1 /* @(#)e_pow.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 /* pow(x,y) return x**y
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.
20 * 2. Perform y*log2(x) = n+y' by simulating multi-precision
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
27 * 3. (anything except 1) ** NAN is NAN
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
33 * 9. +-1 ** +-INF is 1
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)
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 <float.h>
59 #include <math.h>
60
61 #include "math_private.h"
62
63 static const double
64 bp[] = {1.0, 1.5,},
65 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
66 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
67 zero = 0.0,
68 one = 1.0,
69 two = 2.0,
70 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
71 huge = 1.0e300,
72 tiny = 1.0e-300,
73 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
74 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
75 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
76 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
77 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
78 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
79 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
80 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
81 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
82 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
83 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
84 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
85 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
86 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
87 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
88 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
89 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
90 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
91 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
92 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
93 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
94 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
95
96 double
pow(double x,double y)97 pow(double x, double y)
98 {
99 double z,ax,z_h,z_l,p_h,p_l;
100 double yy1,t1,t2,r,s,t,u,v,w;
101 int32_t i,j,k,yisint,n;
102 int32_t hx,hy,ix,iy;
103 u_int32_t lx,ly;
104
105 EXTRACT_WORDS(hx,lx,x);
106 EXTRACT_WORDS(hy,ly,y);
107 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
108
109 /* y==zero: x**0 = 1 */
110 if((iy|ly)==0) return one;
111
112 /* x==1: 1**y = 1, even if y is NaN */
113 if (hx==0x3ff00000 && lx == 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((u_int32_t)(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 */
145 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
146 return (hy>=0)? y: zero;
147 else /* (|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 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 n = (hx>>31)+1;
177
178 /* (x<0)**(non-int) is NaN */
179 if((n|yisint)==0) return (x-x)/(x-x);
180
181 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
182 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
183
184 /* |y| is huge */
185 if(iy>0x41e00000) { /* if |y| > 2**31 */
186 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
187 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
188 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
189 }
190 /* over/underflow if x is not close to one */
191 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
192 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
193 /* now |1-x| is tiny <= 2**-20, suffice to compute
194 log(x) by x-x^2/2+x^3/3-x^4/4 */
195 t = ax-one; /* t has 20 trailing zeros */
196 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
197 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
198 v = t*ivln2_l-w*ivln2;
199 t1 = u+v;
200 SET_LOW_WORD(t1,0);
201 t2 = v-(t1-u);
202 } else {
203 double ss,s2,s_h,s_l,t_h,t_l;
204 n = 0;
205 /* take care subnormal number */
206 if(ix<0x00100000)
207 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
208 n += ((ix)>>20)-0x3ff;
209 j = ix&0x000fffff;
210 /* determine interval */
211 ix = j|0x3ff00000; /* normalize ix */
212 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
213 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
214 else {k=0;n+=1;ix -= 0x00100000;}
215 SET_HIGH_WORD(ax,ix);
216
217 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
218 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
219 v = one/(ax+bp[k]);
220 ss = u*v;
221 s_h = ss;
222 SET_LOW_WORD(s_h,0);
223 /* t_h=ax+bp[k] High */
224 t_h = zero;
225 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
226 t_l = ax - (t_h-bp[k]);
227 s_l = v*((u-s_h*t_h)-s_h*t_l);
228 /* compute log(ax) */
229 s2 = ss*ss;
230 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
231 r += s_l*(s_h+ss);
232 s2 = s_h*s_h;
233 t_h = 3.0+s2+r;
234 SET_LOW_WORD(t_h,0);
235 t_l = r-((t_h-3.0)-s2);
236 /* u+v = ss*(1+...) */
237 u = s_h*t_h;
238 v = s_l*t_h+t_l*ss;
239 /* 2/(3log2)*(ss+...) */
240 p_h = u+v;
241 SET_LOW_WORD(p_h,0);
242 p_l = v-(p_h-u);
243 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
244 z_l = cp_l*p_h+p_l*cp+dp_l[k];
245 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
246 t = (double)n;
247 t1 = (((z_h+z_l)+dp_h[k])+t);
248 SET_LOW_WORD(t1,0);
249 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
250 }
251
252 /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
253 yy1 = y;
254 SET_LOW_WORD(yy1,0);
255 p_l = (y-yy1)*t1+y*t2;
256 p_h = yy1*t1;
257 z = p_l+p_h;
258 EXTRACT_WORDS(j,i,z);
259 if (j>=0x40900000) { /* z >= 1024 */
260 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
261 return s*huge*huge; /* overflow */
262 else {
263 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
264 }
265 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
266 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
267 return s*tiny*tiny; /* underflow */
268 else {
269 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
270 }
271 }
272 /*
273 * compute 2**(p_h+p_l)
274 */
275 i = j&0x7fffffff;
276 k = (i>>20)-0x3ff;
277 n = 0;
278 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
279 n = j+(0x00100000>>(k+1));
280 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
281 t = zero;
282 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
283 n = ((n&0x000fffff)|0x00100000)>>(20-k);
284 if(j<0) n = -n;
285 p_h -= t;
286 }
287 t = p_l+p_h;
288 SET_LOW_WORD(t,0);
289 u = t*lg2_h;
290 v = (p_l-(t-p_h))*lg2+t*lg2_l;
291 z = u+v;
292 w = v-(z-u);
293 t = z*z;
294 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
295 r = (z*t1)/(t1-two)-(w+z*w);
296 z = one-(r-z);
297 GET_HIGH_WORD(j,z);
298 j += (n<<20);
299 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
300 else SET_HIGH_WORD(z,j);
301 return s*z;
302 }
303
304 #if LDBL_MANT_DIG == DBL_MANT_DIG
305 __strong_alias(powl, pow);
306 #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */
307