e_pow.c

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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.
 * ====================================================
 */

/* 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) ** 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"

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;

    /* +-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;
        }
    }

    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;
}