Title: AutoNumerics-Zero: Automated Discovery of State-of-the-Art Mathematical Functions

Abstract:

Transcendental functions, including the exponential function, are fundamental to scientific computing. However, digital hardware lacks native support for these calculations, forcing computers to approximate them through combinations of basic arithmetic operations ${+, -, \times, \div}$. Historically, mathematicians have devised methods like Taylor series over centuries, prioritizing the ability to achieve arbitrary levels of precision. In contrast, modern computing environments typically rely on finite-precision data types, such as float32, where any accuracy exceeding the specific limits of the type is effectively lost. This study investigates whether relaxing the requirement for arbitrary precision can facilitate the discovery of more efficient approximations.

We employ symbolic regression, an evolutionary approach that is well-suited for this task because it can explore arbitrary combinations of operations and optimize non-differentiable objectives, such as minimizing the total number of operations. Our findings demonstrate that this evolutionary process can identify computer programs that surpass established methods, even when starting with no prior mathematical knowledge other than basic arithmetic. Beginning with empty code, symbolic regression generates programs that represent new mathematical expressions. Notably, we identified a 10-operation program that approximates the exponential function to 14 significant figures. This result surpasses the accuracy of previously known approximations of similar complexity by more than six orders of magnitude.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC