We propose improvements in numerical evaluation of symmetric stable density and its partial derivatives with respect to the parameters. They are useful for more reliable evaluation of maximum likelihood estimator and its standard error. Numerical values of the Fisher information matrix of symmetric stable distributions are also given. Our improvements consist of modification of the method of Nolan (1997) for the boundary cases, i.e., in the tail and mode of the densities and in the neighborhood of the Cauchy and the normal distributions.