e_pow.c revision 2116 
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#ifndef lint
14static char rcsid[] = "$Id: e_pow.c,v 1.5 1994/08/18 23:05:51 jtc 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 muti-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
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 NAN
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 "math.h"
63#include "math_private.h"
64
65#ifdef __STDC__
66static const double
67#else
68static double
69#endif
70bp[] = {1.0, 1.5,},
71dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
72dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
73zero    =  0.0,
74one	=  1.0,
75two	=  2.0,
76two53	=  9007199254740992.0,	/* 0x43400000, 0x00000000 */
77huge	=  1.0e300,
78tiny    =  1.0e-300,
79	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
80L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
81L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
82L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
83L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
84L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
85L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
86P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
87P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
88P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
89P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
90P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
91lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
92lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
93lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
94ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
95cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
96cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
97cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
98ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
99ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
100ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
101
102#ifdef __STDC__
103	double __ieee754_pow(double x, double y)
104#else
105	double __ieee754_pow(x,y)
106	double x, y;
107#endif
108{
109	double z,ax,z_h,z_l,p_h,p_l;
110	double y1,t1,t2,r,s,t,u,v,w;
111	int32_t i,j,k,yisint,n;
112	int32_t hx,hy,ix,iy;
113	u_int32_t lx,ly;
114
115	EXTRACT_WORDS(hx,lx,x);
116	EXTRACT_WORDS(hy,ly,y);
117	ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
118
119    /* y==zero: x**0 = 1 */
120	if((iy|ly)==0) return one;
121
122    /* +-NaN return x+y */
123	if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
124	   iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
125		return x+y;
126
127    /* determine if y is an odd int when x < 0
128     * yisint = 0	... y is not an integer
129     * yisint = 1	... y is an odd int
130     * yisint = 2	... y is an even int
131     */
132	yisint  = 0;
133	if(hx<0) {
134	    if(iy>=0x43400000) yisint = 2; /* even integer y */
135	    else if(iy>=0x3ff00000) {
136		k = (iy>>20)-0x3ff;	   /* exponent */
137		if(k>20) {
138		    j = ly>>(52-k);
139		    if((j<<(52-k))==ly) yisint = 2-(j&1);
140		} else if(ly==0) {
141		    j = iy>>(20-k);
142		    if((j<<(20-k))==iy) yisint = 2-(j&1);
143		}
144	    }
145	}
146
147    /* special value of y */
148	if(ly==0) {
149	    if (iy==0x7ff00000) {	/* y is +-inf */
150	        if(((ix-0x3ff00000)|lx)==0)
151		    return  y - y;	/* inf**+-1 is NaN */
152	        else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
153		    return (hy>=0)? y: zero;
154	        else			/* (|x|<1)**-,+inf = inf,0 */
155		    return (hy<0)?-y: zero;
156	    }
157	    if(iy==0x3ff00000) {	/* y is  +-1 */
158		if(hy<0) return one/x; else return x;
159	    }
160	    if(hy==0x40000000) return x*x; /* y is  2 */
161	    if(hy==0x3fe00000) {	/* y is  0.5 */
162		if(hx>=0)	/* x >= +0 */
163		return sqrt(x);
164	    }
165	}
166
167	ax   = fabs(x);
168    /* special value of x */
169	if(lx==0) {
170	    if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
171		z = ax;			/*x is +-0,+-inf,+-1*/
172		if(hy<0) z = one/z;	/* z = (1/|x|) */
173		if(hx<0) {
174		    if(((ix-0x3ff00000)|yisint)==0) {
175			z = (z-z)/(z-z); /* (-1)**non-int is NaN */
176		    } else if(yisint==1)
177			z = -z;		/* (x<0)**odd = -(|x|**odd) */
178		}
179		return z;
180	    }
181	}
182
183    /* (x<0)**(non-int) is NaN */
184    /* CYGNUS LOCAL: This used to be
185	if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);
186       but ANSI C says a right shift of a signed negative quantity is
187       implementation defined.  */
188	if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
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)? huge*huge:tiny*tiny;
198	    if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
199	/* now |1-x| is tiny <= 2**-20, suffice to compute
200	   log(x) by x-x^2/2+x^3/3-x^4/4 */
201	    t = x-1;		/* 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	    SET_LOW_WORD(t1,0);
207	    t2 = v-(t1-u);
208	} else {
209	    double 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; GET_HIGH_WORD(ix,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|<sqrt(3/2) */
219	    else if(j<0xBB67A) k=1;	/* |x|<sqrt(3)   */
220	    else {k=0;n+=1;ix -= 0x00100000;}
221	    SET_HIGH_WORD(ax,ix);
222
223	/* compute s = 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	    s = u*v;
227	    s_h = s;
228	    SET_LOW_WORD(s_h,0);
229	/* t_h=ax+bp[k] High */
230	    t_h = zero;
231	    SET_HIGH_WORD(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 log(ax) */
235	    s2 = s*s;
236	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
237	    r += s_l*(s_h+s);
238	    s2  = s_h*s_h;
239	    t_h = 3.0+s2+r;
240	    SET_LOW_WORD(t_h,0);
241	    t_l = r-((t_h-3.0)-s2);
242	/* u+v = s*(1+...) */
243	    u = s_h*t_h;
244	    v = s_l*t_h+t_l*s;
245	/* 2/(3log2)*(s+...) */
246	    p_h = u+v;
247	    SET_LOW_WORD(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) = (s+..)*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	    SET_LOW_WORD(t1,0);
255	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
256	}
257
258	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
259	if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
260	    s = -one;/* (-ve)**(odd int) */
261
262    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
263	y1  = y;
264	SET_LOW_WORD(y1,0);
265	p_l = (y-y1)*t1+y*t2;
266	p_h = y1*t1;
267	z = p_l+p_h;
268	EXTRACT_WORDS(j,i,z);
269	if (j>=0x40900000) {				/* z >= 1024 */
270	    if(((j-0x40900000)|i)!=0)			/* if z > 1024 */
271		return s*huge*huge;			/* overflow */
272	    else {
273		if(p_l+ovt>z-p_h) return s*huge*huge;	/* overflow */
274	    }
275	} else if((j&0x7fffffff)>=0x4090cc00 ) {	/* z <= -1075 */
276	    if(((j-0xc090cc00)|i)!=0) 		/* z < -1075 */
277		return s*tiny*tiny;		/* underflow */
278	    else {
279		if(p_l<=z-p_h) return s*tiny*tiny;	/* underflow */
280	    }
281	}
282    /*
283     * compute 2**(p_h+p_l)
284     */
285	i = j&0x7fffffff;
286	k = (i>>20)-0x3ff;
287	n = 0;
288	if(i>0x3fe00000) {		/* if |z| > 0.5, set n = [z+0.5] */
289	    n = j+(0x00100000>>(k+1));
290	    k = ((n&0x7fffffff)>>20)-0x3ff;	/* new k for n */
291	    t = zero;
292	    SET_HIGH_WORD(t,n&~(0x000fffff>>k));
293	    n = ((n&0x000fffff)|0x00100000)>>(20-k);
294	    if(j<0) n = -n;
295	    p_h -= t;
296	}
297	t = p_l+p_h;
298	SET_LOW_WORD(t,0);
299	u = t*lg2_h;
300	v = (p_l-(t-p_h))*lg2+t*lg2_l;
301	z = u+v;
302	w = v-(z-u);
303	t  = z*z;
304	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
305	r  = (z*t1)/(t1-two)-(w+z*w);
306	z  = one-(r-z);
307	GET_HIGH_WORD(j,z);
308	j += (n<<20);
309	if((j>>20)<=0) z = scalbn(z,n);	/* subnormal output */
310	else SET_HIGH_WORD(z,j);
311	return s*z;
312}
313