FPGA-Based Implementation of an Error-Controllable and Resource-Efficient Approximation Method for Transcendental Functions

Transcendental functions cannot be expressed algebraically, which brings a big challenge to efficient and accurate approximation. Lookup table (LUT) and piecewise fitting are common traditional methods. However, they either trade approximate accuracy for computation and storage or require unaffordable resources when the expected accuracy is high. In this paper, we have developed a high-precision approximation method, which is error-controllable and resource-efficient. The method originally divides a transcendental function into two parts based on their slope. The steep slope part is approximated by the method of LUT with the interpolation, while the gentle slope part is approximated by the range-addressable lookup table (RALUT) algorithm. The boundary of two parts can be adjusted adaptively according to the expected accuracy. Moreover, we analyzed the error source of our method in detail, and proposed an optimal selection method for table resolution and data bit-width. The proposed algorithm is verified on an actual FPGA board and the results show that the proposed method's error can achieve arbitrarily low. Compared to other methods, the proposed algorithm has a more stable increase in resource consumption as the required accuracy grows, which consumes fewer hardware resources especially at the middle accuracy, with at least 30% LUT slices saved.

Automated summary

This paper proposes a high-precision approximation method for transcendental functions. By dividing the function into two parts based on their slopes, the steep slope part is approximated using an interpolated lookup table, while the gentle slope part is approximated using a range-addressable lookup table algorithm. The boundary between the two parts can be adjusted adaptively according to the desired accuracy. The method achieves arbitrarily low error and consumes fewer hardware resources compared to other methods.