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Mathematical minimalism

Posted on 13 April 2026 by John

Andrzej Odrzywolek recently posted an article on arXiv showing that
you can obtain all the elementary functions from just the function

\operatorname{eml}(x,y) = \exp(x) - \log(y)

and the constant 1. The following equations, taken from the paper's
supplement, show how to bootstrap addition, subtraction,
multiplication, and division from the elm function.

\begin{align*} \exp(z) &\mapsto \operatorname{eml}(z,1) \\ \log(z) &\
mapsto \operatorname{eml}(1,\exp(\operatorname{eml}(1,z))) \\ x - y &
\mapsto \operatorname{eml}(\log(x),\exp(y)) \\ -z &\mapsto (\log 1) -
z \\ x + y &\mapsto x - (-y) \\ 1/z &\mapsto \exp(-\log z) \\ x \cdot
y &\mapsto \exp(\log x + \log y) \end{align*}

See the paper and supplement for how to obtain constants like p and
functions like square and square root, as well as the standard
circular and hyperbolic functions.

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One thought on "Mathematical minimalism"

 1.
    Tomas Singliar
    13 April 2026 at 12:18

    It may seem like a cute exercise in theory of computation, but
    consider the implications for deep neural networks. Stacking emls
    and 1s is exact (to the degree that log and exp are exact), as
    opposed to the universal approximator theorem, which only gives
    epsilon-delta guarantees.

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