1. Title of the Paper AutoNumerics-Zero: Automated Discovery of State-of-the-Art Mathematical Functions 2. Author Information Esteban Real Google DeepMind, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA ereal@google.com 1-650-253-0000 Mirko Rossini Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA mirkorossini@google.com [phone number to be shared separately upon request] Connal de Souza Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA connaldesouza@google.com [phone number to be shared separately upon request] Manav Garg Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA manavgarg@google.com [phone number to be shared separately upon request] Moritz Firsching Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA firsching@google.com [phone number to be shared separately upon request] Quoc V. Le Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA qvl@google.com [phone number to be shared separately upon request] Yao Chen Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA yaoch@google.com [phone number to be shared separately upon request] Akhil Verghese Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA averghese@google.com [phone number to be shared separately upon request] Ekin Dogus Cubuk Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA cubuk@google.com [phone number to be shared separately upon request] David H. Park Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA davidhpark@google.com [phone number to be shared separately upon request] 3. Corresponding Author Esteban Real ereal@google.com 4. Paper 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 {+, -, *, /}, 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. 5. Criteria Satisfied (C) The result is better than the most recent human-created solution to a long-standing or previously unsolved problem of indisputable difficulty in its field. 6. Statement Why Criterion (C) is Satisfied Transcendental functions, such as the exponential, are pervasive in scientific computing. Their numerical calculation, therefore, is a fundamental problem in applied mathematics that has been studied for centuries. Despite this extensive history, mathematical approximations discovered by AutoNumerics-Zero satisfy Criterion (C) by outperforming all known approximations, including human-designed ones. Standard human approximations (such as truncated Taylor series) rely on fixed algebraic templates (such as polynomial forms). In contrast, our method searches the unconstrained space of all computer programs, discovering novel algebraic structures that reuse intermediate computations. Specifically, for the exponential function, AutoNumerics-Zero discovered a 10-operation program that achieves 14 significant figures of accuracy. This program exceeds the accuracy of previously known approximations of this size by more than 6 orders of magnitude. We established this result by comparing against every baseline we could identify, including Taylor expansions, Pade approximants, Minimax/Remez, Chebyshev polynomials, and continued fractions due to Euler, Gauss, and Macon. Our paper guarantees this claim through rigorous mathematical proofs of error bounds. Beyond that 10-operation program, our paper demonstrates AutoNumerics-Zero more generally: not only did it consistently achieve higher accuracies across all other discovered operation sizes for the exponential (Figure 4), but it also discovered state-of-the-art approximations across several distinct functions. Namely, we evaluated cases that illustrate challenges to asymptotic expansion, such as oscillatory behavior (Airy function) and a boundary singularity (modified Bessel function). Ultimately, these results demonstrate that by optimizing directly for practical finite-accuracy targets, unconstrained program search can systematically discover more efficient algorithms than traditional mathematical derivation. 7. Full Citation Esteban Real, Mirko Rossini, Connal de Souza, Manav Garg, Moritz Firsching, Quoc V. Le, Yao Chen, Akhil Verghese, Ekin Dogus Cubuk, and David H. Park. "AutoNumerics-Zero: Automated Discovery of State-of-the-Art Mathematical Functions." To appear in the Proceedings of the 43rd International Conference on Machine Learning (ICML 2026), Seoul, South Korea. Published by Proceedings of Machine Learning Research (PMLR). 8. Prize Money Breakdown Any prize money, if any, is to be divided equally among the co-authors. 9. Required Statement Indicating Why this Entry Could be the "Best" As detailed in Question 6, the mathematical programs automatically discovered by AutoNumerics-Zero outperform the best-known baselines in computing several transcendental functions. Beyond these direct numerical results, we highlight the following aspects of this entry. a) Rigorous mathematical verification: Unlike standard symbolic regression, we do not rely solely on empirical testing. We mathematically prove guaranteed global error bounds for our most salient discovered programs. Importantly, these bounds target the maximum relative error, the accepted practice in the numerical literature (and more stringent than the average error). The proof strategy combines mathematical analysis with automated provers to account for intermediate rounding, guaranteeing forward stability. b) Amortization of compute: While the evolutionary search incurs an upfront compute cost, this one-time offline investment can be amortized by repeated reuse of discovered results. As a practical extension of our core findings above, we also demonstrate in the paper that this search method can be specialized to discover hardware-aware programs: in that case, the programs maximize execution speed on a target CPU architecture while guaranteeing less than 1 ULP of error (the accepted floating-point numerics standard). Such gains may prove helpful in high-performance scientific computing where transcendental functions are evaluated trillions of times. c) Zero-knowledge program discovery: Unlike superoptimization methods that optimize existing implementations, AutoNumerics-Zero starts from empty code to automate the algebraic discovery of mathematical forms. It searches on bounded intervals (subsequently generalized via range reduction) without human mathematical heuristics, asymptotic expansions, or derivatives. The search uses simple random mutations (e.g., "inserting a random operation at a random point") and only basic operations (e.g., limited to {+,-,*,/}). In particular, it does not train on a data corpus, providing a complementary approach to LLM methods. Thus, the search process is not biased by known solutions, leading to novel, compact, and stable programs that reuse intermediate computations for efficiency. In this sense, this entry demonstrates a pure realization of the power of evolutionary computation. More generally, AutoNumerics-Zero aims to establish a new paradigm for the construction of numerical approximations, based on symbolic regression. For centuries, mathematicians have focused on manual derivations capable of arbitrary accuracy, a requirement that modern computing typically discards. By instead optimizing directly for finite-accuracy targets, this entry demonstrates that evolutionary search can systematically exploit this paradigm shift to discover more efficient solutions. 10. Evolutionary Computation Type Symbolic Regression, combining Genetic Programming (GP) and Evolution Strategies (ES). 11. Publication Date In press. This paper has been unconditionally accepted for publication at ICML 2026. The paper attached here is the final camera-ready version, which has already been submitted to the conference. The conference no longer accepts modifications to the camera ready version, as per publicly visible ICML rules. Thus, this entry fulfills the "in press" condition. As evidence of the paper's acceptance, the corresponding author (ereal at google dot com) is forwarding to the organizers (goodman at msu dot edu) an email with subject "Humies entry: AutoNumerics-Zero - evidence of in-press status". This forwarded email, originally written by the ICML organizers, shows the paper's acceptance. Further, we expect that the conference will imminently make the paper's review visible to the general public through OpenReview, which will confirm again the paper's accepted status.