Single operator generating all elementary functions

Short description: Single operator generating all elementary functions

All elementary functions from a single operator

All elementary functions from a single operator is a mathematical result demonstrating that a single binary operation, together with a constant, can generate the full set of elementary functions typically used in scientific computation. The operator, defined as $\mathrm{e}\mathrm{m}\mathrm{l}(x,y)=e^x-\ln (y)$, is referred to as the EML (Exp–Minus–Log) operator.

The work shows that this operator, combined with the constant 1, is sufficient to reproduce arithmetic operations, transcendental functions, and mathematical constants such as $e$, $\pi$, and $i$.

Background

Elementary functions—including exponentials, logarithms, trigonometric functions, and algebraic operations—form the foundation of mathematics, science, and engineering. Traditionally, these functions are treated as distinct primitives.

In digital logic, it is well known that a single gate such as NAND can generate all Boolean operations. However, no analogous single primitive was previously known for continuous mathematics.

Earlier reductions, such as expressing multiplication via logarithms or trigonometric functions via Euler's formula, $e^{i\theta }=\cos \theta +i\sin \theta$, reduced complexity but did not eliminate the need for multiple operations.

Main result

The central result establishes that:

• The operator $\mathrm{e}\mathrm{m}\mathrm{l}(x,y)=e^x-\ln (y)$, together with the constant 1, forms a complete basis for elementary functions.
• All standard operations can be constructed from repeated applications of this operator.

Examples:

• Exponential: $e^x=\mathrm{e}\mathrm{m}\mathrm{l}(x,1)$
• Logarithm: $\ln x=\mathrm{e}\mathrm{m}\mathrm{l}(1,\mathrm{e}\mathrm{m}\mathrm{l}(\mathrm{e}\mathrm{m}\mathrm{l}(1,x),1))$

This implies that a "two-button calculator" (EML + constant 1) is theoretically sufficient to perform all standard scientific calculations.

Structure and representation

Expressions built using the EML operator form binary trees with a simple grammar: $S\to 1\mid \mathrm{e}\mathrm{m}\mathrm{l}(S,S)$.

This representation:

• Produces uniform expression structures
• Is equivalent to full binary trees and Catalan combinatorics
• Enables interpretation of formulas as circuits composed of identical elements

Methodology

The operator was discovered through:

• Systematic reduction of a standard set of 36 calculator primitives
• Iterative "ablation" testing to remove redundant operations
• Numerical and symbolic verification using symbolic regression techniques

A bootstrapping process progressively reconstructed all functions from simpler components until only the EML operator and constant remained.

Related operators

Variants of the EML operator were identified:

• $\mathrm{e}\mathrm{d}\mathrm{l}(x,y)=\frac{e^x}{\ln (y)}$ (requires constant $e$)
• $\ln (x)-e^y$ (with constant $-\mathrm{\infty }$)

These suggest a broader class of "continuous Sheffer operators."

Applications

Symbolic computation

EML provides a uniform representation of mathematical expressions, simplifying symbolic manipulation and transformation.

Compiler design

Expressions can be translated into EML form and executed using a single instruction, enabling:

• Stack-based evaluation
• Potential hardware implementations

Analog and circuit computing

EML expressions can be interpreted as circuits, analogous to digital logic built from NAND gates.

Symbolic regression

The uniform structure enables gradient-based optimization:

• EML trees can be trained to fit data
• Exact formulas can sometimes be recovered from numerical observations

Significance

The result demonstrates that:

• The apparent diversity of elementary functions is not fundamental
• A single operation can generate all standard mathematical expressions
• Continuous mathematics admits a structural simplicity analogous to Boolean logic

Limitations and open questions

• Computation often requires intermediate complex numbers
• A constant (such as 1) is still required
• It remains unknown whether:
  • A single operator without any constant exists
  • A unary (single-input) universal operator exists

See also

• Elementary function
• Symbolic regression
• Boolean algebra
• Euler's formula