One binary operator replaces every calculator button

You know that elementary functions — sin, cos, log, sqrt, exponentiation — each need their own dedicated implementation in hardware, libraries, and calculators. They seem like fundamentally different operations. Boolean logic solved this decades ago with NAND: one gate type builds every digital circuit. Continuous math had no equivalent. You carry a bag of 36+ distinct primitives when, theoretically, far fewer might suffice — but nobody knew the minimum.

Define eml(x,y) = exp(x) − ln(y). Feed it the constant 1 as a seed. Iteratively bootstrap: eml(1,1) = e, eml(x,1) = exp(x), eml(1, eml(eml(1,x),1)) = ln(x), and so on through subtraction, negation, reciprocal, multiplication, division, trigonometric functions via Euler's formula, and all 36 standard calculator operations. The search used algebraically independent transcendental constants (Euler-Mascheroni γ, Glaisher-Kinkelin A) as test inputs with an inverse symbolic calculator to discover candidate formulas. The grammar reduces to S → 1 | eml(S,S) — every expression is a binary tree of identical nodes.

• Single-operator universality — eml(x,y) = exp(x) − ln(y) plus constant 1 replaces all 36 elementary calculator primitives, giving you a minimal grammar S → 1 | eml(S,S) for every elementary expression
• Gradient-based symbolic regression — parameterized EML trees trained with Adam optimizer can recover exact closed-form elementary functions from numerical data at tree depths up to 4
• Cousin operators exist — EDL(x,y) = exp(x)/ln(y) with constant e, and ln(x) − exp(y) with constant −∞, also form complete bases, showing EML is not unique
• Constructive proof — the paper provides explicit EML expressions for all 36 primitives, verifiable in Mathematica, IEEE754 C, NumPy, and PyTorch
• Analog circuit mapping — every EML expression is a uniform binary tree of identical nodes, directly mappable to analog circuits with a single EML gate element

Who It Is For

If you work in symbolic computation, computer algebra, or theoretical mathematics, this is a foundational result worth studying. If you build numerical libraries or scientific computing tools, the symbolic regression angle (trainable EML trees for formula discovery) is the most immediately relevant application. Not useful if you need efficient numerical computation — the expression blow-up makes ...

Worth Exploring?

Worth reading for the theoretical result alone — it is genuinely surprising that a single binary operator suffices for all elementary math. The symbolic regression application (Section 4.3) is experimental: the author demonstrates exact recovery of elementary functions at shallow tree depths, but the EML compiler itself is described as an "unoptimized prototype" and direct searches timed out for several functions. This is an early-stage research result, not a production tool. Watch for follow-up work on the ternary variant that requires no constant.