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 #if defined(LIBM_SCCS) && !defined(lint)
14 __RCSID("$NetBSD: e_pow.c,v 1.17 2016/08/27 10:01:08 christos Exp $");
15 #endif
16
17 /* __ieee754_pow(x,y) return x**y
18 *
19 * n
20 * Method: Let x = 2 * (1+f)
21 * 1. Compute and return log2(x) in two pieces:
22 * log2(x) = w1 + w2,
23 * where w1 has 53-24 = 29 bit trailing zeros.
24 * 2. Perform y*log2(x) = n+y' by simulating multi-precision
25 * arithmetic, where |y'|<=0.5.
26 * 3. Return x**y = 2**n*exp(y'*log2)
27 *
28 * Special cases:
29 * 1. (anything) ** 0 is 1
30 * 2. (anything) ** 1 is itself
31 * 3. (anything) ** NAN is NAN except 1 ** NAN = 1
32 * 4. NAN ** (anything except 0) is NAN
33 * 5. +-(|x| > 1) ** +INF is +INF
34 * 6. +-(|x| > 1) ** -INF is +0
35 * 7. +-(|x| < 1) ** +INF is +0
36 * 8. +-(|x| < 1) ** -INF is +INF
37 * 9. +-1 ** +-INF is 1
38 * 10. +0 ** (+anything except 0, NAN) is +0
39 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
40 * 12. +0 ** (-anything except 0, NAN) is +INF
41 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
42 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
43 * 15. +INF ** (+anything except 0,NAN) is +INF
44 * 16. +INF ** (-anything except 0,NAN) is +0
45 * 17. -INF ** (anything) = -0 ** (-anything)
46 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
47 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
48 *
49 * Accuracy:
50 * pow(x,y) returns x**y nearly rounded. In particular
51 * pow(integer,integer)
52 * always returns the correct integer provided it is
53 * representable.
54 *
55 * Constants :
56 * The hexadecimal values are the intended ones for the following
57 * constants. The decimal values may be used, provided that the
58 * compiler will convert from decimal to binary accurately enough
59 * to produce the hexadecimal values shown.
60 */
61
62 #include "namespace.h"
63 #include "math.h"
64 #include "math_private.h"
65
66 static const double
67 bp[] = {1.0, 1.5,},
68 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
69 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
70 zero = 0.0,
71 one = 1.0,
72 two = 2.0,
73 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
74 huge = 1.0e300,
75 tiny = 1.0e-300,
76 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
77 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
78 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
79 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
80 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
81 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
82 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
83 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
84 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
85 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
86 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
87 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
88 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
89 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
90 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
91 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
92 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
93 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
94 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
95 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
96 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
97 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
98
99 double
__ieee754_pow(double x,double y)100 __ieee754_pow(double x, double y)
101 {
102 double z,ax,z_h,z_l,p_h,p_l;
103 double yy1,t1,t2,r,s,t,u,v,w;
104 int32_t i,j,k,yisint,n;
105 int32_t hx,hy,ix,iy;
106 u_int32_t lx,ly;
107
108 EXTRACT_WORDS(hx,lx,x);
109 EXTRACT_WORDS(hy,ly,y);
110 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
111
112 /* y==zero: x**0 = 1 */
113 if((iy|ly)==0) return one;
114
115 /* x==1: 1**y = 1, even if y is NaN */
116 if (hx==0x3ff00000 && lx == 0) return one;
117
118 /* y!=zero: result is NaN if either arg is NaN */
119 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
120 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
121 return (x+0.0)+(y+0.0);
122
123 /* determine if y is an odd int when x < 0
124 * yisint = 0 ... y is not an integer
125 * yisint = 1 ... y is an odd int
126 * yisint = 2 ... y is an even int
127 */
128 yisint = 0;
129 if(hx<0) {
130 if(iy>=0x43400000) yisint = 2; /* even integer y */
131 else if(iy>=0x3ff00000) {
132 k = (iy>>20)-0x3ff; /* exponent */
133 if(k>20) {
134 j = ly>>(52-k);
135 if((uint32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
136 } else if(ly==0) {
137 j = iy>>(20-k);
138 if((j<<(20-k))==iy) yisint = 2-(j&1);
139 }
140 }
141 }
142
143 /* special value of y */
144 if(ly==0) {
145 if (iy==0x7ff00000) { /* y is +-inf */
146 if(((ix-0x3ff00000)|lx)==0)
147 return one; /* (-1)**+-inf is 1 */
148 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
149 return (hy>=0)? y: zero;
150 else /* (|x|<1)**-,+inf = inf,0 */
151 return (hy<0)?-y: zero;
152 }
153 if(iy==0x3ff00000) { /* y is +-1 */
154 if(hy<0) return one/x; else return x;
155 }
156 if(hy==0x40000000) return x*x; /* y is 2 */
157 if(hy==0x3fe00000) { /* y is 0.5 */
158 if(hx>=0) /* x >= +0 */
159 return __ieee754_sqrt(x);
160 }
161 }
162
163 ax = fabs(x);
164 /* special value of x */
165 if(lx==0) {
166 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
167 z = ax; /*x is +-0,+-inf,+-1*/
168 if(hy<0) z = one/z; /* z = (1/|x|) */
169 if(hx<0) {
170 if(((ix-0x3ff00000)|yisint)==0) {
171 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
172 } else if(yisint==1)
173 z = -z; /* (x<0)**odd = -(|x|**odd) */
174 }
175 return z;
176 }
177 }
178
179 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
180 n = (hx>>31)+1;
181 but ANSI C says a right shift of a signed negative quantity is
182 implementation defined. */
183 n = ((u_int32_t)hx>>31)-1;
184
185 /* (x<0)**(non-int) is NaN */
186 if((n|yisint)==0) return (x-x)/(x-x);
187
188 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
189 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
190
191 /* |y| is huge */
192 if(iy>0x41e00000) { /* if |y| > 2**31 */
193 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
194 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
195 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
196 }
197 /* over/underflow if x is not close to one */
198 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
199 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
200 /* now |1-x| is tiny <= 2**-20, suffice to compute
201 log(x) by x-x^2/2+x^3/3-x^4/4 */
202 t = ax-one; /* t has 20 trailing zeros */
203 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
204 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
205 v = t*ivln2_l-w*ivln2;
206 t1 = u+v;
207 SET_LOW_WORD(t1,0);
208 t2 = v-(t1-u);
209 } else {
210 double ss,s2,s_h,s_l,t_h,t_l;
211 n = 0;
212 /* take care subnormal number */
213 if(ix<0x00100000)
214 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
215 n += ((ix)>>20)-0x3ff;
216 j = ix&0x000fffff;
217 /* determine interval */
218 ix = j|0x3ff00000; /* normalize ix */
219 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
220 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
221 else {k=0;n+=1;ix -= 0x00100000;}
222 SET_HIGH_WORD(ax,ix);
223
224 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
225 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
226 v = one/(ax+bp[k]);
227 ss = u*v;
228 s_h = ss;
229 SET_LOW_WORD(s_h,0);
230 /* t_h=ax+bp[k] High */
231 t_h = zero;
232 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
233 t_l = ax - (t_h-bp[k]);
234 s_l = v*((u-s_h*t_h)-s_h*t_l);
235 /* compute log(ax) */
236 s2 = ss*ss;
237 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
238 r += s_l*(s_h+ss);
239 s2 = s_h*s_h;
240 t_h = 3.0+s2+r;
241 SET_LOW_WORD(t_h,0);
242 t_l = r-((t_h-3.0)-s2);
243 /* u+v = ss*(1+...) */
244 u = s_h*t_h;
245 v = s_l*t_h+t_l*ss;
246 /* 2/(3log2)*(ss+...) */
247 p_h = u+v;
248 SET_LOW_WORD(p_h,0);
249 p_l = v-(p_h-u);
250 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
251 z_l = cp_l*p_h+p_l*cp+dp_l[k];
252 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
253 t = (double)n;
254 t1 = (((z_h+z_l)+dp_h[k])+t);
255 SET_LOW_WORD(t1,0);
256 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
257 }
258
259 /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
260 yy1 = y;
261 SET_LOW_WORD(yy1,0);
262 p_l = (y-yy1)*t1+y*t2;
263 p_h = yy1*t1;
264 z = p_l+p_h;
265 EXTRACT_WORDS(j,i,z);
266 if (j>=0x40900000) { /* z >= 1024 */
267 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
268 return s*huge*huge; /* overflow */
269 else {
270 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
271 }
272 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
273 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
274 return s*tiny*tiny; /* underflow */
275 else {
276 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
277 }
278 }
279 /*
280 * compute 2**(p_h+p_l)
281 */
282 i = j&0x7fffffff;
283 k = (i>>20)-0x3ff;
284 n = 0;
285 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
286 n = j+(0x00100000>>(k+1));
287 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
288 t = zero;
289 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
290 n = ((n&0x000fffff)|0x00100000)>>(20-k);
291 if(j<0) n = -n;
292 p_h -= t;
293 }
294 t = p_l+p_h;
295 SET_LOW_WORD(t,0);
296 u = t*lg2_h;
297 v = (p_l-(t-p_h))*lg2+t*lg2_l;
298 z = u+v;
299 w = v-(z-u);
300 t = z*z;
301 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
302 r = (z*t1)/(t1-two)-(w+z*w);
303 z = one-(r-z);
304 GET_HIGH_WORD(j,z);
305 j += (n<<20);
306 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
307 else SET_HIGH_WORD(z,j);
308 return s*z;
309 }
310