# All elementary functions from a single binary operator

**arXiv:** 2603.21852 [cs.SC]
**Authors:** Andrzej Odrzywołek
**Published:** 2026-03-23 (v1), revised 2026-04-04 (v2)
**URL:** https://arxiv.org/abs/2603.21852

## Abstract

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

## Key Points

1. **EML Operator:** eml(x,y) = exp(x) - ln(y) is a single binary operator that, together with constant 1, can generate all elementary functions
2. **Scientific Calculator Reduction:** A two-button calculator (1, eml) suffices for everything a full 36-button scientific calculator can do
3. **Binary Tree Grammar:** Every EML expression is a binary tree with grammar S → 1 | eml(S,S)
4. **Symbolic Regression:** EML trees can be trained with gradient methods to recover exact closed-form expressions from data
5. **Discovery Method:** Found through systematic exhaustive search and ablation testing
6. **Related Operators:** EDL (exp(x)/ln(y) with constant e) and -eml(y,x) (with constant -∞) are related variants

## Methods

- Started with 36 primitives (constants, functions, operations) from standard scientific calculator
- Iteratively removed elements and verified if remaining set could reconstruct all originals
- Used hybrid numeric bootstrapping verification with algebraically independent transcendental constants
- Search complexity up to K=9 (RPN program length)
- Verified with Mathematica SymbolicRegression package and Rust implementation

## Results

- Progressive reduction: Base-36 → Wolfram (7) → Calc 3 (6) → Calc 2 (4) → Calc 1 (4) → Calc 0 (3) → EML (2)
- EML expression depths range from 1 (exp) to 8 (multiplication)
- Constants: e (depth 3), π (depth 193), i (depth 131)
- Symbolic recovery success rate: 100% at depth 2, ~25% at depths 3-4, <1% at depth 5

## Applications

1. **EML Compiler:** Converts formulas to pure EML form for symbolic/numerical evaluation
2. **Analog Circuits:** EML as building block for analog computing
3. **Symbolic Regression:** Master formula approach with gradient-based optimization
4. **Neural Networks:** EML trees as interpretable architectures

## Open Questions

-是否存在不需要区分常量的二元 Sheffer 算子？
-是否存在一元 Sheffer 算子，同时作为神经激活函数和初等函数生成器？
-是否存在具有更好性质（非指数渐近、无域问题）的类似算子？

## Code & Data

- Repository: https://zenodo.org/records/19183008
- SymbolicRegressionPackage with Mathematica and Rust implementations
