1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
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
8 * is preserved.
9 * ====================================================
10 */
11
12/* __ieee754_pow(x,y) return x**y
13 *
14 * n
15 * Method: Let x = 2 * (1+f)
16 * 1. Compute and return log2(x) in two pieces:
17 * log2(x) = w1 + w2,
18 * where w1 has 53-24 = 29 bit trailing zeros.
19 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
20 * arithmetic, where |y'|<=0.5.
21 * 3. Return x**y = 2**n*exp(y'*log2)
22 *
23 * Special cases:
24 * 1. +-1 ** anything is 1.0
25 * 2. +-1 ** +-INF is 1.0
26 * 3. (anything) ** 0 is 1
27 * 4. (anything) ** 1 is itself
28 * 5. (anything) ** NAN is NAN
29 * 6. NAN ** (anything except 0) is NAN
30 * 7. +-(|x| > 1) ** +INF is +INF
31 * 8. +-(|x| > 1) ** -INF is +0
32 * 9. +-(|x| < 1) ** +INF is +0
33 * 10 +-(|x| < 1) ** -INF is +INF
34 * 11. +0 ** (+anything except 0, NAN) is +0
35 * 12. -0 ** (+anything except 0, NAN, odd integer) is +0
36 * 13. +0 ** (-anything except 0, NAN) is +INF
37 * 14. -0 ** (-anything except 0, NAN, odd integer) is +INF
38 * 15. -0 ** (odd integer) = -( +0 ** (odd integer) )
39 * 16. +INF ** (+anything except 0,NAN) is +INF
40 * 17. +INF ** (-anything except 0,NAN) is +0
41 * 18. -INF ** (anything) = -0 ** (-anything)
42 * 19. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
43 * 20. (-anything except 0 and inf) ** (non-integer) is NAN
44 *
45 * Accuracy:
46 * pow(x,y) returns x**y nearly rounded. In particular
47 * pow(integer,integer)
48 * always returns the correct integer provided it is
49 * representable.
50 *
51 * Constants :
52 * The hexadecimal values are the intended ones for the following
53 * constants. The decimal values may be used, provided that the
54 * compiler will convert from decimal to binary accurately enough
55 * to produce the hexadecimal values shown.
56 */
57
58#include "math_libm.h"
59#include "math_private.h"
60
61#if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
62/* C4756: overflow in constant arithmetic */
63#pragma warning ( disable : 4756 )
64#endif
65
66#ifdef __WATCOMC__ /* Watcom defines huge=__huge */
67#undef huge
68#endif
69
70static const double
71bp[] = {1.0, 1.5,},
72dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
73dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
74zero = 0.0,
75one = 1.0,
76two = 2.0,
77two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
78huge = 1.0e300,
79tiny = 1.0e-300,
80 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
81L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
82L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
83L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
84L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
85L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
86L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
87P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
88P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
89P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
90P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
91P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
92lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
93lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
94lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
95ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
96cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
97cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
98cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
99ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
100ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
101ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
102
103double attribute_hidden __ieee754_pow(double x, double y)
104{
105 double z,ax,z_h,z_l,p_h,p_l;
106 double y1,t1,t2,r,s,t,u,v,w;
107 int32_t i,j,k,yisint,n;
108 int32_t hx,hy,ix,iy;
109 u_int32_t lx,ly;
110
111 EXTRACT_WORDS(hx,lx,x);
112 /* x==1: 1**y = 1 (even if y is NaN) */
113 if (hx==0x3ff00000 && lx==0) {
114 return x;
115 }
116 ix = hx&0x7fffffff;
117
118 EXTRACT_WORDS(hy,ly,y);
119 iy = hy&0x7fffffff;
120
121 /* y==zero: x**0 = 1 */
122 if((iy|ly)==0) return one;
123
124 /* +-NaN return x+y */
125 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
126 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
127 return x+y;
128
129 /* determine if y is an odd int when x < 0
130 * yisint = 0 ... y is not an integer
131 * yisint = 1 ... y is an odd int
132 * yisint = 2 ... y is an even int
133 */
134 yisint = 0;
135 if(hx<0) {
136 if(iy>=0x43400000) yisint = 2; /* even integer y */
137 else if(iy>=0x3ff00000) {
138 k = (iy>>20)-0x3ff; /* exponent */
139 if(k>20) {
140 j = ly>>(52-k);
141 if((j<<(52-k))==ly) yisint = 2-(j&1);
142 } else if(ly==0) {
143 j = iy>>(20-k);
144 if((j<<(20-k))==iy) yisint = 2-(j&1);
145 }
146 }
147 }
148
149 /* special value of y */
150 if(ly==0) {
151 if (iy==0x7ff00000) { /* y is +-inf */
152 if (((ix-0x3ff00000)|lx)==0)
153 return one; /* +-1**+-inf is 1 (yes, weird rule) */
154 if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
155 return (hy>=0) ? y : zero;
156 /* (|x|<1)**-,+inf = inf,0 */
157 return (hy<0) ? -y : zero;
158 }
159 if(iy==0x3ff00000) { /* y is +-1 */
160 if(hy<0) return one/x; else return x;
161 }
162 if(hy==0x40000000) return x*x; /* y is 2 */
163 if(hy==0x3fe00000) { /* y is 0.5 */
164 if(hx>=0) /* x >= +0 */
165 return __ieee754_sqrt(x);
166 }
167 }
168
169 ax = fabs(x);
170 /* special value of x */
171 if(lx==0) {
172 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
173 z = ax; /*x is +-0,+-inf,+-1*/
174 if(hy<0) z = one/z; /* z = (1/|x|) */
175 if(hx<0) {
176 if(((ix-0x3ff00000)|yisint)==0) {
177 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
178 } else if(yisint==1)
179 z = -z; /* (x<0)**odd = -(|x|**odd) */
180 }
181 return z;
182 }
183 }
184
185 /* (x<0)**(non-int) is NaN */
186 if(((((u_int32_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(((((u_int32_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,n); /* subnormal output */
308 else SET_HIGH_WORD(z,j);
309 return s*z;
310}
311
312/*
313 * wrapper pow(x,y) return x**y
314 */
315#ifndef _IEEE_LIBM
316double pow(double x, double y)
317{
318 double z = __ieee754_pow(x, y);
319 if (_LIB_VERSION == _IEEE_|| isnan(y))
320 return z;
321 if (isnan(x)) {
322 if (y == 0.0)
323 return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */
324 return z;
325 }
326 if (x == 0.0) {
327 if (y == 0.0)
328 return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */
329 if (isfinite(y) && y < 0.0)
330 return __kernel_standard(x,y,23); /* pow(0.0,negative) */
331 return z;
332 }
333 if (!isfinite(z)) {
334 if (isfinite(x) && isfinite(y)) {
335 if (isnan(z))
336 return __kernel_standard(x, y, 24); /* pow neg**non-int */
337 return __kernel_standard(x, y, 21); /* pow overflow */
338 }
339 }
340 if (z == 0.0 && isfinite(x) && isfinite(y))
341 return __kernel_standard(x, y, 22); /* pow underflow */
342 return z;
343}
344#else
345strong_alias(__ieee754_pow, pow)
346#endif
347libm_hidden_def(pow)
348