All Elementary Functions From a Single Binary Operator

Why It Matters

A universal continuous‑math operator streamlines symbolic computation and opens the door to hardware implementations that mirror Boolean logic’s simplicity, potentially accelerating scientific‑calculator functions and automated formula discovery.

Key Takeaways

• EML operator uses exp and log to express any elementary function
• All calculator operations derive from eml(x,y) with constant 1
• Binary tree grammar S→1|eml(S,S) yields uniform expression structure
• Gradient‑based symbolic regression can recover exact formulas using shallow EML trees
• Potential hardware simplification mirrors Boolean gate universality for continuous math

Pulse Analysis

The discovery of a single binary operator that can reproduce every elementary function marks a paradigm shift in computational mathematics. While digital hardware relies on a universal two‑input gate for Boolean logic, continuous mathematics has long required a mosaic of distinct operations. Odrzywolek’s eml(x,y)=exp(x)−ln(y) collapses this diversity into one primitive, allowing constants such as e, π, and i, as well as addition, multiplication, and exponentiation, to be expressed through repeated applications of the same node. This uniformity simplifies the underlying grammar to a single production rule, S→1|eml(S,S), turning complex formulas into binary trees of identical elements.

Beyond theoretical elegance, the EML framework offers practical advantages for symbolic regression and machine learning. By treating eml trees as differentiable circuits, standard optimizers like Adam can be employed to fit data and, when the target relationship is elementary, recover the exact closed‑form expression at shallow tree depths. This capability reduces the search space dramatically compared with traditional genetic programming or neural‑symbolic hybrids, delivering faster convergence and interpretable results. Researchers can now leverage a single, gradient‑friendly operator to explore a vast function space without handcrafted feature engineering.

The implications extend to hardware design, where a universal continuous‑math gate could be integrated into ASICs or FPGA cores, mirroring the simplicity of Boolean universality. Such processors would compute trigonometric, logarithmic, and algebraic functions with a uniform instruction set, potentially lowering power consumption and latency for scientific calculators, embedded systems, and high‑performance computing workloads. As the community explores extensions—such as incorporating complex arithmetic or higher‑dimensional data—the eml operator may become a foundational building block for the next generation of mathematical hardware and AI‑driven discovery tools.