#include "math_libm.h"
#include "math_private.h"

Include dependency graph for e_pow.c:

Functions
double attribute_hidden	__ieee754_pow (double x, double y)

Variables
static const double	bp [] = {1.0, 1.5,}
static const double	dp_h [] = { 0.0, 5.84962487220764160156e-01,}
static const double	dp_l [] = { 0.0, 1.35003920212974897128e-08,}
static const double	zero = 0.0
static const double	one = 1.0
static const double	two = 2.0
static const double	two53 = 9007199254740992.0
static const double	huge = 1.0e300
static const double	tiny = 1.0e-300
static const double	L1 = 5.99999999999994648725e-01
static const double	L2 = 4.28571428578550184252e-01
static const double	L3 = 3.33333329818377432918e-01
static const double	L4 = 2.72728123808534006489e-01
static const double	L5 = 2.30660745775561754067e-01
static const double	L6 = 2.06975017800338417784e-01
static const double	P1 = 1.66666666666666019037e-01
static const double	P2 = -2.77777777770155933842e-03
static const double	P3 = 6.61375632143793436117e-05
static const double	P4 = -1.65339022054652515390e-06
static const double	P5 = 4.13813679705723846039e-08
static const double	lg2 = 6.93147180559945286227e-01
static const double	lg2_h = 6.93147182464599609375e-01
static const double	lg2_l = -1.90465429995776804525e-09
static const double	ovt = 8.0085662595372944372e-0017
static const double	cp = 9.61796693925975554329e-01
static const double	cp_h = 9.61796700954437255859e-01
static const double	cp_l = -7.02846165095275826516e-09
static const double	ivln2 = 1.44269504088896338700e+00
static const double	ivln2_h = 1.44269502162933349609e+00
static const double	ivln2_l = 1.92596299112661746887e-08

Function Documentation

double attribute_hidden __ieee754_pow	(	double	x,
		double	y
	)

Definition at line 99 of file e_pow.c.

References __ieee754_sqrt, bp, cp, cp_h, cp_l, dp_h, dp_l, EXTRACT_WORDS, fabs(), GET_HIGH_WORD, huge, i, ivln2, ivln2_h, ivln2_l, j, k, L1, L2, L3, L4, L5, L6, lg2, lg2_h, lg2_l, one, ovt, P1, P2, P3, P4, P5, scalbn, SET_HIGH_WORD, SET_LOW_WORD, tiny, two, two53, and zero.

{
        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);
    /* x==1: 1**y = 1 (even if y is NaN) */
        if (hx==0x3ff00000 && lx==0) {
                return x;
        }
        ix = hx&0x7fffffff;
        EXTRACT_WORDS(hy,ly,y);
        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 one;         /* +-1**+-inf is 1 (yes, weird rule) */
                if (ix >= 0x3ff00000)   /* (|x|>1)**+-inf = inf,0 */
                    return (hy>=0) ? y : zero;
                /* (|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;
}