All elementary functions from a single binary operator

Article Summary

This paper introduces a single binary operator, eml(x,y) = exp(x) - ln(y), which together with the constant 1 is sufficient to express all standard elementary mathematical functions. The operator was discovered through systematic search and proven to cover the full set of scientific calculator operations via compositions encoded as binary trees. A key application is gradient-based symbolic regression: EML trees act as trainable circuits that can recover closed-form functions from numerical data.

Discussion

• The top comment calls this “one of the most significant discoveries in years” and immediately imagines fitting multidimensional functions or wave functions (e.g., the Schrödinger equation) via gradient descent on EML trees.
• Several commenters draw analogies to other minimalist universal systems: the Iota combinator, NOR-completeness (Peirce’s arrow), FRACTRAN, and lambda calculus, questioning what distinguishes EML’s completeness over those.
• A technically critical comment argues the derivations for -x and x^-1 rely on ln(0) = Infinity and are only valid under extended arithmetic conventions, not strict real analysis.
• Community members quickly built interactive tools: one shared a Marimo/SymPy notebook to interactively derive and verify EML expressions from the paper’s diagram.
• Commenters used the paper as a live LLM benchmark: ChatGPT and Gemini succeeded in composing EML expressions on the first try, while Claude required extra prompting and Grok only produced depth estimates.

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