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Program extraction in exact real arithmetic

2014
*
Mathematical Structures in Computer Science
*

They considered a proof involving coinduction of the proposition that any two

doi:10.1017/s0960129513000327
fatcat:fq2pz63ahzhbph72o65axpvecq
*reals**in*[−1, 1] have their average*in*the same interval, and informally*extract*a Haskell*program*from this proof, which works ... Berger and Seisenberger recently elaborated the idea for*extraction*out of proofs involving (only) abstract*reals*. ... Conclusion We presented a formal proof of the existence of the average of two*real*numbers*in*[−1, 1], as a case study*in*constructive*exact**real**arithmetic*. ...##
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From coinductive proofs to exact real arithmetic: theory and applications

2011
*
Logical Methods in Computer Science
*

The

doi:10.2168/lmcs-7(1:8)2011
fatcat:xvobbwkxgfcwfjlqixqqhgmzrq
*extracted**programs*construct and combine*exact**real*number algorithms with respect to the binary signed digit representation of*real*numbers. ... Based on a new coinductive characterization of continuous functions we*extract*certified*programs*for*exact**real*number computation from constructive proofs. ... FROM COINDUCTIVE PROOFS TO*EXACT**REAL**ARITHMETIC*: THEORY AND APPLICATIONS 5*In*the first example object variables are of sort R while*in*the second example they are of sort D (hence R and D are the "universal ...##
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Constructive analysis, types and exact real numbers

2007
*
Mathematical Structures in Computer Science
*

We will present some of the problems and solutions of

doi:10.1017/s0960129506005834
fatcat:kdehtcealnbszcfuszmcbzpdhu
*exact**real**arithmetic*varying from concrete implementations, representation and algorithms to various models for*real*computation. ... We then put these models*in*a uniform framework using realisability, opening the door for the use of type theoretic and coalgebraic constructions both*in*computing and reasoning about these computations ...*Program**Extraction*Correctness is an important issue*in*the implementation of computable analysis (which*in*practice currently mostly amounts to the implementation of*exact**real**arithmetic*.) ...##
###
Error-free computer solution of certain systems of linear equations

1987
*
Journal of Computational and Applied Mathematics
*

Then it

doi:10.1016/0377-0427(87)90002-1
fatcat:c2236bfa75dtzfayf2ltkobuku
*extracts*the*exact*solution x from x* if the error*in*the approximation x * is sufficiently small. ...*In*this article we propose a procedure which generates the*exact*solution for the system Ax = b, where A is an integral nonsingular matrix and b is an integral vector, by improving the initial floating-point ... Numerical examples The following three examples demonstrate the*extraction*procedure. The*program*is coded*in*FORTRAN using double precision*arithmetic*and was run on a CDC Cyber 730 System. e! ...##
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The world's shortest correct exact real arithmetic program?

2012
*
Information and Computation
*

*In*this paper we present what is thought to be the world's shortest correct

*exact*

*real*

*arithmetic*

*program*. ...

*In*addition the

*program*presented here allows beginners to the field to easily experiment with a practical implementation

*in*order to understand some of the issues involved. ... Introduction Various attempts have been made to validate

*exact*

*arithmetics*before. Ménissier provides a hand proof of an

*exact*

*arithmetic*[9] . ...

##
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Subject Index

2007
*
Journal of Discrete Algorithms
*

Itai's randomized broadcasting algorithm, 323
Railway optimization
Track assignment, 250
Rational

doi:10.1016/s1570-8667(07)00076-7
fatcat:wfqxglrznfb6do3wyittd5pfbi
*Exact**arithmetic*on the Stern-Brocot tree, 356*Real*roots*Real*roots of univariate polynomials ...*extracting*motifs from biological weighted se- quences, 229 Multilinear*Exact**arithmetic*on the Stern-Brocot tree, 356 Multiple patterns Searching for a set of correlated patterns, 149 Natural language ...##
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Speeding up Exact Real Arithmetic on Fast Binary Cauchy Sequences by Using Memoization Based on Quantized Precision

2017
*
Journal of Information Processing
*

*Exact*

*Real*

*Arithmetic*on Fast Binary Cauchy Sequences (FBCSs) provides us a simple and fairly fast way to obtain numerical results of arbitrary precision. ... without sacrificing the properties of the

*arithmetic*to be

*exact*

*arithmetic*. ...

*Exact*

*real*

*arithmetic*offers a simple way of

*programming*to obtain arbitrarily accurate answers for computable

*real*numbers [7] . ...

##
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On using floating-point computations to help an exact linear arithmetic decision procedure
[article]

2009
*
arXiv
*
pre-print

We consider the decision problem for quantifier-free formulas whose atoms are linear inequalities interpreted over the

arXiv:0904.3525v1
fatcat:vgmwnfhd7ngz7ey43tmi5a6qra
*reals*or rationals. ... State-of-the-art SMT solvers use simplex implementations over rational numbers, which perform well for typical problems arising from model-checking and*program*analysis (sparse inequalities, small coefficients ... to perform computations*in**exact**arithmetic*. ...##
###
On Using Floating-Point Computations to Help an Exact Linear Arithmetic Decision Procedure
[chapter]

2009
*
Lecture Notes in Computer Science
*

We consider the decision problem for quantifier-free formulas whose atoms are linear inequalities interpreted over the

doi:10.1007/978-3-642-02658-4_42
fatcat:f5um44i5qja4hpwkwhpqjwk2ge
*reals*or rationals. ... State-of-the-art SMT solvers use simplex implementations over rational numbers, which perform well for typical problems arising from model-checking and*program*analysis (sparse inequalities, small coefficients ... to perform computations*in**exact**arithmetic*. ...##
###
Implementing Real Numbers With RZ

2008
*
Electronical Notes in Theoretical Computer Science
*

of

doi:10.1016/j.entcs.2008.03.027
fatcat:jahsbjrqdvanrfgjudjgmpkzpi
*exact**real**arithmetic*. ... RZ is a tool which translates axiomatizations of mathematical structures to*program*specifications using the realizability interpretation of logic. ... As mentioned earlier, efficient implementations of*exact**real**arithmetic*compute*in*stages. ...##
###
From Coinductive Proofs to Exact Real Arithmetic
[chapter]

2009
*
Lecture Notes in Computer Science
*

This is a pilot study

doi:10.1007/978-3-642-04027-6_12
fatcat:yixkpnppfbef5mblyenhh54ysa
*in*using proof-theoretic methods for obtaining certified algorithms*in**exact**real**arithmetic*. ... We give a coinductive characterisation of the set of continuous functions defined on a compact*real*interval, and*extract*certified*programs*that construct and combine*exact**real*number algorithms with ... Note also that*in*the proofs we used*arithmetic*operations on the*reals*and their*arithmetic*laws without implementing or proving them. ...##
###
A coinductive approach to verified exact real number computation

2009
*
Electronic Communications of the EASST
*

We present an approach to verified

doi:10.14279/tuj.eceasst.23.331
dblp:journals/eceasst/BergerL09
fatcat:yorfrgpl35gv5btt3bbcg4rfyy
*programs*for*exact**real*number computation that is based on inductive and coinductive definitions and*program**extraction*from proofs. ... We informally discuss the theoretical background of this method and give examples of*extracted**programs*implementing the translation between the representation by fast converging rational Cauchy sequences ... The generation of provably correct*programs**in**exact**real*number computation is the focus of this paper. ...##
###
Symbolic execution of floating-point computations

2006
*
Software testing, verification & reliability
*

*In*this paper, we address the peculiarities of the symbolic execution of

*program*with oatingpoint numbers. ... Issues

*in*the symbolic execution of this kind of

*programs*are carefully examined and a constraint solver is described that supports constraints over oating-point numbers. ... Gotlieb et al. 25] applied Constraint Logic

*Programming*over nite domains to solve constraints

*extracted*from imperative

*programs*

*in*the tool INKA 26]. ...

##
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A generic lazy evaluation scheme for exact geometric computations

2011
*
Science of Computer Programming
*

It also relies on multi-precision

doi:10.1016/j.scico.2010.09.003
fatcat:nf7ndiyxonaard2ajoxzqzfmpa
*arithmetic*as well as interval*arithmetic*. We apply our approach to the whole geometry kernel of Cgal. ... Our approach is generic and extensible*in*the sense that it is possible to make it a library that users can apply to their own geometric objects and primitives. ... Indeed, the crucial*exactness*assumption is not satisfied when floating-point*arithmetic*is used to implement operations on*real*numbers. ...##
###
Proofs, Programs, Processes
[chapter]

2010
*
Lecture Notes in Computer Science
*

Typical applications of such processes are computations

doi:10.1007/978-3-642-13962-8_5
fatcat:rcvrnid7evgwdcntvdbblrcz64
*in**exact**real**arithmetic*. ... As an example we show how to*extract*a*program*computing the average of two*real*numbers w.r.t. to the binary signed digit representation. ... Typical applications of such processes are computations*in**exact**real**arithmetic*. ...
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