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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

#161

Post by defmacr0 »

DoctorOetker wrote: post #91
canjobear wrote: post #44

Earlier quoted context omitted.

> 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? Because the EML basis makes simple functions (like +) hard to express. Not to diminish this very cool discovery!

consider how a wavefunction might be stored for numerically searching the ground state of a molecule, or a crystal lattice, humans appreciate 2D imagery that is about 1 kilopixel x 1 kilopixel; so expanding this to 3 dimensions that means the wavefunction would have to store 10 ^ 9 complex numbers (easily 4 GB at 16 bit precision for real and imaginary compnents so 4 bytes per complex number), do we really believe th…

> This also re-opens a lot of "party pooper" results in mathematics: impossibility of representing solutions to general quintic (fine print: if we restrict ourselves to arithmetic and roots/radicals).

Solving the quintic in terms of transcendental functions has already been done

https://en.wikipedia.org/wiki/Bring_radical#The_Hermite%E2%8...

Re: All elementary functions from a single binary operator

#163

Post by noobermin »

xpe wrote: post #41
noobermin wrote: post #32

I don't mean to shit on their interesting result, but exp or ln are not really that elementary themselves... it's still an interesting result, but there's a reason that all approximations are done using series of polynomials (taylor expansion).

> but there's a reason that all approximations are done using series of polynomials (taylor expansion). "All" is a tall claim. Have a look at https://perso.ens-lyon.fr/jean-michel.muller/FP5.pdf for example. Jump to slide 18: > Forget about Taylor series > Taylor series are local best approximations: they cannot compete on a whole interval. There is no need to worry about "sh-tt-ng" on their result when there is so m…

Sorry, re-reading this, I should have said "most". As the other reply mentions, Pade approx. are also well liked for numerical methods.

I personally mostly do my everyday work using taylor expansion (mostly explicit numerical methods in comp. EM because they're cheaper these days and it's simpler to write down) so it's what first comes to mind.

Re: All elementary functions from a single binary operator

#164

Post by noobermin »

traes wrote: post #53
noobermin wrote: post #32

I don't mean to shit on their interesting result, but exp or ln are not really that elementary themselves... it's still an interesting result, but there's a reason that all approximations are done using series of polynomials (taylor expansion).

Elementary function is a technical term that this paper uses correctly, not a generic prescription of simplicity. See https://en.wikipedia.org/wiki/Elementary_function .

Then this is a good math paper. Everyone asking for "elm gates" (if such a thing even is possible or efficient) ought to relax a bit.

Re: All elementary functions from a single binary operator

#165

Post by bfrankline »

traes wrote: post #53
noobermin wrote: post #32

I don't mean to shit on their interesting result, but exp or ln are not really that elementary themselves... it's still an interesting result, but there's a reason that all approximations are done using series of polynomials (taylor expansion).

Elementary function is a technical term that this paper uses correctly, not a generic prescription of simplicity. See https://en.wikipedia.org/wiki/Elementary_function .

In numerical analysis, elementary function membership, like special function membership, is ambiguous. In many circumstances, it’s entirely reasonable to describe the natural logarithm as a special function.

Re: All elementary functions from a single binary operator

#166

Post by lioeters »

StilesCrisis wrote: post #130
entaloneralie wrote: post #22

This is amazing! I love seeing FRACTRAN-shaped things on the homepage :) This reminds me of how 1-bit stacks are encoded in binary: A stack of zeros and ones can be encoded in a single number by keeping with bit-shifting and incrementing. Pushing a 0 onto the stack is equivalent to doubling the number. Pushing a 1 is equivalent to doubling and adding 1. Popping is equivalent to dividing by 2, where the remainder is t…

Did you just explain base-2 numbers on the HN forums as if it were novel?

You missed the part where the number is being used as a 1-bit stack. They never claimed novelty, it's just a neat technique that might be unfamiliar to most people.

Re: All elementary functions from a single binary operator

#168

Post by mah4k4l »

Sharlin wrote: post #150
mah4k4l wrote: post #143

According to Gemini 3.1 Pro this would shoot the current weather forecasting power through the roof (and math processing in general): The plan is to use this new "structurally flawless mathematical primitive" EML (this is all beyond me, was just having some fun trying to make it cook things together) in TPUs made out of logarithmic number system circuits. EML would have DAGs to help with the exponential bloat problem…

Congrats, you made a hallucination machine successfully hallucinate?

Here's my NoteboojLM podcast on the subject :-) Sick.

https://notebooklm.google.com/notebook/e0a54a54-c644-4c89-9d...

Re: All elementary functions from a single binary operator

#169

Post by adornKey »

vbezhenar wrote: post #52
simplesighman wrote: post #17

> For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations I read the paper. Is there a table covering all other math operations translated to eml(x,y) form?

Didn't read the paper, but it was easy for me to derive constants 0, 1, e and functions x + y, x - y, exp(x), ln(x), x * y, x / y. So seems to be enough for everything. Very elegant.

Although x + y is surprisingly more complicated than you'd expect at first. The construction first goes for exp(x) and ln(x) then to x - y and finally uses -y to get to x + y.

Re: All elementary functions from a single binary operator

#170

Post by XCSme »

theanonymousone wrote: post #111
eugene3306 wrote: post #61

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 knowl…

I changed the prompt to this: """ Consider a mathematical function EML defined as `eml(x,y)=exp(x)−ln(y)` Please produce `sin(x)/x` as a composition on EMLs and constant number 1 (one). """

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