Systematic study of limb-count and exponent-grouping trade-offs

Determine the trade-offs between the number of floating-point limbs, the exponent-grouping parameter M—including the M = 32 configuration with four limbs under the fast-path error budget—and the fraction of inputs that reach the accurate path in correctly rounded implementations of the branch-free floating-point Payne–Hanek range-reduction algorithm.

Background

The paper presents a branch-free floating-point variation of the Payne–Hanek range-reduction algorithm. Its table stores several floating-point limbs representing scaled pieces of 2^N/π. Using four limbs satisfies the worst-case error budget required when no subsequent rounding test is available, while a correctly rounded implementation can use three limbs on its fast path because rare difficult cases are redirected to an accurate path.

The number of limbs and the exponent-grouping parameter M create competing implementation trade-offs. Fewer limbs reduce arithmetic work, whereas larger exponent groups reduce the lookup-table footprint but weaken the quotient estimate and reduce the available truncation-error margin. The paper specifically notes that four limbs permit M = 32 under the fast-path budget, potentially halving the table, but does not analyze how these choices affect the proportion of inputs requiring the accurate path. A systematic characterization of these trade-offs is therefore explicitly left unresolved.