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{{Short description|Single operator generating all elementary functions}}

== All elementary functions from a single operator ==

'''All elementary functions from a single operator''' <ref>Odrzywołek, Andrzej. All elementary functions from a single operator. https://arxiv.org/abs/2603.21852 </ref> 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 
<math>\mathrm{eml}(x,y) = e^x - \ln(y)</math>, 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 <math>e</math>, <math>\pi</math>, and <math>i</math>.

== 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,
<math>e^{i\theta} = \cos\theta + i\sin\theta</math>,
reduced complexity but did not eliminate the need for multiple operations.

== Main result ==

The central result establishes that:
* The operator <math>\mathrm{eml}(x,y) = e^x - \ln(y)</math>, 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: <math>e^x = \mathrm{eml}(x,1)</math>
* Logarithm: <math>\ln x = \mathrm{eml}\bigl(1,\mathrm{eml}(\mathrm{eml}(1,x),1)\bigr)</math>

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:
<math>S \to 1 \mid \mathrm{eml}(S,S)</math>.

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:
* <math>\mathrm{edl}(x,y) = \frac{e^x}{\ln(y)}</math> (requires constant <math>e</math>)
* <math>\ln(x) - e^y</math> (with constant <math>-\infty</math>)

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]]

== References ==

{{Sourceattribution|Single operator generating all elementary functions|1}}