A researcher has identified a universal mathematical primitive that could simplify the way computers handle complex calculus and symbolic computation. Andrzej Odrzywołek, in a recent paper published on the arXiv repository, demonstrates that a single binary operator, defined as eml(x,y)=exp(x)-ln(y), can generate the entire standard repertoire of a scientific calculator.
Historically, computing elementary functions like sine, cosine, logarithms, and square roots required a diverse set of distinct operations. Odrzywołek’s discovery bridges this gap by showing that when paired with the constant 1, this specific operator acts as a foundational building block for all common transcendental and algebraic functions.
A new grammar for calculation
The discovery relies on what Odrzywołek calls the EML (Exp-Minus-Log) form. By utilizing this operator, every complex mathematical expression can be represented as a uniform binary tree. This creates a simplified grammar where the entire system follows a basic rule: S -> 1 | eml(S,S).
This uniform structure provides more than just a theoretical curiosity. Odrzywołek demonstrated that this architecture enables gradient-based symbolic regression. By treating these EML trees as trainable circuits, standard machine learning optimizers—such as Adam—can recover exact, closed-form mathematical expressions from numerical data.
In his tests, Odrzywołek successfully recovered exact formulas from data at shallow tree depths of up to four. While the architecture is flexible enough to fit arbitrary data, its primary utility lies in its ability to identify the precise, underlying mathematical laws when the data is rooted in elementary functions.
The researcher identified this operator through a systematic, exhaustive search of mathematical possibilities. The findings offer a hardware-analog for continuous mathematics, similar to how a single two-input gate functions as the primitive for all Boolean logic in digital computing.
Odrzywołek has made the code associated with his research available via Zenodo, allowing the scientific community to test the EML framework for symbolic regression tasks. The paper, which underwent revision in early April, suggests that this approach could streamline symbolic computation packages and improve the efficiency of automated mathematical modeling.
