libquadmath/math/hypotq.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 /* From e_hypotl.c -- long double version of e_hypot.c.
  13  * Conversion to long double by Jakub Jelinek, jakub@redhat.com.
  14  * Conversion to __float128 by FX Coudert, fxcoudert@gcc.gnu.org.
  15  */
  16
  17 /* hypotq(x,y)
  18  *
  19  * Method :
  20  *      If (assume round-to-nearest) z=x*x+y*y
  21  *      has error less than sqrtl(2)/2 ulp, than
  22  *      sqrtl(z) has error less than 1 ulp (exercise).
  23  *
  24  *      So, compute sqrtl(x*x+y*y) with some care as
  25  *      follows to get the error below 1 ulp:
  26  *
  27  *      Assume x>y>0;
  28  *      (if possible, set rounding to round-to-nearest)
  29  *      1. if x > 2y  use
  30  *              x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
  31  *      where x1 = x with lower 64 bits cleared, x2 = x-x1; else
  32  *      2. if x <= 2y use
  33  *              t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
  34  *      where t1 = 2x with lower 64 bits cleared, t2 = 2x-t1,
  35  *      y1= y with lower 64 bits chopped, y2 = y-y1.
  36  *
  37  *      NOTE: scaling may be necessary if some argument is too
  38  *            large or too tiny
  39  *
  40  * Special cases:
  41  *      hypotq(x,y) is INF if x or y is +INF or -INF; else
  42  *      hypotq(x,y) is NAN if x or y is NAN.
  43  *
  44  * Accuracy:
  45  *      hypotq(x,y) returns sqrtl(x^2+y^2) with error less
  46  *      than 1 ulps (units in the last place)
  47  */
  48
  49 #include "quadmath-imp.h"
  50
  51 __float128
  52 hypotq (__float128 x, __float128 y)
  53 {
  54   __float128 a, b, t1, t2, y1, y2, w;
  55   int64_t j, k, ha, hb;
  56
  57   GET_FLT128_MSW64(ha,x);
  58   ha &= 0x7fffffffffffffffLL;
  59   GET_FLT128_MSW64(hb,y);
  60   hb &= 0x7fffffffffffffffLL;
  61   if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
  62   SET_FLT128_MSW64(a,ha);       /* a <- |a| */
  63   SET_FLT128_MSW64(b,hb);       /* b <- |b| */
  64   if((ha-hb)>0x78000000000000LL) {return a+b;} /* x/y > 2**120 */
  65   k=0;
  66   if(ha > 0x5f3f000000000000LL) {       /* a>2**8000 */
  67      if(ha >= 0x7fff000000000000LL) {   /* Inf or NaN */
  68          uint64_t low;
  69          w = a+b;                       /* for sNaN */
  70          GET_FLT128_LSW64(low,a);
  71          if(((ha&0xffffffffffffLL)|low)==0) w = a;
  72          GET_FLT128_LSW64(low,b);
  73          if(((hb^0x7fff000000000000LL)|low)==0) w = b;
  74          return w;
  75      }
  76      /* scale a and b by 2**-9600 */
  77      ha -= 0x2580000000000000LL;
  78      hb -= 0x2580000000000000LL;        k += 9600;
  79      SET_FLT128_MSW64(a,ha);
  80      SET_FLT128_MSW64(b,hb);
  81   }
  82   if(hb < 0x20bf000000000000LL) {       /* b < 2**-8000 */
  83       if(hb <= 0x0000ffffffffffffLL) {  /* subnormal b or 0 */
  84           uint64_t low;
  85         GET_FLT128_LSW64(low,b);
  86         if((hb|low)==0) return a;
  87         t1=0;
  88         SET_FLT128_MSW64(t1,0x7ffd000000000000LL); /* t1=2^16382 */
  89         b *= t1;
  90         a *= t1;
  91         k -= 16382;
  92         GET_FLT128_MSW64 (ha, a);
  93         GET_FLT128_MSW64 (hb, b);
  94         if (hb > ha)
  95           {
  96             t1 = a;
  97             a = b;
  98             b = t1;
  99             j = ha;
 100             ha = hb;
 101             hb = j;
 102           }
 103       } else {          /* scale a and b by 2^9600 */
 104           ha += 0x2580000000000000LL;   /* a *= 2^9600 */
 105         hb += 0x2580000000000000LL;     /* b *= 2^9600 */
 106         k -= 9600;
 107         SET_FLT128_MSW64(a,ha);
 108         SET_FLT128_MSW64(b,hb);
 109       }
 110   }
 111     /* medium size a and b */
 112   w = a-b;
 113   if (w>b) {
 114       t1 = 0;
 115       SET_FLT128_MSW64(t1,ha);
 116       t2 = a-t1;
 117       w  = sqrtq(t1*t1-(b*(-b)-t2*(a+t1)));
 118   } else {
 119       a  = a+a;
 120       y1 = 0;
 121       SET_FLT128_MSW64(y1,hb);
 122       y2 = b - y1;
 123       t1 = 0;
 124       SET_FLT128_MSW64(t1,ha+0x0001000000000000LL);
 125       t2 = a - t1;
 126       w  = sqrtq(t1*y1-(w*(-w)-(t1*y2+t2*b)));
 127   }
 128   if(k!=0) {
 129       uint64_t high;
 130       t1 = 1.0Q;
 131       GET_FLT128_MSW64(high,t1);
 132       SET_FLT128_MSW64(t1,high+(k<<48));
 133       w *= t1;
 134       math_check_force_underflow_nonneg (w);
 135       return w;
 136   } else return w;
 137 }