// sqrt using Newton's method, originally from w06 and w09 notes from class
// using frexp and linear approximation
//
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
double mysqrt( double a)
{
  if( a <= 0) return 0;

  double f; int e; f = frexp(a,&e); // a = f*pow(2,e), 0.5 <= f < 1.0
  printf( "f = %g, e = %i\n", f, e); // debug

 // if e is odd, multiply f by 2 and subtract 1 from e to make e even
 // then use Newton to get x = square-root of f, then answer is ldexp(x,e/2)
 //
  if( abs(e) % 2 == 1  ) { // or: (e/2)*2 != e,  or: e & 1 == 1
   f *= 2; --e;
   printf( "f = %g, e = %i\n", f, e); // debug
  }

 // linear approximation for initial estimate
 //
  double prev, x = 0.75 + (1.414-0.707)/(2.0-0.5)*(f-0.5); // 0.5 <= f < 2.0

  for( int count = 0; count < 50; ++count) 
  {
     prev = x; // save copy of x
     x = (x + f/x)/2; // update x
     printf( "x = %.18g\n", x); //debug
     if( x == prev) break; // converged
  }

  return ldexp(x,e/2);
}

int main( void)
{
  double a = 2; scanf( "%lf", &a);
  double x = mysqrt(a);
  double y = sqrt(a);
  printf( "a = %.18g\n", a);
  printf( "x = %.18g\n", x);
  printf( "y = %.18g\n", y);
  printf( "d = %.18g\n", x - y);
  return 0;
}