Polynomial-time solution for finding intersection of two REs, which output the intersection in RE(can be approximate)

Question

There is a way to find intersection of REs :

First convert REs to DFAs, and then define a DFA represents intersection of REs. By converting the defined DFA to RE we can get the RE which represents intersection of REs.

However the algorithm is PSPACE-complete, I wonder if there exists a better algorithm. A approximate method is also accepted.