1 
2 #ifndef lint
3 static  char sccsid[] = "@(#)e_pow.c 1.5 04/04/22 SMI";
4 #endif
5 
6 /*
7  * ====================================================
8  * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
9  *
10  * Permission to use, copy, modify, and distribute this
11  * software is freely granted, provided that this notice
12  * is preserved.
13  * ====================================================
14  */
15 
16 /* __ieee754_pow(x,y) return x**y
17  *
18  *		      n
19  * Method:  Let x =  2   * (1+f)
20  *	1. Compute and return log2(x) in two pieces:
21  *		log2(x) = w1 + w2,
22  *	   where w1 has 53-24 = 29 bit trailing zeros.
23  *	2. Perform y*log2(x) = n+y' by simulating muti-precision
24  *	   arithmetic, where |y'|<=0.5.
25  *	3. Return x**y = 2**n*ieee_exp(y'*log2)
26  *
27  * Special cases:
28  *	1.  (anything) ** 0  is 1
29  *	2.  (anything) ** 1  is itself
30  *	3.  (anything) ** NAN is NAN
31  *	4.  NAN ** (anything except 0) is NAN
32  *	5.  +-(|x| > 1) **  +INF is +INF
33  *	6.  +-(|x| > 1) **  -INF is +0
34  *	7.  +-(|x| < 1) **  +INF is +0
35  *	8.  +-(|x| < 1) **  -INF is +INF
36  *	9.  +-1         ** +-INF is NAN
37  *	10. +0 ** (+anything except 0, NAN)               is +0
38  *	11. -0 ** (+anything except 0, NAN, odd integer)  is +0
39  *	12. +0 ** (-anything except 0, NAN)               is +INF
40  *	13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
41  *	14. -0 ** (odd integer) = -( +0 ** (odd integer) )
42  *	15. +INF ** (+anything except 0,NAN) is +INF
43  *	16. +INF ** (-anything except 0,NAN) is +0
44  *	17. -INF ** (anything)  = -0 ** (-anything)
45  *	18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
46  *	19. (-anything except 0 and inf) ** (non-integer) is NAN
47  *
48  * Accuracy:
49  *	pow(x,y) returns x**y nearly rounded. In particular
50  *			pow(integer,integer)
51  *	always returns the correct integer provided it is
52  *	representable.
53  *
54  * Constants :
55  * The hexadecimal values are the intended ones for the following
56  * constants. The decimal values may be used, provided that the
57  * compiler will convert from decimal to binary accurately enough
58  * to produce the hexadecimal values shown.
59  */
60 
61 #include "fdlibm.h"
62 
63 #ifdef __STDC__
64 static const double
65 #else
66 static double
67 #endif
68 bp[] = {1.0, 1.5,},
69 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
70 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
71 zero    =  0.0,
72 one	=  1.0,
73 two	=  2.0,
74 two53	=  9007199254740992.0,	/* 0x43400000, 0x00000000 */
75 huge	=  1.0e300,
76 tiny    =  1.0e-300,
77 	/* poly coefs for (3/2)*(ieee_log(x)-2s-2/3*s**3 */
78 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
79 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
80 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
81 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
82 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
83 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
84 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
85 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
86 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
87 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
88 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
89 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
90 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
91 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
92 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
93 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
94 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
95 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
96 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
97 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
98 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
99 
100 #ifdef __STDC__
__ieee754_pow(double x,double y)101 	double __ieee754_pow(double x, double y)
102 #else
103 	double __ieee754_pow(x,y)
104 	double x, y;
105 #endif
106 {
107 	double z,ax,z_h,z_l,p_h,p_l;
108 	double y1,t1,t2,r,s,t,u,v,w;
109 	int i0,i1,i,j,k,yisint,n;
110 	int hx,hy,ix,iy;
111 	unsigned lx,ly;
112 
113 	i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
114 	hx = __HI(x); lx = __LO(x);
115 	hy = __HI(y); ly = __LO(y);
116 	ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
117 
118     /* y==zero: x**0 = 1 */
119 	if((iy|ly)==0) return one;
120 
121     /* +-NaN return x+y */
122 	if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
123 	   iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
124 		return x+y;
125 
126     /* determine if y is an odd int when x < 0
127      * yisint = 0	... y is not an integer
128      * yisint = 1	... y is an odd int
129      * yisint = 2	... y is an even int
130      */
131 	yisint  = 0;
132 	if(hx<0) {
133 	    if(iy>=0x43400000) yisint = 2; /* even integer y */
134 	    else if(iy>=0x3ff00000) {
135 		k = (iy>>20)-0x3ff;	   /* exponent */
136 		if(k>20) {
137 		    j = ly>>(52-k);
138 		    if((j<<(52-k))==ly) yisint = 2-(j&1);
139 		} else if(ly==0) {
140 		    j = iy>>(20-k);
141 		    if((j<<(20-k))==iy) yisint = 2-(j&1);
142 		}
143 	    }
144 	}
145 
146     /* special value of y */
147 	if(ly==0) {
148 	    if (iy==0x7ff00000) {	/* y is +-inf */
149 	        if(((ix-0x3ff00000)|lx)==0)
150 		    return  y - y;	/* inf**+-1 is NaN */
151 	        else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
152 		    return (hy>=0)? y: zero;
153 	        else			/* (|x|<1)**-,+inf = inf,0 */
154 		    return (hy<0)?-y: zero;
155 	    }
156 	    if(iy==0x3ff00000) {	/* y is  +-1 */
157 		if(hy<0) return one/x; else return x;
158 	    }
159 	    if(hy==0x40000000) return x*x; /* y is  2 */
160 	    if(hy==0x3fe00000) {	/* y is  0.5 */
161 		if(hx>=0)	/* x >= +0 */
162 		return ieee_sqrt(x);
163 	    }
164 	}
165 
166 	ax   = ieee_fabs(x);
167     /* special value of x */
168 	if(lx==0) {
169 	    if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
170 		z = ax;			/*x is +-0,+-inf,+-1*/
171 		if(hy<0) z = one/z;	/* z = (1/|x|) */
172 		if(hx<0) {
173 		    if(((ix-0x3ff00000)|yisint)==0) {
174 			z = (z-z)/(z-z); /* (-1)**non-int is NaN */
175 		    } else if(yisint==1)
176 			z = -z;		/* (x<0)**odd = -(|x|**odd) */
177 		}
178 		return z;
179 	    }
180 	}
181 
182 	n = (hx>>31)+1;
183 
184     /* (x<0)**(non-int) is NaN */
185 	if((n|yisint)==0) return (x-x)/(x-x);
186 
187 	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
188 	if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
189 
190     /* |y| is huge */
191 	if(iy>0x41e00000) { /* if |y| > 2**31 */
192 	    if(iy>0x43f00000){	/* if |y| > 2**64, must o/uflow */
193 		if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
194 		if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
195 	    }
196 	/* over/underflow if x is not close to one */
197 	    if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
198 	    if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
199 	/* now |1-x| is tiny <= 2**-20, suffice to compute
200 	   ieee_log(x) by x-x^2/2+x^3/3-x^4/4 */
201 	    t = ax-one;		/* t has 20 trailing zeros */
202 	    w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
203 	    u = ivln2_h*t;	/* ivln2_h has 21 sig. bits */
204 	    v = t*ivln2_l-w*ivln2;
205 	    t1 = u+v;
206 	    __LO(t1) = 0;
207 	    t2 = v-(t1-u);
208 	} else {
209 	    double ss,s2,s_h,s_l,t_h,t_l;
210 	    n = 0;
211 	/* take care subnormal number */
212 	    if(ix<0x00100000)
213 		{ax *= two53; n -= 53; ix = __HI(ax); }
214 	    n  += ((ix)>>20)-0x3ff;
215 	    j  = ix&0x000fffff;
216 	/* determine interval */
217 	    ix = j|0x3ff00000;		/* normalize ix */
218 	    if(j<=0x3988E) k=0;		/* |x|<ieee_sqrt(3/2) */
219 	    else if(j<0xBB67A) k=1;	/* |x|<ieee_sqrt(3)   */
220 	    else {k=0;n+=1;ix -= 0x00100000;}
221 	    __HI(ax) = ix;
222 
223 	/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
224 	    u = ax-bp[k];		/* bp[0]=1.0, bp[1]=1.5 */
225 	    v = one/(ax+bp[k]);
226 	    ss = u*v;
227 	    s_h = ss;
228 	    __LO(s_h) = 0;
229 	/* t_h=ax+bp[k] High */
230 	    t_h = zero;
231 	    __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18);
232 	    t_l = ax - (t_h-bp[k]);
233 	    s_l = v*((u-s_h*t_h)-s_h*t_l);
234 	/* compute ieee_log(ax) */
235 	    s2 = ss*ss;
236 	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
237 	    r += s_l*(s_h+ss);
238 	    s2  = s_h*s_h;
239 	    t_h = 3.0+s2+r;
240 	    __LO(t_h) = 0;
241 	    t_l = r-((t_h-3.0)-s2);
242 	/* u+v = ss*(1+...) */
243 	    u = s_h*t_h;
244 	    v = s_l*t_h+t_l*ss;
245 	/* 2/(3log2)*(ss+...) */
246 	    p_h = u+v;
247 	    __LO(p_h) = 0;
248 	    p_l = v-(p_h-u);
249 	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */
250 	    z_l = cp_l*p_h+p_l*cp+dp_l[k];
251 	/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
252 	    t = (double)n;
253 	    t1 = (((z_h+z_l)+dp_h[k])+t);
254 	    __LO(t1) = 0;
255 	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
256 	}
257 
258     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
259 	y1  = y;
260 	__LO(y1) = 0;
261 	p_l = (y-y1)*t1+y*t2;
262 	p_h = y1*t1;
263 	z = p_l+p_h;
264 	j = __HI(z);
265 	i = __LO(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 	    __HI(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 	__LO(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 	j  = __HI(z);
305 	j += (n<<20);
306 	if((j>>20)<=0) z = ieee_scalbn(z,n);	/* subnormal output */
307 	else __HI(z) += (n<<20);
308 	return s*z;
309 }
310