Single binary operator generates all elementary functions
Original: All elementary functions from a single binary operator
Why This Matters
First unified primitive for continuous mathematics; may advance symbolic computation and machine learning interpretability.
Researcher Andrzej Odrzywołek discovered that a single binary operator eml(x,y)=exp(x)-ln(y), combined with constant 1, can generate all elementary mathematical functions including trigonometric, logarithmic, and algebraic operations.
In a paper submitted to arXiv in March 2026, computer scientist Andrzej Odrzywołek demonstrates that all standard scientific calculator functions can be derived from a single binary operator called EML (Exp-Minus-Log). The operator is defined as eml(x,y)=exp(x)-ln(y). Combined only with the constant 1, this operator generates mathematical constants (e, pi, i) and all arithmetic operations (addition, subtraction, multiplication, division, exponentiation) as well as transcendental and algebraic functions. For example, exp(x) equals eml(x,1) and ln(x) can be expressed as eml(1,eml(eml(1,x),1)). The discovery was made through exhaustive computational search. In EML form, every mathematical expression becomes a binary tree of identical nodes following a simple grammar: S→1|eml(S,S). The uniform structure enables gradient-based symbolic regression using standard optimizers like Adam, allowing recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The research includes supplementary information and code made publicly available.