C++: comparison precision

Question

Who can explain this output? One less zero, and it's "false"

#include <iostream>
using namespace std;

int main() {
float a = 2.0000001;
int b = 2;
if(a == b){
cout << "true!";
} else {
cout << "false";
}
return 0;
}

Answer 1

Is this true independently from the compiler?

Answer 2

Also, can you tell me if this code precision can be improved?
https://code.sololearn.com/chgZD5KdzV9m/?ref=app

Answer 3

I implemented (well... pasted and copied), setting up also cout.precision():

https://code.sololearn.com/co8uU4xnKKks/?ref=app

Anyway, results are not different by my factorial version....