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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

#61

Post by eugene3306 »

This makes a good benchmark LLMs:

``` look at this paper: https://arxiv.org/pdf/2603.21852

now please produce 2x+y as a composition on EMLs ```

Opus(paid) - claimed that "2" is circular. Once I told it that ChatGPT have already done this, finished successfully.

ChatGPT(free) - did it from the first try.

Grok - produced estimation of the depth of the formula.

Gemini - success

Deepseek - Assumed some pre-existing knowledge on what EML is. Unable to fetch the pdf from the link, unable to consume pdf from "Attach file"

Kimi - produced long output, stopped and asked to upgrade

GLM - looks ok

Re: All elementary functions from a single binary operator

#62

Post by testaccount28 »

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,
and there is no complex number z that satisfies this equation. indeed if we laboriously expand the given formula for z (the left branch of the tree), we see that it goes through ln(0), and compound expressions.

x^-1 has the same problem.

both formulae work ...sort of... if we allow ln(0) = Infinity and some other moxie, such as x / Infinity = 0 for all finite x.

Re: All elementary functions from a single binary operator

#63

Post by testaccount28 »

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!

Re: All elementary functions from a single binary operator

#65

Post by dang »

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?

Re: All elementary functions from a single binary operator

#66

Post by tripletao »

gopalv wrote: post #58
ikrima wrote: post #46

Earlier quoted context omitted.

From my experience of working in this problem domain for the last year, I'd say it is pretty powerful but the "too good to be true part" comes from that EML buys elegance through exponential expression blow-up. Multiplication alone requires depth-8 trees with 41+ leaves i.e. minimal operator vocabulary trades off against expression length. There's likely an information-theoretic sweet spot between these extremes. It'…

> Multiplication alone requires depth-8 trees with 41+ leaves i.e. minimal operator vocabulary trades off against expression length. That is sort of comparable to how NAND simplify scaling. Division is hell on gates. The single component was the reason scaling went like it did. There was only one gate structure which had to improve to make chips smaller - if a chip used 3 different kinds, then the scaling would've re…

I'm not sure what you mean by this? It's true that any Boolean operation can be expressed in terms of two-input NAND gates, but that's almost never how real IC designers work. A typical standard cell library has lots of primitives, including all common gates and up to entire flip-flops and RAMs, each individually optimized at a transistor level. Realization with NAND2 and nothing else would be possible, but much less efficient.

Efficient numerical libraries likewise contain lots of redundancy. For example, sqrt(x) is mathematically equivalent to pow(x, 0.5), but sqrt(x) is still typically provided separately and faster. Anyone who thinks that eml() function is supposed to lead directly to more efficient computation has missed the point of this (interesting) work.

Re: All elementary functions from a single binary operator

#68

Post by brucehoult »

gilgoomesh wrote: post #38
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…

> 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 computational EML tree? Same reason all boolean logic isn't performed with combinations of NAND – it's computationally inefficient. Polynomials are (for their expressivity) very quick to compute.

The Cray 1 was built 100% from NOR gates and SRAM.

Re: All elementary functions from a single binary operator

#69

Post by paulpauper »

traes wrote: post #55
paulpauper wrote: post #35

Earlier quoted context omitted.

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…

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.

Re: All elementary functions from a single binary operator

#70

Post by NooneAtAll3 »

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…

yeah, it's annoying that author talks about RPN notation, but only gives found formulas in form of images

looks like it computes ln(1)=0, then computes e-ln(0)=+inf, then computes e-ln(+inf)=-inf

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