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Ideally, bit-width analysis methods should be able to find the most appropriate bit-widths to achieve the optimum bit-width-to-error tradeoff for variables and constants in high level DSP algorithms when they are implemented into hardware. The tradeoff enables the fixed-point hardware implementation to be area efficient but still within the allowed error tolerance. Unfortunately, almost all the existing static bit-width analysis methods are Interval Arithmetic (IA) based that may overestimate bit-widths and enable fairly pessimistic bit-width-to-error tradeoff. We have developed an automated and efficient bit-width optimization methodology that is Affine Arithmetic (AA) based. Experiments have proven that, compared to the previous static analysis methods, our methodology not only dramatically reduces the fractional bit-width by more than 35% but also slightly reduces the integer bit-width. In addition, our probabilistic error analysis method further enlarges the bit-width-to-error tradeoff.