WhyExp-Minus-Log (eml): Towards an Explanation Based on Optimization and Symmetry

It has been recently shown that, similarly to the fact that every Boolean operation can be performed if we only have gates of a single type—NAND or NOR—many elementary functions can be obtained by using a constant 1 and a single function eml(x,y)=exp(x)−ln(y). This function is known as exp-minus-log, eml, for short. Just like the use of NAND gates led to computation speed-up, this new discovery has the potential for a similar speed-up. There are several other functions with the same property, but among them, eml is the fastest to compute. In this paper, we provide mathematical arguments that, in some reasonable sense, eml is probably optimal. In these arguments, we use a symmetry-based approach and its relation to optimization.