1
2#ifndef lint
3static  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__
64static const double
65#else
66static double
67#endif
68bp[] = {1.0, 1.5,},
69dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
70dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
71zero    =  0.0,
72one	=  1.0,
73two	=  2.0,
74two53	=  9007199254740992.0,	/* 0x43400000, 0x00000000 */
75huge	=  1.0e300,
76tiny    =  1.0e-300,
77	/* poly coefs for (3/2)*(ieee_log(x)-2s-2/3*s**3 */
78L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
79L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
80L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
81L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
82L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
83L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
84P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
85P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
86P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
87P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
88P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
89lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
90lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
91lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
92ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
93cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
94cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
95cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
96ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
97ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
98ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
99
100#ifdef __STDC__
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