{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T13:35:35Z","timestamp":1790688935940,"version":"4.1.0"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100001665","name":"French National Agency for Research","doi-asserted-by":"crossref","award":["(ANR-20-CE46-0009)"],"award-info":[{"award-number":["(ANR-20-CE46-0009)"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Stochastic rounding (SR) offers an alternative to the deterministic IEEE-754 floating-point rounding modes. In some applications such as PDEs, ODEs, and neural networks, SR empirically improves the numerical behavior and convergence to accurate solutions while the theoretical background remains partial. Recent works by Ipsen, Zhou, Higham, and Mary have computed SR probabilistic error bounds for basic linear algebra kernels. For example, the inner product SR probabilistic bound of the forward error is proportional to [Formula: see text] instead of [Formula: see text] for the default rounding mode. To compute the bounds, these works show that the errors accumulated in computation form a martingale. This paper proposes an alternative framework to characterize SR errors based on the computation of the variance. We pinpoint common error patterns in numerical algorithms and propose a lemma that bounds their variance. For each probability and through the Bienaym\u00e9\u2013Chebyshev inequality, this bound leads to a better probabilistic error bound in several situations. Our method has the advantage of providing a tight probabilistic bound for all algorithms fitting our model. We show how the method can be applied to give SR error bounds for the inner product and Horner polynomial evaluation.<\/jats:p>","DOI":"10.1137\/22m1510819","type":"journal-article","created":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T06:55:08Z","timestamp":1696488908000},"page":"C255-C275","source":"Crossref","is-referenced-by-count":9,"title":["Stochastic Rounding Variance and Probabilistic Bounds: A New Approach"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5057-0170","authenticated-orcid":true,"given":"El-Mehdi","family":"El Arar","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris-Saclay, UVSQ, Li-PaRAD, Saint-Quentin, en Yvelines, France."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Devan","family":"Sohier","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Saclay, UVSQ, Li-PaRAD, Saint-Quentin, en Yvelines, France."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pablo","family":"de Oliveira Castro","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Saclay, UVSQ, Li-PaRAD, Saint-Quentin, en Yvelines, France."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eric","family":"Petit","sequence":"additional","affiliation":[{"name":"Intel Corporation, Oregon, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,10,5]]},"reference":[{"key":"ref1","unstructured":"Graphcore, Mixed Precision Arithmetic for AI: A Hardware Perspective, https:\/\/docs.graphcore.ai\/projects\/ai-float-white-paper\/en\/latest\/index.html (25 August 2021)."},{"key":"ref2","unstructured":"G. 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