Synthesis of fixed-point programs

Abstract

Several problems in the implementations of control systems, signal-processing systems, and scientific computing systems reduce to compiling a polynomial expression over the reals into an imperative program using fixed-point arithmetic. Fixed-point arithmetic only approximates real values, and its operators do not have the fundamental properties of real arithmetic, such as associativity. Consequently, a naive compilation process can yield a program that significantly deviates from the real polynomial, whereas a different order of evaluation can result in a program that is close to the real value on all inputs in its domain.We present a compilation scheme for real-valued arithmetic expressions to fixed-point arithmetic programs. Given a real-valued polynomial expression t, we find an expression t that is equivalent to t over the reals, but whose implementation as a series of fixed-point operations minimizes the error between the fixed-point value and the value of t over the space of all inputs. We show that the corresponding decision problem, checking whether there is an implementation t of t whose error is less than a given constant, is NP-hard. We then propose a solution technique based on genetic programming. Our technique evaluates the fitness of each candidate program using a static analysis based on affine arithmetic. We show that our tool can significantly reduce the error in the fixed-point implementation on a set of linear control system benchmarks. For example, our tool found implementations whose errors are only one half of the errors in the original fixed-point expressions.

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].…”

Automatic Verification of Finite Precision Implementations of Linear Controllers

“…On the other hand, a realizationΣ(1, 1, α, 0) -i.e., z k+1 = z k + y k , u k = αz k , of the controller would introduce a reduced computational error when finite precision arithmetics is used [10]. Thus, controller specification (2) may result in two different software implementations due to the use of different code generation tools.…”

“…High-assurance control software for cyber physical systems has received much attention recently (e.g., [32,3,24,23,22,10,12]). Focusing on robust controller implementation, [32] and [22] provide simulation-based robustness analysis tools, while [3,24,10,12] studies issues related to fixed-point controller design.…”

“…where c i v i is the exact expression as (10), and err abs is the absolute error term bounded by b abs such that |err abs | ≤ b abs . err rel is the relative error term which is related to the variables {v i } (i.e., symbolic values).…”

“…Focusing on robust controller implementation, [32] and [22] provide simulation-based robustness analysis tools, while [3,24,10,12] studies issues related to fixed-point controller design. [4] presents a theorem proving method to verify the control related properties of Simulink models.…”