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/* @(#)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[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 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 __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 */

        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|<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(((((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;

}