The EML Operator: A Single Primitive for Continuous Mathematics

In the realm of digital logic, a single two-input gate like NAND can suffice for all Boolean operations. However, continuous mathematics has traditionally required multiple operations to compute elementary functions such as sin, cos, and log. I introduce a novel binary operator, EML (Exp-Minus-Log), defined as eml(x, y) = exp(x) - ln(y), which, alongside the constant 1, can generate the full suite of functions found in a scientific calculator. This includes constants like e, π, and i, arithmetic operations, and transcendental functions. For instance, ex can be expressed as eml(x, 1), and ln x as eml(1, eml(eml(1, x), 1)). This discovery was made through a systematic exhaustive search, demonstrating that EML can constructively suffice for the scientific calculator basis.

The concept of a single reusable primitive is not new, with examples like the NAND gate in logic and the operational amplifier in engineering. These primitives offer significant value for understanding and pedagogy, despite debates about their conceptual necessity. In mathematics, elementary functions are foundational, yet they have never been reduced to a single operator until now. The EML operator allows for the expression of any standard real elementary function through repeated applications, using the constant 1 to neutralize the logarithmic term. This approach simplifies the structure to a binary tree of identical nodes, enabling gradient-based symbolic regression and the potential for exact recovery of closed-form functions from numerical data.

The search for such a primitive operator involved constructing increasingly primitive calculators, ultimately leading to the EML operator. This operator, surprisingly simple, can express all necessary functions with just two buttons: the binary operator EML and the constant 1. This reduction is as far as possible, as at least one binary operator and one terminal symbol are required.

The implications of the EML operator extend beyond theoretical interest. It offers a new perspective on the structure of elementary functions and their computation, with potential applications in symbolic regression and data fitting. The discovery of EML as a Sheffer-type element in continuous mathematics is unexpected and opens new avenues for exploration in mathematical computation and education.