Elementary function

An elementary function is one that can be built from a small set of basic building blocks using finitely many compositions and arithmetic operations.

The standard definition

Elementary functions are closed under:

• Addition, subtraction, multiplication, division
• Composition
• The inverse operation (where it exists)

Starting from:

• Constants (including rationals and transcendentals like π, e)
• Polynomial functions
• The exponential function exp
• The logarithm ln
• Trigonometric functions and their inverses

Why this class matters

Almost every function you see in introductory calculus, physics, and engineering is elementary. Closed-form integrals like ∫ sin(x)·exp(x) dx land in this class. Non-elementary functions (like the error function erf or the gamma function Γ) are the exception.

Why they're the target for EML

Because so much of applied mathematics stays inside this class, showing that a single operator spans it all is a meaningful compression. It's not just a mathematical curiosity, it has direct implications for compilers, circuit design, and symbolic regression.