ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How 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 recovers a rounded away low part

Example

In binary64 nearest-even arithmetic, take $a = 1$ and $b = 2^{- 53}$ . The call to TwoSum error-free transformation succeeds and returns $(s, e) = (1, 2^{- 53})$ . It preserves $s + e = a + b$ exactly although the leading rounded sum loses the low part.

Facts & Assumptions

Given: binary64 nearest-even arithmetic with $p = 53$ , $e_{\min} = - 1022$ , $e_{\max} = 1023$ , $a = 1$ and $b = 2^{- 53}$ .

[F1]

TwoSum error-free transformation evaluates the six separately rounded operations displayed in its definition.

[F2]

TwoSum returns the exact rounding error of a floating point sum proves $s + e = a + b$ for every successful call in a binary format of precision at least two.

[F3]

Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add requires zero or normal exact intermediates and nearest-even rounding.

Verification

1.1givenF1F3algebra

The spacing above $1$ is $2^{- 52}$ , so $1 + 2^{- 53}$ is the midpoint between $1$ and $1 + 2^{- 52}$ . The even-significand endpoint is $1$ ; hence $s = 1$ . The next exact operands give $z = 1 - 1 = 0$ , $v = 1 - 0 = 1$ , $w = 1 - 1 = 0$ , and $t = 2^{- 53} - 0 = 2^{- 53}$ . Each is representable, so each subtraction is exact.

2.1step 1.1F1F2F3algebra∎

The final exact intermediate is $w + t = 2^{- 53}$ , hence $e = 2^{- 53}$ . The first exact sum $1 + 2^{- 53}$ and every later nonzero exact intermediate ( $1$ or $2^{- 53}$ ) are normal and within range; the zero intermediates are allowed by [F3]. Thus the call succeeds, its format satisfies [F2], and $s + e = 1 + 2^{- 53} = a + b$ exactly, also directly verified by these outputs.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Higham, The Accuracy of Floating Point Summation (standard reference, not scraped)
Ogita, Rump, and Oishi, Accurate Sum and Dot Product (standard reference, not scraped)