Re: EML: All elementary functions from a single operator

On Wed, Apr 15, 2026 at 6:33 PM Richard Fateman <[email protected]> wrote:

> Seems to me to be uninteresting for any practical application
>

Not just practical applications ... I don't see how it's of theoretical
interest either.


> 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().
>

Well, yes, but then you're back to standard operators. His claim is that
somehow squeezing everything into one function is a good idea.

I can't see how.

            -s


> 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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>

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