Current GHC optimises quotients, but not remainders by powers of two. For instance, even even :: Word -> Bool requires division:

evenWord :: Word -> Bool
evenWord = even

results in CMM, containing

if (I64[R1 + 7] % 2 != 0) goto c2sK; else goto c2sQ;

My proposal is to optimise MO_{S,U}_Rem by powers of two in CmmOpt.cmmMachOpFoldM, similar to existing cases for MO_{S,U}_Quot. Something like this may do:

MO_U_Rem rep
   | Just _ <- exactLog2 n ->
         Just (cmmMachOpFold dflags (MO_And rep) [x, CmmLit (CmmInt (n - 1) rep)])
MO_S_Rem rep
   | Just p <- exactLog2 n ->
          Just (cmmMachOpFold dflags (MO_Sub rep)
            [x, cmmMachOpFold dflags (MO_Shl rep)
              [cmmMachOpFold dflags (MO_S_Quot rep) [x, y], CmmLit (CmmInt p rep)]])

Function even is used by default definition of stimes (Data.Semigroup.Internal.stimesDefault). If <> is cheap, division may become a bottleneck.

It is also used by GHC.Real.^. Here is a simple benchmark:

v :: Int
v = maximum [ x ^ (6 :: Word)  | x <- [1..100000000 :: Int] ]

It takes 4.5 seconds on my PC before the patch above (-O2) and only 2.2 seconds after.