% % (c) The AQUA Project, Glasgow University, 1994-1996 % \section[PrelNum]{Module @PrelNum@} The class Num and the type Integer \begin{code} {-# OPTIONS -fno-implicit-prelude #-} module PrelNum where import {-# SOURCE #-} PrelErr import PrelBase import PrelList import PrelEnum import PrelShow infixl 7 * infixl 6 +, - default () -- Double isn't available yet, -- and we shouldn't be using defaults anyway \end{code} %********************************************************* %* * \subsection{Standard numeric class} %* * %********************************************************* \begin{code} class (Eq a, Show a) => Num a where (+), (-), (*) :: a -> a -> a negate :: a -> a abs, signum :: a -> a fromInteger :: Integer -> a fromInt :: Int -> a -- partain: Glasgow extension x - y = x + negate y negate x = 0 - x fromInt (I# i#) = fromInteger (S# i#) -- Go via the standard class-op if the -- non-standard one ain't provided \end{code} A few small numeric functions \begin{code} subtract :: (Num a) => a -> a -> a {-# INLINE subtract #-} subtract x y = y - x ord_0 :: Num a => a ord_0 = fromInt (ord '0') \end{code} %********************************************************* %* * \subsection{Instances for @Int@} %* * %********************************************************* \begin{code} instance Num Int where (+) x y = plusInt x y (-) x y = minusInt x y negate x = negateInt x (*) x y = timesInt x y abs n = if n geInt 0 then n else (negateInt n) signum n | n ltInt 0 = negateInt 1 | n eqInt 0 = 0 | otherwise = 1 fromInteger n = integer2Int n fromInt n = n \end{code} \begin{code} -- These can't go in PrelBase with the defn of Int, because -- we don't have pairs defined at that time! quotRemInt :: Int -> Int -> (Int, Int) a@(I# _) quotRemInt b@(I# _) = (a quotInt b, a remInt b) -- OK, so I made it a little stricter. Shoot me. (WDP 94/10) divModInt :: Int -> Int -> (Int, Int) divModInt x@(I# _) y@(I# _) = (x divInt y, x modInt y) -- Stricter. Sorry if you don't like it. (WDP 94/10) \end{code} %********************************************************* %* * \subsection{The @Integer@ type} %* * %********************************************************* \begin{code} data Integer = S# Int# -- small integers | J# Int# ByteArray# -- large integers \end{code} Convenient boxed Integer PrimOps. \begin{code} zeroInteger :: Integer zeroInteger = S# 0# int2Integer :: Int -> Integer {-# INLINE int2Integer #-} int2Integer (I# i) = S# i integer2Int :: Integer -> Int integer2Int (S# i) = I# i integer2Int (J# s d) = case (integer2Int# s d) of { n# -> I# n# } addr2Integer :: Addr# -> Integer {-# INLINE addr2Integer #-} addr2Integer x = case addr2Integer# x of (# s, d #) -> J# s d toBig (S# i) = case int2Integer# i of { (# s, d #) -> J# s d } toBig i@(J# _ _) = i \end{code} %********************************************************* %* * \subsection{Dividing @Integers@} %* * %********************************************************* \begin{code} quotRemInteger :: Integer -> Integer -> (Integer, Integer) quotRemInteger a@(S# (-2147483648#)) b = quotRemInteger (toBig a) b quotRemInteger (S# i) (S# j) = case quotRemInt (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j ) quotRemInteger i1@(J# _ _) i2@(S# _) = quotRemInteger i1 (toBig i2) quotRemInteger i1@(S# _) i2@(J# _ _) = quotRemInteger (toBig i1) i2 quotRemInteger (J# s1 d1) (J# s2 d2) = case (quotRemInteger# s1 d1 s2 d2) of (# s3, d3, s4, d4 #) -> (J# s3 d3, J# s4 d4) divModInteger a@(S# (-2147483648#)) b = divModInteger (toBig a) b divModInteger (S# i) (S# j) = case divModInt (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j) divModInteger i1@(J# _ _) i2@(S# _) = divModInteger i1 (toBig i2) divModInteger i1@(S# _) i2@(J# _ _) = divModInteger (toBig i1) i2 divModInteger (J# s1 d1) (J# s2 d2) = case (divModInteger# s1 d1 s2 d2) of (# s3, d3, s4, d4 #) -> (J# s3 d3, J# s4 d4) remInteger :: Integer -> Integer -> Integer remInteger ia 0 = error "Prelude.Integral.rem{Integer}: divide by 0" remInteger a@(S# (-2147483648#)) b = remInteger (toBig a) b remInteger (S# a) (S# b) = S# (remInt# a b) {- Special case doesn't work, because a 1-element J# has the range -(2^32-1) -- 2^32-1, whereas S# has the range -2^31 -- (2^31-1) remInteger ia@(S# a) (J# sb b) | sb ==# 1# = S# (remInt# a (word2Int# (integer2Word# sb b))) | sb ==# -1# = S# (remInt# a (0# -# (word2Int# (integer2Word# sb b)))) | 0# <# sb = ia | otherwise = S# (0# -# a) -} remInteger ia@(S# _) ib@(J# _ _) = remInteger (toBig ia) ib remInteger (J# sa a) (S# b) = case int2Integer# b of { (# sb, b #) -> case remInteger# sa a sb b of { (# sr, r #) -> S# (sr *# (word2Int# (integer2Word# sr r))) }} remInteger (J# sa a) (J# sb b) = case remInteger# sa a sb b of (# sr, r #) -> J# sr r quotInteger :: Integer -> Integer -> Integer quotInteger ia 0 = error "Prelude.Integral.quot{Integer}: divide by 0" quotInteger a@(S# (-2147483648#)) b = quotInteger (toBig a) b quotInteger (S# a) (S# b) = S# (quotInt# a b) {- Special case disabled, see remInteger above quotInteger (S# a) (J# sb b) | sb ==# 1# = S# (quotInt# a (word2Int# (integer2Word# sb b))) | sb ==# -1# = S# (quotInt# a (0# -# (word2Int# (integer2Word# sb b)))) | otherwise = zeroInteger -} quotInteger ia@(S# _) ib@(J# _ _) = quotInteger (toBig ia) ib quotInteger (J# sa a) (S# b) = case int2Integer# b of { (# sb, b #) -> case quotInteger# sa a sb b of (# sq, q #) -> J# sq q } quotInteger (J# sa a) (J# sb b) = case quotInteger# sa a sb b of (# sg, g #) -> J# sg g \end{code} \begin{code} gcdInteger :: Integer -> Integer -> Integer gcdInteger a@(S# (-2147483648#)) b = gcdInteger (toBig a) b gcdInteger a b@(S# (-2147483648#)) = gcdInteger a (toBig b) gcdInteger (S# a) (S# b) = S# (gcdInt# a b) gcdInteger ia@(S# a) ib@(J# sb b) | a ==# 0# = abs ib | sb ==# 0# = abs ia | otherwise = S# (gcdIntegerInt# sb b a) gcdInteger ia@(J# sa a) ib@(S# b) | sa ==# 0# = abs ib | b ==# 0# = abs ia | otherwise = S# (gcdIntegerInt# sa a b) gcdInteger (J# sa a) (J# sb b) = case gcdInteger# sa a sb b of (# sg, g #) -> J# sg g lcmInteger :: Integer -> Integer -> Integer lcmInteger a 0 = zeroInteger lcmInteger 0 b = zeroInteger lcmInteger a b = (divExact aa (gcdInteger aa ab)) * ab where aa = abs a ab = abs b divExact :: Integer -> Integer -> Integer divExact a@(S# (-2147483648#)) b = divExact (toBig a) b divExact (S# a) (S# b) = S# (quotInt# a b) divExact (S# a) (J# sb b) = S# (quotInt# a (sb *# (word2Int# (integer2Word# sb b)))) divExact (J# sa a) (S# b) = case int2Integer# b of (# sb, b #) -> case divExactInteger# sa a sb b of (# sd, d #) -> J# sd d divExact (J# sa a) (J# sb b) = case divExactInteger# sa a sb b of (# sd, d #) -> J# sd d \end{code} %********************************************************* %* * \subsection{The @Integer@ instances for @Eq@, @Ord@} %* * %********************************************************* \begin{code} instance Eq Integer where (S# i) == (S# j) = i ==# j (S# i) == (J# s d) = cmpIntegerInt# s d i ==# 0# (J# s d) == (S# i) = cmpIntegerInt# s d i ==# 0# (J# s1 d1) == (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) ==# 0# (S# i) /= (S# j) = i /=# j (S# i) /= (J# s d) = cmpIntegerInt# s d i /=# 0# (J# s d) /= (S# i) = cmpIntegerInt# s d i /=# 0# (J# s1 d1) /= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) /=# 0# ------------------------------------------------------------------------ instance Ord Integer where (S# i) <= (S# j) = i <=# j (J# s d) <= (S# i) = cmpIntegerInt# s d i <=# 0# (S# i) <= (J# s d) = cmpIntegerInt# s d i >=# 0# (J# s1 d1) <= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <=# 0# (S# i) > (S# j) = i ># j (J# s d) > (S# i) = cmpIntegerInt# s d i ># 0# (S# i) > (J# s d) = cmpIntegerInt# s d i <# 0# (J# s1 d1) > (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) ># 0# (S# i) < (S# j) = i <# j (J# s d) < (S# i) = cmpIntegerInt# s d i <# 0# (S# i) < (J# s d) = cmpIntegerInt# s d i ># 0# (J# s1 d1) < (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <# 0# (S# i) >= (S# j) = i >=# j (J# s d) >= (S# i) = cmpIntegerInt# s d i >=# 0# (S# i) >= (J# s d) = cmpIntegerInt# s d i <=# 0# (J# s1 d1) >= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) >=# 0# compare (S# i) (S# j) | i ==# j = EQ | i <=# j = LT | otherwise = GT compare (J# s d) (S# i) = case cmpIntegerInt# s d i of { res# -> if res# <# 0# then LT else if res# ># 0# then GT else EQ } compare (S# i) (J# s d) = case cmpIntegerInt# s d i of { res# -> if res# ># 0# then LT else if res# <# 0# then GT else EQ } compare (J# s1 d1) (J# s2 d2) = case cmpInteger# s1 d1 s2 d2 of { res# -> if res# <# 0# then LT else if res# ># 0# then GT else EQ } \end{code} %********************************************************* %* * \subsection{The @Integer@ instances for @Num@} %* * %********************************************************* \begin{code} instance Num Integer where (+) i1@(S# i) i2@(S# j) = case addIntC# i j of { (# r, c #) -> if c ==# 0# then S# r else toBig i1 + toBig i2 } (+) i1@(J# _ _) i2@(S# _) = i1 + toBig i2 (+) i1@(S# _) i2@(J# _ _) = toBig i1 + i2 (+) (J# s1 d1) (J# s2 d2) = case plusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d (-) i1@(S# i) i2@(S# j) = case subIntC# i j of { (# r, c #) -> if c ==# 0# then S# r else toBig i1 - toBig i2 } (-) i1@(J# _ _) i2@(S# _) = i1 - toBig i2 (-) i1@(S# _) i2@(J# _ _) = toBig i1 - i2 (-) (J# s1 d1) (J# s2 d2) = case minusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d (*) i1@(S# i) i2@(S# j) = case mulIntC# i j of { (# r, c #) -> if c ==# 0# then S# r else toBig i1 * toBig i2 } (*) i1@(J# _ _) i2@(S# _) = i1 * toBig i2 (*) i1@(S# _) i2@(J# _ _) = toBig i1 * i2 (*) (J# s1 d1) (J# s2 d2) = case timesInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d negate (S# (-2147483648#)) = 2147483648 negate (S# i) = S# (negateInt# i) negate (J# s d) = J# (negateInt# s) d -- ORIG: abs n = if n >= 0 then n else -n abs (S# (-2147483648#)) = 2147483648 abs (S# i) = case abs (I# i) of I# j -> S# j abs n@(J# s d) = if (s >=# 0#) then n else J# (negateInt# s) d signum (S# i) = case signum (I# i) of I# j -> S# j signum (J# s d) = let cmp = cmpIntegerInt# s d 0# in if cmp ># 0# then S# 1# else if cmp ==# 0# then S# 0# else S# (negateInt# 1#) fromInteger x = x fromInt (I# i) = S# i \end{code} %********************************************************* %* * \subsection{The @Integer@ instance for @Enum@} %* * %********************************************************* \begin{code} instance Enum Integer where succ x = x + 1 pred x = x - 1 toEnum n = int2Integer n fromEnum n = integer2Int n {-# INLINE enumFrom #-} {-# INLINE enumFromThen #-} {-# INLINE enumFromTo #-} {-# INLINE enumFromThenTo #-} enumFrom x = build (\c _ -> enumDeltaIntegerFB c x 1) enumFromThen x y = build (\c _ -> enumDeltaIntegerFB c x (y-x)) enumFromTo x lim = build (\c n -> enumDeltaToIntegerFB c n x 1 lim) enumFromThenTo x y lim = build (\c n -> enumDeltaToIntegerFB c n x (y-x) lim) enumDeltaIntegerFB :: (Integer -> b -> b) -> Integer -> Integer -> b enumDeltaIntegerFB c x d = x c enumDeltaIntegerFB c (x+d) d enumDeltaIntegerList :: Integer -> Integer -> [Integer] enumDeltaIntegerList x d = x : enumDeltaIntegerList (x+d) d enumDeltaToIntegerFB c n x delta lim | delta >= 0 = up_fb c n x delta lim | otherwise = dn_fb c n x delta lim enumDeltaToIntegerList x delta lim | delta >= 0 = up_list x delta lim | otherwise = dn_list x delta lim up_fb c n x delta lim = go (x::Integer) where go x | x > lim = n | otherwise = x c go (x+delta) dn_fb c n x delta lim = go (x::Integer) where go x | x < lim = n | otherwise = x c go (x+delta) up_list x delta lim = go (x::Integer) where go x | x > lim = [] | otherwise = x : go (x+delta) dn_list x delta lim = go (x::Integer) where go x | x < lim = [] | otherwise = x : go (x+delta) {-# RULES "enumDeltaInteger" enumDeltaIntegerFB (:) = enumDeltaIntegerList "enumDeltaToInteger" enumDeltaToIntegerFB (:) [] = enumDeltaToIntegerList #-} \end{code} %********************************************************* %* * \subsection{The @Integer@ instances for @Show@} %* * %********************************************************* \begin{code} instance Show Integer where showsPrec x = showSignedInteger x showList = showList__ (showsPrec 0) showSignedInteger :: Int -> Integer -> ShowS showSignedInteger p n r | n < 0 && p > 6 = '(':jtos n (')':r) | otherwise = jtos n r jtos :: Integer -> String -> String jtos i rs | i < 0 = '-' : jtos' (-i) rs | otherwise = jtos' i rs where jtos' :: Integer -> String -> String jtos' n cs | n < 10 = chr (fromInteger n + (ord_0::Int)) : cs | otherwise = jtos' q (chr (integer2Int r + (ord_0::Int)) : cs) where (q,r) = n quotRemInteger 10 \end{code}