The problem of reliability analysis of dynamical systems is considered. Failure event in the problem is formulated as the exceedance of a performance variable over a prescribed threshold level. Saddlepoint approximation technique provides a choice to estimate the Cumulative Distribution Function (CDF) of the performance variable. The failure probability is obtained as the value of the complement of the CDF at a specified threshold. The method requires finding the saddlepoint from a simple algebraic equation that depends on the Cumulant Generating Function (CGF) of the performance variable. Two different methods, which respectively use Taylor series expansion and statistical averaging, are investigated for estimating the CGF of the performance variable based on its random samples. A ten-storey shear building model subjected to white noise excitation is used to show the preference of a combination of these two methods in terms of accuracy and efficiency.