Can someone explain how is this different from lambda calculus, it seems like you can derive the same in both. I don't understand both well enough and hence the question.
Lambda calculus talks about computable functions, where the types of the inputs are typically something discrete, like `Bool` or `Nat`. Here, the domain is the real numbers.
This is amazing! I love seeing FRACTRAN-shaped things on the homepage :) This reminds me of how 1-bit stacks are encoded in binary:
A stack of zeros and ones can be encoded in a single number by keeping with bit-shifting and incrementing.
Pushing a 0 onto the stack is equivalent to doubling the number.
Pushing a 1 is equivalent to doubling and adding 1.
Popping is equivalent to dividing by 2, where the remainder is the number.
I use something not too far off for my daily a programming based on a similar idea:
Rejoice is a concatenative programming language in which data is encoded as multisets that compose by multiplication. Think Fractran, without the rule-searching, or Forth without a stack.
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 :)
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
> 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.
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.
> 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.
> 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.
> A calculator with just two buttons, EML and the digit 1, can compute
everything a full scientific calculator does
Reminds me of the Iota combinator, one of the smallest formal systems that can be combined to produce a universal Turing machine, meaning it can express all of computation.
derivation of -x seems wrong. we can look at the execution trace on a stack machine, but it's actually not hard to see. starting from the last node before the output, we see that the tree has the form
and the claim is that, after it's expanded, z will be such that this whole thing is equal to -x. but with some algebra, this is happening only if
e^z = 0,
and there is no complex number z that satisfies this equation. indeed if we laboriously expand the given formula for z (the left branch of the tree), we see that it goes through ln(0), and compound expressions.
x^-1 has the same problem.
both formulae work ...sort of... if we allow ln(0) = Infinity and some other moxie, such as x / Infinity = 0 for all finite x.
> EML-compiled formulas work flawlessly in symbolic Mathematica and IEEE754 floating-point… This is because some formulas internally might rely on the following properties of extended reals: ln 0 = −∞, e^(−∞) = 0.
And then follows with:
> But EML expressions in general do not work ‘out of the box’ in pure Python/Julia or numerical Mathematica.
Thus, the paper’s completeness claim depends on a non-standard arithmetic convention (ln(0) = -∞), not just complex numbers as it primarily advertises. While the paper is transparent about this, it is however, buried on page 11 rather than foregrounded as a core caveat. Your comment deserves credit for flagging it.
> using EML trees as trainable circuits ..., I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4
That's awesome. I always wondered if there is some way to do this.
For completeness, there is also Peirce’s arrow aka NOR operation which is functionally complete. Fun applications iirc VMProtect copy protection system has an internal VM based on NOR.
Not sure it really compares to NAND() and the likes.
Simply because bool algebra doesn't have that many functions and all of them are very simple to implement.
A complex bool function made out of NANDs (or the likes) is little more complex than the same made out of the other operators.
Implementing even simple real functions out of eml() seems to me to add a lot of computational complexity even with both exp() and ln() implemented in hardware in O(1).
I think about stuff sum(), div() and mod().
Of course, I might be badly wrong as I am not a mathematician (not even by far).
But I don't see, at the moment, the big win on this.
I think what you want is the supplementary information, part II "completeness proof sketch" on page 12. You already spotted the formulas for "exp" and real natural "L"og; then x - y = eml(L(x), exp(y)) and from there apparently it is all "standard" identities. They list the arithmetic operators then some constants, the square root, and exponentials, then the trig stuff is on the next page.
You can find this link on the right side of the arxiv page:
Didn't read the paper, but it was easy for me to derive constants 0, 1, e and functions x + y, x - y, exp(x), ln(x), x * y, x / y. So seems to be enough for everything. Very elegant.
I was curious about that too. Gemini actually gave a decent list. Trig functions come from Euler's identity:
e^ix = cos x + i sin x
which means:
e^-ix = cos -x + i sin -x
= cos x - i sin x
so adding them together:
e^ix + e^-ix = 2 cos x
cos x = (e*ix - e^-ix) / 2
So I guess the real part of that.
Multiplication, division, addition and subtraction are all straightforward. So are hyperbolic trig functions. All other trig functions can be derived as per above.
Yes, the point is to be 'helpful' and 'erudite' and to point out the "Significance Statement" rather than the patronizing "Read the paper" which is obviously completely beyond the faculties of 99% of of hn readers, and where the inclusion of the 'special section' is not going to be obvious. I missed it on first scan.
Dreadfully slow for integer math but probably some similar performance to something like a CORDIC for specific operations. If you can build an FPU that does exp() and ln() really fast, it's simple binary tree traversal to find the solution.
You already have an FPU that approximates exp() and ln() really fast, because float<->integer conversions approximate the power 2 functions respectively. Doing it accurately runs face-first into the tablemaker's dilemma, but you could do this with just 2 conversions, 2 FMAs (for power adjustments), and a subtraction per. A lot of cases would be even faster. Whether that's worth it will be situational.
If you've never worked through a derivation/explanation of the Y combinator, definitely find one (there are many across the internet) and work through it until the light bulb goes off. It's pretty incredible, it almost seems like "matter ex nihilo" which shouldn't work, and yet does.
It's one of those facts that tends to blow minds when it's first encountered, I can see why one would name a company after it.
> No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations.
I was taught that these were all hypergeometric functions. What distinction is being drawn here?
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.
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.
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).
It's basically using the "-" embedded in the definition of the eml operator.
Table 4 shows the "size" of the operators when fully expanded to "eml" applications, which is quite large for +, -, ×, and /.
Here's one approach which agrees with the minimum sizes they present:
eml(x, y ) = exp(x) − ln(y) # 1 + x + y
eml(x, 1 ) = exp(x) # 2 + x
eml(1, y ) = e - ln(y) # 2 + y
eml(1, exp(e - ln(y))) = ln(y) # 6 + y; construction from eq (5)
ln(1) = 0 # 7
After you have ln and exp, you can invert their applications in the eml function
eml(ln x, exp y) = x - y # 9 + x + y
Using a subtraction-of-subtraction to get addition leads to the cost of "27" in Table 4; I'm not sure what formula leads to 19 but I'm guessing it avoids the expensive construction of 0 by using something simpler that cancels:
I don't think this is ever making it past the editor of any journal, let alone peer review.
Elementary functions such as exponentiation, logarithms and trigonometric functions are
the standard vocabulary of STEM education. Each comes with its own rules and a dedicated button on a scientific calculator;
What?
and No comparable primitive has been known for continuous mathematics: computing elementary
functions such as sin, cos, √
, and log has always required multiple distinct operations.
Here we show that a single binary operator
Yeah, this is done by using tables and series. His method does not actually facilitate the computation of these functions.
There is no such things as "continuous mathematics". Maybe he meant to say continuous function?
Looking at page 14, it looks like he reinvented the concept of the vector valued function or something. The whole thing is rediscovering something that already exists.
The principal result is "all elementary functions can be represented by this function and constant 1". I'm not sure if this was known before. Applications are another matter, but I suspect interesting ones do exist.
This preprint was written by a researcher at an accredited university with a PhD in physics. I'm sure they know what a vector valued function is.
The point of this paper is not to revolutionize how a scientific calculator functions overnight, its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application, analogous to how a NAND or NOR gate creates all of the discrete logic gates. Hence, "continuous mathematics" as opposed to discrete mathematics. It seems to me you're being overly negative without solid reasoning.
its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application,
But he didn't show this though. I skimmed the paper many times. He creates multiple branches of these trees in the last page, so it's not truly a single nested operation.
Well, it is still the case, even if not explicitly shown. Personally I think it almost boils down to school math, with some details around complex logarithms; the rest seems to be simpler.
A stack of zeros and ones can be encoded in a single number by keeping with bit-shifting and incrementing.
I use something not too far off for my daily a programming based on a similar idea:Rejoice is a concatenative programming language in which data is encoded as multisets that compose by multiplication. Think Fractran, without the rule-searching, or Forth without a stack.
https://wiki.xxiivv.com/site/rejoice
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 :)
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
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.
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.
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.
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".
Because the EML basis makes simple functions (like +) hard to express.
Not to diminish this very cool discovery!
It seems like a neat parlour trick, indeed. But significant discovery?
> Submitted on 23 Mar 2026 (v1), last revised 4 Apr 2026 (this version, v2)
Reminds me of the Iota combinator, one of the smallest formal systems that can be combined to produce a universal Turing machine, meaning it can express all of computation.
x^-1 has the same problem.
both formulae work ...sort of... if we allow ln(0) = Infinity and some other moxie, such as x / Infinity = 0 for all finite x.
looks like it computes ln(1)=0, then computes e-ln(0)=+inf, then computes e-ln(+inf)=-inf
> EML-compiled formulas work flawlessly in symbolic Mathematica and IEEE754 floating-point… This is because some formulas internally might rely on the following properties of extended reals: ln 0 = −∞, e^(−∞) = 0.
And then follows with:
> But EML expressions in general do not work ‘out of the box’ in pure Python/Julia or numerical Mathematica.
Thus, the paper’s completeness claim depends on a non-standard arithmetic convention (ln(0) = -∞), not just complex numbers as it primarily advertises. While the paper is transparent about this, it is however, buried on page 11 rather than foregrounded as a core caveat. Your comment deserves credit for flagging it.
−z = 1 − (e − ((e − 1) − z))
That's awesome. I always wondered if there is some way to do this.
Quick google seach brings up https://github.com/pr701/nor_vm_core, which has a basic idea
``` look at this paper: https://arxiv.org/pdf/2603.21852
now please produce 2x+y as a composition on EMLs ```
Opus(paid) - claimed that "2" is circular. Once I told it that ChatGPT have already done this, finished successfully.
ChatGPT(free) - did it from the first try.
Grok - produced estimation of the depth of the formula.
Gemini - success
Deepseek - Assumed some pre-existing knowledge on what EML is. Unable to fetch the pdf from the link, unable to consume pdf from "Attach file"
Kimi - produced long output, stopped and asked to upgrade
GLM - looks ok
Simply because bool algebra doesn't have that many functions and all of them are very simple to implement.
A complex bool function made out of NANDs (or the likes) is little more complex than the same made out of the other operators.
Implementing even simple real functions out of eml() seems to me to add a lot of computational complexity even with both exp() and ln() implemented in hardware in O(1). I think about stuff sum(), div() and mod().
Of course, I might be badly wrong as I am not a mathematician (not even by far).
But I don't see, at the moment, the big win on this.
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 read the paper. Is there a table covering all other math operations translated to eml(x,y) form?
You can find this link on the right side of the arxiv page:
https://arxiv.org/src/2603.21852v2/anc/SupplementaryInformat...
Multiplication, division, addition and subtraction are all straightforward. So are hyperbolic trig functions. All other trig functions can be derived as per above.
Posts like these are the reason i check HN every day
It’s a derivation of the Y combinator from ruby lambdas
It's one of those facts that tends to blow minds when it's first encountered, I can see why one would name a company after it.
More on topic:
> No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations.
I was taught that these were all hypergeometric functions. What distinction is being drawn here?
Exp and ln, isn't the operation its own inverse depending on the parameter? What a neat find.
This is a function from ℝ² to ℝ. It can't be its own inverse; what would that mean?
Although you also need to encode where to put the input.
The real question is what emoji to use for eml when written out.
Some Emil or another, I suppose. Maybe the one from Ratatouille, or maybe this one: https://en.wikipedia.org/wiki/Emil_i_L%C3%B6nneberga
I'm kidding, of course. You can encode anything in bits this way.
See https://en.wikipedia.org/wiki/Elementary_function.
"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.
Table 4 shows the "size" of the operators when fully expanded to "eml" applications, which is quite large for +, -, ×, and /.
Here's one approach which agrees with the minimum sizes they present:
After you have ln and exp, you can invert their applications in the eml function Using a subtraction-of-subtraction to get addition leads to the cost of "27" in Table 4; I'm not sure what formula leads to 19 but I'm guessing it avoids the expensive construction of 0 by using something simpler that cancels: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?
For larger or negative inputs you get a NaN because ECMAScript has limited precision and doesn't handle imaginary numbers.
exp(a) = eml(a, 1) ln(a)=eml(1,eml(eml(1,a),1))
Plugging those in is an excercise to the reader
But no...
This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding.
Elementary functions such as exponentiation, logarithms and trigonometric functions are the standard vocabulary of STEM education. Each comes with its own rules and a dedicated button on a scientific calculator;
What?
and No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, √ , and log has always required multiple distinct operations. Here we show that a single binary operator
Yeah, this is done by using tables and series. His method does not actually facilitate the computation of these functions.
There is no such things as "continuous mathematics". Maybe he meant to say continuous function?
Looking at page 14, it looks like he reinvented the concept of the vector valued function or something. The whole thing is rediscovering something that already exists.
The point of this paper is not to revolutionize how a scientific calculator functions overnight, its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application, analogous to how a NAND or NOR gate creates all of the discrete logic gates. Hence, "continuous mathematics" as opposed to discrete mathematics. It seems to me you're being overly negative without solid reasoning.
But he didn't show this though. I skimmed the paper many times. He creates multiple branches of these trees in the last page, so it's not truly a single nested operation.