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/* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */

/* @(#)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.

 * ====================================================

 */

#if defined(LIBM_SCCS) && !defined(lint)

static char rcsid[] = "$Id: e_pow.c,v 1.6 1994/09/13 00:40:33 jtc Exp $";

#endif

/* __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 "math.h"

#include "math_private.h"

#ifdef __STDC__

static const double

#else

static double

#endif

bp[] = {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;

	u_int32_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 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 */

	if(((((u_int32_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|

	    else if(j<0xBB67A) k=1;	/* |x|

	    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(((((u_int32_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,n);	/* subnormal output */

	else SET_HIGH_WORD(z,j);

	return s*z;

}