APM-EML: Adaptive Parallel Macros Extension of the Exponential-Minus-Log Operator for Efficient Symbolic Regression

Odrzywolek (2026) demonstrated that the single binary operator eml (x, y) = eˣ - ln (y), together with the constant 1, suffices to express all elementary functions as uniform binary trees. While mathematically elegant, this representation suffers from severe expression blow-up: basic arithmetic requires tree depths of 8 or more. We propose APM-EML (Adaptive Parallel Macros Extension of the EML operator), a layered extension that wraps verified EML subtrees as reusable macro nodes. This preserves the theoretical universality of EML while dramatically reducing effective search depth. We further extend APM-EML with four capabilities: learnable constants via gradient-free optimization, multi-variable support, adaptive macro discovery, and parallel genetic programming with multi-restart search for hard problems. With parallel GP on a 48-thread CPU server, per-problem operator specialization, and strict convergence tolerance (MSE < 10^-10), APM-EML achieves 17/17 mathematically exact recovery (all final MSE < 10^-29) on the combined Nguyen 1-12 and 5-equation AI Feynman benchmark in 27. 7 seconds of total wall-clock time, matching PySR's 100% accuracy at ~2. 6x lower wall time on the same 48-thread server and strongly outperforming gplearn (6/17, 35%). With a CuPy-backed GPU implementation and 6 concurrent workers on a single NVIDIA A100, APM-EML also converges on E = (1/2) m v² at the first restart across five data scales from 10² to 10⁶ points in 132 seconds of total wall-clock time — a 10, 000x increase in data size produces only a 2. 93x increase in wall time. APM-EML autonomously rediscovers the canonical EML identity xʸ = exp (y ln x) on Nguyen-11 and learns 4*pi as a single parameter in Coulomb's law.