Automated Precision Analysis: A Polynomial Algebraic Approach

Abstract

When migrating an algorithm onto hardware, the potential saving that can be obtained by tuning the precision used in the algorithm to meet a range or error specification is often overlooked; the major reason is that it is hard to choose a number system which can guarantee any such specification can be met. Instead, the problem is mitigated by opting to use IEEE standard single or double precision so as to be 'no worse' than a software implementation. However, the flexibility in the number representation is one of the key factors that can only be exploited on FPGAs, unlike GPUs and general purpose processors, and hence ignoring this potential significantly limits the performance achievable on an FPGA. To this end, this paper describes a tool which analyses algorithms with given input ranges under a finite precision to provide information that could be used to tune the hardware to the algorithm specifications. We demonstrate the proposed procedure on an iteration of the conjugate gradient algorithm, achieving a reduction in slices of over 40% when meeting the same error specification found by traditional methods. We also show it achieves comparable bounds to recent literature in a small fraction of the execution time, with greater scalability.

Bounding Variable Values and Round-Off Effects Using Handelman Representations

“…This paper elaborates on previous work by the authors published in [10] with a more detailed description of the search heuristic, several new tests to illustrate the various contributions of this work, as well as new comparisons against existing literature, and a broader discussion of its potential for word-length optimisation and limitations. A summary of the main contributions of this work are as follows:…”

“…This theorem can be applied to find lower and upper bounds to satisfyγ lower ≤ f (δ) ≤γ upper by considering that we are trying to show that the functions f (δ) −γ lower andγ upper − f (δ) are non-negative over the compact set of inequalities specifying the bounds on δ given in (10). By Theorem 1, this is equivalent to showing f (δ) −γ lower has a Handelman representation of the form (11), or similarly satisfying (12) for the upper bound.…”

“…A polynomial p ghr is non-negative over the compact set S, defined in (10), if and only if p ghr can be represented by a Generalised Handelman Representation of the form (13).…”

Mixed Precision Processing in Reconfigurable Systems

“…The search for minimum area-delay product usually involves a non-convex integer programming problem with number of variables proportional to the complexity of the data-path, an exhaustive search is usually infeasible and thus the result is not guaranteed to be optimal. Although the polynomial algebraic approach reduced the problem to a convex integer programming problem [15]. However, an exhaustive search is still infeasible because the variables can only take integer values.…”

“…By combining the two models, one can search for the design with a sufficient accuracy and a minimum area-delay product. Common accuracy modelling approaches include simulation approach [7], interval arithmetic [8], backward propagation analysis [9], affine arithmetic [10], [11], [12], [13], SAT-Modulo theory [14] and the polynomial algebraic approach [15]. The search for minimum area-delay product usually involves a non-convex integer programming problem with number of variables proportional to the complexity of the data-path, an exhaustive search is usually infeasible and thus the result is not guaranteed to be optimal.…”