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 π and functions like square and square root, as well as the standard circular and hyperbolic functions.

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