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All elementary functions from a single binary operator

arxiv.org

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Re: All elementary functions from a single binary operator

#71

Post by lioeters »

dang wrote: post #65
DoctorOetker wrote: post #30

EDIT: please change the article link to the most recent version (as of now still v2), it is currently pointing to the v1 version which misses the figures. I'm still reading this, but if this checks out, this is one of the most significant discoveries in years. Why use splines or polynomials or haphazardly chosen basis functions if you can just fit (gradient descent) your data or wave functions to the proper computati…

What URL should we change it to?

The URL is already pointing at v2, which apparently is the newest one requested by the comment above.

> Submitted on 23 Mar 2026 (v1), last revised 4 Apr 2026 (this version, v2)

Re: All elementary functions from a single binary operator

#72

Post by thaumasiotes »

hyperhello wrote: post #37
thaumasiotes wrote: post #25

Earlier quoted context omitted.

> isn't the operation its own inverse depending on the parameter? This is a function from ℝ² to ℝ. It can't be its own inverse; what would that mean?

eml(1,eml(x,1)) = eml(eml(1,x),1) = exp(ln(x)) = ln(exp(x)) = x

But f(x) = eml(1, x) and g(x) = eml(x, 1) are different operations. What operation are you saying is supposed to be its own inverse?

Re: All elementary functions from a single binary operator

#73

Post by avmich »

paulpauper wrote: post #35
zephen wrote: post #15

Judging by the title, I thought I would have a good laugh, like when the doctor discovered numerical integration and published a paper. But no... This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding.

I don't think this is ever making it past the editor of any journal, let alone peer review. Elementary functions such as exponentiation, logarithms and trigonometric functions are the standard vocabulary of STEM education. Each comes with its own rules and a dedicated button on a scientific calculator; What? and No comparable primitive has been known for continuous mathematics: computing elementary functions such as…

The principal result is "all elementary functions can be represented by this function and constant 1". I'm not sure if this was known before. Applications are another matter, but I suspect interesting ones do exist.

Re: All elementary functions from a single binary operator

#74

Post by Rubicund »

testaccount28 wrote: post #63
testaccount28 wrote: post #62

derivation of -x seems wrong. we can look at the execution trace on a stack machine, but it's actually not hard to see. starting from the last node before the output, we see that the tree has the form eml(z, eml(x, 1)) = e^z - ln(eml(x, 1)) = e^z - ln(e^x) = e^z - x and the claim is that, after it's expanded, z will be such that this whole thing is equal to -x. but with some algebra, this is happening only if e^z = 0…

ah, the paper acknowledges this. my bad for jumping to the diagrams!

On page 11, the paper explicitly states:

> EML-compiled formulas work flawlessly in symbolic Mathematica and IEEE754 floating-point… This is because some formulas internally might rely on the following properties of extended reals: ln 0 = −∞, e^(−∞) = 0.

And then follows with:

> But EML expressions in general do not work ‘out of the box’ in pure Python/Julia or numerical Mathematica.

Thus, the paper’s completeness claim depends on a non-standard arithmetic convention (ln(0) = -∞), not just complex numbers as it primarily advertises. While the paper is transparent about this, it is however, buried on page 11 rather than foregrounded as a core caveat. Your comment deserves credit for flagging it.

Re: All elementary functions from a single binary operator

#75

Post by avmich »

paulpauper wrote: post #69
traes wrote: post #55

Earlier quoted context omitted.

This preprint was written by a researcher at an accredited university with a PhD in physics. I'm sure they know what a vector valued function is. The point of this paper is not to revolutionize how a scientific calculator functions overnight, its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application, analogous to how a NAND or NOR g…

its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application, But he didn't show this though. I skimmed the paper many times. He creates multiple branches of these trees in the last page, so it's not truly a single nested operation.

Well, it is still the case, even if not explicitly shown. Personally I think it almost boils down to school math, with some details around complex logarithms; the rest seems to be simpler.

Re: All elementary functions from a single binary operator

#77

Post by dang »

lioeters wrote: post #71
dang wrote: post #65

Earlier quoted context omitted.

What URL should we change it to?

The URL is already pointing at v2, which apparently is the newest one requested by the comment above. > Submitted on 23 Mar 2026 (v1), last revised 4 Apr 2026 (this version, v2)

Thanks! that's what it looked like to me too.

Re: All elementary functions from a single binary operator

#80

Post by bollu »

evnix wrote: post #78

Can someone explain how is this different from lambda calculus, it seems like you can derive the same in both. I don't understand both well enough and hence the question.

Lambda calculus talks about computable functions, where the types of the inputs are typically something discrete, like `Bool` or `Nat`. Here, the domain is the real numbers.
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