e_hypot.c

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/* @(#)e_hypot.c 1.3 95/01/18 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/* __ieee754_hypot(x,y)
 *
 * Method :                  
 *  If (assume round-to-nearest) z=x*x+y*y 
 *  has error less than sqrt(2)/2 ulp, than 
 *  sqrt(z) has error less than 1 ulp (exercise).
 *
 *  So, compute sqrt(x*x+y*y) with some care as 
 *  follows to get the error below 1 ulp:
 *
 *  Assume x>y>0;
 *  (if possible, set rounding to round-to-nearest)
 *  1. if x > 2y  use
 *      x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
 *  where x1 = x with lower 32 bits cleared, x2 = x-x1; else
 *  2. if x <= 2y use
 *      t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
 *  where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1, 
 *  y1= y with lower 32 bits chopped, y2 = y-y1.
 *      
 *  NOTE: scaling may be necessary if some argument is too 
 *        large or too tiny
 *
 * Special cases:
 *  hypot(x,y) is INF if x or y is +INF or -INF; else
 *  hypot(x,y) is NAN if x or y is NAN.
 *
 * Accuracy:
 *  hypot(x,y) returns sqrt(x^2+y^2) with error less 
 *  than 1 ulps (units in the last place) 
 */

#include "fdlibm.h"

#ifdef __STDC__
    double __ieee754_hypot(double x, double y)
#else
    double __ieee754_hypot(x,y)
    double x, y;
#endif
{
    double a=x,b=y,t1,t2,y1,y2,w;
    int j,k,ha,hb;

    ha = __HI(x)&0x7fffffff;    /* high word of  x */
    hb = __HI(y)&0x7fffffff;    /* high word of  y */
    if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
    __HI(a) = ha;   /* a <- |a| */
    __HI(b) = hb;   /* b <- |b| */
    if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */
    k=0;
    if(ha > 0x5f300000) {   /* a>2**500 */
       if(ha >= 0x7ff00000) {   /* Inf or NaN */
           w = a+b;         /* for sNaN */
           if(((ha&0xfffff)|__LO(a))==0) w = a;
           if(((hb^0x7ff00000)|__LO(b))==0) w = b;
           return w;
       }
       /* scale a and b by 2**-600 */
       ha -= 0x25800000; hb -= 0x25800000;  k += 600;
       __HI(a) = ha;
       __HI(b) = hb;
    }
    if(hb < 0x20b00000) {   /* b < 2**-500 */
        if(hb <= 0x000fffff) {  /* subnormal b or 0 */  
        if((hb|(__LO(b)))==0) return a;
        t1=0;
        __HI(t1) = 0x7fd00000;  /* t1=2^1022 */
        b *= t1;
        a *= t1;
        k -= 1022;
        } else {        /* scale a and b by 2^600 */
            ha += 0x25800000;   /* a *= 2^600 */
        hb += 0x25800000;   /* b *= 2^600 */
        k -= 600;
        __HI(a) = ha;
        __HI(b) = hb;
        }
    }
    /* medium size a and b */
    w = a-b;
    if (w>b) {
        t1 = 0;
        __HI(t1) = ha;
        t2 = a-t1;
        w  = sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
    } else {
        a  = a+a;
        y1 = 0;
        __HI(y1) = hb;
        y2 = b - y1;
        t1 = 0;
        __HI(t1) = ha+0x00100000;
        t2 = a - t1;
        w  = sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b)));
    }
    if(k!=0) {
        t1 = 1.0;
        __HI(t1) += (k<<20);
        return t1*w;
    } else return w;
}