eml_sr: Symbolic Regression via the EML Operator

Introduction

eml_sr is a Rust library for symbolic regression — searching for a closed-form mathematical formula that fits a numerical dataset. Its search space is built around the EML operator, a genuine mathematical discovery by Andrzej Odrzywołek (Jagiellonian University):

eml(x, y) = e^x - ln(y)

Odrzywołek proved that this single binary operator, combined with only the constant 1, is sufficient to reconstruct the standard repertoire of a scientific calculator — arithmetic, elementary transcendental functions, and constants like e, π, and i. The proof is published on arXiv: All elementary functions from a single binary operator.

eml_sr uses EML as one operator inside a practical Rust symbolic-regression engine, alongside conventional "fast-path" operators (Sin, Cos, Exp, Log, Sqrt, Square, Cube, and standard arithmetic). See Read This First: Status & Honesty Notes below for exactly what is implemented today versus what is still a research direction.

Read This First: Status & Honesty Notes

This section exists because earlier revisions of this README described capabilities more confidently than the current code supports. Please read it before deciding how to use this project.

What is implemented and working today:

• A parallel (Rayon) breadth-first / beam-search engine (src/engine/bfs.rs) that searches over increasingly complex expression trees, built from EML plus the standard operator set.
• Levenberg–Marquardt local optimization of numeric constants (src/engine/optimizer.rs) — a conventional technique, not something unique to EML.
• A Pareto front of formulas trading off accuracy vs. complexity, and a Python SmartSearcher wrapper (smart_search.py) that retries the search on transformed targets (y, y², ln(y), 1/y) to catch functions nested inside sqrt/log/division.
• Rust and Python (PyO3) APIs, published on crates.io and PyPI. This part has been exercised against real physics formulas (a subset of the Feynman symbolic-regression benchmark) and found exact or near-exact matches for several of them.

What is not implemented, despite being described in earlier versions of this document as the project's core idea:

• Continuous gradient optimization over a "master" EML tree (training a large homogeneous EML tree with an optimizer like Adam, then snapping weights to 0/1 to reveal a formula) — this is the method described in Odrzywołek's own paper (validated there only at shallow tree depth, ≤4), and it is not what the current Rust engine does. The current engine is a conventional discrete combinatorial search, comparable in spirit to other symbolic-regression tools (e.g. PySR, gplearn), with EML available as one operator among several.
• Compiling arbitrary formulas down to pure-EML instruction sequences, and any EML virtual machine / analog-circuit / VLSI compiler — these are described as potential applications of the underlying math in the sections below, not as features this codebase provides.
• In practice, across the physics formulas this project has been tested against, the discovered formulas almost never use the EML operator directly — the cheaper "fast-path" operators (Exp, Log, Sqrt, Square, Divide, ...) are preferred by the search's complexity-penalized scoring, because they are unary and thus structurally cheaper than reconstructing the same function through EML. EML currently pays off mainly for expressions with the exact shape eᴬ − ln(B), where A and B differ.

None of this means the underlying math is wrong — it's independently verified (see citation above). It means: what you get by installing eml_sr today is a working, conventional Rust symbolic-regression engine that happens to include EML as an operator, not yet a realization of the "continuous optimization over a universal operator" vision. If that gradient-based approach is what you're looking for, it does not exist here yet.

Why EML and standard operators?

EML alone can represent any elementary function, but doing so can require deeply nested trees (e.g. reconstructing sin(x) purely from eml compositions), and search cost grows combinatorially with tree depth. So by default eml_sr also registers cheap, purpose-built unary/binary operators (Exp, Log, Sqrt, Sin, Cos, Tan, ArcSin/Cos/Tan, Square, Cube, and standard arithmetic) so the search can reach common functions in a single node instead of many. EML stays in the operator set as a general fallback — useful in particular for exp(...) − ln(...)-shaped relationships that don't have a dedicated shortcut.

If you want to force a search that can only use EML (no shortcuts), the library supports a compile-time "Pure EML" build — see docs/CONTRIBUTING.md. Be aware this makes the search dramatically slower and deeper, as documented in docs/STATUS.md.

Scientific Foundation and Authors

Andrzej Odrzywołek, a theoretical physicist at the Institute of Theoretical Physics at the Jagiellonian University (Krakow, Poland), discovered the EML operator through a systematic exhaustive search and proved constructively that it — combined with the constant 1 — suffices to generate:

• Basic arithmetic operations (+, -, ×, /).
• All elementary functions (sin, cos, log, powers...).
• Fundamental constants of mathematics such as e, π, and the imaginary unit i.

Full reference: All elementary functions from a single binary operator, arXiv:2603.21852. A follow-up paper by Tomasz Stachowiak, Algebraic structure behind Odrzywołek's EML operator, arXiv:2604.23893, examines the group-theoretic structure behind it. This eml_sr project is an independent engineering effort that uses the EML operator; it is not authored by or officially affiliated with either paper's authors.

Potential Applications of EML (Conceptual — Not Implemented Here)

The math itself opens interesting doors, discussed below for context. eml_sr does not currently implement any of the following — they are included so readers understand why EML is interesting, not as a feature list.

• Breaking the AI "Black Box": training a neural network on an EML tree with a standard optimizer and snapping weights to exact values (0 or 1) could, in principle, produce closed-form formulas instead of opaque weights. This is the method demonstrated (at shallow depth) in Odrzywołek's paper — not something this Rust engine does.
• EML Compiler / Single-Instruction Stack Machine: because every elementary-function expression can in principle be rewritten as nested EML instructions, one could compile any formula into a "one-instruction" stack machine — useful for formal verification. No such compiler exists in this repository.
• VLSI / Analog Computing: a uniform binary-tree structure of identical EML units could, in principle, be mapped to repeated analog circuit elements instead of designing bespoke circuits per operation. Purely conceptual here.
• Minimal grammar for parsing/storage: the grammar S -> 1 | eml(S, S) is extremely simple, which could simplify storage/parsing of mathematical expressions. eml_sr's own internal representation does not use this minimal form — it uses a mixed operator tree, for the performance reasons explained above.

Quick Start

1. Installation

Python Users:

pip install eml_sr

Rust Users:

cargo add eml_sr

2. Basic Usage (Python)

from eml_sr import Searcher

# Your data
X = [[1.0], [2.0], [3.0]]
y = [2.5, 4.5, 6.5]  # f(x) = 2x + 0.5

# Search for the formula
searcher = Searcher()
result = searcher.fit(X, y)

print(f"Formula: {result.formula}")
# Output: Formula: (v_{0} * 2.0) + 0.5

3. Basic Usage (Rust)

use eml_sr::{Searcher, SearchConfig};

fn main() {
    let searcher = Searcher::new(SearchConfig::default());
    let xs = vec![1.0, 2.0, 3.0];
    let ys = vec![2.5, 4.5, 6.5];

    if let Ok(result) = searcher.find_function(&xs, &ys) {
        println!("Found formula: {}", result.formula);
    }
}

Project Status & Safety

For detailed information about current capabilities, supported platforms, measured (not aspirational) performance numbers, and safety warnings regarding memory usage (OOM), see docs/STATUS.md.

Development & Contributing

If you want to build from source, run benchmarks, or contribute to the core engine, see docs/CONTRIBUTING.md.

Note: eml_sr is an actively developed, working symbolic-regression engine. It is not yet a full realization of the continuous-optimization / gradient-training vision described above — see "Read This First" for the honest current state.