The standard integral equation approach is used to extract the bridge function and other correlation functions of simple fluids. To do this, we firstly use the hybridized mean spherical approximation (HMSA), which involves a renormalization of the indirect correlation function. Secondly, we propose a new division scheme for the pair potential founded on the merging process of a test particle into the system. Thirdly, we take advantage of the thermodynamic consistent condition put forward in the HMSA to optimize the depth of the long-range part of the pair potential inside the core, making the procedure autonomous and state dependent. The method is tested on the pure Lennard-Jones fluid in supercritical conditions. Very good agreement is obtained for structural quantities and thermodynamic properties as compared to the Monte Carlo (MC) results for the bridge function and to the molecular dynamics (MD) results for the thermodynamic properties. (C) 2001 American Institute of Physics.