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EML Operator

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EMLFunction

The EML operator, where "EML" is short for exp-minus-log, is defined by Odrzywołek (2026) as the binary operator

E(x,y)=exp(x)-lny.
(1)

As a real-valued function of two variables, the EML operator is plotted above on a portion of the domain y>0.

Odrzywołek (2026) showed constructively that, together with the constant 1, this operator can express a fixed scientific-calculator basis of elementary functions and operations by repeated composition. For example,

exp(x)=E(x,1)
(2)
lnx=E(1,E(E(1,x),1)).
(3)

Expressions using only E and 1 are generated by the grammar

S->1|E(S,S),
(4)

where S denotes an expression and | denotes alternation. Thus each such expression is a binary tree with a single type of internal node. In this sense, the EML operator is analogous to the NAND operator for Boolean logic, with the qualification that the construction is for the specified basis of elementary functions and may use complex intermediate values.

See also

Binary Operator, Binary Tree, Elementary Function, Exponential Function, Logarithm, NAND

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References

Odrzywołek, A. "All Elementary Functions from a Single Binary Operator." 4 Apr 2026. https://arxiv.org/abs/2603.21852.Odrzywołek, A. "SymbolicRegressionPackage: Basic Building Blocks for Symbolic Regression." 2026. https://github.com/VA00/SymbolicRegressionPackage.Stachowiak, T. "Algebraic Structure Behind Odrzywołek's EML Operator." 26 Apr 2026. https://arxiv.org/abs/2604.23893.

Cite this as:

Weisstein, Eric W. "EML Operator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EMLOperator.html

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