Optimizing Mitchell's Method for Approximate Logarithmic Addition via Base Selection with Application to Back-Propagation

Mitchell's method typically is used to compute approximate base-2 antilogarithms using minimal hardware (no ROM). Another use of identical hardware approximates the function (called the “addition logarithm”) that computes the sum of values represented by the Logarithmic Number System (LNS). Since Mitchell's method was originally conceived for base-2 antilogarithms, the previous usage of Mitchell's method for LNS addition also considered only base-2. For applications where approximate arithmetic is viable, there is little need for compatibility with standard base-2 arithmetic, such as IEEE-754. Therefore, the LNS base, $b$ , is a design parameter that can be optimized without increasing the hardware cost. Previous optimization of the LNS base (trading range and precision) was implemented with ROM (ideal accuracy), but no previous work has optimized the base for a ROM-less (lower accuracy) Mitchell LNS. Assuming a uniform distribution, we derive $b\approx 2.09$ as optimal for Mitchell LNS addition; however, in actual applications, like machine learning, searching for the application-specific base yields a low-cost Mitchell implementation of back-propagation that converges as successfully as a more expensive IEEE-754 implementation.