All elementary functions from a single binary operator • Hacker News Board

Re: All elementary functions from a single binary operator

#41

Post by xpe » Mon, Apr 13, 2026, 4:38 AM UTC

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 much to learn about other approximation techniques.

Re: All elementary functions from a single binary operator

#42

Post by eru » Mon, Apr 13, 2026, 4:39 AM UTC

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…

> I'm still reading this, but if this checks out, this is one of the most significant discoveries in years.

It seems like a neat parlour trick, indeed. But significant discovery?

Re: All elementary functions from a single binary operator

#43

Post by eru » Mon, Apr 13, 2026, 4:41 AM UTC

tripdout wrote: post #12

Interesting, but is the required combination of EML gates less complex than using other primitives?

In general, no.

Re: All elementary functions from a single binary operator

#44

Post by canjobear » Mon, Apr 13, 2026, 4:43 AM UTC

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?

Because the EML basis makes simple functions (like +) hard to express.

Not to diminish this very cool discovery!

Re: All elementary functions from a single binary operator

#45

Post by [deleted] » Mon, Apr 13, 2026, 4:45 AM UTC

thaumasiotes wrote: post #25

hyperhello wrote: post #14

> eml(x,y)=exp(x)-ln(y) Exp and ln, isn't the operation its own inverse depending on the parameter? What a neat find.

> 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?

[deleted]

Re: All elementary functions from a single binary operator

#46

Post by ikrima » Mon, Apr 13, 2026, 4:47 AM UTC

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's interesting to see his EML approach whereas mine was more on generating a context sensitive homoiconic grammar.

I've had lots of success combining spectral neural nets (GNNs, FNOs, Neural Tangent Kernels) with symbolic regression and using Operad Theory and Category Theory as my guiding mathematical machinery

Re: All elementary functions from a single binary operator

#47

Post by nostrademons » Mon, Apr 13, 2026, 4:48 AM UTC

selcuka wrote: post #5

So, like brainf*ck (the esoteric programming language), but for maths?

More like lambda calculus, but for continuous functions.

Re: All elementary functions from a single binary operator

#48

Post by woopsn » Mon, Apr 13, 2026, 4:49 AM UTC

thaumasiotes wrote: post #25

hyperhello wrote: post #14

> eml(x,y)=exp(x)-ln(y) Exp and ln, isn't the operation its own inverse depending on the parameter? What a neat find.

> 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?

It's a kind of superposition representation a la Kolmogorov-Arnold, a learnable functional basis for elementary functions g(x,y)=f(x) - f^{-1}(y) in this sense with f=exp.

Re: All elementary functions from a single binary operator

#49

Post by [deleted] » Mon, Apr 13, 2026, 4:50 AM UTC

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…

[deleted]

Re: All elementary functions from a single binary operator

#50

Post by Nevermark » Mon, Apr 13, 2026, 4:52 AM UTC

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.

They are done with transistors though. Transistors form an efficient, single element, universal digital basis.

And are a much less arbitrary choice than NAND, vs. NOR, XOR, etc.

Using transistors as conceptual digital logic primitives, where power dissipation isn't a thing, Pass Logic is "The Way".