/* @(#)e_pow.c 5.1 93/09/24 *//* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice  * is preserved. * ==================================================== *//* __ieee754_pow(x,y) return x**y * *		      n * Method:  Let x =  2   * (1+f) *	1. Compute and return log2(x) in two pieces: *		log2(x) = w1 + w2, *	   where w1 has 53-24 = 29 bit trailing zeros. *	2. Perform y*log2(x) = n+y' by simulating muti-precision  *	   arithmetic, where |y'|<=0.5. *	3. Return x**y = 2**n*exp(y'*log2) * * Special cases: *	1.  (anything) ** 0  is 1 *	2.  (anything) ** 1  is itself *	3.  (anything) ** NAN is NAN *	4.  NAN ** (anything except 0) is NAN *	5.  +-(|x| > 1) **  +INF is +INF *	6.  +-(|x| > 1) **  -INF is +0 *	7.  +-(|x| < 1) **  +INF is +0 *	8.  +-(|x| < 1) **  -INF is +INF *	9.  +-1         ** +-INF is NAN *	10. +0 ** (+anything except 0, NAN)               is +0 *	11. -0 ** (+anything except 0, NAN, odd integer)  is +0 *	12. +0 ** (-anything except 0, NAN)               is +INF *	13. -0 ** (-anything except 0, NAN, odd integer)  is +INF *	14. -0 ** (odd integer) = -( +0 ** (odd integer) ) *	15. +INF ** (+anything except 0,NAN) is +INF *	16. +INF ** (-anything except 0,NAN) is +0 *	17. -INF ** (anything)  = -0 ** (-anything) *	18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) *	19. (-anything except 0 and inf) ** (non-integer) is NAN * * Accuracy: *	pow(x,y) returns x**y nearly rounded. In particular *			pow(integer,integer) *	always returns the correct integer provided it is  *	representable. * * Constants : * The hexadecimal values are the intended ones for the following  * constants. The decimal values may be used, provided that the  * compiler will convert from decimal to binary accurately enough  * to produce the hexadecimal values shown. */#include "fdlibm.h"#ifndef _DOUBLE_IS_32BITS#ifdef __STDC__static const double #elsestatic double #endifbp[] = {1.0, 1.5,},dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */zero    =  0.0,one	=  1.0,two	=  2.0,two53	=  9007199254740992.0,	/* 0x43400000, 0x00000000 */huge	=  1.0e300,tiny    =  1.0e-300,	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/#ifdef __STDC__	double __ieee754_pow(double x, double y)#else	double __ieee754_pow(x,y)	double x, y;#endif{	double z,ax,z_h,z_l,p_h,p_l;	double y1,t1,t2,r,s,t,u,v,w;	__int32_t i,j,k,yisint,n;	__int32_t hx,hy,ix,iy;	__uint32_t lx,ly;	EXTRACT_WORDS(hx,lx,x);	EXTRACT_WORDS(hy,ly,y);	ix = hx&0x7fffffff;  iy = hy&0x7fffffff;    /* y==zero: x**0 = 1 */	if((iy|ly)==0) return one; 	    /* +-NaN return x+y */	if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||	   iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 		return x+y;	    /* determine if y is an odd int when x < 0     * yisint = 0	... y is not an integer     * yisint = 1	... y is an odd int     * yisint = 2	... y is an even int     */	yisint  = 0;	if(hx<0) {		    if(iy>=0x43400000) yisint = 2; /* even integer y */	    else if(iy>=0x3ff00000) {		k = (iy>>20)-0x3ff;	   /* exponent */		if(k>20) {		    j = ly>>(52-k);		    if((j<<(52-k))==ly) yisint = 2-(j&1);		} else if(ly==0) {		    j = iy>>(20-k);		    if((j<<(20-k))==iy) yisint = 2-(j&1);		}	    }			}     /* special value of y */	if(ly==0) { 		    if (iy==0x7ff00000) {	/* y is +-inf */	        if(((ix-0x3ff00000)|lx)==0)		    return  y - y;	/* inf**+-1 is NaN */	        else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */		    return (hy>=0)? y: zero;	        else			/* (|x|<1)**-,+inf = inf,0 */		    return (hy<0)?-y: zero;	    } 	    if(iy==0x3ff00000) {	/* y is  +-1 */		if(hy<0) return one/x; else return x;	    }	    if(hy==0x40000000) return x*x; /* y is  2 */	    if(hy==0x3fe00000) {	/* y is  0.5 */		if(hx>=0)	/* x >= +0 */		return __ieee754_sqrt(x);		    }	}	ax   = fabs(x);    /* special value of x */	if(lx==0) {	    if(ix==0x7ff00000||ix==0||ix==0x3ff00000){		z = ax;			/*x is +-0,+-inf,+-1*/		if(hy<0) z = one/z;	/* z = (1/|x|) */		if(hx<0) {		    if(((ix-0x3ff00000)|yisint)==0) {			z = (z-z)/(z-z); /* (-1)**non-int is NaN */		    } else if(yisint==1) 			z = -z;		/* (x<0)**odd = -(|x|**odd) */		}		return z;	    }	}        /* (x<0)**(non-int) is NaN */    /* CYGNUS LOCAL: This used to be	if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);       but ANSI C says a right shift of a signed negative quantity is       implementation defined.  */	if(((((__uint32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);    /* |y| is huge */	if(iy>0x41e00000) { /* if |y| > 2**31 */	    if(iy>0x43f00000){	/* if |y| > 2**64, must o/uflow */		if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;		if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;	    }	/* over/underflow if x is not close to one */	    if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;	    if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;	/* now |1-x| is tiny <= 2**-20, suffice to compute 	   log(x) by x-x^2/2+x^3/3-x^4/4 */	    t = x-1;		/* t has 20 trailing zeros */	    w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));	    u = ivln2_h*t;	/* ivln2_h has 21 sig. bits */	    v = t*ivln2_l-w*ivln2;	    t1 = u+v;	    SET_LOW_WORD(t1,0);	    t2 = v-(t1-u);	} else {	    double s2,s_h,s_l,t_h,t_l;	    n = 0;	/* take care subnormal number */	    if(ix<0x00100000)		{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }	    n  += ((ix)>>20)-0x3ff;	    j  = ix&0x000fffff;	/* determine interval */	    ix = j|0x3ff00000;		/* normalize ix */	    if(j<=0x3988E) k=0;		/* |x|<sqrt(3/2) */	    else if(j<0xBB67A) k=1;	/* |x|<sqrt(3)   */	    else {k=0;n+=1;ix -= 0x00100000;}	    SET_HIGH_WORD(ax,ix);	/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */	    u = ax-bp[k];		/* bp[0]=1.0, bp[1]=1.5 */	    v = one/(ax+bp[k]);	    s = u*v;	    s_h = s;	    SET_LOW_WORD(s_h,0);	/* t_h=ax+bp[k] High */	    t_h = zero;	    SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));	    t_l = ax - (t_h-bp[k]);	    s_l = v*((u-s_h*t_h)-s_h*t_l);	/* compute log(ax) */	    s2 = s*s;	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));	    r += s_l*(s_h+s);	    s2  = s_h*s_h;	    t_h = 3.0+s2+r;	    SET_LOW_WORD(t_h,0);	    t_l = r-((t_h-3.0)-s2);	/* u+v = s*(1+...) */	    u = s_h*t_h;	    v = s_l*t_h+t_l*s;	/* 2/(3log2)*(s+...) */	    p_h = u+v;	    SET_LOW_WORD(p_h,0);	    p_l = v-(p_h-u);	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */	    z_l = cp_l*p_h+p_l*cp+dp_l[k];	/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */	    t = (double)n;	    t1 = (((z_h+z_l)+dp_h[k])+t);	    SET_LOW_WORD(t1,0);	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);	}	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */	if(((((__uint32_t)hx>>31)-1)|(yisint-1))==0)	    s = -one;/* (-ve)**(odd int) */    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */	y1  = y;	SET_LOW_WORD(y1,0);	p_l = (y-y1)*t1+y*t2;	p_h = y1*t1;	z = p_l+p_h;	EXTRACT_WORDS(j,i,z);	if (j>=0x40900000) {				/* z >= 1024 */	    if(((j-0x40900000)|i)!=0)			/* if z > 1024 */		return s*huge*huge;			/* overflow */	    else {		if(p_l+ovt>z-p_h) return s*huge*huge;	/* overflow */	    }	} else if((j&0x7fffffff)>=0x4090cc00 ) {	/* z <= -1075 */	    if(((j-0xc090cc00)|i)!=0) 		/* z < -1075 */		return s*tiny*tiny;		/* underflow */	    else {		if(p_l<=z-p_h) return s*tiny*tiny;	/* underflow */	    }	}    /*     * compute 2**(p_h+p_l)     */	i = j&0x7fffffff;	k = (i>>20)-0x3ff;	n = 0;	if(i>0x3fe00000) {		/* if |z| > 0.5, set n = [z+0.5] */	    n = j+(0x00100000>>(k+1));	    k = ((n&0x7fffffff)>>20)-0x3ff;	/* new k for n */	    t = zero;	    SET_HIGH_WORD(t,n&~(0x000fffff>>k));	    n = ((n&0x000fffff)|0x00100000)>>(20-k);	    if(j<0) n = -n;	    p_h -= t;	} 	t = p_l+p_h;	SET_LOW_WORD(t,0);	u = t*lg2_h;	v = (p_l-(t-p_h))*lg2+t*lg2_l;	z = u+v;	w = v-(z-u);	t  = z*z;	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));	r  = (z*t1)/(t1-two)-(w+z*w);	z  = one-(r-z);	GET_HIGH_WORD(j,z);	j += (n<<20);	if((j>>20)<=0) z = scalbn(z,(int)n);	/* subnormal output */	else SET_HIGH_WORD(z,j);	return s*z;}#endif /* defined(_DOUBLE_IS_32BITS) */