Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Abstract

Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation. Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor ExpansionsAlexey Solovyev, Charles Jacobsen, Zvonimir Rakamarić, and Ganesh Gopalakrishnan School of Computing, University of Utah, Salt Lake City, UT 84112, USA {monad,charlesj,zvonimir,ganesh}@cs.utah.edu Abstract. Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation.

Automatic Verification of Finite Precision Implementations of Linear Controllers

“…For example, = 2 −53 and δ = 2 −1075 for the double precision (i.e., 64 bits) rounding to the nearest [33]. Combining the two equations (11) and (12), we have the following model for the floating-point operations:…”

“…There has been static analysis techniques (e.g., [5,13,16]) developed for the analysis of finite precision numerical programs, but they focus on verifying properties such as numerical stability, the absence of buffer overflow and the absence of arithmetic exception rather than verifying the equivalence between code and a dynamical system model as the specification of the controller. Finally, there has been software verification work using the model extraction technique [8,19,20,34,29], and the floating-point roundoff error estimation has been studied in [11,9,33].…”

Daisy - Framework for Analysis and Optimization of Numerical Programs (Tool Paper)

“…For instance, the tools Fluctuat [22], Rosa [14], Gappa [17], FPTaylor [41], Real2Float [31] and PRECiSA [34] automatically provide sound error bounds on floating-point (and some also on fixed-point) roundoff errors. Such a static error analysis is a pre-requisite for any optimization technique providing rigorous results, such as recent ones which choose a mixed-precision assignment [10] or an errorminimizing rewriting of the non-associative finite-precision arithmetic [15,37].…”

“…To make it userfriendly, we adopt the input format of Rosa, which is a real-valued functional domain-specific language in Scala. Unlike other tools today, which have custom input formats [41] or use prefix notation [12], Daisy's input is easily human readable 1 and natural to use. Daisy is itself written in the Scala programming language [35] and has limited and optional dependencies, making it portable and easy to install.…”

“…-static dataflow analysis for finite-precision roundoff errors [14] with mixedprecision support and additional support for the dReal SMT solver [21], -FPTaylor's optimization-based absolute error analysis [41], -transcendental function support, for dataflow analysis following [13], -interval subdivision, used by Fluctuat [22] to obtain tighter error bounds, -rewriting optimization based on genetic programming [15].…”