A Lightweight Rational Approximation for Inverse Square Root with Newton-Compatible Initialization
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2026
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| _version_ | 1866901743123562496 |
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| author | Trinity Labo |
| author_facet | Trinity Labo |
| contents | <div>This work presents a lightweight rational approximation for the inverse square root function together with a Newton-compatible initialization strategy. The method is built around a low-order rational approximation on the normalized interval [1, 2], combined with exact power-of-two scaling to cover the practical range [0.25, 4].</div> <div> </div> <div>The construction is designed for low arithmetic cost, simple implementation, and predictable error behavior. On the normalized interval, the proposed approximation achieves an empirical maximum relative error of about 3.43e-4, while preserving a one-sided error structure that is useful for stable numerical handling. After a single Newton refinement step, the error is reduced to the 1e-7 scale.</div> <div> </div> <div>Rather than targeting formal optimality, this note focuses on practical usefulness: compact evaluation, explicit coefficients, and direct compatibility with fast inverse-square-root workflows. Potential applications include graphics, physics simulation, ray tracing, distance computation, and lightweight numerical kernels.</div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19019977 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Lightweight Rational Approximation for Inverse Square Root with Newton-Compatible Initialization Trinity Labo <div>This work presents a lightweight rational approximation for the inverse square root function together with a Newton-compatible initialization strategy. The method is built around a low-order rational approximation on the normalized interval [1, 2], combined with exact power-of-two scaling to cover the practical range [0.25, 4].</div> <div> </div> <div>The construction is designed for low arithmetic cost, simple implementation, and predictable error behavior. On the normalized interval, the proposed approximation achieves an empirical maximum relative error of about 3.43e-4, while preserving a one-sided error structure that is useful for stable numerical handling. After a single Newton refinement step, the error is reduced to the 1e-7 scale.</div> <div> </div> <div>Rather than targeting formal optimality, this note focuses on practical usefulness: compact evaluation, explicit coefficients, and direct compatibility with fast inverse-square-root workflows. Potential applications include graphics, physics simulation, ray tracing, distance computation, and lightweight numerical kernels.</div> |
| title | A Lightweight Rational Approximation for Inverse Square Root with Newton-Compatible Initialization |
| url | https://doi.org/10.5281/zenodo.19019977 |