One issue that arises with rounding modes in a pure lazy language like Haskell is that it's pretty hard to nail down the semantics. The "round towards negative infinity" mode is a dynamic hardware feature, available by setting a rounding mode flag in the FPU. If you execute some hunk of code with that flag having different values, you will in general get different results. So if you try to do this in Haskell, you have to decide how to expose it without violating purity. This would break things: roundWith :: RoundMode -> ( () -> a) -> a consider let x = 0.1 + 0.2 in roundWith NegInfty (\_ -> x) vs roundWith NegInfty (\_ -> 0.1 + 0.2) So you need to make different numeric types. This is what Ed Kmett's "rounding" cabal package does, albeit in a much more general way where the new floating types are parameterized by both rounding mode and precision. The bottom line is that IEEE Floating Point numbers are actually a monad. There is NaN, which makes them a Maybe, or given that there are many NaNs perhaps an Either. And there is the rounding mode, which is sort of a Reader. As an aside, changing the rounding mode flags can be surprisingly slow on modern CPUs. One trick for doing interval arithmetic efficiently is to set the flags once, say to TowardInf. Then x*y computes the product of x and y rounding towards positive infinity while -((-x)*y) computes the same product rounding towards negative infinity without fiddling with the rounding mode. So for adding intervals, instead of Interval x0 x1 + Interval y0 y1 = Interval (x0+y0) (x1+y1) Interval x0 x1 * Interval y0 y1 | x0 >= 0 && y0 >= 0 = Interval (x0*y0) (x1*y1) -- etc but with some added hair to fiddle with the rounding modes, one leaves the rounding mode fixed and writes Interval x0 x1 + Interval y0 y1 = Interval (-(-x0-y0)) (x1+y1) Interval x0 x1 * Interval y0 y1 | x0 >= 0 && y0 >= 0 = Interval (-((-x0)*y0)) (x1*y1) -- etc As a final throwaway, allowing (Interval x0 x1) where x0>x1, i.e., improper intervals, gives a very pleasant system with a surprisingly nice and rich semantics. For example, Interval 1 2 - Interval 2 1 = Interval 0 0 which makes for a group rather than just a semigroup. The system is known variously as modal intervals, Kaucher intervals, directed interval arithmetic, or generalized interval arithmetic, and can be used to automatically prove interesting arithmetic properties full of ∀s and ∃s. -- Barak A. Pearlmutter Hamilton Institute & Dept Comp Sci, NUI Maynooth, Co. Kildare, Ireland http://www.bcl.hamilton.ie/~barak/