src/math/e_hypot.c

/* @(#)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.
 * ====================================================
 */

/* 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 <math.h>
#include "math_private.h"

double
hypot(double x, double y)
{
        double a=x,b=y,t1,t2,y1,y2,w;
        int32_t j,k,ha,hb;

        GET_HIGH_WORD(ha,x);
        ha &= 0x7fffffff;
        GET_HIGH_WORD(hb,y);
        hb &= 0x7fffffff;
        if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
        SET_HIGH_WORD(a,ha);    /* a <- |a| */
        SET_HIGH_WORD(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 */
               uint32_t low;
               w = a+b;                 /* for sNaN */
               GET_LOW_WORD(low,a);
               if(((ha&0xfffff)|low)==0) w = a;
               GET_LOW_WORD(low,b);
               if(((hb^0x7ff00000)|low)==0) w = b;
               return w;
           }
           /* scale a and b by 2**-600 */
           ha -= 0x25800000; hb -= 0x25800000;  k += 600;
           SET_HIGH_WORD(a,ha);
           SET_HIGH_WORD(b,hb);
        }
        if(hb < 0x20b00000) {   /* b < 2**-500 */
            if(hb <= 0x000fffff) {      /* subnormal b or 0 */
                uint32_t low;
                GET_LOW_WORD(low,b);
                if((hb|low)==0) return a;
                t1=0;
                SET_HIGH_WORD(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;
                SET_HIGH_WORD(a,ha);
                SET_HIGH_WORD(b,hb);
            }
        }
    /* medium size a and b */
        w = a-b;
        if (w>b) {
            t1 = 0;
            SET_HIGH_WORD(t1,ha);
            t2 = a-t1;
            w  = sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
        } else {
            a  = a+a;
            y1 = 0;
            SET_HIGH_WORD(y1,hb);
            y2 = b - y1;
            t1 = 0;
            SET_HIGH_WORD(t1,ha+0x00100000);
            t2 = a - t1;
            w  = sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b)));
        }
        if(k!=0) {
            uint32_t high;
            t1 = 1.0;
            GET_HIGH_WORD(high,t1);
            SET_HIGH_WORD(t1,high+(k<<20));
            return t1*w;
        } else return w;
}