## Kahan’s algorithm for a correct discriminant computation at last formally proven, in

Venue: | n o 2, February 2009 |

Citations: | 6 - 3 self |

### BibTeX

@INPROCEEDINGS{Boldo_kahan’salgorithm,

author = {Sylvie Boldo},

title = {Kahan’s algorithm for a correct discriminant computation at last formally proven, in},

booktitle = {n o 2, February 2009},

year = {},

pages = {220--225}

}

### OpenURL

### Abstract

Abstract—This article tackles Kahan’s algorithm to compute accurately the discriminant. This is a known difficult problem, and this algorithm leads to an error bounded by 2 ulps of the floating-point result. The proofs involved are long and tricky and even trickier than expected as the test involved may give a result different from the result of the same test without rounding. We give here the total demonstration of the validity of this algorithm, and we provide sufficient conditions to guarantee that neither overflow nor underflow will jeopardize the result. The IEEE-754 double-precision program is annotated using the Why platform and the proof obligations are done using the Coq automatic proof checker. Index Terms—Floating point, discriminant, formal proof, Why platform, Coq.

### Citations

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(Show Context)
Citation Context ...sion with correct rounding [4], . Horner’s rule under assumptions (such as elementary function evaluation) [5], . accurate summation under tough assumptions (all numbers are nonnegative, for example) =-=[6]-=-, and . accurate discriminant computation [7]. We here are interested in the last algorithm. The reason is that the pen-and-paper proofs provided are described as “far longer and trickier” than the al... |

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Citation Context ...ified computation to get correct, accurate, or consistent results. This category includes, for example . expansion algorithms that allow multiprecision computations using only the floating-point unit =-=[1]-=-, [2], . CRlibm that computes correctly rounded elementary functions on IEEE double precision [3], . multiprecision algorithms on higher precision with correct rounding [4], . Horner’s rule under assu... |

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Citation Context ...bility in the proof of the correctness of this program. This is the reason why we guarantee it using formal proofs. The formalization of the floating-point numbers and arithmetic is the following one =-=[9]-=-: a float is a pair of signed integers ðn; eÞ with both n and e bounded and has a value equal to n 2e . This formalization has permitted proving both old and new results [10]. As we want to prove the ... |

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Citation Context ...lementary functions on IEEE double precision [3], . multiprecision algorithms on higher precision with correct rounding [4], . Horner’s rule under assumptions (such as elementary function evaluation) =-=[5]-=-, . accurate summation under tough assumptions (all numbers are nonnegative, for example) [6], and . accurate discriminant computation [7]. We here are interested in the last algorithm. The reason is ... |

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Citation Context ...le under assumptions (such as elementary function evaluation) [5], . accurate summation under tough assumptions (all numbers are nonnegative, for example) [6], and . accurate discriminant computation =-=[7]-=-. We here are interested in the last algorithm. The reason is that the pen-and-paper proofs provided are described as “far longer and trickier” than the algorithms and programs and Kahan deferred thei... |

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Citation Context ... computation to get correct, accurate, or consistent results. This category includes, for example . expansion algorithms that allow multiprecision computations using only the floating-point unit [1], =-=[2]-=-, . CRlibm that computes correctly rounded elementary functions on IEEE double precision [3], . multiprecision algorithms on higher precision with correct rounding [4], . Horner’s rule under assumptio... |

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Citation Context ...ample . expansion algorithms that allow multiprecision computations using only the floating-point unit [1], [2], . CRlibm that computes correctly rounded elementary functions on IEEE double precision =-=[3]-=-, . multiprecision algorithms on higher precision with correct rounding [4], . Horner’s rule under assumptions (such as elementary function evaluation) [5], . accurate summation under tough assumption... |