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.h"
59 #include "math_private.h"
60
61 static const double
62 bp[] = {1.0, 1.5,},
63 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
64 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
65 zero = 0.0,
66 one = 1.0,
67 two = 2.0,
68 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
69 huge = 1.0e300,
70 tiny = 1.0e-300,
71 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
72 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
73 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
74 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
75 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
76 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
77 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
78 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
79 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
80 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
81 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
82 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
83 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
84 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
85 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
86 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
87 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
88 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
89 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
90 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
91 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
92 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
93
__ieee754_pow(double x,double y)94 double __ieee754_pow(double x, double y)
95 {
96 double z,ax,z_h,z_l,p_h,p_l;
97 double _y1,t1,t2,r,s,t,u,v,w;
98 int32_t i,j,k,yisint,n;
99 int32_t hx,hy,ix,iy;
100 u_int32_t lx,ly;
101
102 EXTRACT_WORDS(hx,lx,x);
103 /* x==1: 1**y = 1 (even if y is NaN) */
104 if (hx==0x3ff00000 && lx==0) {
105 return x;
106 }
107 ix = hx&0x7fffffff;
108
109 EXTRACT_WORDS(hy,ly,y);
110 iy = hy&0x7fffffff;
111
112 /* y==zero: x**0 = 1 */
113 if((iy|ly)==0) return one;
114
115 /* +-NaN return x+y */
116 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
117 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
118 return x+y;
119
120 /* determine if y is an odd int when x < 0
121 * yisint = 0 ... y is not an integer
122 * yisint = 1 ... y is an odd int
123 * yisint = 2 ... y is an even int
124 */
125 yisint = 0;
126 if(hx<0) {
127 if(iy>=0x43400000) yisint = 2; /* even integer y */
128 else if(iy>=0x3ff00000) {
129 k = (iy>>20)-0x3ff; /* exponent */
130 if(k>20) {
131 j = ly>>(52-k);
132 if((j<<(52-k))==ly) yisint = 2-(j&1);
133 } else if(ly==0) {
134 j = iy>>(20-k);
135 if((j<<(20-k))==iy) yisint = 2-(j&1);
136 }
137 }
138 }
139
140 /* special value of y */
141 if(ly==0) {
142 if (iy==0x7ff00000) { /* y is +-inf */
143 if (((ix-0x3ff00000)|lx)==0)
144 return one; /* +-1**+-inf is 1 (yes, weird rule) */
145 if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
146 return (hy>=0) ? y : zero;
147 /* (|x|<1)**-,+inf = inf,0 */
148 return (hy<0) ? -y : zero;
149 }
150 if(iy==0x3ff00000) { /* y is +-1 */
151 if(hy<0) return one/x; else return x;
152 }
153 if(hy==0x40000000) return x*x; /* y is 2 */
154 if(hy==0x3fe00000) { /* y is 0.5 */
155 if(hx>=0) /* x >= +0 */
156 return __ieee754_sqrt(x);
157 }
158 }
159
160 ax = fabs(x);
161 /* special value of x */
162 if(lx==0) {
163 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
164 z = ax; /*x is +-0,+-inf,+-1*/
165 if(hy<0) z = one/z; /* z = (1/|x|) */
166 if(hx<0) {
167 if(((ix-0x3ff00000)|yisint)==0) {
168 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
169 } else if(yisint==1)
170 z = -z; /* (x<0)**odd = -(|x|**odd) */
171 }
172 return z;
173 }
174 }
175
176 /* (x<0)**(non-int) is NaN */
177 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
178
179 /* |y| is huge */
180 if(iy>0x41e00000) { /* if |y| > 2**31 */
181 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
182 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
183 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
184 }
185 /* over/underflow if x is not close to one */
186 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
187 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
188 /* now |1-x| is tiny <= 2**-20, suffice to compute
189 log(x) by x-x^2/2+x^3/3-x^4/4 */
190 t = x-1; /* t has 20 trailing zeros */
191 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
192 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
193 v = t*ivln2_l-w*ivln2;
194 t1 = u+v;
195 SET_LOW_WORD(t1,0);
196 t2 = v-(t1-u);
197 } else {
198 double s2,s_h,s_l,t_h,t_l;
199 n = 0;
200 /* take care subnormal number */
201 if(ix<0x00100000)
202 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
203 n += ((ix)>>20)-0x3ff;
204 j = ix&0x000fffff;
205 /* determine interval */
206 ix = j|0x3ff00000; /* normalize ix */
207 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
208 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
209 else {k=0;n+=1;ix -= 0x00100000;}
210 SET_HIGH_WORD(ax,ix);
211
212 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
213 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
214 v = one/(ax+bp[k]);
215 s = u*v;
216 s_h = s;
217 SET_LOW_WORD(s_h,0);
218 /* t_h=ax+bp[k] High */
219 t_h = zero;
220 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
221 t_l = ax - (t_h-bp[k]);
222 s_l = v*((u-s_h*t_h)-s_h*t_l);
223 /* compute log(ax) */
224 s2 = s*s;
225 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
226 r += s_l*(s_h+s);
227 s2 = s_h*s_h;
228 t_h = 3.0+s2+r;
229 SET_LOW_WORD(t_h,0);
230 t_l = r-((t_h-3.0)-s2);
231 /* u+v = s*(1+...) */
232 u = s_h*t_h;
233 v = s_l*t_h+t_l*s;
234 /* 2/(3log2)*(s+...) */
235 p_h = u+v;
236 SET_LOW_WORD(p_h,0);
237 p_l = v-(p_h-u);
238 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
239 z_l = cp_l*p_h+p_l*cp+dp_l[k];
240 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
241 t = (double)n;
242 t1 = (((z_h+z_l)+dp_h[k])+t);
243 SET_LOW_WORD(t1,0);
244 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
245 }
246
247 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
248 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
249 s = -one;/* (-ve)**(odd int) */
250
251 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
252 _y1 = y;
253 SET_LOW_WORD(_y1,0);
254 p_l = (y-_y1)*t1+y*t2;
255 p_h = _y1*t1;
256 z = p_l+p_h;
257 EXTRACT_WORDS(j,i,z);
258 if (j>=0x40900000) { /* z >= 1024 */
259 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
260 return s*huge*huge; /* overflow */
261 else {
262 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
263 }
264 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
265 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
266 return s*tiny*tiny; /* underflow */
267 else {
268 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
269 }
270 }
271 /*
272 * compute 2**(p_h+p_l)
273 */
274 i = j&0x7fffffff;
275 k = (i>>20)-0x3ff;
276 n = 0;
277 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
278 n = j+(0x00100000>>(k+1));
279 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
280 t = zero;
281 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
282 n = ((n&0x000fffff)|0x00100000)>>(20-k);
283 if(j<0) n = -n;
284 p_h -= t;
285 }
286 t = p_l+p_h;
287 SET_LOW_WORD(t,0);
288 u = t*lg2_h;
289 v = (p_l-(t-p_h))*lg2+t*lg2_l;
290 z = u+v;
291 w = v-(z-u);
292 t = z*z;
293 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
294 r = (z*t1)/(t1-two)-(w+z*w);
295 z = one-(r-z);
296 GET_HIGH_WORD(j,z);
297 j += (n<<20);
298 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
299 else SET_HIGH_WORD(z,j);
300 return s*z;
301 }
302
303 /*
304 * wrapper pow(x,y) return x**y
305 */
306 #ifndef _IEEE_LIBM
pow(double x,double y)307 double pow(double x, double y)
308 {
309 double z = __ieee754_pow(x, y);
310 if (_LIB_VERSION == _IEEE_|| isnan(y))
311 return z;
312 if (isnan(x)) {
313 if (y == 0.0)
314 return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */
315 return z;
316 }
317 if (x == 0.0) {
318 if (y == 0.0)
319 return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */
320 if (isfinite(y) && y < 0.0)
321 return __kernel_standard(x,y,23); /* pow(0.0,negative) */
322 return z;
323 }
324 if (!isfinite(z)) {
325 if (isfinite(x) && isfinite(y)) {
326 if (isnan(z))
327 return __kernel_standard(x, y, 24); /* pow neg**non-int */
328 return __kernel_standard(x, y, 21); /* pow overflow */
329 }
330 }
331 if (z == 0.0 && isfinite(x) && isfinite(y))
332 return __kernel_standard(x, y, 22); /* pow underflow */
333 return z;
334 }
335 #else
336 strong_alias(__ieee754_pow, pow)
337 #endif
338 libm_hidden_def(pow)
339