一个简单运算符可生成所有初等函数，让科学计算统一为树形结构。
All elementary functions from a single binary operator
- 用单一运算 eml(x,y)=exp(x)-ln(y) 构造所有初等函数
- 在深度4的树结构下可从数据中精确恢复公式
- 适合需要符号回归或公式发现的研究者
单个二元门足以实现数字硬件中的全部布尔逻辑。然而，在连续数学中，尚无类似通用原语：计算 sin、cos、sqrt、log 等初等函数始终需多个不同操作。本文证明，仅用一个二元运算 eml(x,y)=exp(x)-ln(y) 和常数1，即可生成标准科学计算器的所有功能。包括常数 e、π、i；加减乘除与幂运算；以及常见超越函数和代数函数。例如：exp(x)=eml(x,1)，ln(x)=eml(1,eml(eml(1,x),1))。该运算符的存在出乎意料，通过系统穷举搜索发现，并经构造性证明其完备性。所有表达式在EML（Exp-Minus-Log）形式下均为相同节点组成的二叉树，语法简洁如：S -> 1 | eml(S,S)。这种统一结构支持基于梯度的符号回归：使用Adam优化器训练EML树，可在树深不超过4时，从数值数据中精确恢复闭式初等函数。该架构虽能拟合任意数据，但当真实规律为初等函数时，可实现精确公式还原。
原文摘要 · Abstract (English)
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.
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