5 ms·

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

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 computational EML tree? Got a multidimensional and multivariate function to model (with random samples or a full map)? Just do gradient descent and convert it to approximant EML trees. Perform gradient descent on EML function tree "phi" so that the derivatives in the Schroedinger equation match. But as I said, still reading, this sounds too good to be true, but I have witnessed such things before :)

gilgoomesh 7mo ago
> 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.
- Nevermark 7mo ago
  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".
  - samus 7mo ago
    Single transistors aren't yet logic gates by themselves; they are amplifiers with a very specific gain function that makes it possible to use them as switches. Logic gates usually consist of at least two transistors. See https://en.wikipedia.org/wiki/CMOS https://en.wikipedia.org/wiki/CMOS for an example of how it is done in CMOS technology.
    - adrian_b 7mo ago
      That is correct. Moreover, the amplifying function must exist at least in some gates, to restore the logic levels, but there is no need for it to exist in all gates. For instance, any logic circuit can be implemented using AND gates and/or OR gates made with diodes, which have no gain, i.e. no amplifying function, together with 1-transistor inverters, which provide both the NOT function and the gain needed to restore the logic levels. The logic functions such as NAND can be expressed in several way using simpler components, which correspond directly with the possible hardware implementations. Nowadays, the most frequent method of logic implementation is by using parallel and series connections of switches, for which the MOS transistors are well suited. Another way to express the logic functions is by using the minimum and maximum functions, which correspond directly with diode-based circuits. All the logic functions can also be expressed using the 2 operations of the binary finite field, addition (a.k.a. XOR) and multiplication (a.k.a. AND). This does not lead to simpler hardware implementations, but it can simplify some theoretical work, by using algebraic results. Actually this kind of expressing logic is the one that should have been properly named as Boolean logic, as the contribution of George Boole has been precisely replacing "false" and "true" with "0" and "1" and reinterpreting the classic logic operations as arithmetic operations. It is very weird to see in some programming languages a data type named Boolean, whose possible values are "false" and "true", instead of the historically correct Boolean values, "0" and "1", which can also be much more useful in programming than "false" and "true", by simplifying expressions, especially in array operations (which is why array-oriented languages like APL use "0" and "1", not "false" and "true").
    - Nevermark 7mo ago
      "Pass transistors" are transistors being operated in pass/impedance switch mode. Pass logic. Digital. [0] This is extremely basic digital circuit design. You can create digital circuits as compositions of gates. But you can often implement the same logic, with fewer transistors, using pass logic. Pass logic is also great for asynchronous digital design. [0] https://en.wikipedia.org/wiki/Pass_transistor_logic https://en.wikipedia.org/wiki/Pass_transistor_logic
  - throw-qqqqq 7mo ago
    > Transistors form an efficient, single element, universal digital basis But transistors can be N or P-channel, so it’s not a single logical primitive, like e.g. NAND-gates.
    - Nevermark 7mo ago
      True. NAND also use both N and P-channel transistors, and have to use at least the same number, but frequently many more transistors for anything non-trivial. But as you point out, as a computational basis, pass transistors have two variants.
- brucehoult 7mo ago
  The Cray 1 was built 100% from NOR gates and SRAM.
  - IAmBroom 7mo ago
    50 years ago.
    - brucehoult 7mo ago
      Correct. Your point being? Digital logic didn't change.
eru 7mo ago
> 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?
canjobear 7mo ago
> 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!
- DoctorOetker 7mo ago
  consider how a wavefunction might be stored for numerically searching the ground state of a molecule, or a crystal lattice, humans appreciate 2D imagery that is about 1 kilopixel x 1 kilopixel; so expanding this to 3 dimensions that means the wavefunction would have to store 10 ^ 9 complex numbers (easily 4 GB at 16 bit precision for real and imaginary compnents so 4 bytes per complex number), do we really believe that a DAG variant of the EML construction would consume a bigger value to represent the analytically correct solution? do we really believe that the 4GB DAG variant of EML would produce a less accurate representation (i.e. less fidelity with the schroedinger equation?) If the ground state hydrogen atom is any indication, my money is on EML-style constructions, not naive 3D arrays modelling the wavefunction by brute force. This also re-opens a lot of "party pooper" results in mathematics: impossibility of representing solutions to general quintic (fine print: if we restrict ourselves to arithmetic and roots/radicals). In mathematics and physics there have been a lot of "party pooper" results which later found more profound and interesting positive results by properly rephrasing the question. A negative result for a myopic question isn't very informative on its own.
  - defmacr0 7mo ago
    > This also re-opens a lot of "party pooper" results in mathematics: impossibility of representing solutions to general quintic (fine print: if we restrict ourselves to arithmetic and roots/radicals). Solving the quintic in terms of transcendental functions has already been done https://en.wikipedia.org/wiki/Bring_radical#The_Hermite%E2%80%93Kronecker%E2%80%93Brioschi_characterization https://en.wikipedia.org/wiki/Bring_radical#The_Hermite%E2%8...
ikrima 7mo ago
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
- canjobear 7mo ago
  Link to your work?
- gopalv 7mo ago
  > 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.
  - tripletao 7mo ago
    I'm not sure what you mean by this? It's true that any Boolean operation can be expressed in terms of two-input NAND gates, but that's almost never how real IC designers work. A typical standard cell library has lots of primitives, including all common gates and up to entire flip-flops and RAMs, each individually optimized at a transistor level. Realization with NAND2 and nothing else would be possible, but much less efficient. Efficient numerical libraries likewise contain lots of redundancy. For example, sqrt(x) is mathematically equivalent to pow(x, 0.5), but sqrt(x) is still typically provided separately and faster. Anyone who thinks that eml() function is supposed to lead directly to more efficient computation has missed the point of this (interesting) work.
deleted 7mo ago
[deleted]
dang 7mo ago
What URL should we change it to?
- lioeters 7mo ago
  The URL is already pointing at v2, which apparently is the newest one requested by the comment above. > Submitted on 23 Mar 2026 (v1), last revised 4 Apr 2026 (this version, v2)
  - dang 7mo ago
    Thanks! that's what it looked like to me too.
    - DoctorOetker 7mo ago
      I agree: I don't understand what happened, but the first "View PDF" resulted in a PDF where the hyperlinks to the figures didn't work. Upon closer inspection it wasn't v1 at all, thats a PNAS article. I am unable to remove the EDIT:... line in my original comment at this time...
      - dang 7mo ago
        No worries! We definitely want the best link so such comments are helpful.
- fc417fc802 7mo ago
  IMO arxiv links should pretty much always be to the abstract (ie .../abs/...) as opposed to particular versions of the html or pdf.
siddboots 7mo ago
While I'm really enjoying this paper, I think you are way overstating the significance here. This is mathematically interesting, and conceptually elegant, but there is nothing in this paper that suggests a competitive regression or optimisation approach. I might have misunderstood, but from the two "Why do X when you can do just Y with EML" sentences, I think you are describing symbolic regression, which has been around for quite some time and is a serious grown-up technique these days. But even the best symbolic regression tools do not typically "replace" other regression approaches.
PaulHoule 7mo ago
I can't say I'm surprised at this result at all, in fact I'm surprised something like this wasn't already known.
jmyeet 7mo ago
This isn't all that significant to anyone who has done Calculus 2 and knows about Taylor's Series. All this really says is that the Taylor's expansions of e^x and ln x are sufficient to express to express trig functions, which is trivially true from Euler's formula as long as you're in the complex domain. Arithmetic operations follow from the fact that e^x and ln x are inverses, in particular that e^ln(x) = x. Taylor's series seem a bit like magic when you first see them but then you get to Real Analysis and find out there are whole classes of functions that they can't express. This paper is interesting but it's not revolutionary.
- DoctorOetker 7mo ago
  There is a huge number of people who understand Taylor series, know how to compute them, and the things you can do with Taylor (and other kinds of) expansions. Yet none of us identified that this binary operation spans that lot of them, but I'm willing to read references to predating observations of the same kind. The author does mention alternative systems (incomplete in some specific sense) in the paper. I did however keep thinking there was a lot of attention to trying to include special constants even though we don't know that much about these constants yet, while comparatively little attention went to say elliptic integrals etc. When aiming for a wide catch, you'd include some of those esoteric functions, or erf() etc... I also wished they had attempted to find a representation for derivative and integrals.
cryptonector 7mo ago
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.
- ted_dunning 7mo ago
  You should mark sarcasm as subtle as this.
SideQuark 7mo ago
The compute, energy, and physical cost of this versus a simple x+y is easily an order of magnitude. It will not replace anything in computing, except maybe fringe experiments.

gilgoomesh 7mo ago

> 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.

Nevermark 7mo ago
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".
- samus 7mo ago
  Single transistors aren't yet logic gates by themselves; they are amplifiers with a very specific gain function that makes it possible to use them as switches. Logic gates usually consist of at least two transistors. See https://en.wikipedia.org/wiki/CMOS https://en.wikipedia.org/wiki/CMOS for an example of how it is done in CMOS technology.
  - adrian_b 7mo ago
    That is correct. Moreover, the amplifying function must exist at least in some gates, to restore the logic levels, but there is no need for it to exist in all gates. For instance, any logic circuit can be implemented using AND gates and/or OR gates made with diodes, which have no gain, i.e. no amplifying function, together with 1-transistor inverters, which provide both the NOT function and the gain needed to restore the logic levels. The logic functions such as NAND can be expressed in several way using simpler components, which correspond directly with the possible hardware implementations. Nowadays, the most frequent method of logic implementation is by using parallel and series connections of switches, for which the MOS transistors are well suited. Another way to express the logic functions is by using the minimum and maximum functions, which correspond directly with diode-based circuits. All the logic functions can also be expressed using the 2 operations of the binary finite field, addition (a.k.a. XOR) and multiplication (a.k.a. AND). This does not lead to simpler hardware implementations, but it can simplify some theoretical work, by using algebraic results. Actually this kind of expressing logic is the one that should have been properly named as Boolean logic, as the contribution of George Boole has been precisely replacing "false" and "true" with "0" and "1" and reinterpreting the classic logic operations as arithmetic operations. It is very weird to see in some programming languages a data type named Boolean, whose possible values are "false" and "true", instead of the historically correct Boolean values, "0" and "1", which can also be much more useful in programming than "false" and "true", by simplifying expressions, especially in array operations (which is why array-oriented languages like APL use "0" and "1", not "false" and "true").
  - Nevermark 7mo ago
    "Pass transistors" are transistors being operated in pass/impedance switch mode. Pass logic. Digital. [0] This is extremely basic digital circuit design. You can create digital circuits as compositions of gates. But you can often implement the same logic, with fewer transistors, using pass logic. Pass logic is also great for asynchronous digital design. [0] https://en.wikipedia.org/wiki/Pass_transistor_logic https://en.wikipedia.org/wiki/Pass_transistor_logic
- throw-qqqqq 7mo ago
  > Transistors form an efficient, single element, universal digital basis But transistors can be N or P-channel, so it’s not a single logical primitive, like e.g. NAND-gates.
  - Nevermark 7mo ago
    True. NAND also use both N and P-channel transistors, and have to use at least the same number, but frequently many more transistors for anything non-trivial. But as you point out, as a computational basis, pass transistors have two variants.
brucehoult 7mo ago
The Cray 1 was built 100% from NOR gates and SRAM.
- IAmBroom 7mo ago
  50 years ago.
  - brucehoult 7mo ago
    Correct. Your point being? Digital logic didn't change.

eru 7mo ago

> 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?

canjobear 7mo ago

> 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!

DoctorOetker 7mo ago
consider how a wavefunction might be stored for numerically searching the ground state of a molecule, or a crystal lattice, humans appreciate 2D imagery that is about 1 kilopixel x 1 kilopixel; so expanding this to 3 dimensions that means the wavefunction would have to store 10 ^ 9 complex numbers (easily 4 GB at 16 bit precision for real and imaginary compnents so 4 bytes per complex number), do we really believe that a DAG variant of the EML construction would consume a bigger value to represent the analytically correct solution? do we really believe that the 4GB DAG variant of EML would produce a less accurate representation (i.e. less fidelity with the schroedinger equation?) If the ground state hydrogen atom is any indication, my money is on EML-style constructions, not naive 3D arrays modelling the wavefunction by brute force. This also re-opens a lot of "party pooper" results in mathematics: impossibility of representing solutions to general quintic (fine print: if we restrict ourselves to arithmetic and roots/radicals). In mathematics and physics there have been a lot of "party pooper" results which later found more profound and interesting positive results by properly rephrasing the question. A negative result for a myopic question isn't very informative on its own.
- defmacr0 7mo ago
  > This also re-opens a lot of "party pooper" results in mathematics: impossibility of representing solutions to general quintic (fine print: if we restrict ourselves to arithmetic and roots/radicals). Solving the quintic in terms of transcendental functions has already been done https://en.wikipedia.org/wiki/Bring_radical#The_Hermite%E2%80%93Kronecker%E2%80%93Brioschi_characterization https://en.wikipedia.org/wiki/Bring_radical#The_Hermite%E2%8...

ikrima 7mo ago

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

canjobear 7mo ago
Link to your work?
gopalv 7mo ago
> 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.
- tripletao 7mo ago
  I'm not sure what you mean by this? It's true that any Boolean operation can be expressed in terms of two-input NAND gates, but that's almost never how real IC designers work. A typical standard cell library has lots of primitives, including all common gates and up to entire flip-flops and RAMs, each individually optimized at a transistor level. Realization with NAND2 and nothing else would be possible, but much less efficient. Efficient numerical libraries likewise contain lots of redundancy. For example, sqrt(x) is mathematically equivalent to pow(x, 0.5), but sqrt(x) is still typically provided separately and faster. Anyone who thinks that eml() function is supposed to lead directly to more efficient computation has missed the point of this (interesting) work.

deleted 7mo ago

[deleted]

dang 7mo ago

What URL should we change it to?

lioeters 7mo ago
The URL is already pointing at v2, which apparently is the newest one requested by the comment above. > Submitted on 23 Mar 2026 (v1), last revised 4 Apr 2026 (this version, v2)
- dang 7mo ago
  Thanks! that's what it looked like to me too.
  - DoctorOetker 7mo ago
    I agree: I don't understand what happened, but the first "View PDF" resulted in a PDF where the hyperlinks to the figures didn't work. Upon closer inspection it wasn't v1 at all, thats a PNAS article. I am unable to remove the EDIT:... line in my original comment at this time...
    - dang 7mo ago
      No worries! We definitely want the best link so such comments are helpful.
fc417fc802 7mo ago
IMO arxiv links should pretty much always be to the abstract (ie .../abs/...) as opposed to particular versions of the html or pdf.

siddboots 7mo ago

While I'm really enjoying this paper, I think you are way overstating the significance here. This is mathematically interesting, and conceptually elegant, but there is nothing in this paper that suggests a competitive regression or optimisation approach. I might have misunderstood, but from the two "Why do X when you can do just Y with EML" sentences, I think you are describing symbolic regression, which has been around for quite some time and is a serious grown-up technique these days. But even the best symbolic regression tools do not typically "replace" other regression approaches.

PaulHoule 7mo ago

I can't say I'm surprised at this result at all, in fact I'm surprised something like this wasn't already known.

jmyeet 7mo ago

This isn't all that significant to anyone who has done Calculus 2 and knows about Taylor's Series. All this really says is that the Taylor's expansions of e^x and ln x are sufficient to express to express trig functions, which is trivially true from Euler's formula as long as you're in the complex domain. Arithmetic operations follow from the fact that e^x and ln x are inverses, in particular that e^ln(x) = x. Taylor's series seem a bit like magic when you first see them but then you get to Real Analysis and find out there are whole classes of functions that they can't express. This paper is interesting but it's not revolutionary.

DoctorOetker 7mo ago
There is a huge number of people who understand Taylor series, know how to compute them, and the things you can do with Taylor (and other kinds of) expansions. Yet none of us identified that this binary operation spans that lot of them, but I'm willing to read references to predating observations of the same kind. The author does mention alternative systems (incomplete in some specific sense) in the paper. I did however keep thinking there was a lot of attention to trying to include special constants even though we don't know that much about these constants yet, while comparatively little attention went to say elliptic integrals etc. When aiming for a wide catch, you'd include some of those esoteric functions, or erf() etc... I also wished they had attempted to find a representation for derivative and integrals.

cryptonector 7mo ago

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.

ted_dunning 7mo ago
You should mark sarcasm as subtle as this.

SideQuark 7mo ago

The compute, energy, and physical cost of this versus a simple x+y is easily an order of magnitude. It will not replace anything in computing, except maybe fringe experiments.