One Button's Magic: How the EML Operator Tames All Math Functions

Imagine a scientific calculator covered in buttons: sin, cos, tan, log, √, π, e... Each button represents an independent set of mathematical rules. Now imagine someone telling you: all those buttons are redundant.

You only need two—a key labeled "EML" and the constant "1"—to do everything the calculator can do. Compute a sine? Press EML a few times. Square root? EML. The value of π? EML again.

It sounds like magic, but it's real.

The "Broken Calculator" Problem

Mathematics has a fun puzzle called the "broken calculator": if most of your calculator's keys are broken, can you still compute what you need?

Andrzej Odrzywołek of Jagiellonian University in Poland pushed this question to its limit. He wasn't satisfied with mild versions like "only four arithmetic operations" or "only exponentials and logarithms." He wanted to know: how few operations suffice?

The answer: one binary operator plus one constant.

The operator is called EML ("Exp-Minus-Log"):

eml(x, y) = exp(x) - ln(y)

That's it: e to the power x, minus the natural log of y. Together with the constant 1, this is the entire toolkit.

How the Magic Works

Exponential function

The most direct case:

exp(x) = eml(x, 1)

Since ln(1) = 0, eml(x, 1) = exp(x). One step.

Natural logarithm

Slightly more involved, requiring four levels of nesting:

ln(z) = eml(1, eml(eml(1, z), 1))

The inner layer is eml(1, z) = e¹ - ln(z) = e - ln(z); wrapped again... trust the algebra—it does equal ln(z).

Multiplication

Multiplication needs nesting of depth 8. The logic: with exp and ln available, the logarithmic identity turns multiplication back into addition:

x × y = exp(ln(x) + ln(y))

Since EML can produce both exp and ln, it can produce multiplication.

Constants: e, π, i

Even mathematical constants emerge from EML and 1:

• e = eml(1, 1)
• π can be reached via ln(-1) = iπ
• i (the imaginary unit) hides in the branch choice of the complex logarithm

All of this from eml(x,y) = exp(x) - ln(y) and the number 1.

Why It Matters

1. A "minimal generating set" for mathematics

In digital circuits, there's a classic result: all Boolean logic can be built from a single NAND gate. But for continuous mathematics—real and complex numbers—people long assumed no such "universal brick" existed. You seemed to need addition, multiplication, exponentials, logs, trig functions... each playing its own role.

Odrzywołek's finding shows: continuous math has its NAND gate too. EML is the single atomic operation; everything else is a derivative of it.

2. A unified data structure

In the EML world, every mathematical expression becomes a binary tree. Every node has the same shape—an EML operation—differing only in leaves (some are 1, some are input variables).

All elementary functions share one grammar:

S → 1 | eml(S, S)

This is the unified representation computer scientists dream of: sin(x), √x, e^x are all instances of the same structure in EML form, opening new doors for symbolic computation, formula discovery, and code optimization.

3. Implications for AI

The paper's second half applies EML trees to symbolic regression: given data points, find the formula that generates them. Traditional methods search a huge function space; the unified EML structure compresses it dramatically.

Using the standard Adam optimizer (the deep learning workhorse) on EML trees of depth at most 4, the author recovered exact closed-form elementary functions from numerical data. Even better, the same architecture can fit arbitrary data, but when the underlying law really is an elementary function, it automatically discovers the exact formula.

This evokes the question of activation function choice in neural networks. ReLU became popular partly because it's simple, differentiable, and composable. Could EML become a new "differentiable computational primitive," especially in applications needing symbolic interpretability?

The Intuition Behind the Math

Why exp(x) - ln(y) of all things?

Odrzywołek says he found it through "systematic exhaustive search." But there is intuition:

exp and ln are inverse operations, like addition/subtraction or multiplication/division. When they appear in the same expression, they create tension—one grows, the other shrinks—and that asymmetry may be the source of rich behavior.

Subtraction is non-commutative: eml(x,y) ≠ eml(y,x). This gives expression trees "directionality": the same nodes in different orders produce different results. That freedom is essential for building complex functions.

The paper also mentions two "cousins" of EML:

• EDL: exp(x) / ln(y), requiring the constant e
• -EML: ln(x) - exp(y), requiring the constant -∞

So EML isn't the only answer—but it's the simplest, needing only the constant 1.

Some Reflections

Teaching possibilities

If EML ever went mainstream, what would math education look like? Imagine schoolchildren not memorizing multiplication tables or wrangling sin and cos, but learning a "universal language of computation" where every operation is an arrangement of one basic brick.

That's radical, but intriguing. Real teaching must respect cognitive development—concrete before abstract—but EML at least provides a conceptual anchor: whatever function you learn, remember it's a different face of the same root.

The philosophy of computation

There's also a philosophical point: the relationship between simplicity and complexity.

We assume complex things (a calculator's full functionality) need complex parts (36 buttons). But mathematics keeps telling us: complexity often emerges from the iterated application of simple rules. From Conway's Game of Life to the Mandelbrot set, from NAND gates to the EML operator, the world seems to love generating "much" from "little."

It recalls physicists' urge toward a "theory of everything." EML isn't physics' theory of everything, but it displays a similar minimalist aesthetic: one primitive, explaining it all.

About the Author

Andrzej Odrzywołek is a researcher at the Institute of Theoretical Physics, Jagiellonian University, with a background spanning astrophysics, general relativity, and computational mathematics. Previous work covers black hole accretion, supernova explosions, neutrino astronomy, and even a playful paper on "how to build the perfect igloo."

The EML paper extends his line of research on "numerical constant identification"—he has long been interested in reverse-engineering formulas from numbers, including a 2020 arXiv preprint on standards for identifying numerical constants.

Interestingly, the discovery was entirely accidental. The paper states: "The existence of such an operator was not anticipated; I found it by systematic exhaustive search."

Many major breakthroughs in science happen this way: not derived from theory, but popping out of "let's try it" computer searches.

Conclusion

Next time you pick up a scientific calculator, or call Math.sin() or Math.log() in code, consider: these seemingly independent functions can all grow out of one simple combination.

eml(x, y) = exp(x) - ln(y)

One formula, plus the number 1. That's everything.

Mathematics' beauty of simplicity sometimes hides in exactly such unexpected places.

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Reference paper:

• Odrzywołek, A. (2026). *All elementary functions from a single binary operator*. arXiv:2603.21852
• Paper link: https://arxiv.org/abs/2603.21852

Further reading:

• The Sheffer stroke (the NAND-gate equivalent in Boolean logic)
• Wolfram's search for a "theory of everything"
• Symbolic regression and AI-driven scientific discovery

*"The true sign of understanding something is being able to explain it in simple language."*

*— A physicist who liked playing the bongo drums*