Fast Mixed-Precision Real Evaluation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913985137213440 |
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| author | Yadrov, Artem Panchekha, Pavel |
| author_facet | Yadrov, Artem Panchekha, Pavel |
| contents | Evaluating real-valued expressions to high precision is a key building block in computational mathematics, physics, and numerics. A typical implementation evaluates the whole expression in a uniform precision, doubling that precision until a sufficiently-accurate result is achieved. This is wasteful: usually only a few operations really need to be performed at high precision, and the bulk of the expression could be computed much faster. However, such non-uniform precision assignments have, to date, been impractical to compute. We propose a fast new algorithm for deriving such precision assignments. The algorithm leverages results computed at lower precisions to analytically determine a mixed-precision assignment that will result in a sufficiently-accurate result. Our implementation, Reval, achieves an average speed-up of 1.72x compared to the state-of-the-art Sollya tool, with the speed-up increasing to 5.21x on the most difficult input points. An examination of the precisions used with and without precision tuning shows that the speed-up results from assigning lower precisions for the majority of operations, though additional optimizations enabled by the non-uniform precision assignments also play a role. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07468 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast Mixed-Precision Real Evaluation Yadrov, Artem Panchekha, Pavel Mathematical Software Numerical Analysis Evaluating real-valued expressions to high precision is a key building block in computational mathematics, physics, and numerics. A typical implementation evaluates the whole expression in a uniform precision, doubling that precision until a sufficiently-accurate result is achieved. This is wasteful: usually only a few operations really need to be performed at high precision, and the bulk of the expression could be computed much faster. However, such non-uniform precision assignments have, to date, been impractical to compute. We propose a fast new algorithm for deriving such precision assignments. The algorithm leverages results computed at lower precisions to analytically determine a mixed-precision assignment that will result in a sufficiently-accurate result. Our implementation, Reval, achieves an average speed-up of 1.72x compared to the state-of-the-art Sollya tool, with the speed-up increasing to 5.21x on the most difficult input points. An examination of the precisions used with and without precision tuning shows that the speed-up results from assigning lower precisions for the majority of operations, though additional optimizations enabled by the non-uniform precision assignments also play a role. |
| title | Fast Mixed-Precision Real Evaluation |
| topic | Mathematical Software Numerical Analysis |
| url | https://arxiv.org/abs/2410.07468 |