libm/e_hypot.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /* __ieee754_hypot(x,y)
  13  *
  14  * Method :
  15  *      If (assume round-to-nearest) z=x*x+y*y
  16  *      has error less than sqrt(2)/2 ulp, than
  17  *      sqrt(z) has error less than 1 ulp (exercise).
  18  *
  19  *      So, compute sqrt(x*x+y*y) with some care as
  20  *      follows to get the error below 1 ulp:
  21  *
  22  *      Assume x>y>0;
  23  *      (if possible, set rounding to round-to-nearest)
  24  *      1. if x > 2y  use
  25  *              x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
  26  *      where x1 = x with lower 32 bits cleared, x2 = x-x1; else
  27  *      2. if x <= 2y use
  28  *              t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
  29  *      where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
  30  *      y1= y with lower 32 bits chopped, y2 = y-y1.
  31  *
  32  *      NOTE: scaling may be necessary if some argument is too
  33  *            large or too tiny
  34  *
  35  * Special cases:
  36  *      hypot(x,y) is INF if x or y is +INF or -INF; else
  37  *      hypot(x,y) is NAN if x or y is NAN.
  38  *
  39  * Accuracy:
  40  *      hypot(x,y) returns sqrt(x^2+y^2) with error less
  41  *      than 1 ulps (units in the last place)
  42  */
  43
  44 #include "math.h"
  45 #include "math_private.h"
  46
  47 double __ieee754_hypot(double x, double y)
  48 {
  49         double a=x,b=y,t1,t2,_y1,y2,w;
  50         int32_t j,k,ha,hb;
  51
  52         GET_HIGH_WORD(ha,x);
  53         ha &= 0x7fffffff;
  54         GET_HIGH_WORD(hb,y);
  55         hb &= 0x7fffffff;
  56         if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
  57         SET_HIGH_WORD(a,ha);    /* a <- |a| */
  58         SET_HIGH_WORD(b,hb);    /* b <- |b| */
  59         if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */
  60         k=0;
  61         if(ha > 0x5f300000) {   /* a>2**500 */
  62            if(ha >= 0x7ff00000) {       /* Inf or NaN */
  63                u_int32_t low;
  64                w = a+b;                 /* for sNaN */
  65                GET_LOW_WORD(low,a);
  66                if(((ha&0xfffff)|low)==0) w = a;
  67                GET_LOW_WORD(low,b);
  68                if(((hb^0x7ff00000)|low)==0) w = b;
  69                return w;
  70            }
  71            /* scale a and b by 2**-600 */
  72            ha -= 0x25800000; hb -= 0x25800000;  k += 600;
  73            SET_HIGH_WORD(a,ha);
  74            SET_HIGH_WORD(b,hb);
  75         }
  76         if(hb < 0x20b00000) {   /* b < 2**-500 */
  77             if(hb <= 0x000fffff) {      /* subnormal b or 0 */
  78                 u_int32_t low;
  79                 GET_LOW_WORD(low,b);
  80                 if((hb|low)==0) return a;
  81                 t1=0;
  82                 SET_HIGH_WORD(t1,0x7fd00000);   /* t1=2^1022 */
  83                 b *= t1;
  84                 a *= t1;
  85                 k -= 1022;
  86             } else {            /* scale a and b by 2^600 */
  87                 ha += 0x25800000;       /* a *= 2^600 */
  88                 hb += 0x25800000;       /* b *= 2^600 */
  89                 k -= 600;
  90                 SET_HIGH_WORD(a,ha);
  91                 SET_HIGH_WORD(b,hb);
  92             }
  93         }
  94     /* medium size a and b */
  95         w = a-b;
  96         if (w>b) {
  97             t1 = 0;
  98             SET_HIGH_WORD(t1,ha);
  99             t2 = a-t1;
 100             w  = __ieee754_sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
 101         } else {
 102             a  = a+a;
 103             _y1 = 0;
 104             SET_HIGH_WORD(_y1,hb);
 105             y2 = b - _y1;
 106             t1 = 0;
 107             SET_HIGH_WORD(t1,ha+0x00100000);
 108             t2 = a - t1;
 109             w  = __ieee754_sqrt(t1*_y1-(w*(-w)-(t1*y2+t2*b)));
 110         }
 111         if(k!=0) {
 112             u_int32_t high;
 113             t1 = 1.0;
 114             GET_HIGH_WORD(high,t1);
 115             SET_HIGH_WORD(t1,high+(k<<20));
 116             return t1*w;
 117         } else return w;
 118 }