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Innocuous Double Rounding of Basic Arithmetic Operations

Pierre Roux
2014 Journal of Formalized Reasoning  
It belongs to the folklore in the floating-point arithmetic community that double rounding is innocuous for the basic arithmetic operations (addition, division, multiplication, and square root) as soon  ...  Double rounding practically happens, for instance, when implementing the IEEE754 binary32 format with an arithmetic unit performing operations only in the larger binary64 format, such as done in the PowerPC  ...  That is, the second rounding • 2 is innocuous for basic arithmetic operations.  ... 
doi:10.6092/issn.1972-5787/4359 dblp:journals/jfrea/Roux14 fatcat:bfxvvtm7y5fzlfnt2obriixut4

Innocuous Double Rounding of Basic Arithmetic Operations

Pierre Roux
unpublished
It belongs to the folklore in the floating-point arithmetic community that double rounding is innocuous for the basic arithmetic operations (addition, division, multiplication , and square root) as soon  ...  Double rounding practically happens, for instance, when implementing the IEEE754 binary32 format with an arithmetic unit performing operations only in the larger binary64 format, such as done in the PowerPC  ...  Innocuous Double Rounding of Basic Arithmetic Operations · 133 Definition 1.  ... 
fatcat:pgkuzbpysjcghdv54fggkabkqa

Some Formal Tools for Computer Arithmetic: Flocq and Gappa

Sylvie Boldo, Guillaume Melquiond
2021 2021 IEEE 28th Symposium on Computer Arithmetic (ARITH)  
Their purpose is to help the user in writing proofs regarding computer arithmetic, e.g., certifying a bound on a round-off error, while aiming at a high level of guarantee.  ...  Flocq is a library of mathematical definitions and theorems for the Coq proof assistant; Gappa is meant to compute bounds of values and errors, while producing the corresponding formal proof.  ...  This work was supported by the NuSCAP (ANR-20-CE48-0014) project of the French national research agency (ANR).  ... 
doi:10.1109/arith51176.2021.00031 fatcat:ibiw7molinc37ebxd4whml44pe

A Formally-Verified C Compiler Supporting Floating-Point Arithmetic

S. Boldo, J.-H Jourdan, X. Leroy, G. Melquiond
2013 2013 IEEE 21st Symposium on Computer Arithmetic  
Floating-point arithmetic is known to be tricky: roundings, formats, exceptional values.  ...  In this paper, we report on our recent success in formally specifying and proving correct CompCert's compilation of floating-point arithmetic.  ...  The basic axiom for algorithms and the basic goal for hardware components was still that all the operations are correctly rounded.  ... 
doi:10.1109/arith.2013.30 dblp:conf/arith/BoldoJLM13 fatcat:enjkv2q5sje6vhkpauim4vgoza

Verified Compilation of Floating-Point Computations

Sylvie Boldo, Jacques-Henri Jourdan, Xavier Leroy, Guillaume Melquiond
2014 Journal of automated reasoning  
Floating-point arithmetic is known to be tricky: roundings, formats, exceptional values.  ...  In this paper, we report on our recent success in formally specifying and proving correct CompCert's compilation of floating-point arithmetic.  ...  Acknowledgments This work was supported by the Verasco project (ANR-11-INSE-003) of Agence Nationale de la Recherche.  ... 
doi:10.1007/s10817-014-9317-x fatcat:xqmxtjlkmnc4vfmhcyltd4xiqe

The pitfalls of verifying floating-point computations

David Monniaux
2008 ACM Transactions on Programming Languages and Systems  
Current critical systems commonly use a lot of floating-point computations, and thus the testing or static analysis of programs containing floating-point operators has become a priority.  ...  However, correctly defining the semantics of common implementations of floating-point is tricky, because semantics may change with many factors beyond source-code level, such as choices made by compilers  ...  For the sake of a better understanding, in Section 2, we recall the basics of IEEE-754 arithmetic.  ... 
doi:10.1145/1353445.1353446 fatcat:nsle4apatrg7tcsiseeyz5c42i

Recent progress in exact geometric computation

C. Li, S. Pion, C.K. Yap
2005 The Journal of Logic and Algebraic Programming  
Computational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue.  ...  . ୋ This paper is based on a talk presented at the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real  ...  For surveys of multi-precision numbers, see [101, 39] . All big number packages support the four basic arithmetic operations (+, −, ×, ÷).  ... 
doi:10.1016/j.jlap.2004.07.006 fatcat:yzdzf5sefneahn2d6exsaqi2ri

Deductive Verification of Floating-Point Java Programs in KeY [chapter]

Rosa Abbasi, Jonas Schiffl, Eva Darulova, Mattias Ulbrich, Wolfgang Ahrendt
2021 Lecture Notes in Computer Science  
This is unfortunate, as floating-point arithmetic is particularly unintuitive to reason about due to rounding as well as the presence of the special values infinity and 'Not a Number' (NaN).  ...  Our support in the KeY verifier handles arithmetic via floating-point decision procedures inside SMT solvers and transcendental functions via axiomatization.  ...  Our approach attempts to get the best of both worlds by distinguishing between basic floating-point arithmetic, i. e., elementary operations and comparisons, and more complex functions which do not have  ... 
doi:10.1007/978-3-030-72013-1_13 fatcat:shfgyytxprfyhhpgchgpr3zyxu

Revisiting "What Every Computer Scientist Should Know About Floating-point Arithmetic" [article]

Vincent Lafage
2020 arXiv   pre-print
The differences between the sets in which ideal arithmetics takes place and the sets of floating point numbers are outlined.  ...  A set of classical problems in correct numerical evaluation is presented, to increase the awareness of newcomers to the field.  ...  (EFT) of arithmetics: given one operation • and two floating point numbers a and b, the rounded operation a • b would produce an approximate result s, but the corresponding EFT will produce a sum of two  ... 
arXiv:2012.02492v1 fatcat:4eknhnijgfcm3anlptquypscwu

Numerical 'health check' for scientific codes: the CADNA approach

N.S. Scott, F. Jézéquel, C. Denis, J.-M. Chesneaux
2007 Computer Physics Communications  
One important source of error that is difficult to detect and control is round-off error propagation which originates from the use of finite precision arithmetic.  ...  In doing so we hope to stimulate a greater awareness of round-off error propagation and present a practical means by which it can be analyzed and managed.  ...  The authors are grateful to Maurice Clint and Charlotte Froese Fischer for a critical reading of the manuscript and for constructive comments.  ... 
doi:10.1016/j.cpc.2007.01.005 fatcat:k46feswzvjgivg5la6jq5ts5e4

Deductive Verification of Floating-Point Java Programs in KeY [article]

Rosa Abbasi Boroujeni, Jonas Schiffl, Eva Darulova, Mattias Ulbrich, Wolfgang Ahrendt
2021 arXiv   pre-print
This is unfortunate, as floating-point arithmetic is particularly unintuitive to reason about due to rounding as well as the presence of the special values infinity and 'Not a Number' (NaN).  ...  Our support in the KeY verifier handles arithmetic via floating-point decision procedures inside SMT solvers and transcendental functions via axiomatization.  ...  Our approach attempts to get the best of both worlds by distinguishing between basic floating-point arithmetic, i. e., elementary operations and comparisons, and more complex functions which do not have  ... 
arXiv:2101.08733v1 fatcat:gbpg7i5v2fanje46m7nfj52rli

Emulation of a FMA and Correctly Rounded Sums: Proved Algorithms Using Rounding to Odd

Sylvie Boldo, Guillaume Melquiond
2008 IEEE transactions on computers  
By using it for some intermediate values instead of rounding to nearest, correctly rounded results can be obtained at the end of computations.  ...  We present an algorithm for emulating the fused multiply-and-add operator.  ...  Double rounding can also be made innocuous by introducing a new rounding mode and using it for the first rounding.  ... 
doi:10.1109/tc.2007.70819 fatcat:pjdojhubyfhefhl3n5i6ouy32e

A unified Coq framework for verifying C programs with floating-point computations

Tahina Ramananandro, Paul Mountcastle, Benoît Meister, Richard Lethin
2016 Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs - CPP 2016  
rounding errors and energy-efficient approximations of square root and sine.  ...  the formal semantics of CompCert Clight and the Flocq formal specification of IEEE 754 floating-point arithmetic for the verification of properties of floating-point computations in C programs.  ...  The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressly or implied, of the DARPA or the U.S.  ... 
doi:10.1145/2854065.2854066 dblp:conf/cpp/RamananandroMML16 fatcat:3p5fjcmis5bqvnjy4mdusjr7mu

The Numerical Reliability of Econometric Software

B. D McCullough, H. D Vinod
1999 Journal of Economic Literature  
Accuracy refers to the error of an approximation, whereas precision refers to the accuracy with which basic arithmetic operations are performed.  ...  Whether via truncation or roundoff, error is introduced into most any result of an arithmetic operation.  ...  The more quantitative journals can devote space to publication of more sophisticated benchmarks or to discussion of software; for example, the Journal of Economic and Social Measurement has a special issue  ... 
doi:10.1257/jel.37.2.633 fatcat:3nbir2ssxbeyfjigj6idps5ngi

Algorithm 812: BPOLY: An object-oriented library of numerical algorithms for polynomials in Bernstein form

Yi-Feng Tsai, Rida T. Farouki
2001 ACM Transactions on Mathematical Software  
By invoking the class environment and operator overloading, each polynomial in an expression is interpreted as an object compatible with the arithmetic operations and other common functions (subdvision  ...  A series of empirical tests indicates that the library functions are typically very accurate and reliable, even for polynomials of surprisingly high degree.  ...  Arithmetic operations and other basic functions on these polynomials are then defined, receiving "instances" of the objects as input, and returning new instances as outputs.  ... 
doi:10.1145/383738.383743 fatcat:weegintqbfg4jh6n5cvlhnpbba
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