aiqbench analysis
AutoNumerics-Zero demonstrates that evolutionary symbolic regression can autonomously discover novel mathematical approximations for transcendental functions like the exponential, outperforming centuries-old Taylor series methods in efficiency and accuracy within finite-precision constraints. By optimizing for operation count rather than arbitrary precision, this approach yields compact programs that achieve superior significant figures without prior mathematical knowledge. This validates the potential of automated discovery techniques to redefine fundamental computational primitives in scientific computing.
AutoNumerics-Zero: Automated Discovery of State-of-the-Art Mathematical Functions
arXiv:2312.08472v2 Announce Type: replace-cross Abstract: Transcendental functions, such as the exponential, are central to scientific computing, yet they cannot be natively calculated by digital hardware. Instead, computers must approximate these functions by combining basic operations, such as $\{+, -, \times, \div\}$, using methods like Taylor series. These methods were developed over centuries by mathematicians, who focused on approaches that could attain arbitrary accuracy. However, computers can handle most applications by using only finite-precision types, like float32, where any accuracy beyond the type's precision is effectively discarded. We explore, therefore, whether forgoing arbitrary accuracy can lead to the discovery of more efficient approximations. The evolutionary method of symbolic regression is particularly suitable, as it can search for arbitrary operation combinations and can optimize non-differentiable objectives, such as the number of operations used. Our results show that evolution can discover computer programs that outperform established methods in this setting, despite having no prior mathematical knowledge beyond the calculation of the basic operations. Starting from empty code, symbolic regression constructs programs representing novel mathematical expressions. In particular, we discovered a 10-operation program that approximates the exponential function to 14 significant figures, exceeding the accuracy of previously known approximations of this size by more than 6 orders of magnitude.
Abstract:Transcendental functions, such as the exponential, are central to scientific computing, yet they cannot be natively calculated by digital hardware. Instead, computers must approximate these functions by combining basic operations, such as $\{+, -, \times, ÷\}$, using methods like Taylor series. These methods were developed over centuries by mathematicians, who focused on approaches that could attain arbitrary accuracy. However, computers can handle most applications by using only finite-precision types, like float32, where any accuracy beyond the type's precision is effectively discarded. We explore, therefore, whether forgoing arbitrary accuracy can lead to the discovery of more efficient approximations. The evolutionary method of symbolic regression is particularly suitable, as it can search for arbitrary operation combinations and can optimize non-differentiable objectives, such as the number of operations used. Our results show that evolution can discover computer programs that outperform established methods in this setting, despite having no prior mathematical knowledge beyond the calculation of the basic operations. Starting from empty code, symbolic regression constructs programs representing novel mathematical expressions. In particular, we discovered a 10-operation program that approximates the exponential function to 14 significant figures, exceeding the accuracy of previously known approximations of this size by more than 6 orders of magnitude.
Neural and Evolutionary Computing (cs.NE); Machine Learning (cs.LG); Numerical Analysis (math.NA)
I.2.2; I.2.6; G.1.2
arXiv:2312.08472 [cs.NE]
(or arXiv:2312.08472v2 [cs.NE] for this version)
https://doi.org/10.48550/arXiv.2312.08472 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 303 words
