One Operator to Rule Them All: exp(x)−ln(y) Generates Every Calculator Function

In digital logic, a single universal gate (like NAND) can build any Boolean circuit. Continuous mathematics had no equivalent - until now. Researcher Andrzej Odrzywolek demonstrates that a single binary operator, eml(x, y) = exp(x) − ln(y), combined with the constant 1, can construct every standard elementary function: trig, logarithms, square roots, arithmetic, and complex constants like pi and i.

The construction is concrete. Exponentiation is just eml(x, 1). Natural log requires a few nested calls. Every expression reduces to a binary tree of identical eml nodes, yielding a grammar as minimal as S → 1 | eml(S, S). The operator was discovered through exhaustive search, not intuition - its existence was not predicted.

Beyond elegance, the result has a practical angle: EML trees can serve as trainable circuits for symbolic regression. Using Adam optimization on shallow trees (depth ≤ 4), the paper shows exact recovery of closed-form formulas from numerical data. The architecture can fit arbitrary data, but when the underlying law is elementary, it can reconstruct the precise symbolic expression.