Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
TwoSum error-free transformation
Definition
For real floating-point a,b compute s=fl(a+b), z=fl(s-a), e=fl((a-(s-z))+(b-z)), with every operation separately rounded. Return the first range failure and operation index if an exact intermediate is nonzero subnormal, overflowed, or otherwise invalid; otherwise return (s,e). The fixed sequence terminates. Its general exact-transform claim is a separate theorem; this definition alone does not prove exact recovery.
Facts & Assumptions
Given: a finite input of the stated type and the operationwise arithmetic model.
Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add: each successful elementary operation returns a finite value, and an invalid or out-of-range operation returns a failure status.
Proof
Every loop index ranges over the finite input length or matrix dimensions; any branch test in the definition is a comparison of already supplied finite values.
At each operation the model [F1] either returns a finite next value or the defined failure status, and every loop has its stated finite bound. Thus execution terminates with the specified output or first failure.
Depends on
Used by
Dependency tree · two levels
2 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
- Fundamentals of Numerical Computation, floating-point arithmetic (standard reference, not scraped)
- Higham, The Accuracy of Floating Point Summation (standard reference, not scraped)
- Ogita, Rump, and Oishi, Accurate Sum and Dot Product (standard reference, not scraped)