GoKawiilNot all elementary functions can be expressed with exp-minus-log April 14, 2026
By Robert Smith
All Elementary Functions from a Single Operator is a paper by Andrzej Odrzywołek that has been making rounds on the internet lately, being called everything from a “breakthrough” to “groundbreaking”. Some are going as far as to suggest that the entire foundations of computer engineering and machine learning should be re-built as a result of this. The paper says that the function
$$ E(x,y) := \exp x - \log y $$
together with variables and the constant $1$, which we will call EML terms, are sufficient to express all elementary functions, and proceeds to give constructions for many constants and functions, from addition to $\pi$ to hyperbolic trigonometry.
I think the result is neat and thought-provoking. Odrzywołek is explicit about his definition of “elementary function”. His Table 1 fixes “elementary” as 36 specific symbols, and under that definition his theorem is correct and clever, so long as we accept some of his modifications to the conventional $\log$ function and do arithmetic with infinities.
My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage. Odrzywołek recognizes this, saying little more than “[t]hat generality is not needed here” and that his work takes “the ordinary scientific-calculator point of view”. He does not offer further commentary.
What is this more general setting, and does his claim still hold? In modern pure mathematics, dating back to the 19th century, the definition of “elementary function” has been well established. We’ll get to a definition shortly, but to cut to the chase, the titular result does not hold in this setting. As such, in layman’s terms, I do not consider the “Exp-Minus-Log” function to be the continuous analog of the Boolean NAND gate or the universal quantum CCNOT/CSWAP gates.
The rough TL;DR is this: Elementary functions typically include arbitrary polynomial root functions, and EML terms cannot express them. Below, I’ll give a relatively technical argument that EML terms are not sufficient to express what I consider standard elementary functions.
To avoid any confusion, the purpose of this blog post is manifold:
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