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__FBSDID("$FreeBSD$");
14
15/* __ieee754_pow(x,y) return x**y
16 *
17 *		      n
18 * Method:  Let x =  2   * (1+f)
19 *	1. Compute and return log2(x) in two pieces:
20 *		log2(x) = w1 + w2,
21 *	   where w1 has 53-24 = 29 bit trailing zeros.
22 *	2. Perform y*log2(x) = n+y' by simulating multi-precision
23 *	   arithmetic, where |y'|<=0.5.
24 *	3. Return x**y = 2**n*exp(y'*log2)
25 *
26 * Special cases:
27 *	1.  (anything) ** 0  is 1
28 *	2.  (anything) ** 1  is itself
29 *	3.  (anything) ** NAN is NAN except 1 ** NAN = 1
30 *	4.  NAN ** (anything except 0) is NAN
31 *	5.  +-(|x| > 1) **  +INF is +INF
32 *	6.  +-(|x| > 1) **  -INF is +0
33 *	7.  +-(|x| < 1) **  +INF is +0
34 *	8.  +-(|x| < 1) **  -INF is +INF
35 *	9.  +-1         ** +-INF is 1
36 *	10. +0 ** (+anything except 0, NAN)               is +0
37 *	11. -0 ** (+anything except 0, NAN, odd integer)  is +0
38 *	12. +0 ** (-anything except 0, NAN)               is +INF
39 *	13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
40 *	14. -0 ** (odd integer) = -( +0 ** (odd integer) )
41 *	15. +INF ** (+anything except 0,NAN) is +INF
42 *	16. +INF ** (-anything except 0,NAN) is +0
43 *	17. -INF ** (anything)  = -0 ** (-anything)
44 *	18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
45 *	19. (-anything except 0 and inf) ** (non-integer) is NAN
46 *
47 * Accuracy:
48 *	pow(x,y) returns x**y nearly rounded. In particular
49 *			pow(integer,integer)
50 *	always returns the correct integer provided it is
51 *	representable.
52 *
53 * Constants :
54 * The hexadecimal values are the intended ones for the following
55 * constants. The decimal values may be used, provided that the
56 * compiler will convert from decimal to binary accurately enough
57 * to produce the hexadecimal values shown.
58 */
59
60#include "math.h"
61#include "math_private.h"
62
63static const double
64bp[] = {1.0, 1.5,},
65dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
66dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
67zero    =  0.0,
68one	=  1.0,
69two	=  2.0,
70two53	=  9007199254740992.0,	/* 0x43400000, 0x00000000 */
71huge	=  1.0e300,
72tiny    =  1.0e-300,
73	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
74L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
75L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
76L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
77L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
78L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
79L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
80P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
81P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
82P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
83P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
84P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
85lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
86lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
87lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
88ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
89cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
90cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
91cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
92ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
93ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
94ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
95
96double
97__ieee754_pow(double x, double y)
98{
99	double z,ax,z_h,z_l,p_h,p_l;
100	double y1,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    /* y!=zero: result is NaN if either arg is NaN */
116	if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
117	   iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
118		return (x+0.0)+(y+0.0);
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((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    /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
177	n = (hx>>31)+1;
178       but ANSI C says a right shift of a signed negative quantity is
179       implementation defined.  */
180	n = ((u_int32_t)hx>>31)-1;
181
182    /* (x<0)**(non-int) is NaN */
183	if((n|yisint)==0) return (x-x)/(x-x);
184
185	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
186	if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
187
188    /* |y| is huge */
189	if(iy>0x41e00000) { /* if |y| > 2**31 */
190	    if(iy>0x43f00000){	/* if |y| > 2**64, must o/uflow */
191		if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
192		if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
193	    }
194	/* over/underflow if x is not close to one */
195	    if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
196	    if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
197	/* now |1-x| is tiny <= 2**-20, suffice to compute
198	   log(x) by x-x^2/2+x^3/3-x^4/4 */
199	    t = ax-one;		/* t has 20 trailing zeros */
200	    w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
201	    u = ivln2_h*t;	/* ivln2_h has 21 sig. bits */
202	    v = t*ivln2_l-w*ivln2;
203	    t1 = u+v;
204	    SET_LOW_WORD(t1,0);
205	    t2 = v-(t1-u);
206	} else {
207	    double ss,s2,s_h,s_l,t_h,t_l;
208	    n = 0;
209	/* take care subnormal number */
210	    if(ix<0x00100000)
211		{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
212	    n  += ((ix)>>20)-0x3ff;
213	    j  = ix&0x000fffff;
214	/* determine interval */
215	    ix = j|0x3ff00000;		/* normalize ix */
216	    if(j<=0x3988E) k=0;		/* |x|<sqrt(3/2) */
217	    else if(j<0xBB67A) k=1;	/* |x|<sqrt(3)   */
218	    else {k=0;n+=1;ix -= 0x00100000;}
219	    SET_HIGH_WORD(ax,ix);
220
221	/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
222	    u = ax-bp[k];		/* bp[0]=1.0, bp[1]=1.5 */
223	    v = one/(ax+bp[k]);
224	    ss = u*v;
225	    s_h = ss;
226	    SET_LOW_WORD(s_h,0);
227	/* t_h=ax+bp[k] High */
228	    t_h = zero;
229	    SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
230	    t_l = ax - (t_h-bp[k]);
231	    s_l = v*((u-s_h*t_h)-s_h*t_l);
232	/* compute log(ax) */
233	    s2 = ss*ss;
234	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
235	    r += s_l*(s_h+ss);
236	    s2  = s_h*s_h;
237	    t_h = 3.0+s2+r;
238	    SET_LOW_WORD(t_h,0);
239	    t_l = r-((t_h-3.0)-s2);
240	/* u+v = ss*(1+...) */
241	    u = s_h*t_h;
242	    v = s_l*t_h+t_l*ss;
243	/* 2/(3log2)*(ss+...) */
244	    p_h = u+v;
245	    SET_LOW_WORD(p_h,0);
246	    p_l = v-(p_h-u);
247	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */
248	    z_l = cp_l*p_h+p_l*cp+dp_l[k];
249	/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
250	    t = (double)n;
251	    t1 = (((z_h+z_l)+dp_h[k])+t);
252	    SET_LOW_WORD(t1,0);
253	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
254	}
255
256    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
257	y1  = y;
258	SET_LOW_WORD(y1,0);
259	p_l = (y-y1)*t1+y*t2;
260	p_h = y1*t1;
261	z = p_l+p_h;
262	EXTRACT_WORDS(j,i,z);
263	if (j>=0x40900000) {				/* z >= 1024 */
264	    if(((j-0x40900000)|i)!=0)			/* if z > 1024 */
265		return s*huge*huge;			/* overflow */
266	    else {
267		if(p_l+ovt>z-p_h) return s*huge*huge;	/* overflow */
268	    }
269	} else if((j&0x7fffffff)>=0x4090cc00 ) {	/* z <= -1075 */
270	    if(((j-0xc090cc00)|i)!=0) 		/* z < -1075 */
271		return s*tiny*tiny;		/* underflow */
272	    else {
273		if(p_l<=z-p_h) return s*tiny*tiny;	/* underflow */
274	    }
275	}
276    /*
277     * compute 2**(p_h+p_l)
278     */
279	i = j&0x7fffffff;
280	k = (i>>20)-0x3ff;
281	n = 0;
282	if(i>0x3fe00000) {		/* if |z| > 0.5, set n = [z+0.5] */
283	    n = j+(0x00100000>>(k+1));
284	    k = ((n&0x7fffffff)>>20)-0x3ff;	/* new k for n */
285	    t = zero;
286	    SET_HIGH_WORD(t,n&~(0x000fffff>>k));
287	    n = ((n&0x000fffff)|0x00100000)>>(20-k);
288	    if(j<0) n = -n;
289	    p_h -= t;
290	}
291	t = p_l+p_h;
292	SET_LOW_WORD(t,0);
293	u = t*lg2_h;
294	v = (p_l-(t-p_h))*lg2+t*lg2_l;
295	z = u+v;
296	w = v-(z-u);
297	t  = z*z;
298	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
299	r  = (z*t1)/(t1-two)-(w+z*w);
300	z  = one-(r-z);
301	GET_HIGH_WORD(j,z);
302	j += (n<<20);
303	if((j>>20)<=0) z = scalbn(z,n);	/* subnormal output */
304	else SET_HIGH_WORD(z,j);
305	return s*z;
306}
307