Abstract
Abstract 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.
Highlights
As a method of representing real numbers, floating point can support a wide dynamic range and high precision of values
A novel affine approximation method, Approximation Affine based on Space Extreme Estimation (AASEE), is proposed to reduce the uncertainty of multiplication and achieve an accurate and efficient range analysis of multiplication in this paper
It can be seen that the area, which is calculated by AASEE, is less than that by as that of trivial range estimation (AATRE) and AACHA, and the area difference between them is increasing with the target precision
Summary
As a method of representing real numbers, floating point can support a wide dynamic range and high precision of values. AA cannot provide an exact affine form for nonlinear operations To solve this problem, Stolfi and de Figueiredo [22] proposed affine approximation methods for multiplication, which include trivial range estimation (AATRE) and Chebyshev approximation (AACHA). The accumulation of the uncertainty of all signals in the computational chain may result in an error explosion, which is unacceptable in application Such overestimation obviously cannot satisfy the accuracy requirement of the system, which limits the application of AATRE in large systems. Since LTI operations are accurately covered by AA, the proposed method is applied in the field of the range analysis of word-length optimization in this paper. A novel affine approximation method, Approximation Affine based on Space Extreme Estimation (AASEE), is proposed to reduce the uncertainty of multiplication and achieve an accurate and efficient range analysis of multiplication in this paper.
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