Why is the formula for relative approximation error with respect to the current approximation?

I have to approximate the root of the equation $x^3 -4x - 9 = 0$ by using the Bisection method from my numerical methods class. The question also states that I have to iterate the process until my relative approximation error gets below a specified error ($\epsilon_s$) of 0.1%. I have no problem of solving the question, but what I wonder is the step of finding the relative approximate error.

The formula for calculating relative approximate error is: $$\epsilon_a = \frac{current\,approximate-previous\,approximation}{current\,approximation} $$

I wonder why relative approximate error isn't calculated by dividing by the previous approximation instead? I think it is a way to see how improved our current approximation is compared to the previous one.

Let's say you have some quantity $x\ne 0$ and an approximation $\tilde x$. The relative error is given by $$ |\varepsilon_x| = \frac{|x-\tilde x|}{|x|}. $$

Normally, we do not have access to $x$, but we are able to get an upper bound for $|x -\tilde x|$. For this reason, one usually computes $\varepsilon_{\tilde x}$ instead. This is justified by the fact that

$$ |\varepsilon_{\tilde x}| = |\varepsilon_x|-|\varepsilon_x|^2+|\varepsilon_x|^3-\cdots $$