All elementary functions from a single binary operator

2026/03/23 by Andrzej Odrzywołek · 58 voices · 1 citation
#cs.SC #cs.LG

paper · pdf

Abstract

A single two-input gate suffices for all of Boolean logic in digital hardware. 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. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

Citations

• KAN: Kolmogorov-Arnold Networks
• An aperiodic monotile
• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• An algorithm for the machine calculation of complex Fourier series

Cited by

• EML-AirComp: Layered Over-the-Air Computation from a Single Nomographic Gate

Discussions

• All elementary functions from a single binary operator [hn, 858 points, 298 comments]
• Physicist has written a fascinating big beautiful paper.Let’s not be afraid to call it what it is - groundbreaking. arxiv.org/abs/2603.21852 [bsky, 376 points, 19 comments]
• All elementary functions from a single binary operator [lobsters, 57 points, 7 comments]
• This is so cool! Given only the "exponential minus log" function, elm(x,y)=exp(x)-ln(y), and the constant 1, you can perform +, -, x, ÷, exp, ln, trig, powers, roots, etc. You can also obtain e and π! [bsky, 35 points, 2 comments]
• This paper claims you can define all elementary functions using just two primitive operations: the constant 1, and the binary function eml(x, y) := e^x - log(y). It's probably true but it may need wor [bsky, 26 points, 1 comments]
• All elementary functions from a single binary operator arxiv.org/abs/2603.21852 Amusant ! [bsky, 16 points, 1 comments]
• Extremely cool paper 😎 All elementary functions from a single binary operator arxiv.org/abs/2603.21852 [bsky, 9 points, 0 comments]
• context [bsky, 6 points, 0 comments]
• 2/2 arxiv.org/abs/2603.21852 [bsky, 5 points, 0 comments]
• The ramifications of this paper for foundations of physics are mind blowing arxiv.org/abs/2603.218... [bsky, 3 points, 0 comments]
• Dá-me pena não entender nada de matemáticas porque esta fim de semana dous amigos meus estavam flipando com isto. arxiv.org/abs/2603.21852 [bsky, 3 points, 1 comments]
• @cubiclogic.bsky.social Ένας και μόνο μαθηματικός τελεστής αρκεί για να προκύψουν όλες οι βασικές μαθηματικές συναρτήσεις και σταθερές arxiv.org/pdf/2603.218... [bsky, 2 points, 1 comments]
• PS: Here's something you might like: arxiv.org/abs/2603.21852 [bsky, 2 points, 2 comments]
• Planning on building a scientific calculator with only two buttons. arxiv.org/pdf/2603.21852 [bsky, 2 points, 1 comments]
• All elementary functions from a single binary operator [bsky, 1 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 [bsky, 1 points, 1 comments]
• eml(x, y) = exp(x) − ln(y) 面白そうな論文。数学におけるNANDゲートってeml(x, y)と定数1の組み合わせで表せるぜって主張らしい e^x = eml(x, 1) ln(x) = eml(1, eml(eml(1, x), 1)) というような具合 arxiv.org/abs/2603.21852 >RT: 2043756177083826377 [bsky, 1 points, 1 comments]
• みてた arxiv.org/abs/2603.21852 [bsky, 1 points, 0 comments]
• While NAND gives all boolean operators, arxiv.org/pdf/2603.21852 derives all elementary functions from eml(x,y) = exp(x) - ln(y) [bsky, 1 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 https://news.ycombinator.com/item?id=47746610 [bsky, 1 points, 1 comments]
• emlと真理保存性について考えるんだけど、eml は S→1 ∣ eml(S,S) という形で、真理保存性の問題に不動点としての構造を与えたことになる気がするんだけどどうなんだろう arxiv.org/pdf/2603.21852 www.jstage.jst.go.jp/article/jpss... [bsky, 1 points, 1 comments]
• Daily Dystopia Break: Mmmmmm Math arxiv.org/abs/2603.21852 [bsky, 0 points, 1 comments]
• 🌐単一の二項演算子から得られるすべての基本関数 https://arxiv.org/abs/2603.21852 via #HackerNews [bsky, 0 points, 0 comments]
• arxiv.org/abs/2603.21852 すごい でもこれがうつくしいかと言うと [bsky, 0 points, 2 comments]
• All elementary functions from a single binary operator https://lobste.rs/s/rnx6fh #math [bsky, 0 points, 0 comments]
• One [binary operator] to rule them all, One [binary operator] to find them, One [binary operator] to bring them all, And in the darkness bind them Pretty cool #computerscience advancement, actually. N [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 (http://news.ycombinator.com/item?id=47746610) [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 (http://news.ycombinator.com/item?id=47746610) [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 (http://news.ycombinator.com/item?id=47746610) [bsky, 0 points, 0 comments]
• 2項演算子と定数からよく知られる初等関数や定数（超越数を含む）を導出できるというアイデアが話題らしい。eml(x, y) = exp(x) - ln(y)、edl(x, y) = exp(x) / ln(y)、-eml(y, x) = ln(x) - exp(y)について、{eml, 1}、{edl, e}、{-eml, -∞}がそれにあたるとのこと。著者は論理演算にとってのNANDにたとえている [bsky, 0 points, 0 comments]
• https://arxiv.org/abs/2603.21852 単一の2入力演算子 eml(x,y)=exp(x)-ln(y) と定数1だけで、科学計算の基本関数を生成できると示しています。 e、π、iに加え、四則演算や指数・対数・三角関数まで表せます。 EML木は記号回帰にも使え、浅い深さなら数値データから厳密な式復元が可能です。 [bsky, 0 points, 0 comments]
• Discover how a single binary operator can generate all elementary functions! This innovative approach could reshape our understanding of mathematical functions and their interrelations. Dive into the [bsky, 0 points, 0 comments]
• 🚨🚨🚨Breaking: Our world was created by a Monkey god, randomly typing E and 1. 🤷‍♂️ One operator is all you need All elementary functions from a single operator arxiv.org/pdf/2603.21852 @drbriankeat [bsky, 0 points, 1 comments]
• All elementary functions from a single binary operator view on hacker news [bsky, 0 points, 0 comments]
• Tout définir avec elm(x,y) = exp(x) - ln(y) ? https://arxiv.org/abs/2603.21852v2 #math [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https:// arxiv.org/abs/2603.21852 # arxiv [mastodon, 0 points, 0 comments]
• All elementary functions from a single binary operator https:// arxiv.org/abs/2603.21852 # HackerNews # elementaryfunctions # binaryoperator # mathresearch # arxiv # innovation # programming [mastodon, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 (http://news.ycombinator.com/item?id=47746610) [bsky, 0 points, 0 comments]
• (1,E(E(1,E(1,E(E(1,1),1))),1)),E(E(1,E(E(1,E(1,E(E(1,E(E(1,E(E(1,E(1,E(E(1,1),1))),1)),E(1,1))),1))),1)),1)),1)),1),1),1))),1))),1)),E(E(E(1,E(E(1,E(1,E(E(1,1),1))),1)),E(E(1,E(E(1,E(1,E(E(1,E(E(1,E(E [bsky, 0 points, 0 comments]
• arxiv.org/abs/2603.21852 => github.com/oaustegard/e... [bsky, 0 points, 1 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 (https://news.ycombinator.com/item?id=47746610) [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 (https://news.ycombinator.com/item?id=47746610) [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 (https://news.ycombinator.com/item?id=47746610) [bsky, 0 points, 0 comments]
• Andrzej Odrzywo{\l}ek: All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 https://arxiv.org/pdf/2603.21852 https://arxiv.org/html/2603.21852 [bsky, 0 points, 0 comments]
• 'A single two-input gate suffices for all of Boolean logic in digital hardware.' arxiv.org/abs/2603.21852 [bsky, 0 points, 1 comments]
• 1/2 Un solo operador binario genera todas las funciones elementales Investigadores demuestran que con un único operador binario es posible expresar cualquier función elemental (suma, producto, exponen [bsky, 0 points, 1 comments]
• @davidbessis.bsky.social I'm listening to your book at the moment, good stuff! Can you help me understand why arxiv.org/abs/2603.21852 is dumb? [bsky, 0 points, 1 comments]
• https://bsky.app/profile/hackernews.com.web.brid.gy/post/3mjebmxzt3gf2 [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator #HackerNews https://arxiv.org/abs/2603.21852 [bsky, 0 points, 0 comments]
• arxiv.org/pdf/2603.21852 [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 https://news.ycombinator.com/item?id=47746610 [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 comments #arxiv.org [bsky, 0 points, 0 comments]
• All elementary functions from a single binary operator View Article | Join the HN Conversation Summary of HN discussion 🧵👇 [bsky, 0 points, 1 comments]
• All elementary functions from a single binary operator https://arxiv.org/abs/2603.21852 [comments] [344 points] [bsky, 0 points, 0 comments]
• 📰 All elementary functions from a single binary operator 🔗 https://arxiv.org/abs/2603.21852 💬 Discuss on HN [bsky, 0 points, 0 comments]
• "Here we show that the operator eml(x, y) = exp(x) − ln(y), together with the constant 1, does exactly that: it reconstructs arithmetic, all standard elementary transcendental functions, and constants [bsky, 0 points, 0 comments]
• When your calculator only needs just two buttons, EML(x,y) = exp(x)- ln (y) and the digit 1 for calculate everything a full scientific calculator does. arxiv.org/abs/2603.21852 [bsky, 0 points, 0 comments]

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