-- | The finite nimbers are a field of characteristic 2.  There is no field homomorphism from the rationals to the nimbers, so @'fromRational'@ is always an error.
instance Fractional Nimber where
  fromRational _ = error "Cannot map from field of characteristic 0 to characteristic 2"
  recip 0 = error "Divide by zero"
  recip 1 = 1
  recip n =
    let m = floorLog @Int $ floorLog n -- D = 2^2^m is the largest Fermat 2-power less than or equal to n
        a = n `shiftR` bit m -- n = aD+b
        aD = a `shiftL` bit m
        b = n .^. aD
        semiD = bit (bit m - 1) -- semimultiple of D
     in (aD + a + b) / (semiD * a * a + b * (a + b))

-- | The finite nimbers are a field of characteristic 2.  There is no field homomorphism from the rationals to the nimbers, so @'fromRational'@ is always an error.
instance Fractional Nimber where
  fromRational _ = error "Cannot map from field of characteristic 0 to characteristic 2"
  recip 0 = error "Divide by zero"
  recip 1 = 1
  recip n =
    let m = floorLog @Int $ floorLog n -- D = 2^2^m is the largest Fermat 2-power less than or equal to n
        a = n `shiftR` bit m -- n = aD+b
        aD = a `shiftL` bit m
        b = n .^. aD
        semiD = bit (bit m - 1) -- semimultiple of D
     in (aD + a + b) / (semiD * sqr a + b * (a + b))