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.
Subnormal and overflow branches return range failure
Example
Use binary64 and the no-subnormal success policy. Put N_min=2^-1022 and the least positive subnormal 2^-1074. Inputs N_min and -(N_min+2^-1074) are normal, but their exact sum is -2^-1074, so the only tree sum in pairwise summation returns range-failure. Also N_max+N_max overflows at the root and returns range-failure. These are policy branches, not claims that IEEE hardware lacks gradual underflow.
Facts & Assumptions
Given: binary64 normal endpoints and the no-subnormal range rule.
Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add: nonzero subnormal and overflow results return range-failure.
Balanced pairwise summation with range-failure status: each internal sum is rounded and the first failure is returned.
Verification
The exact first sum is 2^-1022-(2^-1022+2^-1074)=-2^-1074, a nonzero subnormal; [F1] therefore returns range-failure.
The exact sum N_max+N_max exceeds the largest finite normal magnitude, so the root triggers the same status under [F2].
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
- Fundamentals of Numerical Computation, floating-point arithmetic (standard reference, not scraped)
- Stuart and Voss, Matrix Analysis and Algorithms, Warwick lecture notes (standard reference, not scraped)