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TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedprecheck pass
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TwoProduct with fused multiply-add returns the exact product residual

Statement

Let a,b be real floating-point numbers successfully processed by TwoProduct product and residual computation using fused multiply-add, and assume the exact residual ab−p is representable as zero or a normal number. Then the returned pair satisfies p+e=ab exactly.

Facts & Assumptions

Given: the returned product p and fused multiply-add residual.

[F1]

TwoProduct product and residual computation using fused multiply-add: p=fl(ab) and e=fma(a,b,-p).

[F2]

Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add: fused multiply-add rounds the single exact expression ab+c to nearest.

Proof

1.1givenF1F2algebra

The exact value supplied to the FMA is ab-p. By the representability assumption and correct rounding in [F2], e=ab-p exactly.

2.1step 1.1algebra∎

Adding p gives p+e=ab. The addition here is in the real-number conclusion, not an extra rounded operation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources