eml / symbolic regression / elementary functions

A new paper shows a single binary operator eml(x,y)=exp(x)-ln(y), together with the constant 1, can generate the full suite of elementary functions (e, pi, i, +, -, *, /, ^, sin, cos, sqrt, log, etc.). The author found the operator by exhaustive search and gives constructive encodings (for example exp(x)=eml(x,1) and ln(x)=eml(1,eml(eml(1,x),1))). Representing expressions as uniform binary trees (grammar S -> 1 | eml(S,S)) yields a simple circuit template that the paper leverages for gradient-based symbolic regression: training EML trees with Adam can recover exact closed-form elementary functions from numerical data at shallow depths. Code and supplementary materials are provided. This unifies continuous mathematical primitives into a single operator and suggests new compact formalisms for symbolic computation and model discovery.

A researcher shows that a single binary operator eml(x,y)=exp(x)-ln(y), plus the constant 1, can generate the full set of elementary functions used by scientific calculators. Andrzej Odrzywołek provides explicit constructions for constants (e, π, i), arithmetic (add, subtract, multiply, divide), exponentiation, roots, logs, trig, and other transcendental functions using only eml and 1, and proves sufficiency by constructive exhaustive search. The paper also introduces EML-form expressions as uniform binary trees (grammar S -> 1 | eml(S,S)), and demonstrates gradient-based symbolic regression using trainable EML trees with Adam to exactly recover closed-form elementary functions from numerical data at shallow depths. Code and supplementary materials are provided.

Researchers claim a single binary operator can generate all elementary functions (sin, cos, sqrt, log, etc.), challenging the historical need for multiple primitives in continuous mathematics. The Hacker News discussion links the arXiv paper and raises practical and theoretical questions — comparisons with hypergeometric representations, implications for hardware math coprocessors, and whether an optimized EML (elementary-math-logic) implementation could replace traditional math units. Commenters note connections to functional-programming primitives (Y combinator), esoteric languages, and how basic arithmetic (addition, negation) could be derived once exponent and log are available. The result matters for theoretical foundations of computation and could inspire new software libraries or specialized hardware for numeric computation.

A mathematician shows that a single binary operator eml(x,y)=exp(x)-ln(y), plus the constant 1, can generate all conventional elementary functions used in scientific calculators. Andrzej Odrzywołek proves constructively that eml encodes constants (e, pi, i), arithmetic (add, multiply, divide, exponentiation) and transcendental/algebraic functions via binary trees where every node is the same operator, yielding the grammar S -> 1 | eml(S,S). He found eml through exhaustive search and demonstrates practical use: treating EML trees as trainable circuits optimized with Adam, he recovers exact closed-form elementary functions from numerical data at shallow depths and fits arbitrary data. Code and supplementary materials are provided. This offers a minimal, uniform representation for symbolic computation and gradient-based symbolic regression.