- class (C a, C a) => C a where
- fastSplitFraction :: (RealFrac a, C a, C b) => (a -> Int) -> (Int -> a) -> a -> (b, a)
- fixSplitFraction :: (C a, C b, Ord a) => (b, a) -> (b, a)
- fastFraction :: (RealFrac a, C a) => (a -> a) -> a -> a
- preludeFraction :: (RealFrac a, C a) => a -> a
- fixFraction :: (C a, Ord a) => a -> a
- splitFractionInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> (Int, a)
- floorInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int
- ceilingInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int
- roundInt :: (C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int
- approxRational :: (C a, C a) => a -> a -> Rational
- powersOfTwo :: C a => [a]
- pairsOfPowersOfTwo :: (C a, C b) => [(a, b)]
- genericFloor :: (Ord a, C a, C b) => a -> b
- genericCeiling :: (Ord a, C a, C b) => a -> b
- genericTruncate :: (Ord a, C a, C b) => a -> b
- genericRound :: (Ord a, C a, C b) => a -> b
- genericFraction :: (Ord a, C a) => a -> a
- genericSplitFraction :: (Ord a, C a, C b) => a -> (b, a)
- genericPosFloor :: (Ord a, C a, C b) => a -> b
- genericPosCeiling :: (Ord a, C a, C b) => a -> b
- genericHalfPosFloorDigits :: (Ord a, C a, C b) => a -> ((a, b), [Bool])
- genericPosRound :: (Ord a, C a, C b) => a -> b
- genericPosFraction :: (Ord a, C a) => a -> a
- genericPosSplitFraction :: (Ord a, C a, C b) => a -> (b, a)
There are probably more laws, but some laws are
(fromInteger.fst.splitFraction) a + (snd.splitFraction) a === a ceiling (toRational x) === ceiling x :: Integer truncate (toRational x) === truncate x :: Integer floor (toRational x) === floor x :: Integer
If there wouldn't be
Real.C a and
ToInteger.C b constraints,
we could also use this class for splitting ratios of polynomials.
As an aside, let me note the similarities
splitFraction x and
x divMod 1 (if that were defined).
In particular, it might make sense to unify the rounding modes somehow.
IEEEFloat-specific calls are removed here (cf.
so probably nobody will actually use this default definition.
fraction doesn't return the integer part of the number.
This also removes a type ambiguity if the integer part is not needed.
The new methods
They always round to
This means that the fraction is always non-negative and
is always smaller than 1.
This is more useful in practice and
can be generalised to more than real numbers.
T denominator type supports
T can provide
e.g. fractions of polynomials.
However the ''integral'' part would not be of type class
Note: All of these methods can be defined exclusively with functions from Ord and Ring. We could write a power-of-two-algorithm like the one for finding the number of digits of an Integer in FixedPoint-fractions module. This would even be reasonably efficient. I think the module should be renamed to RealRing, and the superclass constraint should be lifted from Field to Ring.
We might also add a round method, that rounds 0.5 always up or always down. This is much more efficient in inner loops and is acceptable or even preferable for many applications.
The ToInteger constraint can be lifted to Ring.
TODO: Should be moved to a continued fraction module.
generic implementation of round functions
The generic rounding functions need a number of operations proportional to the number of binary digits of the integer portion. If operations like multiplication with two and comparison need time proportional to the number of binary digits, then the overall rounding requires quadratic time.