The first-order reliability method (FORM) is efficient but may not be accurate for nonlinear limit-state functions. The second-order reliability method (SORM) is more accurate but less efficient. To maintain both high accuracy and efficiency, we propose a new second-order reliability analysis method with first-order efficiency. The method first performs the FORM to identify the most probable point (MPP). Then, the associated limit-state function is decomposed into additive univariate functions at the MPP. Each univariate function is further approximated by a quadratic function. The cumulant generating function of the approximated limit-state function is then available so that saddlepoint approximation can be easily applied in computing the probability of failure. The accuracy of the new method is comparable to that of the SORM, and its efficiency is in the same order of magnitude as the FORM.