We study the dynamics of cold molecule formation via photo- or magneto-association of quantum degenerate atomic gases for the case when the field configuration is defined by the quasi-linear level crossing Demkov-Kunike model, which is characterized by a bell-shaped pulse and finite variation of the frequency detuning. We generalize the approach developed for the Landau-Zener model and propose a cubic polynomial equation for the asymptotic transition probability at t → + ∞ applicable in the large detuning regime of the interaction process. The proposed approximation applies to all values of the Rabi frequency; hence, it presents a unified description of previously noticed weak, moderate, and strong interaction limits of the fast sweep regime. This cubic equation for the transition probability provides improvement of previous results by an order of the magnitude.© (2010) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.