Not all elementary functions can be expressed with “exp-minus-log”

The claim and the buzz

It has been reported that a recent paper by Andrzej Odrzywołek — circulating under the title All Elementary Functions from a Single Operator — argues that a single two-argument operator E(x,y) := exp(x) − log(y), together with variables and the constant 1, suffices to build every “elementary” function. The claim has been called everything from “a breakthrough” to “groundbreaking,” and some have even suggested, allegedly, that it should force a rethink of parts of computer engineering and machine learning. The author is explicit about his setup: he pins “elementary” down to a 36-symbol “scientific calculator basis,” and under that narrow definition many constructions in the paper check out.

The counterexample: absolute value

But here’s the kicker. A concise critique published on StyleWarning points out a simple, devastating counterexample: the absolute value function |x|. Every EML term (an expression built from E, variables, and 1) is real-analytic on its domain — a fact proved by structural induction because exp and (where defined) log preserve analyticity. Yet |x| is not differentiable at 0, so it cannot be real-analytic there; therefore no EML term can equal |x| for all real x. Odrzywołek can dodge this only by changing the standard meanings of exp or log, or by moving to extended reals — moves that are, to many, significant and nonstandard.

Complex domain and why this matters

The critique goes further: even in the complex plane there is no branch of any EML term that equals |z| on a nonempty open set, so the issue isn’t merely a real-line technicality. The emotional moment is clear — a flashy unifying claim runs headlong into a middle-school-level function. Is E(x,y) really the continuous analogue of NAND or a universal quantum gate? Not without some serious caveats. Under the paper’s restricted definition the theorem is clever and correct; but in the broader, conventional sense of “elementary functions,” it has been reported that the title overstates the result. That nuance matters — especially if the next step is rebuilding toolchains or rewriting textbooks.

Sources: stylewarning.com, Hacker News