{-# LANGUAGE NoImplicitPrelude #-} module Algebra.RealRing where import qualified Algebra.RealRing98 as RealRing98 import qualified Algebra.PrincipalIdealDomain as PID import qualified Algebra.Field as Field import qualified Algebra.Ring as Ring import qualified Algebra.ToRational as ToRational import qualified Algebra.ToInteger as ToInteger import qualified Algebra.Absolute as Absolute import qualified Algebra.OrderDecision as OrdDec import Algebra.OrderDecision ((=?), ) import Algebra.Field (fromRational, ) import Algebra.RealIntegral (quotRem, ) import Algebra.IntegralDomain (divMod, even, ) import Algebra.Ring ((*), fromInteger, one, ) import Algebra.Additive ((+), (-), negate, zero, ) import Algebra.ZeroTestable (isZero, ) import Algebra.ToInteger (fromIntegral, ) import qualified Number.Ratio as Ratio import Number.Ratio (T((:%)), Rational) import Data.Int (Int, Int8, Int16, Int32, Int64, ) import Data.Word (Word, Word8, Word16, Word32, Word64, ) import qualified GHC.Float as GHC import Data.List as List import Data.Tuple.HT (mapFst, mapPair, ) import Prelude (Integer, Float, Double, ) import qualified Prelude as P import NumericPrelude.Base {- | Minimal complete definition: 'splitFraction' or 'floor' There are probably more laws, but some laws are > splitFraction x === (fromInteger (floor x), fraction x) > fromInteger (floor x) + fraction x === x > floor x <= x x < floor x + 1 > ceiling x - 1 < x x <= ceiling x > 0 <= fraction x fraction x < 1 > - ceiling x === floor (-x) > truncate x === signum x * floor (abs x) > ceiling (toRational x) === ceiling x :: Integer > truncate (toRational x) === truncate x :: Integer > floor (toRational x) === floor x :: Integer The 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. Many people will associate rounding with fractional numbers, and thus they are surprised about the superclass being @Ring@ not @Field@. The reason is that all of these methods can be defined exclusively with functions from @Ord@ and @Ring@. The implementations of 'genericFloor' and other functions demonstrate that. They implement power-of-two-algorithms like the one for finding the number of digits of an 'Integer' in FixedPoint-fractions module. They are even reasonably efficient. I am still uncertain whether it was a good idea to add instances for @Integer@ and friends, since calling @floor@ or @fraction@ on an integer may well indicate a bug. The rounding functions are just the identity function and 'fraction' is constant zero. However, I decided to associate our class with @Ring@ rather than @Field@, after I found myself using repeated subtraction and testing rather than just calling @fraction@, just in order to get the constraint @(Ring a, Ord a)@ that was more general than @(RealField a)@. For the results of the rounding functions we have chosen the constraint @Ring@ instead of @ToInteger@, since this is more flexible to use, but it still signals to the user that only integral numbers can be returned. This is so, because the plain @Ring@ class only provides @zero@, @one@ and operations that allow to reach all natural numbers but not more. As an aside, let me note the similarities between @splitFraction x@ and @divMod x 1@ (if that were defined). In particular, it might make sense to unify the rounding modes somehow. 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 'Number.Ratio.T' denominator type supports 'Algebra.IntegralDomain.divMod', every 'Number.Ratio.T' can provide 'fraction' and 'splitFraction', e.g. fractions of polynomials. However the @Ring@ constraint for the ''integral'' part of 'splitFraction' is too weak in order to generate polynomials. After all, I am uncertain whether this would be useful or not. Can there be a separate class for 'fraction', 'splitFraction', 'floor' and 'ceiling' since they do not need reals and their ordering? 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. -} class (Absolute.C a, Ord a) => C a where {-# MINIMAL splitFraction | floor #-} splitFraction :: (Ring.C b) => a -> (b,a) fraction :: a -> a ceiling, floor :: (Ring.C b) => a -> b truncate :: (Ring.C b) => a -> b round :: (ToInteger.C b) => a -> b splitFraction x = (floor x, fraction x) fraction x = x - fromInteger (floor x) floor x = fromInteger (fst (splitFraction x)) ceiling x = - floor (-x) -- truncate x = signum x * floor (abs x) truncate x = if x>=0 then floor x else ceiling x {- The ToInteger constraint can be lifted to Ring if use Integer temporarily. I expect this would not be efficient in many cases. -} round x = let (n,r) = splitFraction x in case compare (2*r) one of LT -> n EQ -> if even n then n else n+1 GT -> n+1 {- | This function rounds to the closest integer. For @fraction x == 0.5@ it rounds away from zero. This function is not the result of an ingenious mathematical insight, but is simply a kind of rounding that is the fastest on IEEE floating point architectures. -} roundSimple :: (C a, Ring.C b) => a -> b roundSimple x = let (n,r) = splitFraction x in case compare (2*r) one of LT -> n EQ -> if x<0 then n else n+1 GT -> n+1 instance (ToInteger.C a, PID.C a) => C (Ratio.T a) where splitFraction (x:%y) = (fromIntegral q, r:%y) where (q,r) = divMod x y instance C Int where {-# INLINE splitFraction #-} {-# INLINE fraction #-} {-# INLINE floor #-} {-# INLINE ceiling #-} {-# INLINE round #-} {-# INLINE truncate #-} splitFraction x = (fromIntegral x, zero) fraction _ = zero floor x = fromIntegral x ceiling x = fromIntegral x round x = fromIntegral x truncate x = fromIntegral x instance C Integer where {-# INLINE splitFraction #-} {-# INLINE fraction #-} {-# INLINE floor #-} {-# INLINE ceiling #-} {-# INLINE round #-} {-# INLINE truncate #-} splitFraction x = (fromInteger x, zero) fraction _ = zero floor x = fromInteger x ceiling x = fromInteger x round x = fromInteger x truncate x = fromInteger x instance C Float where {-# INLINE splitFraction #-} {-# INLINE fraction #-} {-# INLINE floor #-} {-# INLINE ceiling #-} {-# INLINE round #-} {-# INLINE truncate #-} splitFraction = fastSplitFraction GHC.float2Int GHC.int2Float fraction = RealRing98.fastFraction (GHC.int2Float . GHC.float2Int) floor = fromInteger . P.floor ceiling = fromInteger . P.ceiling round = fromInteger . P.round truncate = fromInteger . P.truncate instance C Double where {-# INLINE splitFraction #-} {-# INLINE fraction #-} {-# INLINE floor #-} {-# INLINE ceiling #-} {-# INLINE round #-} {-# INLINE truncate #-} splitFraction = fastSplitFraction GHC.double2Int GHC.int2Double fraction = RealRing98.fastFraction (GHC.int2Double . GHC.double2Int) floor = fromInteger . P.floor ceiling = fromInteger . P.ceiling round = fromInteger . P.round truncate = fromInteger . P.truncate {-# INLINE fastSplitFraction #-} fastSplitFraction :: (P.RealFrac a, Absolute.C a, Ring.C b) => (a -> Int) -> (Int -> a) -> a -> (b,a) fastSplitFraction trunc toFloat x = fixSplitFraction $ if toFloat minBound <= x && x <= toFloat maxBound then case trunc x of n -> (fromIntegral n, x - toFloat n) else case P.properFraction x of (n,f) -> (fromInteger n, f) {-# INLINE fixSplitFraction #-} fixSplitFraction :: (Ring.C a, Ring.C b, Ord a) => (b,a) -> (b,a) fixSplitFraction (n,f) = -- if x>=0 || f==0 if f>=0 then (n, f) else (n-1, f+1) {-# INLINE fixFraction #-} fixFraction :: (Ring.C a, Ord a) => a -> a fixFraction y = if y>=0 then y else y+1 {- mapM_ (\n -> let x = fromInteger n / 10 in print (x, floorInt GHC.double2Int GHC.int2Double x)) [-20,-19..20] -} {-# INLINE splitFractionInt #-} splitFractionInt :: (Ring.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> (Int, a) splitFractionInt trunc toFloat x = let n = trunc x in fixSplitFraction (n, x - toFloat n) {-# INLINE floorInt #-} floorInt :: (Ring.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int floorInt trunc toFloat x = let n = trunc x in if x >= toFloat n then n else pred n {-# INLINE ceilingInt #-} ceilingInt :: (Ring.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int ceilingInt trunc toFloat x = let n = trunc x in if x <= toFloat n then n else succ n {-# INLINE roundInt #-} roundInt :: (Field.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int roundInt trunc toFloat x = let half = 0.5 -- P.fromRational halfUp = x+half n = floorInt trunc toFloat halfUp in if toFloat n == halfUp && P.odd n then pred n else n {-# INLINE roundSimpleInt #-} roundSimpleInt :: (Field.C a, Absolute.C a, Ord a) => (a -> Int) -> (Int -> a) -> a -> Int roundSimpleInt trunc _toFloat x = trunc (x + Absolute.signum x * 0.5) {- RULES maybe used, when Prelude implementations become more efficient "NP.round :: Float -> Int" round = P.round :: Float -> Int; "NP.truncate :: Float -> Int" truncate = P.truncate :: Float -> Int; "NP.floor :: Float -> Int" floor = P.floor :: Float -> Int; "NP.ceiling :: Float -> Int" ceiling = P.ceiling :: Float -> Int; "NP.round :: Double -> Int" round = P.round :: Double -> Int; "NP.truncate :: Double -> Int" truncate = P.truncate :: Double -> Int; "NP.floor :: Double -> Int" floor = P.floor :: Double -> Int; "NP.ceiling :: Double -> Int" ceiling = P.ceiling :: Double -> Int; -} -- these rules will also be needed for Int16 et.al. {-# RULES "NP.round :: Float -> Int" round = roundInt GHC.float2Int GHC.int2Float; "NP.roundSimple :: Float -> Int" round = roundSimpleInt GHC.float2Int GHC.int2Float; "NP.truncate :: Float -> Int" truncate = GHC.float2Int ; "NP.floor :: Float -> Int" floor = floorInt GHC.float2Int GHC.int2Float; "NP.ceiling :: Float -> Int" ceiling = ceilingInt GHC.float2Int GHC.int2Float; "NP.round :: Double -> Int" round = roundInt GHC.double2Int GHC.int2Double; "NP.roundSimple :: Double -> Int" round = roundSimpleInt GHC.double2Int GHC.int2Double; "NP.truncate :: Double -> Int" truncate = GHC.double2Int ; "NP.floor :: Double -> Int" floor = floorInt GHC.double2Int GHC.int2Double; "NP.ceiling :: Double -> Int" ceiling = ceilingInt GHC.double2Int GHC.int2Double; "NP.splitFraction :: Float -> (Int, Float)" splitFraction = splitFractionInt GHC.float2Int GHC.int2Float; "NP.splitFraction :: Double -> (Int, Double)" splitFraction = splitFractionInt GHC.double2Int GHC.int2Double; #-} -- generated by GenerateRules.hs {-# RULES "NP.round :: a -> Int8" round = (P.fromIntegral :: Int -> Int8) . round; "NP.roundSimple :: a -> Int8" roundSimple = (P.fromIntegral :: Int -> Int8) . roundSimple; "NP.truncate :: a -> Int8" truncate = (P.fromIntegral :: Int -> Int8) . truncate; "NP.floor :: a -> Int8" floor = (P.fromIntegral :: Int -> Int8) . floor; "NP.ceiling :: a -> Int8" ceiling = (P.fromIntegral :: Int -> Int8) . ceiling; "NP.round :: a -> Int16" round = (P.fromIntegral :: Int -> Int16) . round; "NP.roundSimple :: a -> Int16" roundSimple = (P.fromIntegral :: Int -> Int16) . roundSimple; "NP.truncate :: a -> Int16" truncate = (P.fromIntegral :: Int -> Int16) . truncate; "NP.floor :: a -> Int16" floor = (P.fromIntegral :: Int -> Int16) . floor; "NP.ceiling :: a -> Int16" ceiling = (P.fromIntegral :: Int -> Int16) . ceiling; "NP.round :: a -> Int32" round = (P.fromIntegral :: Int -> Int32) . round; "NP.roundSimple :: a -> Int32" roundSimple = (P.fromIntegral :: Int -> Int32) . roundSimple; "NP.truncate :: a -> Int32" truncate = (P.fromIntegral :: Int -> Int32) . truncate; "NP.floor :: a -> Int32" floor = (P.fromIntegral :: Int -> Int32) . floor; "NP.ceiling :: a -> Int32" ceiling = (P.fromIntegral :: Int -> Int32) . ceiling; "NP.round :: a -> Int64" round = (P.fromIntegral :: Int -> Int64) . round; "NP.roundSimple :: a -> Int64" roundSimple = (P.fromIntegral :: Int -> Int64) . roundSimple; "NP.truncate :: a -> Int64" truncate = (P.fromIntegral :: Int -> Int64) . truncate; "NP.floor :: a -> Int64" floor = (P.fromIntegral :: Int -> Int64) . floor; "NP.ceiling :: a -> Int64" ceiling = (P.fromIntegral :: Int -> Int64) . ceiling; "NP.round :: a -> Word" round = (P.fromIntegral :: Int -> Word) . round; "NP.roundSimple :: a -> Word" roundSimple = (P.fromIntegral :: Int -> Word) . roundSimple; "NP.truncate :: a -> Word" truncate = (P.fromIntegral :: Int -> Word) . truncate; "NP.floor :: a -> Word" floor = (P.fromIntegral :: Int -> Word) . floor; "NP.ceiling :: a -> Word" ceiling = (P.fromIntegral :: Int -> Word) . ceiling; "NP.round :: a -> Word8" round = (P.fromIntegral :: Int -> Word8) . round; "NP.roundSimple :: a -> Word8" roundSimple = (P.fromIntegral :: Int -> Word8) . roundSimple; "NP.truncate :: a -> Word8" truncate = (P.fromIntegral :: Int -> Word8) . truncate; "NP.floor :: a -> Word8" floor = (P.fromIntegral :: Int -> Word8) . floor; "NP.ceiling :: a -> Word8" ceiling = (P.fromIntegral :: Int -> Word8) . ceiling; "NP.round :: a -> Word16" round = (P.fromIntegral :: Int -> Word16) . round; "NP.roundSimple :: a -> Word16" roundSimple = (P.fromIntegral :: Int -> Word16) . roundSimple; "NP.truncate :: a -> Word16" truncate = (P.fromIntegral :: Int -> Word16) . truncate; "NP.floor :: a -> Word16" floor = (P.fromIntegral :: Int -> Word16) . floor; "NP.ceiling :: a -> Word16" ceiling = (P.fromIntegral :: Int -> Word16) . ceiling; "NP.round :: a -> Word32" round = (P.fromIntegral :: Int -> Word32) . round; "NP.roundSimple :: a -> Word32" roundSimple = (P.fromIntegral :: Int -> Word32) . roundSimple; "NP.truncate :: a -> Word32" truncate = (P.fromIntegral :: Int -> Word32) . truncate; "NP.floor :: a -> Word32" floor = (P.fromIntegral :: Int -> Word32) . floor; "NP.ceiling :: a -> Word32" ceiling = (P.fromIntegral :: Int -> Word32) . ceiling; "NP.round :: a -> Word64" round = (P.fromIntegral :: Int -> Word64) . round; "NP.roundSimple :: a -> Word64" roundSimple = (P.fromIntegral :: Int -> Word64) . roundSimple; "NP.truncate :: a -> Word64" truncate = (P.fromIntegral :: Int -> Word64) . truncate; "NP.floor :: a -> Word64" floor = (P.fromIntegral :: Int -> Word64) . floor; "NP.ceiling :: a -> Word64" ceiling = (P.fromIntegral :: Int -> Word64) . ceiling; "NP.splitFraction :: a -> (Int8,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Int8) . splitFraction; "NP.splitFraction :: a -> (Int16,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Int16) . splitFraction; "NP.splitFraction :: a -> (Int32,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Int32) . splitFraction; "NP.splitFraction :: a -> (Int64,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Int64) . splitFraction; "NP.splitFraction :: a -> (Word,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Word) . splitFraction; "NP.splitFraction :: a -> (Word8,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Word8) . splitFraction; "NP.splitFraction :: a -> (Word16,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Word16) . splitFraction; "NP.splitFraction :: a -> (Word32,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Word32) . splitFraction; "NP.splitFraction :: a -> (Word64,a)" splitFraction = mapFst (P.fromIntegral :: Int -> Word64) . splitFraction; #-} {- | TODO: Should be moved to a continued fraction module. -} approxRational :: (ToRational.C a, C a) => a -> a -> Rational approxRational rat eps = simplest (rat-eps) (rat+eps) where simplest x y | y < x = simplest y x | x == y = xr | x > 0 = simplest' n d n' d' | y < 0 = - simplest' (-n') d' (-n) d | otherwise = 0 :% 1 where xr@(n:%d) = ToRational.toRational x (n':%d') = ToRational.toRational y simplest' n d n' d' -- assumes 0 < n%d < n'%d' | isZero r = q :% 1 | q /= q' = (q+1) :% 1 | otherwise = (q*n''+d'') :% n'' where (q,r) = quotRem n d (q',r') = quotRem n' d' (n'':%d'') = simplest' d' r' d r -- * generic implementation of round functions powersOfTwo :: (Ring.C a) => [a] powersOfTwo = iterate (2*) one pairsOfPowersOfTwo :: (Ring.C a, Ring.C b) => [(a,b)] pairsOfPowersOfTwo = zip powersOfTwo powersOfTwo {- | 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. -} genericFloor :: (Ord a, Ring.C a, Ring.C b) => a -> b genericFloor a = if a>=zero then genericPosFloor a else negate $ genericPosCeiling $ negate a genericCeiling :: (Ord a, Ring.C a, Ring.C b) => a -> b genericCeiling a = if a>=zero then genericPosCeiling a else negate $ genericPosFloor $ negate a genericTruncate :: (Ord a, Ring.C a, Ring.C b) => a -> b genericTruncate a = if a>=zero then genericPosFloor a else negate $ genericPosFloor $ negate a genericRound :: (Ord a, Ring.C a, Ring.C b) => a -> b genericRound a = if a>=zero then genericPosRound a else negate $ genericPosRound $ negate a genericFraction :: (Ord a, Ring.C a) => a -> a genericFraction a = if a>=zero then genericPosFraction a else fixFraction $ negate $ genericPosFraction $ negate a genericSplitFraction :: (Ord a, Ring.C a, Ring.C b) => a -> (b,a) genericSplitFraction a = if a>=zero then genericPosSplitFraction a else fixSplitFraction $ mapPair (negate, negate) $ genericPosSplitFraction $ negate a genericPosFloor :: (Ord a, Ring.C a, Ring.C b) => a -> b genericPosFloor a = snd $ foldr (\(pa,pb) acc@(accA,accB) -> let newA = accA+pa in if newA>a then acc else (newA,accB+pb)) (zero,zero) $ takeWhile ((a>=) . fst) $ pairsOfPowersOfTwo genericPosCeiling :: (Ord a, Ring.C a, Ring.C b) => a -> b genericPosCeiling a = snd $ (\(ps,u:_) -> foldr (\(pa,pb) acc@(accA,accB) -> let newA = accA-pa in if newA>=a then (newA,accB-pb) else acc) u ps) $ span ((a>) . fst) $ (zero,zero) : pairsOfPowersOfTwo {- genericPosFloorDigits :: (Ord a, Ring.C a, Ring.C b) => a -> ((a,b), [Bool]) genericPosFloorDigits a = List.mapAccumR (\acc@(accA,accB) (pa,pb) -> let newA = accA+pa b = newA<=a in (if b then (newA,accB+pb) else acc, b)) (zero,zero) $ takeWhile ((a>=) . fst) $ pairsOfPowersOfTwo -} genericHalfPosFloorDigits :: (Ord a, Ring.C a, Ring.C b) => a -> ((a,b), [Bool]) genericHalfPosFloorDigits a = List.mapAccumR (\acc@(accA,accB) (pa,pb) -> let newA = accA+pa b = newA<=a in (if b then (newA,accB+pb) else acc, b)) (zero,zero) $ takeWhile ((a>=) . fst) $ zip powersOfTwo (zero:powersOfTwo) genericPosRound :: (Ord a, Ring.C a, Ring.C b) => a -> b genericPosRound a = let a2 = 2*a ((ai,bi), ds) = genericHalfPosFloorDigits a2 in if ai==a2 then case ds of True : True : _ -> bi+one _ -> bi else case ds of True : _ -> bi+one _ -> bi genericPosFraction :: (Ord a, Ring.C a) => a -> a genericPosFraction a = foldr (\p acc -> if p>acc then acc else acc-p) a $ takeWhile (a>=) $ powersOfTwo genericPosSplitFraction :: (Ord a, Ring.C a, Ring.C b) => a -> (b,a) genericPosSplitFraction a = foldr (\(pb,pa) acc@(accB,accA) -> if pa>accA then acc else (accB+pb,accA-pa)) (zero,a) $ takeWhile ((a>=) . snd) $ pairsOfPowersOfTwo {- | Needs linear time with respect to the number of digits. This and other functions using OrderDecision like @floor@ where argument and result are the same may be moved to a new module. -} decisionPosFraction :: (OrdDec.C a, Ring.C a) => a -> a decisionPosFraction a0 = (\ps -> foldr (\p cont a -> (a=?p) (a-p) a) (error "decisionPosFraction: end of list should never be reached") ps a0) $ concatMap (reverse . flip take powersOfTwo) powersOfTwo {- Works but needs quadratic time with respect to the number of digits. I feel that there must be something more efficient. -} decisionPosFractionSqrTime :: (OrdDec.C a, Ring.C a) => a -> a decisionPosFractionSqrTime a0 = (\ps -> foldr (\p cont a -> (a=?p) (a-p) a) (error "decisionPosFraction: end of list should never be reached") ps a0) $ concatMap reverse $ inits powersOfTwo