Piecewise Parabolic Approximate Computation Based on an Error-Flattened Segmenter and a Novel Quantizer

Abstract

This paper proposes a novel Piecewise Parabolic Approximate Computation method for hardware function evaluation, which mainly incorporates an error-flattened segmenter and an implementation quantizer. Under a required software maximum absolute error (MAE), the segmenter adaptively selects a minimum number of parabolas to approximate the objective function. By completely imitating the circuit’s behavior before actual implementation, the quantizer calculates the minimum quantization bit width to ensure a non-redundant fixed-point hardware architecture with an MAE of 1 unit of least precision (ulp), eliminating the iterative design time for the circuits. The method causes the number of segments to reach the theoretical limit, and has great advantages in the number of segments and the size of the look-up table (LUT). To prove the superiority of the proposed method, six common functions were implemented by the proposed method under TSMC-90 nm technology. Compared to the state-of-the-art piecewise quadratic approximation methods, the proposed method has advantages in the area with roughly the same delay. Furthermore, a unified function-evaluation unit was also implemented under TSMC-90 nm technology.

QPA: A Quantization-Aware Piecewise Polynomial Approximation Methodology for Hardware-Efficient Implementations

“…These features can be used to calculate the truncation errors in hardware design. There are also methods like [17] and [19] that did not use quantization-aware fitting but considered the truncation error after the end of the segmentation algorithm. It makes these methods must define the target MAE twice, which is overconstrained because the extra quantization step will not only break the error-flattened characteristics but also will take more segments to achieve the same 𝑟𝑀𝐴𝐸 compared with using a quantization-aware fitting algorithm.…”

“…6(a) shows the detailed polynomial calculation circuit used by [17], and Fig. 6(b) is the detailed polynomial calculation circuit used by [19]. In Fig.…”

“…𝑠𝑝 is an array containing 𝑠𝑔 Note that the segmentation algorithm in this flow implies that the longer the fitting interval is, the worse the 𝑟𝑀𝐴𝐸 is, which is not always true when the coefficients and the intermediate results are truncated. However, considering the trade-off of segmentation performance and calculation speed, this dichotomy segmentation algorithm is still good enough and has been widely used by many existing state-of-the-art errorflattened piecewise approximation methods, like [17]- [19].…”

“…In addition to these PWL methods, there are also PWQ methods using parabolas to approximate the target function. Based on the PLAC method, An et al [19] proposed a PWQ method to adapt to higher accuracy fitting scenarios, but the fitting algorithm they use is still not the best. To minimize the maximum absolute error (MAE) of fitting, the best algorithm is the Remez algorithm which can find the theoretical min-max fitting polynomial of the target function [20], [21].…”

“…Another shortcoming of methods proposed in [16]- [19] is that although they considered the quantization effect, they truncate the polynomial coefficients by direct rounding. Tawfik et al [23] used a mixed-integer linear programming (MILP) method and got a significant increase in accuracy over direct rounding.…”