All elementary functions from a single operator

Abstract

All elementary functions from a single operator

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, $\mathrm{eml}(x,y)=\exp (x)-\ln (y)$, together with the constant $1$, generates the standard repertoire of a scientific calculator. This includes constants such as $e$, $\pi$, and $i$; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, $\exp (x)=\mathrm{eml}(x,1),\ln (x)=\mathrm{eml}(1,\mathrm{eml}(\mathrm{eml}(1,x),1))$, and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as $S->1|\mathrm{eml}(S,S)$. This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to $4$. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

Authors

• Andrzej Odrzywołek