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

Re: All elementary functions from a single binary operator

#201

Post by cryptonector » Mon, Apr 13, 2026, 3:59 PM 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…

Given this amazing work, an efficient EML operator HW implementation could revolutionize a bunch of things. So the next thing might be an efficient EML HW implementation.

Re: All elementary functions from a single binary operator

#202

Post by marcosdumay » Mon, Apr 13, 2026, 4:06 PM UTC

peyton wrote: post #96

meindnoch wrote: post #95

Earlier quoted context omitted.

Are you under the impression that CPUs are made exclusively from NAND gates? You can't be serious.

Might’ve gotten mixed up with CMOS dominance, or I’m ignorant.

Just to add a bit, but modern digital circuits are almost exclusively MOS, but even the "complementary" bit isn't universal in a large IC.

Re: All elementary functions from a single binary operator

#203

Post by xigoi » Mon, Apr 13, 2026, 4:11 PM UTC

nurettin wrote: post #56

The problem with symbolic regression is ln(y) is undefined at 0, so you can't freely generate expressions with it. We need to guard it with something like ln(1+y*y) or ln(1+|y|) or return undefined.

The article uses extended arithmetic where ln(0) = -∞.

Re: All elementary functions from a single binary operator

#204

Post by xigoi » Mon, Apr 13, 2026, 4:13 PM UTC

simplesighman wrote: post #36

nick238 wrote: post #18

Earlier quoted context omitted.

Don't know adding, but multiplication has diagram on the last page of the PDF. xy = eml(eml(1, eml(eml(eml(eml(1, eml(eml(1, eml(1, x)), 1)), eml(1, eml(eml(1, eml(y, 1)), 1))), 1), 1)), 1) From Table 4, I think addition is slightly more complicated?

Thanks for posting that. You had a transcribing typo which was corrected in the ECMAScript below. Here's the calculation for 5 x 7: const eml = (x,y) => Math.exp(x) - Math.log(y); const mul = (x,y) => eml(eml(1,eml(eml(eml(1,eml(eml(1,eml(1,x)),1)),eml(1,eml(eml(1,eml(y,1)),1))),1)),1); console.log(mul(5,7)); > 35.00000000000001 For larger or negative inputs you get a NaN because ECMAScript has limited precision and…

> For larger or negative inputs you get a NaN because ECMAScript has limited precision and doesn't handle imaginary numbers.

This also shows why EML is not practical for computation.

Re: All elementary functions from a single binary operator

#205

Post by drdeca » Mon, Apr 13, 2026, 4:13 PM UTC

gus_massa wrote: post #198

lugao wrote: post #197

Earlier quoted context omitted.

Exactly! If you didn't strictly limit the operator's complexity, you could just smuggle a Turing machine in via bitwise logic and turn the whole thing into a parlor trick. The beauty here is that eml(x,y) is a pure, continuous analytical function with no hidden branching whatsoever. To clarify my earlier point: the author isn't trying to build a practical calculator or generate human-readable algebra. Using exp and l…

> The beauty here is that eml(x,y) is a pure, continuous analytical function with no hidden branching whatsoever. They use the complex version of logarithm, that has a lot of branching problems.

Different sense of “branching”

Re: All elementary functions from a single binary operator

#206

Post by drdeca » Mon, Apr 13, 2026, 4:16 PM UTC

“ Elementary functions, for many students epitomized by the dreaded sine and cosine, ” dreaded?

Re: All elementary functions from a single binary operator

#207

Post by karpathy » Mon, Apr 13, 2026, 4:23 PM UTC

All possible 36 distinct level-2 eml functions of one variable (the first 18 of them with entirely Real outputs, the other 18 with "intermediate" complex-valued components):

https://imgur.com/a/K7AoOFi

Re: All elementary functions from a single binary operator

#208

Post by psychoslave » Mon, Apr 13, 2026, 4:25 PM UTC

fxwin wrote: post #113

psychoslave wrote: post #64

Very nice, though I'm not found of the name. What comes to my mind as an alternative which I would subjectivity finer is "axe". Think axiom or axiology. Anyone with other suggestions? Or even remarks on this one?

i think eml is fine, names should be connected to the thing they represent so 'exponential minus log' makes sense to me

Gust and color are hard to conciliate.

On my side I like direct sementic connections, but find convoluted indirections conflated through lazy sigles strongly repulsive. I can appreciate an acronym that make and the direct connection and playful indirect reference to expanded terms.

Re: All elementary functions from a single binary operator

#209

Post by gus_massa » Mon, Apr 13, 2026, 4:30 PM UTC

drdeca wrote: post #205

gus_massa wrote: post #198

Earlier quoted context omitted.

> The beauty here is that eml(x,y) is a pure, continuous analytical function with no hidden branching whatsoever. They use the complex version of logarithm, that has a lot of branching problems.

Different sense of “branching”

Yep.

Re: All elementary functions from a single binary operator

#210

Post by boutell » Mon, Apr 13, 2026, 4:34 PM UTC

Halfway through I was imagining aliens to whom this operator comes naturally and our math is weird. By the end I found out that we might be those aliens.