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arXiv · 2509.05666

Accuracy of Mathematical Functions in Julia

Mantas Mikaitis·Mosè Giordano·Tejaswa Rizyal
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Abstract

Basic computer arithmetic operations, such as $+$, $\times$, or $\div$ are correctly rounded, whilst mathematical functions such as $e^x$, $\ln(x)$, or $\sin(x)$ in general are not, meaning that separate implementations may provide different results when presented with an exact same input, and that their accuracy may differ. We present a methodology and a software tool that is suited for exhaustive and non-exhaustive testing of mathematical functions available as part of the Julia programming language, in various floating-point formats. The software tool is useful to the users of Julia, to quantise the level of accuracy of the mathematical functions and interpret possible effects of errors on their scientific computation codes that depend on these functions. It is also useful to the developers and maintainers of the functions in Julia Base, to test the modifications to existing functions and to test the accuracy of new functions. The software (a test bench) is designed to be easy to set up for running the accuracy tests in automatic regression testing. Our focus is to provide software that is user friendly and allows to avoid the need for specialised knowledge of floating-point arithmetic or the workings of mathematical functions; users only need to supply a list of formats, choose the rounding modes, and specify the input space search strategies based on how long they can afford the testing to run. We have utilized the test bench to determine the errors of a subset of mathematical functions in Julia 1.12.7, for binary16, binary32, and binary64 IEEE 754 floating-point formats, and found $0.49$ to $0.51$ULPs in binary16, and $0.49$ to $2.4$ULPs of error in binary32 and binary64.

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Mantas Mikaitis, Mosè Giordano, Tejaswa Rizyal. 2025-09-06. Accuracy of Mathematical Functions in Julia. https://arxiv.org/abs/2509.05666

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