// optimalizovana funkce 
double fce(double x,double y) {
  double c = sin(sqrt(x*x+y*y)) ;
  return (1-c*c)/(1+(x*x+y*y)/1000); 
} 

// derivace dle X 
double derivX(double x,double y) {
  double c = sin(sqrt(x*x+y*y)); 

  double derivSqrt = 0.5/sqrt(x*x+y*y); //derivace odmocniny 
  derivSqrt *= 2*x;   
  double derivc = cos(sqrt(x*x+y*y))*derivSqrt; // derivace C - slozena funkce (sin' -> cos) 

  double f = 1 - c*c; 
  double fderiv = - 2*c*derivc; // derivace f

  double g = (1+(x*x+y*y)/1000);
  double gderiv = 2*x/1000;  // derivace g

  return  (fderiv*g -f*gderiv)/(g*g);  // derivace podilu 
} 

double odhadX(double x,double y) {
  return (fce(x+0.000001,y)-fce(x,y))/0.000001;
} 

// derivace dle Y 
double derivY(double x,double y) {
  double c = sin(sqrt(x*x+y*y)); 

  double derivSqrt = 0.5/sqrt(x*x+y*y); //derivace odmocniny 
  derivSqrt *= 2*y;  
  double derivc = cos(sqrt(x*x+y*y))*derivSqrt; // derivace C - slozena funkce (sin' -> cos) 

  // fce = f/g 
  double f = 1 - c*c; 
  double fderiv = - 2*c*derivc; // derivace f 

  double g = (1+(x*x+y*y)/1000); 
  double gderiv = 2*y/1000;  // deriface g 

  return  (fderiv*g -f*gderiv)/(g*g); // derivace podilu 
} 

double odhadY(double x,double y) {
  return (fce(x,y+0.000001)-fce(x,y))/0.000001;
}