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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

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.

My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title does not hold. 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 roots, and EML terms cannot express them. Below, I’ll give a relatively technical but independent argument that EML terms are not sufficient to express what I consider standard elementary functions.

Disclaimer: I audited graduate-level mathematics courses almost 20 years ago, and I am not a professional mathematician. Please email me if my statements are clumsy or incorrect.