0) {
136 j = iy>>(20-k);
137 if((j<<(20-k))==iy) yisint = 2-(j&1);
138 }
139 }
140 }
141
142 /* special value of y */
143 if(ly==0) {
144 if (iy==0x7ff00000) { /* y is +-inf */
145 if(((ix-0x3ff00000)|lx)==0)
146 return one; /* (-1)**+-inf is NaN */
147 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
148 return (hy>=0)? y: zero;
149 else /* (|x|<1)**-,+inf = inf,0 */
150 return (hy<0)?-y: zero;
151 }
152 if(iy==0x3ff00000) { /* y is +-1 */
153 if(hy<0) return one/x; else return x;
154 }
155 if(hy==0x40000000) return x*x; /* y is 2 */
156 if(hy==0x40080000) return x*x*x; /* y is 3 */
157 if(hy==0x40100000) { /* y is 4 */
158 u = x*x;
159 return u*u;
160 }
161 if(hy==0x3fe00000) { /* y is 0.5 */
162 if(hx>=0) /* x >= +0 */
163 return sqrt(x);
164 }
165 }
166
167 ax = fabs(x);
168 /* special value of x */
169 if(lx==0) {
170 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
171 z = ax; /*x is +-0,+-inf,+-1*/
172 if(hy<0) z = one/z; /* z = (1/|x|) */
173 if(hx<0) {
174 if(((ix-0x3ff00000)|yisint)==0) {
175 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
176 } else if(yisint==1)
177 z = -z; /* (x<0)**odd = -(|x|**odd) */
178 }
179 return z;
180 }
181 }
182
183 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
184 n = (hx>>31)+1;
185 but ANSI C says a right shift of a signed negative quantity is
186 implementation defined. */
187 n = ((u_int32_t)hx>>31)-1;
188
189 /* (x<0)**(non-int) is NaN */
190 if((n|yisint)==0) return (x-x)/(x-x);
191
192 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
193 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
194
195 /* |y| is huge */
196 if(iy>0x41e00000) { /* if |y| > 2**31 */
197 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
198 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
199 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
200 }
201 /* over/underflow if x is not close to one */
202 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
203 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
204 /* now |1-x| is tiny <= 2**-20, suffice to compute
205 log(x) by x-x^2/2+x^3/3-x^4/4 */
206 t = ax-one; /* t has 20 trailing zeros */
207 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
208 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
209 v = t*ivln2_l-w*ivln2;
210 t1 = u+v;
211 SET_LOW_WORD(t1,0);
212 t2 = v-(t1-u);
213 } else {
214 double ss,s2,s_h,s_l,t_h,t_l;
215 n = 0;
216 /* take care subnormal number */
217 if(ix<0x00100000)
218 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
219 n += ((ix)>>20)-0x3ff;
220 j = ix&0x000fffff;
221 /* determine interval */
222 ix = j|0x3ff00000; /* normalize ix */
223 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
224 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
225 else {k=0;n+=1;ix -= 0x00100000;}
226 SET_HIGH_WORD(ax,ix);
227
228 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
229 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
230 v = one/(ax+bp[k]);
231 ss = u*v;
232 s_h = ss;
233 SET_LOW_WORD(s_h,0);
234 /* t_h=ax+bp[k] High */
235 t_h = zero;
236 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
237 t_l = ax - (t_h-bp[k]);
238 s_l = v*((u-s_h*t_h)-s_h*t_l);
239 /* compute log(ax) */
240 s2 = ss*ss;
241 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
242 r += s_l*(s_h+ss);
243 s2 = s_h*s_h;
244 t_h = 3.0+s2+r;
245 SET_LOW_WORD(t_h,0);
246 t_l = r-((t_h-3.0)-s2);
247 /* u+v = ss*(1+...) */
248 u = s_h*t_h;
249 v = s_l*t_h+t_l*ss;
250 /* 2/(3log2)*(ss+...) */
251 p_h = u+v;
252 SET_LOW_WORD(p_h,0);
253 p_l = v-(p_h-u);
254 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
255 z_l = cp_l*p_h+p_l*cp+dp_l[k];
256 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
257 t = (double)n;
258 t1 = (((z_h+z_l)+dp_h[k])+t);
259 SET_LOW_WORD(t1,0);
260 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
261 }
262
263 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
264 y1 = y;
265 SET_LOW_WORD(y1,0);
266 p_l = (y-y1)*t1+y*t2;
267 p_h = y1*t1;
268 z = p_l+p_h;
269 EXTRACT_WORDS(j,i,z);
270 if (j>=0x40900000) { /* z >= 1024 */
271 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
272 return s*huge*huge; /* overflow */
273 else {
274 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
275 }
276 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
277 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
278 return s*tiny*tiny; /* underflow */
279 else {
280 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
281 }
282 }
283 /*
284 * compute 2**(p_h+p_l)
285 */
286 i = j&0x7fffffff;
287 k = (i>>20)-0x3ff;
288 n = 0;
289 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
290 n = j+(0x00100000>>(k+1));
291 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
292 t = zero;
293 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
294 n = ((n&0x000fffff)|0x00100000)>>(20-k);
295 if(j<0) n = -n;
296 p_h -= t;
297 }
298 t = p_l+p_h;
299 SET_LOW_WORD(t,0);
300 u = t*lg2_h;
301 v = (p_l-(t-p_h))*lg2+t*lg2_l;
302 z = u+v;
303 w = v-(z-u);
304 t = z*z;
305 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
306 r = (z*t1)/(t1-two)-(w+z*w);
307 z = one-(r-z);
308 GET_HIGH_WORD(j,z);
309 j += (n<<20);
310 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
311 else SET_HIGH_WORD(z,j);
312 return s*z;
313 }
314
315 #if (LDBL_MANT_DIG == 53)
316 openlibm_weak_reference(pow, powl);
317 #endif
318