LUT-ALMs: Trading Off Accuracy and Power for Approximate Logarithmic Multipliers via LUT Optimization

Logarithmic multiplier (LM) converts fixed-point (FxP) input operands to logarithmic numbers and performs multiplication with simple shift and addition operations, which achieves distinct power reduction, yet with significant single-sided errors. This paper proposes to fuse error compensation with logarithmic conversion by using customized look-up tables (LUTs). To avoid the use of large LUTs, partition strategies are designed for the optimization of LUTs. In addition, to effectively balance the accuracy and hardware costs, two iterative algorithms are proposed for generating precision-configurable LUTs. Based on the optimized LUTs, high-accuracy and low-power approximate LMs (LUT-ALMs) are constructed for 8-bit and 16-bit multiplications. Furthermore, to enhance the flexibility of data type and bit width, a mixed-mode LM (MM-ALM) supporting eight multiplication modes is devised. Compared with the exact 8-bit signed multiplier from Synopsys DesignWare (DW) library (DW-Exact), LUT-ALMs show up to 30.96% and 23.99% reductions in the power-delay product (PDP) and area-delay product (ADP), respectively, with a mean relative error distance (MRED) of 3.45%. Compared with state-of-the-art approximate 8- bit signed multipliers, LUT-ALMs form the Pareto front in terms of PDP and MRED. For 16-bit multiplication, LUT-ALMs obtain up to 70.90% and 62.13% savings in PDP and ADP with a MRED of 2.75%, compared with the corresponding DW-Exact. Compared with the corresponding mixed-mode exact multiplier constructed of DW multipliers, MM-ALM performing 8-bit multiplication achieves up to 57.44% and 47.99% reductions in PDP and ADP, respectively. When performing 4-bit multiplications, MM-ALM can save up to 37.16% savings in PDP. With lower hardware over-heads, LUT-ALMs and MM-ALM present comparable accuracy to the corresponding exact designs in the considered convolutional neural networks (CNNs) and image processing applications. The hardware description of the devised LMs is open-sourced athttps://anonymous.4open.science/r/LUT-ALM-5FD1.