Symbolic regression

Symbolic regression is the problem of finding a closed-form mathematical expression that fits a set of observed data points.

Contrast with regular regression

Traditional regression fits coefficients in a fixed model: "what a, b minimize |y − (a·x + b)|²?"

Symbolic regression fits the form itself: "what function f best fits these (x, y) pairs?"

How EML helps

The paper proposes using EML trees as the search space. Every candidate formula is a binary tree of eml calls, a simple combinatorial object that you can:

1. Enumerate (shallow trees first)
2. Mutate (swap subtrees, introduce variables)
3. Optimize numerically (gradient descent on the constants inside each leaf via the master formula)

Results from the paper

• Depth-2 trees: 100% blind recovery on common test targets.
• Depth-3 to depth-4 trees: ~25% recovery.
• Successful runs converge to near machine-epsilon error (≈ 10⁻³²), meaning the formula is rediscovered exactly, not just approximately.

Why a fixed basis helps

Most symbolic-regression engines (like PySR or AI Feynman) use a handful of primitives per tree node, which explodes the search space. EML has one primitive, so the branching factor at each node is much smaller, making the search tractable even though trees must be deeper.