Algebra.RealField
Synopsis
class (C a, C a) => C a where
 splitFraction :: C b => a -> (b, a) fraction :: a -> a ceiling :: C b => a -> b floor :: C b => a -> b truncate :: C b => a -> b round :: C b => a -> b
preludeSplitFraction :: (RealFrac a, C a, C b) => a -> (b, a)
preludeFraction :: (RealFrac a, C a) => a -> a
fractionTrunc :: (C a, Ord a) => (a -> a) -> a -> a
fixFraction :: (C a, Ord a) => a -> a
approxRational :: (C a, C a) => a -> a -> Rational
Documentation
 class (C a, C a) => C a where Source

Minimal complete definition: splitFraction or floor

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 between 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. Prelude.RealFloat) so probably nobody will actually use this default definition.

Henning: New function 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 fraction and splitFraction differ from Prelude.properFraction semantics. They always round to floor. 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. Since every T denominator type supports divMod, every T can provide fraction and splitFraction, e.g. fractions of polynomials. However the ''integral'' part would not be of type class C.

Can there be a separate class for fraction, splitFraction, floor and ceiling since they do not need reals and their ordering?

Methods
 splitFraction :: C b => a -> (b, a) Source
 fraction :: a -> a Source
 ceiling :: C b => a -> b Source
 floor :: C b => a -> b Source
 truncate :: C b => a -> b Source
 round :: C b => a -> b Source
Instances
 C Double C Float C T C T C a => C (T a) (C a, C a) => C (T a)
 preludeSplitFraction :: (RealFrac a, C a, C b) => a -> (b, a) Source
 preludeFraction :: (RealFrac a, C a) => a -> a Source
 fractionTrunc :: (C a, Ord a) => (a -> a) -> a -> a Source
 fixFraction :: (C a, Ord a) => a -> a Source
 approxRational :: (C a, C a) => a -> a -> Rational Source
TODO: Should be moved to a continued fraction module.