Approaching number "e" as a division of integers

#include <iostream>
#include <math.h>
#define e 2.71828
#define RELATIVEERROR 0.0005
#define LIMIT 10000

using namespace std;

int main()
{
    int index = -1;
    for(int i=1; i<LIMIT; i++){
        double product = i * e;
        double error = fabs(round(product) - product) / e;
        cout << i << " " << product << " " << error << endl;
        if(error <= RELATIVEERROR){
            index = i;
            break;
        }
    }
    // I have found the appropriate index, if this exists
    if(index == -1){
        cout << "I cannot find a good approach of e (relativeError <= 0.05%) with integers in the range of " << LIMIT << endl;
    }
    else{
        cout << "A good approach of e (relativeError <= 0.05%) with integers in range of " << LIMIT << " is: " << round(index * e) << " / " << index << " = " << round(index * e) / index << endl;
    }
    return 0;
}