All Elementary Functions from a Single Operator
Author: Andrzej Odrzywołek (2026)
1. Introduction: One operator for all mathematics
The paper presents a remarkable idea: all standard elementary functions can be constructed using just one binary operator and a single constant. The operator, called EML (Exp-Minus-Log), is defined as $E(x,y)=e^x-\ln(y)$. Together with the constant $1$, it can reproduce familiar operations found on a scientific calculator, including addition, subtraction, multiplication, division, exponentiation, logarithms, square roots, and trigonometric functions. The idea is analogous to the NAND gate in digital electronics, which can be used to construct every Boolean logic operation. The result shows how a large collection of mathematical operations can be reduced to an extremely simple computational foundation.
2. How the operator works
The central idea is that exponential and logarithmic functions can be combined to construct other mathematical operations. For example, the exponential function follows immediately from $E(x,1)=e^x$. More complicated functions can be produced through repeated applications of EML. The author used systematic computational searches to discover the operator. Starting with 36 mathematical primitives, he progressively removed operations while testing whether the remaining operations could reproduce them. The final system requires only one binary operator and the constant $1$. Some constructions use complex numbers, which are particularly useful for generating constants such as $i$ and $\pi$.
3. Mathematical results and verification
The paper demonstrates that EML can generate a collection of 36 elementary mathematical primitives, covering arithmetic operations, constants, algebraic functions, trigonometric functions, and inverse trigonometric functions. Every expression can be represented as a binary tree, with the same operation appearing at every internal node. The resulting mathematical grammar is remarkably simple: $S\rightarrow1\mid E(S,S)$. The author combines numerical searches, symbolic verification, and independent numerical checks to validate the resulting expressions. Different functions require different numbers of applications of EML, with some simple functions having very short representations while others require substantially deeper expression trees.
4. Applications and future possibilities
The approach could have applications in symbolic regression, scientific computing, and computer hardware. The paper shows that EML expression trees can be optimized using gradient-based optimization, including the Adam algorithm, making it possible to search for mathematical formulas from numerical data. Instead of merely finding an approximation, the method can potentially recover an exact symbolic expression for an elementary function. The author also describes a prototype compiler capable of translating ordinary mathematical expressions into EML expressions. Possible future applications include specialized processors, FPGA implementations, and minimalist calculators, although practical systems would need to address numerical precision, complex logarithms, and exceptional values.
Key takeaways
SOURCE
Author: Andrzej Odrzywołek (2026)
1. Introduction: One operator for all mathematics
The paper presents a remarkable idea: all standard elementary functions can be constructed using just one binary operator and a single constant. The operator, called EML (Exp-Minus-Log), is defined as $E(x,y)=e^x-\ln(y)$. Together with the constant $1$, it can reproduce familiar operations found on a scientific calculator, including addition, subtraction, multiplication, division, exponentiation, logarithms, square roots, and trigonometric functions. The idea is analogous to the NAND gate in digital electronics, which can be used to construct every Boolean logic operation. The result shows how a large collection of mathematical operations can be reduced to an extremely simple computational foundation.
2. How the operator works
The central idea is that exponential and logarithmic functions can be combined to construct other mathematical operations. For example, the exponential function follows immediately from $E(x,1)=e^x$. More complicated functions can be produced through repeated applications of EML. The author used systematic computational searches to discover the operator. Starting with 36 mathematical primitives, he progressively removed operations while testing whether the remaining operations could reproduce them. The final system requires only one binary operator and the constant $1$. Some constructions use complex numbers, which are particularly useful for generating constants such as $i$ and $\pi$.
3. Mathematical results and verification
The paper demonstrates that EML can generate a collection of 36 elementary mathematical primitives, covering arithmetic operations, constants, algebraic functions, trigonometric functions, and inverse trigonometric functions. Every expression can be represented as a binary tree, with the same operation appearing at every internal node. The resulting mathematical grammar is remarkably simple: $S\rightarrow1\mid E(S,S)$. The author combines numerical searches, symbolic verification, and independent numerical checks to validate the resulting expressions. Different functions require different numbers of applications of EML, with some simple functions having very short representations while others require substantially deeper expression trees.
4. Applications and future possibilities
The approach could have applications in symbolic regression, scientific computing, and computer hardware. The paper shows that EML expression trees can be optimized using gradient-based optimization, including the Adam algorithm, making it possible to search for mathematical formulas from numerical data. Instead of merely finding an approximation, the method can potentially recover an exact symbolic expression for an elementary function. The author also describes a prototype compiler capable of translating ordinary mathematical expressions into EML expressions. Possible future applications include specialized processors, FPGA implementations, and minimalist calculators, although practical systems would need to address numerical precision, complex logarithms, and exceptional values.
Key takeaways
- Universal operator: A single binary operator, EML, together with $1$, can generate a broad collection of elementary functions.
- NAND analogy: EML plays a role somewhat analogous to NAND in Boolean logic.
- 36 primitives: The paper demonstrates the construction of 36 mathematical primitives.
- Simple grammar: Mathematical expressions can be represented as binary trees containing the same operator.
- Computational discovery: Computer-assisted search played an important role in finding the constructions.
- Complex numbers: Complex arithmetic helps generate important constants such as $i$ and $\pi$.
- Symbolic regression: EML provides a framework for discovering mathematical formulas from numerical data.
- Hardware potential: The concept could potentially lead to specialized mathematical processors or simplified computational architectures.
- Deeper question: The work illustrates a fundamental theme in mathematics and computer science: complex computational systems can sometimes emerge from extremely small sets of primitives.
SOURCE
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