float equality comparisons keep surviving code reviewThey pass the tests, because the tests use values that happen to be exactly representable. The failure arrives with real input, months later.
Running a computation twice — once normally, once with intervals — is an expensive way to get an honest error bar, and occasionally the only way.
A single refinement step recovers most of the accuracy lost to a badly conditioned solve, provided the residual is computed in higher precision. That proviso is where implementations usually go wrong.