I wish i had seen this a few days ago i spent days on this before arriving at the following:
It’s completely wrong, for multiple reasons.
First, and immediately, none of the derived total functions are the functions they appear to be, because they are all partial functions.
Worse than that the domain over which these trees are defined is undecidable by Richardson. You can encode Hilbert’s 10th with elementary functions. Which means you can’t algorithmically decide if a tree evaluates to zero so you never know if ln(y) is undefined
So the central claim is done right off the bat.
Secondly and probably more importantly, these trees have exponential blow up problems because even fairly shallow trees involve tetrations of e that don’t cancel. Even in float64 you can‘t add 800+800 without overflow. Clamping it just gives you wrong answers and clamped exp A) isn’t elementary and B) gives bounded growth so can’t generate all elementary functions.
The consequences for his claimed practical application is disastrous. the loss function is NaN everywhere once you get to around depth 6 in his trees. If you clamp it you just get a flat plateau. Even if you had infinite precision the gradient would be unusable and wouldn’t converge.
As a second note, he frequently in the paper talks about using positive reals as inputs. First, positive reals aren’t even part of his generating set (and if they are his whole central claim is wrong, it has uncountably many constants), but even if they are, fundamentally his operations are complex, and you have subtraction so it is not hard to construct trees that should be total but which never the less are undefined at undecideably many points.
Using extended reals doesn’t save it either. There are just different undefined values that lead to undecideability.
None of these arguments are really specific to EML either. Any binary operator that hopes to generate elementary functions must include exp and ln, must be partial and must be able to encode hilbert’s 10th. There is no fix for this.
You might ask why his tests didn’t find any of this but if you look at his code, they do. He just carefully restricts them to a narrow domain where they happen to work and even then he does things like drop imaginary components and filters out undefined results. He doesn‘t even really hide this in the paper, he just dismisses them as implementation details that can be fixed but the problems are structural
So what we are left with is a generator that can generate exp(x) cleanly and essentially nothing else.