text: /* @(#)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.
   * ====================================================
   */
  
  /* 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 multi-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 except 1) ** 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 1
   *	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 <float.h>
  #include <math.h>
  
  #include "math_private.h"
  
  static const double
  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*/
  
  double
  pow(double x, double y)
  {
  	double z,ax,z_h,z_l,p_h,p_l;
  	double yy1,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;
  
      /* x==1: 1**y = 1, even if y is NaN */
  	if (hx==0x3ff00000 && lx == 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((u_int32_t)(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  one;	/* (-1)**+-inf is 1 */
  	        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;
  	    }
  	}
  
  	n = (hx>>31)+1;
  
      /* (x<0)**(non-int) is NaN */
  	if((n|yisint)==0) return (x-x)/(x-x);
  
  	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  	if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
  
      /* |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)? s*huge*huge:s*tiny*tiny;
  	    if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*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 = ax-one;		/* 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 ss,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 ss = 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]);
  	    ss = u*v;
  	    s_h = ss;
  	    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 = ss*ss;
  	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
  	    r += s_l*(s_h+ss);
  	    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 = ss*(1+...) */
  	    u = s_h*t_h;
  	    v = s_l*t_h+t_l*ss;
  	/* 2/(3log2)*(ss+...) */
  	    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) = (ss+..)*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);
  	}
  
      /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
  	yy1  = y;
  	SET_LOW_WORD(yy1,0);
  	p_l = (y-yy1)*t1+y*t2;
  	p_h = yy1*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;
  }
  
  #if	LDBL_MANT_DIG == DBL_MANT_DIG
  __strong_alias(powl, pow);
  #endif	/* LDBL_MANT_DIG == DBL_MANT_DIG */
  