# All elementary functions from a single binary operator

arXiv: https://arxiv.org/abs/2603.21852 · HTML: https://mlsanity.com/paper/2603.21852/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 56 references that are in the map.
- First posted on arXiv 23 Mar 2026.

## Builds on (most cited first, 9 of 9)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2109.14545/) (2021)
- [Teaching Arithmetic to Small Transformers](https://mlsanity.com/paper/2307.03381/) (2023)

## Built on by (most cited first, 5 of 5)

- [EML Trees Are Universal Approximators](https://mlsanity.com/paper/2606.23179/) (2026)
- [Auto-Relational Reasoning](https://mlsanity.com/paper/2604.26507/) (2026)
- [Additive Atomic Forests for Symbolic Function and Antiderivative Discovery](https://mlsanity.com/paper/2605.08130/) (2026)
- [EML-CD: Causal Mechanism Recovery via EML Symbolic Trees in Structure Learning](https://mlsanity.com/paper/2606.05942/) (2026)
- [Diversity of EML-type operators](https://mlsanity.com/paper/2609.11210/) (2026)

## Abstract (arXiv)

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, 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)=eml(x,1), ln(x)=eml(1,eml(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 | 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.