In many instances of fixed-point multiplication, a full precision result is not required. Instead it is sufficient to return a faithfully rounded result. Faithful rounding permits the machine representable number either immediately above or below the full precision result, if the latter is not exactly representable. Multipliers which take full advantage of this freedom can be implemented using less circuit area and consuming less power. The most common implementations internally truncate the partial product array. However, truncation applied to the most common of multiplier architectures, namely Booth architectures, results in non-commutative implementations. The industrial adoption of truncated multipliers is limited by the absence of formal verification of such implementations, since exhaustive simulation is typically infeasible. We present a commutative truncated Booth multiplier architecture and derive closed form necessary and sufficient conditions for faithful rounding. We also provide the bit-vectors giving rise to the worst-case error. We present a formal verification methodology based on ACL2 which scales up to 42 bit multipliers. We synthesize a range of commutative faithfully rounded multipliers and show that truncated booth implementations are up to 31% smaller than externally truncated multipliers.

We present a generic digit serial method (DSM) to compute the digits of a real number V. Bounds on these digits, and on the errors in the associated estimates of V formed from these digits, are derived. To illustrate our results, we derive such bounds for a parameterized family of high-radix algorithms for division and square root. These bounds enable a DSM designer to determine, for example, whether a given choice of parameters allows rapid formation and rounding of its approximation to V.

Maintaining software in the face of evolving dependencies is a challenging problem, and in addition to good release practices there is a need for automatic dependency analysis tools to avoid errors creeping in. Verified software reveals more semantic information in the form of mechanized proofs of functional specifications, and this can be used for dependency analysis. In this paper we present a scheme for automatic dependency analysis of verified software, which for each program checks that the collection of installed libraries is sufficient to guarantee its functional correctness. We illustrate the scheme with a case study of Haskell packages verified in higher order logic. The dependency analysis reduces the burden of maintaining verified Haskell packages by automatically computing version ranges for the packages they depend on, such that any combination provides the functionality required for correct operation.