To effectively analyze the influence of two-wave with diffuse power (TWDP) fading on the achievable error rate performance of binary phase-shift keying (BPSK) signaling, we derive two novel concise asymptotic closed-form bit error rate (BER) formulas. We perform asymptotic analysese based on existing exact and approximate BER formulas, which are obtained from the exact probability density function (PDF) or moment generating function (MGF), and the approximate PDF of TWDP fading. The derived asymptotic closed-form expressions yield explicit insights into the achievable error rate performance in TWDP fading environments. Furthermore, the absolute relative error (ARE) between the exact and approximate coding gains is investigated, from which we also propose a criterion for the order of an approximate PDF, which is more robust than the conventional criterion. Numerical results clearly demonstrate the accuracy of the derived asymptotic formulas, and also support our proposed criterion.