Representable correcting terms for possibly underflowing floating point operations

Abstract

Studying floating point arithmetic, authors have shown that the implemented operations (addition, subtraction, multiplication, division and square root) can compute a result and an exact correcting term using the same format as the inputs. Following a path initiated in 1965, many authors supposed that neither underflow nor overflow occurred in the process. Overflow is not critical as this kind of exception creates persisting non numeric quantities. Underflow may be fatal to the process as it returns wrong numeric values with little warning. Our new conditions guarantee that the correcting term is exact when the result is a number. We have validated our proofs against Coq automatic proof checker. Our development has raised many questions, some of them were expected while other ones were surprising.

Avoiding double roundings in scaled Newton-Raphson division

“…That lemma can be traced back to Kahan [7] or Markstein [9]. The presentation we give here is close to that of Boldo and Daumas [1], [10].…”

“…Consider the following example. Assume the floating-point format being considered is binary32 (that format was called single precision in the previous version of IEEE 754: precision p = 24, extremal exponents e min = −126 and e max = 127), and that RN is round-to-nearest-ties-to-even 1 . Consider the two floating-point input values (the significands are represented in binary):…”

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“…PFF is a Coq library initially developed in [11] and various results have been added since then [16], [17], [12]. It has been purely designed to be a high-level formalization of IEEE-754: only floating-point formats with gradual underflow are supported.…”

Provably faithful evaluation of polynomials

“…We give necessary and sufficient conditions for this error to be representable, even when underflow occurs [3]. Furthermore, we compute the exponent of the exhibited bounded float that represents the error term.…”