## Floating-point arithmetic in the Coq system

Citations: | 6 - 0 self |

### BibTeX

@MISC{Melquiond_floating-pointarithmetic,

author = {Guillaume Melquiond},

title = {Floating-point arithmetic in the Coq system},

year = {}

}

### OpenURL

### Abstract

The process of proving some mathematical theorems can be greatly reduced by relying on numericallyintensive computations with a certified arithmetic. This article presents a formalization of floatingpoint arithmetic that makes it possible to efficiently compute inside the proofs of the Coq system. This certified library is a multi-radix and multi-precision implementation free from underflow and overflow. It provides the basic arithmetic operators and a few elementary functions. 1

### Citations

488 |
Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions
- Bertot, Castéran
- 2004
(Show Context)
Citation Context ...easing. As a consequence, the process of proving mathematical theorems is slowly shifting from a deductive approach to a computational approach. One of these formal systems is the Coq proof assistant =-=[1]-=-, which is based on the calculus of inductive constructions. Its formalism makes it possible to evaluate functions and to use their results inside proofs. This system is therefore a good candidate for... |

73 | Mpfr: A multiple-precision binary floating-point library with correct rounding
- Fousse, Hanrot, et al.
(Show Context)
Citation Context ...min, emax]. For instance, the double precision arithmetic described in the IEEE-754 standard [9] is a radix-2 arithmetic with p = 53, emin = −1049, and emax = 971. A multi-precision library like MPFR =-=[5]-=- works with any precision but still has bounded exponents, though the bounds are sufficiently big so that they do not matter usually. 2.1 Number format The floating-point formalization presented in th... |

37 |
Fast evaluation of elementary mathematical functions with correctly rounded last bit
- Ziv
- 1991
(Show Context)
Citation Context ... few trivial inputs, e.g. 0, one can only compute non-singleton ranges enclosing the exact result. This is nonetheless sufficient in order to get correct rounding, as shown by Ziv’s iterative process =-=[10]-=-. Formalizing this process in Coq, however, depends on theorems that are currently out of scope. So the elementary functions do not return the correctly-rounded result, they return an interval enclosi... |

32 |
A generic library of floating-point numbers and its application to exact computing
- Daumas, Rideau, et al.
- 2001
(Show Context)
Citation Context ...malizations exist, for various proof assistants, and they have been successful in proving numerous facts on floating-point arithmetic. For the sake of conciseness, only two of them will be cited here =-=[3, 7]-=-. The main difference between these libraries and the one described in this paper is in the rationale. They were designed for proving floating-point code, but not for actually computing. For instance,... |

22 | Guaranteed proofs using interval arithmetic
- Daumas, Melquiond, et al.
- 2005
(Show Context)
Citation Context ...of automatic differentiation for doing interval evaluation with Taylor’s order-1 decomposition. As a consequence, these Coq tactics are able to automatically handle a theorem originally proved in PVS =-=[2]-=- by using the exact same formal methods but without relying on an external oracle. This theorem was stating a tight bound on the relative error between Earth local radius a rp(φ) = √ 1 + (1 − f) 2 × t... |

17 |
et al., “An American national standard: IEEE standard for binary floating point arithmetic
- Stevenson
- 1987
(Show Context)
Citation Context ...tissas m that are bounded: |m| < β p . For the same reasons, exponents e are also constrained to a range [emin, emax]. For instance, the double precision arithmetic described in the IEEE-754 standard =-=[9]-=- is a radix-2 arithmetic with p = 53, emin = −1049, and emax = 971. A multi-precision library like MPFR [5] works with any precision but still has bounded exponents, though the bounds are sufficiently... |

12 | Real Number Calculations and Theorem Proving
- Lester
- 2008
(Show Context)
Citation Context .... So some tactics have been developed in order to automatically prove bounds on real-valued expressions. These tactics are based on interval arithmetic and are similar to some existing PVS strategies =-=[8]-=-. They also support bisection search, and a bit of automatic differentiation for doing interval evaluation with Taylor’s order-1 decomposition. As a consequence, these Coq tactics are able to automati... |

10 |
A Purely Functional Library for Modular Arithmetic and Its Application to Certifying Large Prime Numbers
- Grégoire, Théry
- 2006
(Show Context)
Citation Context ...on of the integers as lists in order to speed up radix-2 floating-point computations. The implementation can be sped up even further by replacing the bit lists with binary trees of fixedsize integers =-=[6]-=-. The binary-tree structure allows for divide-and-conquer algorithms, e.g. Karatsuba’s multiplication. Moreover, Coq can use 31-bit machine integers for representing the leafs of the trees. This consi... |

10 |
A machine-checked theory of floating-point arithmetic
- Harrison
- 1999
(Show Context)
Citation Context ...malizations exist, for various proof assistants, and they have been successful in proving numerous facts on floating-point arithmetic. For the sake of conciseness, only two of them will be cited here =-=[3, 7]-=-. The main difference between these libraries and the one described in this paper is in the rationale. They were designed for proving floating-point code, but not for actually computing. For instance,... |