CommitLineData
6ff43c94
PA
1/* @(#)e_pow.c 1.5 04/04/22 SMI */
2/* \$FreeBSD: head/lib/msun/src/e_pow.c 226595 2011-10-21 06:26:07Z das \$ */
b34b60bc
JS
3/*
4 * ====================================================
6ff43c94 5 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
b34b60bc 6 *
b34b60bc 7 * Permission to use, copy, modify, and distribute this
6ff43c94 8 * software is freely granted, provided that this notice
b34b60bc
JS
9 * is preserved.
10 * ====================================================
b34b60bc
JS
11 */
12
6ff43c94 13/* __ieee754_pow(x,y) return x**y
b34b60bc
JS
14 *
15 * n
16 * Method: Let x = 2 * (1+f)
17 * 1. Compute and return log2(x) in two pieces:
18 * log2(x) = w1 + w2,
19 * where w1 has 53-24 = 29 bit trailing zeros.
2fedfd5c 20 * 2. Perform y*log2(x) = n+y' by simulating multi-precision
b34b60bc
JS
21 * arithmetic, where |y'|<=0.5.
22 * 3. Return x**y = 2**n*exp(y'*log2)
23 *
24 * Special cases:
25 * 1. (anything) ** 0 is 1
26 * 2. (anything) ** 1 is itself
2fedfd5c 27 * 3. (anything) ** NAN is NAN except 1 ** NAN = 1
b34b60bc
JS
28 * 4. NAN ** (anything except 0) is NAN
29 * 5. +-(|x| > 1) ** +INF is +INF
30 * 6. +-(|x| > 1) ** -INF is +0
31 * 7. +-(|x| < 1) ** +INF is +0
32 * 8. +-(|x| < 1) ** -INF is +INF
2fedfd5c 33 * 9. +-1 ** +-INF is 1
b34b60bc
JS
34 * 10. +0 ** (+anything except 0, NAN) is +0
35 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
36 * 12. +0 ** (-anything except 0, NAN) is +INF
37 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
38 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
39 * 15. +INF ** (+anything except 0,NAN) is +INF
40 * 16. +INF ** (-anything except 0,NAN) is +0
41 * 17. -INF ** (anything) = -0 ** (-anything)
42 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
43 * 19. (-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)
6ff43c94 48 * always returns the correct integer provided it is
b34b60bc
JS
49 * representable.
50 *
51 * Constants :
6ff43c94
PA
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
b34b60bc
JS
55 * to produce the hexadecimal values shown.
56 */
57
6ff43c94 58#include "math.h"
b34b60bc
JS
59#include "math_private.h"
60
61static const double
62bp[] = {1.0, 1.5,},
63dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
64dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
65zero = 0.0,
66one = 1.0,
67two = 2.0,
68two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
69huge = 1.0e300,
70tiny = 1.0e-300,
71 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
72L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
73L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
74L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
75L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
76L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
77L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
78P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
79P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
80P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
81P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
82P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
83lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
84lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
85lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
86ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
87cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
88cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
89cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
90ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
91ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
92ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
93
94double
6ff43c94 95__ieee754_pow(double x, double y)
b34b60bc
JS
96{
97 double z,ax,z_h,z_l,p_h,p_l;
6ff43c94 98 double y1,t1,t2,r,s,t,u,v,w;
b34b60bc
JS
99 int32_t i,j,k,yisint,n;
100 int32_t hx,hy,ix,iy;
101 u_int32_t lx,ly;
102
103 EXTRACT_WORDS(hx,lx,x);
104 EXTRACT_WORDS(hy,ly,y);
105 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
106
107 /* y==zero: x**0 = 1 */
6ff43c94
PA
108 if((iy|ly)==0) return one;
109
110 /* x==1: 1**y = 1, even if y is NaN */
111 if (hx==0x3ff00000 && lx == 0) return one;
b34b60bc 112
6ff43c94 113 /* y!=zero: result is NaN if either arg is NaN */
b34b60bc 114 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
6ff43c94
PA
115 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
116 return (x+0.0)+(y+0.0);
b34b60bc
JS
117
118 /* determine if y is an odd int when x < 0
119 * yisint = 0 ... y is not an integer
120 * yisint = 1 ... y is an odd int
121 * yisint = 2 ... y is an even int
122 */
123 yisint = 0;
6ff43c94 124 if(hx<0) {
b34b60bc
JS
125 if(iy>=0x43400000) yisint = 2; /* even integer y */
126 else if(iy>=0x3ff00000) {
127 k = (iy>>20)-0x3ff; /* exponent */
128 if(k>20) {
129 j = ly>>(52-k);
6ff43c94 130 if((j<<(52-k))==ly) yisint = 2-(j&1);
b34b60bc
JS
131 } else if(ly==0) {
132 j = iy>>(20-k);
133 if((j<<(20-k))==iy) yisint = 2-(j&1);
134 }
6ff43c94
PA
135 }
136 }
b34b60bc
JS
137
138 /* special value of y */
6ff43c94 139 if(ly==0) {
b34b60bc
JS
140 if (iy==0x7ff00000) { /* y is +-inf */
141 if(((ix-0x3ff00000)|lx)==0)
2fedfd5c 142 return one; /* (-1)**+-inf is 1 */
b34b60bc
JS
143 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
144 return (hy>=0)? y: zero;
145 else /* (|x|<1)**-,+inf = inf,0 */
146 return (hy<0)?-y: zero;
6ff43c94 147 }
b34b60bc
JS
148 if(iy==0x3ff00000) { /* y is +-1 */
149 if(hy<0) return one/x; else return x;
150 }
151 if(hy==0x40000000) return x*x; /* y is 2 */
152 if(hy==0x3fe00000) { /* y is 0.5 */
153 if(hx>=0) /* x >= +0 */
6ff43c94 154 return sqrt(x);
b34b60bc
JS
155 }
156 }
157
158 ax = fabs(x);
159 /* special value of x */
160 if(lx==0) {
161 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
162 z = ax; /*x is +-0,+-inf,+-1*/
163 if(hy<0) z = one/z; /* z = (1/|x|) */
164 if(hx<0) {
165 if(((ix-0x3ff00000)|yisint)==0) {
166 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
6ff43c94 167 } else if(yisint==1)
b34b60bc
JS
168 z = -z; /* (x<0)**odd = -(|x|**odd) */
169 }
170 return z;
171 }
172 }
6ff43c94
PA
173
174 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
b34b60bc 175 n = (hx>>31)+1;
6ff43c94
PA
176 but ANSI C says a right shift of a signed negative quantity is
177 implementation defined. */
178 n = ((u_int32_t)hx>>31)-1;
b34b60bc
JS
179
180 /* (x<0)**(non-int) is NaN */
181 if((n|yisint)==0) return (x-x)/(x-x);
182
183 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
184 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
185
186 /* |y| is huge */
187 if(iy>0x41e00000) { /* if |y| > 2**31 */
188 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
189 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
190 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
191 }
192 /* over/underflow if x is not close to one */
193 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
194 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
6ff43c94 195 /* now |1-x| is tiny <= 2**-20, suffice to compute
b34b60bc
JS
196 log(x) by x-x^2/2+x^3/3-x^4/4 */
197 t = ax-one; /* t has 20 trailing zeros */
198 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
199 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
200 v = t*ivln2_l-w*ivln2;
201 t1 = u+v;
202 SET_LOW_WORD(t1,0);
203 t2 = v-(t1-u);
204 } else {
205 double ss,s2,s_h,s_l,t_h,t_l;
206 n = 0;
207 /* take care subnormal number */
208 if(ix<0x00100000)
209 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
210 n += ((ix)>>20)-0x3ff;
211 j = ix&0x000fffff;
212 /* determine interval */
213 ix = j|0x3ff00000; /* normalize ix */
214 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
215 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
216 else {k=0;n+=1;ix -= 0x00100000;}
217 SET_HIGH_WORD(ax,ix);
218
219 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
220 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
221 v = one/(ax+bp[k]);
222 ss = u*v;
223 s_h = ss;
224 SET_LOW_WORD(s_h,0);
225 /* t_h=ax+bp[k] High */
226 t_h = zero;
227 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
228 t_l = ax - (t_h-bp[k]);
229 s_l = v*((u-s_h*t_h)-s_h*t_l);
230 /* compute log(ax) */
231 s2 = ss*ss;
232 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
233 r += s_l*(s_h+ss);
234 s2 = s_h*s_h;
235 t_h = 3.0+s2+r;
236 SET_LOW_WORD(t_h,0);
237 t_l = r-((t_h-3.0)-s2);
238 /* u+v = ss*(1+...) */
239 u = s_h*t_h;
240 v = s_l*t_h+t_l*ss;
241 /* 2/(3log2)*(ss+...) */
242 p_h = u+v;
243 SET_LOW_WORD(p_h,0);
244 p_l = v-(p_h-u);
245 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
246 z_l = cp_l*p_h+p_l*cp+dp_l[k];
247 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
248 t = (double)n;
249 t1 = (((z_h+z_l)+dp_h[k])+t);
250 SET_LOW_WORD(t1,0);
251 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
252 }
253
6ff43c94
PA
254 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
255 y1 = y;
256 SET_LOW_WORD(y1,0);
257 p_l = (y-y1)*t1+y*t2;
258 p_h = y1*t1;
b34b60bc
JS
259 z = p_l+p_h;
260 EXTRACT_WORDS(j,i,z);
261 if (j>=0x40900000) { /* z >= 1024 */
262 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
263 return s*huge*huge; /* overflow */
264 else {
265 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
266 }
267 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
268 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
269 return s*tiny*tiny; /* underflow */
270 else {
271 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
272 }
273 }
274 /*
275 * compute 2**(p_h+p_l)
276 */
277 i = j&0x7fffffff;
278 k = (i>>20)-0x3ff;
279 n = 0;
280 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
281 n = j+(0x00100000>>(k+1));
282 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
283 t = zero;
284 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
285 n = ((n&0x000fffff)|0x00100000)>>(20-k);
286 if(j<0) n = -n;
287 p_h -= t;
6ff43c94 288 }
b34b60bc
JS
289 t = p_l+p_h;
290 SET_LOW_WORD(t,0);
291 u = t*lg2_h;
292 v = (p_l-(t-p_h))*lg2+t*lg2_l;
293 z = u+v;
294 w = v-(z-u);
295 t = z*z;
296 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
297 r = (z*t1)/(t1-two)-(w+z*w);
298 z = one-(r-z);
299 GET_HIGH_WORD(j,z);
300 j += (n<<20);
301 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
302 else SET_HIGH_WORD(z,j);
303 return s*z;
304}