# BWOLF: Bit-Width Optimization for Statistical Divergence with Logarithmic Functions

### Qian Xu, Guowei Sun and Gang Qu

#### University of Maryland, College Park, USA

## Abstract

Approximate computing is a promising technique in improving the energy efficiency for error-resilient applications such as multimedia, signal processing and neural network. A large amount of reported work is on the design of approximate computation units with truncated data under error constraints. However, they mainly focus on simple arithmetic operations, addition and multiplication to be more specific. In this paper, we study how to apply the truncation method to the floating-point logarithmic operation which is getting increasingly popular. We analyze the tradeoff between the precision of computation and the energy it requires and derive a formula on the most energy efficient implementation of the logarithm unit for a given error variance range. Based on this theoretical result, we propose BWOLF (Bit-Width Optimization for Logarithmic Function), which uses a sequential quadratic programming algorithm to determine the way to truncate data (i.e., bit-width optimization) in a program with logarithm and other arithmetic operations such that the energy consumption is minimized under a fixed error budget. We evaluate the efficacy of BWOLF in energy saving on two widely used applications: Kullback-Leibler Divergence and Bayesian Neural Network. The experimental results validate the correctness of our analysis and show significant amount of energy saving over both the full-precision computation and the uniform truncation method. The energy savings range from 27.18% to 95.92% for different error constraints.

[The authors opted for not publicly sharing a presentation video.]