Correctly round AA conversion to float - #42710
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Convert algebraic interval endpoints through exact rationals, refine until a binary64 rounding cell is determined, and compare exactly at midpoint boundaries. This fixes lazy exact zeros without discarding representable tiny values and covers subnormal, tie, signed-zero, and overflow edges.\n\nFixes sagemath#42545.
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Fixes #42545.
Problem
AlgebraicReal.__float__delegated tofloat(RR(self)). The intermediate 53-bit MPFR value can introduce double rounding, and a lazy algebraic expression that is exactly zero can therefore become a small nonzero Python float. This caused the matrix failure reported in #42545.Changes
QQbarandLazyAlgebraicfloat doctests.This PR contains no matrix or LDLT changes.
Verification
src/sage/rings/qqbar.py: 1688 doctests passed.src/sage/rings/real_lazy.pyx: 286 doctests passed.Truefromis_positive_semidefinite().Checklist