This paper presents a new technique for optimizing formal analysis of Boolean formulas and Linear Temporal Logic (LTL) formulas, namely the formula simplification tables. A formula simplification table is a mathematical table that shows all possible simplifications of the formula under different truth assignments of its variables. The simplification table is constructed using a three-valued logic: besides true and false, the variable can take an unknown value. The advantages of constructing a simplification table of a formula are two-fold. First, it can be used to compute the logical influence weight of each variable in the formula, which is a quantitative score that shows the importance of the variable in affecting the outcome of the formula. Second, it can be used to identify variables that have the highest logical influences on the outcome of the formula. We demonstrate the effectiveness of formula simplification table in the context of software verification by developing an efficient framework for the well-known decentralized monitoring problem.