Using only the functions $\max$ and $\min$ and arithmetic operations (no if clauses), express the amount of possible overlap between two intervals $[a_1,a_2]$ and $[b_1,b_2]$, where $a_1,a_2,b_1,b_2$ are arbitrary real numbers with $a_1<a_2$ and $b_1<b_2$.

Think about intervals on the real number line. Sketch $[a_1,a_2]$ and $[b_1,b_2]$ when these overlap?

What is the left boundary of the overlap in terms of min/max of the 4 numbers?

Assume that intervals overlap, partially or fully, like in the figure above. We can notice that the smallest value in this intersection is always $\max(a_1, b_1),$ and the largest value in this intersection is $\min(a_2,b_2).$ Hence, the amount of overlap of these intervals can be expressed as $\min(a_2,b_2)-\max(a_1, b_1).$
There will be some overlap only if this value is positive, hence the final formula is $\max(0,\min(a_2,b_2)-\max(a_1, b_1)).$