All elementary functions from a single binary operator

The discovery of the EML operator, sufficient to evaluate the standard explicit purely transcendental elementary functions, has led to considerable interest and discussion across multiple scientific disciplines. However, most authors have focused on the binary EML itself, while numerous similar variants with slightly different properties are now known. This article attempts to close this gap by enumerating and classifying them. We also take this opportunity to clarify common misconceptions related to the EML operator. The principal goal, symbolic regression within an architecture as close as possible to proven neural networks which combine matrix multiplication with a single univariate non-linear activation function, remains beyond reach. Instead, we propose a Möbius layer, with rational functions replacing matrix operations, and showcase the recently discovered activation function eml(x,1/x), which allows exp(x) and ln(x) to be recovered separately, and hence all elementary functions to be evaluated within a rational generalization of the neural network.

We present a framework for the simultaneous symbolic recovery of a function and its antiderivative from data. The framework rests on three ideas. First, a derivative algebra: the observation that the product rule $\frac{d}{dx}[f \cdot g] = f'g + fg'$ and the chain rule, applied to a seed set of elementary functions, generate a self-expanding system of function-derivative pairs -- a living library that grows each time a new function is discovered. Second, two complementary primitives -- EML$\,(e^u - \ln v)$, which is theoretically complete for all elementary functions, and SOL$\,(\sin u - \cos v)$, introduced here, which makes trigonometric atoms available at depth1 instead of depth$\sim$8 -- that seed the library with core atoms cheaply. Third, additive atomic forests: finite sums of primitive trees, optionally composed via multiplicative nodes, whose derivatives are fitted to data by continuous optimisation or by exhaustive search over the library. Because differentiation of each atom is determined by construction, the forest simultaneously encodes a symbolic expression $F$ and its derivative $F'$; no symbolic integration step is required. The library is not a fixed object: it self-constructs from a small seed set by recursive application of the product rule, chain rule, and the two primitives, and it can grow as newly discovered functions are folded back in. The larger the library, the richer the expressible class of candidate functions. We give conditional completeness, additive-depth, and analytic simultaneous-recovery results for the framework. Empirically, in our reported runs on 17 classification benchmarks, sparse atom combinations match or exceed XGBoost on 13 datasets while producing interpretable formulas.

The recently introduced EML (Exp-Minus-Log) function acts as continuous analogue of NAND gates, providing a compositional building block capable of representing elementary functions. In this work, we study the expressive power of tree-structured compositions of EML functions. We show that such trees enjoy a universal approximation property for functions in $W^{k, \infty}$ for $k \in \mathbb N$, drawing on classical neural network approximation arguments while exploiting the ability to explicitly construct EML trees that mimic polynomial representations. We further propose a learning algorithm for EML-type trees equipped with fitting parameters, and demonstrate its feasibility in practical optimization problems. Our results establish EML trees as a theoretically grounded framework for function approximation.