Single Binary Operator Generates All Elementary Functions
Original: All elementary functions from a single binary operator
Why This Matters
Provides fundamental mathematical primitive that could simplify computational frameworks
Researcher Andrzej Odrzywołek discovered that eml(x,y)=exp(x)-ln(y) with constant 1 can generate all standard scientific calculator functions including sin, cos, sqrt, log through systematic search and constructive proof.
A breakthrough mathematical discovery shows that all elementary functions can be generated from a single binary operator eml(x,y)=exp(x)-ln(y) plus the constant 1. This includes constants like e, pi, i, and all arithmetic operations (addition, subtraction, multiplication, division, exponentiation) as well as transcendental and algebraic functions. For example, exp(x)=eml(x,1) and ln(x)=eml(1,eml(eml(1,x),1)). The researcher found this through exhaustive search and proved it constructively. Every expression becomes a binary tree with identical nodes, creating the simple grammar S -> 1 | eml(S,S). The uniform structure enables gradient-based symbolic regression using EML trees as trainable circuits with Adam optimizer, demonstrating exact recovery of closed-form functions from numerical data at tree depths up to 4.
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