Generating All Elementary Functions from a Single Binary Operator

EML is a single binary operator defined as eml(x, y) = exp(x) − ln(y).

It is presented that with just this operator and the constant 1, one can express all arithmetic operations (+, −, ×, /, exponentiation), transcendental functions (sin, cos, log, √, etc.), and even complex constants (e, π, i).

For example, basic functions can be reduced to EML combinations, such as exp(x) = eml(x, 1), and every EML expression takes the form of a binary tree consisting of identical nodes.

This uniform structure fits well with symbolic regression. The paper shows that using the EML tree like a trainable circuit, closed-form elementary functions can be recovered through gradient-based optimization such as Adam.

The key points are as follows.

• A single universal operator: Just as NAND serves as the foundation of Boolean logic, EML aims to serve as a universal operator for continuous mathematics.
• Scientific-calculator-level expressiveness: Standard calculator functionality is reconstructed with just one binary operator and one constant.
• Structural simplicity: Every expression is represented as a tree of the same form, which is advantageous for both search and learning.
• Experimental results: It was shown that accurate formula recovery is possible at shallow tree depths (up to 4).

The paper concludes that these properties could open up new possibilities for formula discovery and symbolic AI.