Can the product of logistic ($x_1$) and logistic ($x_2$) be approximated by logistic ($x_1+x_2$)?

Question

Consider two logistic functions that are multiplied $$ f(x_1,x_2) = \frac{e^{x_1}}{1+e^{x_1}} \times\frac{e^{x_2}}{1+e^{x_2}}=\frac{e^{x_1+x_2}}{1+e^{x_1+x_2}+e^{x_1}+e^{x_2}} $$

To what extent (or under which non-trivial conditions) can this be approximated by $$ f(x_1,x_2) \approx \frac{e^{x_1+x_2}}{1+e^{x_1+x_2 }} $$ ?

Answer 1

If $x_1 = z$ and $x_2 = -z$, then it most definitely is not a logistic-shaped curve (it would be bell-shaped). So in that case, it cannot be approximated by such a function.