//  Lecture 2. Example 1. "Numerical derivative"
// Complile: g++ -o numderivative numderivative.cc -lm 

#include <iostream>
using namespace std;
#include <stdio.h>
#include <math.h> 

#define PRECISION float  // Set precision level   


PRECISION f (PRECISION  x ) {

	return exp(x); // function of interest
}  

PRECISION exact_derivative (PRECISION  x ) {

        return exp(x); // its exact analytical derivative
}



PRECISION num_derivative (PRECISION x, PRECISION h ) {

	return (f(x + h) - f(x))/h; // its numerical derivative  

} 

main() { 



	PRECISION x = 1.0; 
	PRECISION h=0.0;

	PRECISION ErrPrev = 100.0;  
	PRECISION Error = 100.0; 

// Sanity check. Comment out.
//	cout << f(x) << endl;


// Output results:

cout << endl; cout << endl;

cout << "--------------------Example--------------------------" << endl; 
cout << "Evaluate (approximate) numerical derivative of a specified function using df/dx ~ [f(x+h) - f(x)]/h. Compare with an exact result.  Make your own conclusions. " << endl; 

cout << endl;

cout << " step size h" << "   Exact value of df/dx   "  <<  "   Approximate df/dx    " <<  "  Relative error (%)  " << endl;

	for (int i=0; i >= -20; i-- ) {

	if (ErrPrev > Error) ErrPrev = Error;

		h = pow(10.0, (PRECISION)(i) ); 

		Error = 100.0*fabs( (exact_derivative(x) -  num_derivative(x, h))/exact_derivative(x) ); 

//		cout << "          " <<  exact_derivative(x) << "        " << h << "           " << num_derivative(x, h) << "             " << 100.0*fabs(exact_derivative(x) -  num_derivative(x, h))/exact_derivative(x) << endl; 

		printf("%5.3e %s %10.7f %s %10.7f %s %7.5f \n", h, "          ", exact_derivative(x), "              ", num_derivative(x, h), "              ", Error ) ;  


	} 
	
	if (ErrPrev > Error ) { 
		cout << endl;
		cout << "It works! (seems like....)"  << endl;   
		cout << endl;
	}	
	
	else {   
		cout << endl; 
		cout << " CONCLUSION: The difference between theory and pratice is larger in practice than in theory "  << endl; 
		cout << endl; 
	} 
}