-- | The only reason this instance exists is to define square roots.  None of the other @'Floating'@ methods apply to @'Nimber'@s.
instance Floating Nimber where
  sqrt 0 = 0
  sqrt 1 = 1
  sqrt 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 sqrt a `shiftL` bit m + sqrt a * sqrt semiD + sqrt b
  pi = error "π is not a nimber"
  exp _ = error "exp undefined for nimbers"
  log _ = error "log undefined for nimbers"
  sin _ = error "Trigonometric functions undefined for nimbers"
  cos _ = error "Trigonometric functions undefined for nimbers"
  tan _ = error "Trigonometric functions undefined for nimbers"
  asin _ = error "Trigonometric functions undefined for nimbers"
  acos _ = error "Trigonometric functions undefined for nimbers"
  atan _ = error "Trigonometric functions undefined for nimbers"
  sinh _ = error "Hyperbolic functions undefined for nimbers"
  cosh _ = error "Hyperbolic functions undefined for nimbers"
  tanh _ = error "Hyperbolic functions undefined for nimbers"
  asinh _ = error "Hyperbolic functions undefined for nimbers"
  acosh _ = error "Hyperbolic functions undefined for nimbers"
  atanh _ = error "Hyperbolic functions undefined for nimbers"