Re: EML: All elementary functions from a single operator

Seems to me to be uninteresting for any practical application
Of more economical use would be to allow
integers, i,
exp(x)    .. which is, I suppose eml(x,1)
 log(y) ... eml(-infinity, y)
and rational operations +,*,-, /  .  maybe  abs(). maybe some single-valued
version of sqrt().

RJF




On Wed, Apr 15, 2026 at 3:17 PM Stavros Macrakis <[email protected]> wrote:

> The representation is as far as I can tell completely useless.
>
> I played around a bit with generating eml expressions. One thing that's
> immediately clear is that the vast majority of expression trees produce
> uninteresting functions.
>
> I don't find it surprising that you can represent all the continuous
> scientific-calculator functions with compositions of *eml*. After
> all, x^y / trig / arctrig / hyperbolic all have well-known definitions in
> terms of exp and log.
>
> What *is* a bit shocking is how hard it is to write, say, the number 2.
> The paper reports that it requires 19 instances of *eml*, probably as
> "+"(1,1), since it takes 19 instances to write "+". And *i*=sqrt(-1)
> requires >55 instances, so those trig functions are going to be enormous.
>
> The *eml *form of standard functions quickly gets huge. Conversely, most
> of the functions you can write with it are utterly uninteresting. You ask
> whether Maxima can simplify various arcane expressions back to their
> elementary forms. Well, let's look at some examples. There are 2,090,918
> trees of depth 4 (or less). Most of them involve nested exponentials,
> like %e-log(%e^%e^x/x-log(%e^%e-log(%e-log(x)))), and I don
>
> As for "symbolic regression", that makes no sense. Most of the search
> space consists of useless functions like
> -(log(%e^%e^x-x*log(%e^%e-log(%e-log(x))))-log(x)-%e). If you allow more
> constants, e.g., 0 and *i*, the trees get less deep, but more
> broad. Might as well have a search space where the density of "interesting"
> functions is large.
>
> I don't find this elegant at all.
>
> In fact, I wonder if it is elaborate April Fool's joke published 5 days
> late.
>
>              -s
>
>
> On Wed, Apr 15, 2026 at 3:19 PM Barton Willis <[email protected]> wrote:
>
>> I kept scanning the paper for a table showing how to express each
>> elementary function in terms of *eml*. I never found one. After that, my
>> interest evaporated. It’s cute, I suppose; useful, I'm not convinced.
>> The “phylogenetic” tree of the elementary functions is a typesetting tour
>> de force, but isn’t the English language already sufficiently weirded?
>> It might be amusing to see whether Maxima can simplify various arcane
>> expressions in *eml* back to their elementary forms. And what about
>> expressing the principal branches? That looks like a nightmare.
>>
>>
>> --Barton
>> ------------------------------
>> *From:* Stavros Macrakis <[email protected]>
>> *Sent:* Wednesday, April 15, 2026 1:17 PM
>> *To:* <[email protected]> <
>> [email protected]>
>> *Subject:* [Maxima-discuss] EML: All elementary functions from a single
>> operator
>>
>> Caution: Non-NU Email
>>
>> The physicist Andrzej Odrzywołek (*adiunkt* at Jagiellonian
>> University),  announced
>> <https://urldefense.com/v3/__https://arxiv.org/html/2603.21852v2__;!!PvXuogZ4sRB2p-tU!CA_SQ_a-ptqXUR52sYmTFsZPoqM1rv8G8RRMpX7n0iMb36Rdj7JVWac5EfgBxFPePSiC6rswnszt5fE$>
>> :
>>
>>  a single binary operator, eml(x,y)=exp(x)-ln(y),  together with the
>> constant 1, generates the standard repertoire of a scientific calculator.
>>
>> and compared *eml* to the Sheffer stroke (*nand*) as a universal
>> operator, in fact he calls it *EML Sheffer*. The intermediate results
>> are in general complex.
>>
>> This has gotten some attention in the press, e.g., in *The Register
>> <https://urldefense.com/v3/__https://www.theregister.com/2026/04/14/two_button_calculator/__;!!PvXuogZ4sRB2p-tU!CA_SQ_a-ptqXUR52sYmTFsZPoqM1rv8G8RRMpX7n0iMb36Rdj7JVWac5EfgBxFPePSiC6rsw8-BevUA$>*
>> .
>>
>> Let's assume that his central claim is true -- it doesn't seem
>> implausible, although I wonder whether his use of the principal branch runs
>> into problems.
>>
>> Still, as far as I can tell, this has little interest for mathematics or
>> symbolic calculation. He claims two applications for it:
>>
>>    1. analog computing "pure-EML form could possibly be implemented
>>    efficiently in FPGA or analog circuits."
>>    2. discovering closed-form expressions from data
>>
>> For analog computing, the combination of *exp *and *log *in the formulas
>> means that intermediate results may have a large range of magnitudes, and
>> accuracy will be hard. EML requires complex arithmetic; he has no concrete
>> proposal on how to implement that in analog circuits -- perhaps as
>> magnitude/phase in an AC circuit? And even something as simple as * x*y* is
>> a 12-node network.
>>
>> For discovering closed-form expressions from data points ("symbolic
>> regression"), he proposes using gradient-based optimizers (like Adam) to
>> train trees to recover closed-form expressions from numerical data, and
>> gives some examples. I am skeptical, because (a) it's not clear to me
>> that having one operator and deep trees is better than having many
>> operators and shallower trees; (b) combining *exp* and *log *is going to
>> create expressions that have large dynamic range and unstable behavior
>> which I'd think would be unsuitable for gradient-based optimization.
>>
>> He also claims
>>
>>  all the above difficulties (edge cases) are not much different from
>> those usually encountered in every kind of floating-point or symbolic
>> computation.
>>
>>
>> Comments?
>>
>> Does anyone see any value in this?
>>
>>               -s
>>
>> All elementary functions from a single operator
>> Andrzej Odrzywołek
>> Institute of Theoretical Physics, Jagiellonian University, 30-348 Krakow,
>> Poland
>> E-mail: [email protected]
>>
>> https://arxiv.org/html/2603.21852v2#S0.SSx1.p1.7
>> <https://urldefense.com/v3/__https://arxiv.org/html/2603.21852v2*S0.SSx1.p1.7__;Iw!!PvXuogZ4sRB2p-tU!CA_SQ_a-ptqXUR52sYmTFsZPoqM1rv8G8RRMpX7n0iMb36Rdj7JVWac5EfgBxFPePSiC6rsw0XFuLag$>
>>
>>
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