A refined affine approximation method of multiplication for range analysis in word-length optimization

Affine arithmetic (AA) is widely used in range analysis in word-length optimization of hardware designs. To reduce the uncertainty in the AA and achieve efficient and accurate range analysis of multiplication, this paper presents a novel refined affine approximation method, Approximation Affine based on Space Extreme Estimation (AASEE). The affine form of multiplication is divided into two parts. The first part is the approximate affine form of the operation. In the second part, the equivalent affine form of the estimated range of the difference, which is introduced by the approximation, is represented by an extra noise symbol. In AASEE, it is proven that the proposed approximate affine form is the closest to the result of multiplication based on linear geometry. The proposed equivalent affine form of AASEE is more accurate since the extreme value theory of multivariable functions is used to minimize the difference between the result of multiplication and the approximate affine form. The computational complexity of AASEE is the same as that of trivial range estimation (AATRE) and lower than that of Chebyshev approximation (AACHA). The proposed affine form of multiplication is demonstrated with polynomial approximation, B-splines, and multivariate polynomial functions. In experiments, the average of the ranges derived by AASEE is 59% and 89% of that by AATRE and AACHA, respectively. The integer bits derived by AASEE are 2 and 1 b less than that by AATRE and AACHA at most, respectively.