Interval arithmetic backend that doesn't set rounding modes?

[sheaf]

I notice you already have implementation of algorithms like 2Sum in fp-ieee. Would it make sense to add a backend to rounded-hw which uses these to compute addition of intervals, instead of relying on switching floating-point register rounding modes (which can be slow)?

I'm adding fused multiply-add to GHC in GHC MR !9996. WIth that, we can also do something like:

prod :: Double -> Double -> ( Double, Double )
prod a b =
  let !p = a * b
      !e = fmsub a b p
  in case compare e 0 of
       LT -> ( nextDown p, p )
       EQ -> ( p, p )
       GT -> ( p, nextUp p )

I'd be interested to see the benchmarks to compare the approaches.