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1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
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/* __ieee754_pow(x,y) return x**y
13 *
14 *		      n
15 * Method:  Let x =  2   * (1+f)
16 *	1. Compute and return log2(x) in two pieces:
17 *		log2(x) = w1 + w2,
18 *	   where w1 has 53-24 = 29 bit trailing zeros.
19 *	2. Perform y*log2(x) = n+y' by simulating muti-precision
20 *	   arithmetic, where |y'|<=0.5.
21 *	3. Return x**y = 2**n*exp(y'*log2)
22 *
23 * Special cases:
24 *	1.  +-1 ** anything  is 1.0
25 *	2.  +-1 ** +-INF     is 1.0
26 *	3.  (anything) ** 0  is 1
27 *	4.  (anything) ** 1  is itself
28 *	5.  (anything) ** NAN is NAN
29 *	6.  NAN ** (anything except 0) is NAN
30 *	7.  +-(|x| > 1) **  +INF is +INF
31 *	8.  +-(|x| > 1) **  -INF is +0
32 *	9.  +-(|x| < 1) **  +INF is +0
33 *	10  +-(|x| < 1) **  -INF is +INF
34 *	11. +0 ** (+anything except 0, NAN)               is +0
35 *	12. -0 ** (+anything except 0, NAN, odd integer)  is +0
36 *	13. +0 ** (-anything except 0, NAN)               is +INF
37 *	14. -0 ** (-anything except 0, NAN, odd integer)  is +INF
38 *	15. -0 ** (odd integer) = -( +0 ** (odd integer) )
39 *	16. +INF ** (+anything except 0,NAN) is +INF
40 *	17. +INF ** (-anything except 0,NAN) is +0
41 *	18. -INF ** (anything)  = -0 ** (-anything)
42 *	19. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
43 *	20. (-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 "math_libm.h"
59#include "math_private.h"
60
61#if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
62/* C4756: overflow in constant arithmetic */
63#pragma warning ( disable : 4756 )
64#endif
65
66static const double
67bp[] = {1.0, 1.5,},
68dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
69dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
70zero    =  0.0,
71one	=  1.0,
72two	=  2.0,
73two53	=  9007199254740992.0,	/* 0x43400000, 0x00000000 */
74huge	=  1.0e300,
75tiny    =  1.0e-300,
76	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
77L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
78L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
79L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
80L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
81L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
82L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
83P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
84P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
85P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
86P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
87P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
88lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
89lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
90lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
91ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
92cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
93cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
94cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
95ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
96ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
97ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
98
99double attribute_hidden __ieee754_pow(double x, double y)
100{
101	double z,ax,z_h,z_l,p_h,p_l;
102	double y1,t1,t2,r,s,t,u,v,w;
103	int32_t i,j,k,yisint,n;
104	int32_t hx,hy,ix,iy;
105	u_int32_t lx,ly;
106
107	EXTRACT_WORDS(hx,lx,x);
108    /* x==1: 1**y = 1 (even if y is NaN) */
109	if (hx==0x3ff00000 && lx==0) {
110		return x;
111	}
112	ix = hx&0x7fffffff;
113
114	EXTRACT_WORDS(hy,ly,y);
115	iy = hy&0x7fffffff;
116
117    /* y==zero: x**0 = 1 */
118	if((iy|ly)==0) return one;
119
120    /* +-NaN return x+y */
121	if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
122	   iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
123		return x+y;
124
125    /* determine if y is an odd int when x < 0
126     * yisint = 0	... y is not an integer
127     * yisint = 1	... y is an odd int
128     * yisint = 2	... y is an even int
129     */
130	yisint  = 0;
131	if(hx<0) {
132	    if(iy>=0x43400000) yisint = 2; /* even integer y */
133	    else if(iy>=0x3ff00000) {
134		k = (iy>>20)-0x3ff;	   /* exponent */
135		if(k>20) {
136		    j = ly>>(52-k);
137		    if((j<<(52-k))==ly) yisint = 2-(j&1);
138		} else if(ly==0) {
139		    j = iy>>(20-k);
140		    if((j<<(20-k))==iy) yisint = 2-(j&1);
141		}
142	    }
143	}
144
145    /* special value of y */
146	if(ly==0) {
147	    if (iy==0x7ff00000) {       /* y is +-inf */
148	        if (((ix-0x3ff00000)|lx)==0)
149		    return one;	        /* +-1**+-inf is 1 (yes, weird rule) */
150	        if (ix >= 0x3ff00000)   /* (|x|>1)**+-inf = inf,0 */
151		    return (hy>=0) ? y : zero;
152	        /* (|x|<1)**-,+inf = inf,0 */
153		return (hy<0) ? -y : zero;
154	    }
155	    if(iy==0x3ff00000) {	/* y is  +-1 */
156		if(hy<0) return one/x; else return x;
157	    }
158	    if(hy==0x40000000) return x*x; /* y is  2 */
159	    if(hy==0x3fe00000) {	/* y is  0.5 */
160		if(hx>=0)	/* x >= +0 */
161		    return __ieee754_sqrt(x);
162	    }
163	}
164
165	ax   = fabs(x);
166    /* special value of x */
167	if(lx==0) {
168	    if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
169		z = ax;			/*x is +-0,+-inf,+-1*/
170		if(hy<0) z = one/z;	/* z = (1/|x|) */
171		if(hx<0) {
172		    if(((ix-0x3ff00000)|yisint)==0) {
173			z = (z-z)/(z-z); /* (-1)**non-int is NaN */
174		    } else if(yisint==1)
175			z = -z;		/* (x<0)**odd = -(|x|**odd) */
176		}
177		return z;
178	    }
179	}
180
181    /* (x<0)**(non-int) is NaN */
182	if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
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)? huge*huge:tiny*tiny;
192	    if(ix>0x3ff00000) return (hy>0)? huge*huge: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 = x-1;		/* 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 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 s = 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	    s = u*v;
221	    s_h = s;
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 = s*s;
230	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
231	    r += s_l*(s_h+s);
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 = s*(1+...) */
237	    u = s_h*t_h;
238	    v = s_l*t_h+t_l*s;
239	/* 2/(3log2)*(s+...) */
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) = (s+..)*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	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
253	if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
254	    s = -one;/* (-ve)**(odd int) */
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
308/*
309 * wrapper pow(x,y) return x**y
310 */
311#ifndef _IEEE_LIBM
312double pow(double x, double y)
313{
314	double z = __ieee754_pow(x, y);
315	if (_LIB_VERSION == _IEEE_|| isnan(y))
316		return z;
317	if (isnan(x)) {
318		if (y == 0.0)
319			return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */
320		return z;
321	}
322	if (x == 0.0) {
323		if (y == 0.0)
324	    		return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */
325		if (isfinite(y) && y < 0.0)
326			return __kernel_standard(x,y,23); /* pow(0.0,negative) */
327		return z;
328	}
329	if (!isfinite(z)) {
330		if (isfinite(x) && isfinite(y)) {
331			if (isnan(z))
332				return __kernel_standard(x, y, 24); /* pow neg**non-int */
333			return __kernel_standard(x, y, 21); /* pow overflow */
334		}
335	}
336	if (z == 0.0 && isfinite(x) && isfinite(y))
337		return __kernel_standard(x, y, 22); /* pow underflow */
338	return z;
339}
340#else
341strong_alias(__ieee754_pow, pow)
342#endif
343libm_hidden_def(pow)
344
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