EML Symbolic Regression Prototype

"All elementary functions from a single operator" Odrzywołek (2026) — arXiv:2603.21852

A single binary operator eml(x, y) = exp(x) − ln(y), together with the constant 1, generates every standard elementary function via the grammar

S → 1 | eml(S, S)

This prototype implements gradient-based symbolic regression over EML trees (the "master formula"), reproducing the exact recovery experiments from §4.3.

Quick start

python -m venv .venv
.venv/bin/pip install torch numpy
.venv/bin/python demo.py

Structure

Path	Purpose
eml/core.py	Scalar operator + closed-form constructions
eml/formula.py	EMLMasterFormula — parameterised binary tree
eml/train.py	Adam + entropy hardening + weight snapping
demo.py	Verification, gradient SR, manual wiring

Visualization companion

Start the companion server, then open http://localhost:8765/ in a browser:

.venv/bin/python -m eml.viz_server   # terminal A
.venv/bin/python demo.py             # terminal B — trees appear live

The browser shows an SVG composition tree for every recovered formula. Use the expression + x input in the top bar to explore interactively:

Expression	x	f(x)	What it is
eml(x, 1)	0	1	exp(0) = 1
eml(x, 1)	1	2.718	exp(1) = e
eml(1, eml(eml(1, x), 1))	1	0	ln(1) = 0 exactly
eml(1, eml(eml(1, x), 1))	2.718	1	ln(e) = 1 exactly
eml(1, eml(eml(1, x), 1))	7.389	2	ln(e²) = 2 exactly
eml(eml(1, eml(eml(1,x),1)), 1)	5	5	exp(ln(x)) = x identity
eml(eml(1, eml(eml(1,x),1)), 1)	42	42	works for any x

Shape labels in the stats bar:

• chain — depth = leaves; pure linear composition (e.g. ln(x): depth 4, leaves 4)
• branching — some depth, more leaves; mixed
• wide — leaves ≥ 2 × depth; balanced, short and broad

Key results

• exp(x) recovered exactly (depth 1, 4 params) in every random restart.
• ln(x) recovered exactly (depth 3, 34 params) in 9/10 perturbed restarts; post-snap MSE ≈ 1.5 × 10⁻³² (machine epsilon squared for complex128).
• Manual wiring of eml(1, eml(eml(1,x), 1)) confirms exact symbolic identity.