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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

#51

Post by canjobear »

ikrima wrote: post #46
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…

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'…

Link to your work?

Re: All elementary functions from a single binary operator

#52

Post by vbezhenar »

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.

Re: All elementary functions from a single binary operator

#53

Post by traes »

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.

Re: All elementary functions from a single binary operator

#54

Post by [deleted] »

curtisf wrote: post #40
BobbyTables2 wrote: post #2

How does one actually add with this?

It's basically using the "-" embedded in the definition of the eml operator. Table 4 shows the "size" of the operators when fully expanded to "eml" applications, which is quite large for +, -, ×, and /. Here's one approach which agrees with the minimum sizes they present: eml(x, y ) = exp(x) − ln(y) # 1 + x + y eml(x, 1 ) = exp(x) # 2 + x eml(1, y ) = e - ln(y) # 2 + y eml(1, exp(e - ln(y))) = ln(y) # 6 + y; construc…

[deleted]

Re: All elementary functions from a single binary operator

#55

Post by traes »

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…

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 gate creates all of the discrete logic gates. Hence, "continuous mathematics" as opposed to discrete mathematics. It seems to me you're being overly negative without solid reasoning.

Re: All elementary functions from a single binary operator

#58

Post by gopalv »

ikrima wrote: post #46
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…

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 required more than one parallel innovation to go (sort of like how LED lighting had to wait for blue).

If you need two or more components, then you have to keep switching tools instead of hammer, hammer, hammer.

Re: All elementary functions from a single binary operator

#59

Post by freehorse »

bzax wrote: post #3
BobbyTables2 wrote: post #2

How does one actually add with this?

Well, once you've derived unary exp and ln you can get subtraction, which then gets you unary negation and you have addition.

And then by using the fact that the exponential turns addition into multiplication, you get multiplication (and subtraction gives division).

Re: All elementary functions from a single binary operator

#60

Post by notorandit »

Not sure it really compares to NAND() and the likes.

Simply because bool algebra doesn't have that many functions and all of them are very simple to implement.

A complex bool function made out of NANDs (or the likes) is little more complex than the same made out of the other operators.

Implementing even simple real functions out of eml() seems to me to add a lot of computational complexity even with both exp() and ln() implemented in hardware in O(1). I think about stuff sum(), div() and mod().

Of course, I might be badly wrong as I am not a mathematician (not even by far).

But I don't see, at the moment, the big win on this.

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