Smoothed complexity of algorithm can explain the practical performance of algorithm more efficiently. Condition number of matrix is a main root to result in large error in solution during the running of Gaussian algorithm. Sankar, et al. performed a smoothed analysis of condition number of symmetric matrix under zero-preserving perturbations. However, the smoothed complexity presented by Sankar, et al. was higher and more complicated. To solve this problem, two key inequalities are presented. The inequalities are used to improving the smoothed complexity of condition number of symmetric matrix. The smoothed analysis of bits of precision needed by using Gaussian algorithm is performed and lower smoothed complexity is presented.