X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=ghc%2Flib%2Fstd%2FPrelNum.lhs;h=48ed0d956373317893d1ffbd8685f51b65d6b9d5;hb=e921b2e307532e0f30eefa88b11a124be592bde4;hp=f70f7269ec0f1170c154ca6067892b77166da508;hpb=f7989a6dea8c43352f363117d9bb07439953ccdc;p=ghc-hetmet.git

diff --git a/ghc/lib/std/PrelNum.lhs b/ghc/lib/std/PrelNum.lhs
index f70f726..48ed0d9 100644
--- a/ghc/lib/std/PrelNum.lhs
+++ b/ghc/lib/std/PrelNum.lhs
@@ -4,6 +4,15 @@
 
 \section[PrelNum]{Module @PrelNum@}
 
+The class
+
+	Num
+
+and the type
+
+	Integer
+
+
 \begin{code}
 {-# OPTIONS -fno-implicit-prelude #-}
 
@@ -15,16 +24,16 @@ import PrelList
 import PrelEnum
 import PrelShow
 
-infixr 8  ^, ^^, **
-infixl 7  %, /, `quot`, `rem`, `div`, `mod`
 infixl 7  *
 infixl 6  +, -
 
+default ()		-- Double isn't available yet, 
+			-- and we shouldn't be using defaults anyway
 \end{code}
 
 %*********************************************************
 %*							*
-\subsection{Standard numeric classes}
+\subsection{Standard numeric class}
 %*							*
 %*********************************************************
 
@@ -41,104 +50,20 @@ class  (Eq a, Show a) => Num a  where
     fromInt (I# i#)	= fromInteger (S# i#)
 					-- Go via the standard class-op if the
 					-- non-standard one ain't provided
+\end{code}
 
-class  (Num a, Ord a) => Real a  where
-    toRational		::  a -> Rational
-
-class  (Real a, Enum a) => Integral a  where
-    quot, rem, div, mod	:: a -> a -> a
-    quotRem, divMod	:: a -> a -> (a,a)
-    toInteger		:: a -> Integer
-    toInt		:: a -> Int -- partain: Glasgow extension
-
-    n `quot` d		=  q  where (q,_) = quotRem n d
-    n `rem` d		=  r  where (_,r) = quotRem n d
-    n `div` d		=  q  where (q,_) = divMod n d
-    n `mod` d		=  r  where (_,r) = divMod n d
-    divMod n d 		=  if signum r == negate (signum d) then (q-1, r+d) else qr
-			   where qr@(q,r) = quotRem n d
-
-class  (Num a) => Fractional a  where
-    (/)			:: a -> a -> a
-    recip		:: a -> a
-    fromRational	:: Rational -> a
-
-    recip x		=  1 / x
-    x / y		= x * recip y
-
-class  (Fractional a) => Floating a  where
-    pi			:: a
-    exp, log, sqrt	:: a -> a
-    (**), logBase	:: a -> a -> a
-    sin, cos, tan	:: a -> a
-    asin, acos, atan	:: a -> a
-    sinh, cosh, tanh	:: a -> a
-    asinh, acosh, atanh :: a -> a
-
-    x ** y		=  exp (log x * y)
-    logBase x y		=  log y / log x
-    sqrt x		=  x ** 0.5
-    tan  x		=  sin  x / cos  x
-    tanh x		=  sinh x / cosh x
-
-class  (Real a, Fractional a) => RealFrac a  where
-    properFraction	:: (Integral b) => a -> (b,a)
-    truncate, round	:: (Integral b) => a -> b
-    ceiling, floor	:: (Integral b) => a -> b
-
-    truncate x		=  m  where (m,_) = properFraction x
-    
-    round x		=  let (n,r) = properFraction x
-    			       m     = if r < 0 then n - 1 else n + 1
-    			   in case signum (abs r - 0.5) of
-    				-1 -> n
-    			 	0  -> if even n then n else m
-    				1  -> m
-    
-    ceiling x		=  if r > 0 then n + 1 else n
-    			   where (n,r) = properFraction x
-    
-    floor x		=  if r < 0 then n - 1 else n
-    			   where (n,r) = properFraction x
-
-class  (RealFrac a, Floating a) => RealFloat a  where
-    floatRadix		:: a -> Integer
-    floatDigits		:: a -> Int
-    floatRange		:: a -> (Int,Int)
-    decodeFloat		:: a -> (Integer,Int)
-    encodeFloat		:: Integer -> Int -> a
-    exponent		:: a -> Int
-    significand		:: a -> a
-    scaleFloat		:: Int -> a -> a
-    isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE
-                        :: a -> Bool
-    atan2	        :: a -> a -> a
-
-
-    exponent x		=  if m == 0 then 0 else n + floatDigits x
-			   where (m,n) = decodeFloat x
-
-    significand x	=  encodeFloat m (negate (floatDigits x))
-			   where (m,_) = decodeFloat x
-
-    scaleFloat k x	=  encodeFloat m (n+k)
-			   where (m,n) = decodeFloat x
-			   
-    atan2 y x
-      | x > 0            =  atan (y/x)
-      | x == 0 && y > 0  =  pi/2
-      | x <  0 && y > 0  =  pi + atan (y/x) 
-      |(x <= 0 && y < 0)            ||
-       (x <  0 && isNegativeZero y) ||
-       (isNegativeZero x && isNegativeZero y)
-                         = -atan2 (-y) x
-      | y == 0 && (x < 0 || isNegativeZero x)
-                          =  pi    -- must be after the previous test on zero y
-      | x==0 && y==0      =  y     -- must be after the other double zero tests
-      | otherwise         =  x + y -- x or y is a NaN, return a NaN (via +)
+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@}
@@ -157,57 +82,228 @@ instance  Num Int  where
 	     | n `eqInt` 0 = 0
 	     | otherwise   = 1
 
-    fromInteger (S# i#) = I# i#
-    fromInteger (J# s# d#)
-      = case (integer2Int# s# d#) of { i# -> I# i# }
+    fromInteger n = integer2Int n
+    fromInt n	  = n
+\end{code}
 
-    fromInt n		= n
 
-instance  Real Int  where
-    toRational x	=  toInteger x % 1
+\begin{code}
+-- These can't go in PrelBase with the defn of Int, because
+-- we don't have pairs defined at that time!
 
-instance  Integral Int	where
-    a@(I# _) `quotRem` b@(I# _)	= (a `quotInt` b, a `remInt` b)
+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)
 
-    -- Following chks for zero divisor are non-standard (WDP)
-    a `quot` b	=  if b /= 0
-		   then a `quotInt` b
-		   else error "Prelude.Integral.quot{Int}: divide by 0"
-    a `rem` b	=  if b /= 0
-		   then a `remInt` b
-		   else error "Prelude.Integral.rem{Int}: divide by 0"
-
-    x `div` y = if x > 0 && y < 0	then quotInt (x-y-1) y
-		else if x < 0 && y > 0	then quotInt (x-y+1) y
-		else quotInt x y
-    x `mod` y = if x > 0 && y < 0 || x < 0 && y > 0 then
-		    if r/=0 then r+y else 0
-	    	else
-		    r
-	      where r = remInt x y
-
-    divMod x@(I# _) y@(I# _) = (x `div` y, x `mod` y)
+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#
 
---OLD:   even x = eqInt (x `mod` 2) 0
---OLD:   odd x  = neInt (x `mod` 2) 0
+int2Integer :: Int -> Integer
+{-# INLINE int2Integer #-}
+int2Integer (I# i) = S# i
 
-    toInteger (I# i)  = int2Integer i  -- give back a full-blown Integer
-    toInt x	      = x
+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{Instances for @Integer@}
+\subsection{Dividing @Integers@}
 %*							*
 %*********************************************************
 
 \begin{code}
-toBig (S# i) = case int2Integer# i of { (# s, d #) -> J# s d }
-toBig i@(J# _ _) = i
+quotRemInteger :: Integer -> Integer -> (Integer, Integer)
+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 (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 (S# a) (S# b) 
+  = S# (remInt# a b)
+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 (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 (S# a) (S# b) 
+  = S# (quotInt# a b)
+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 (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 (S# a) (S# b)
+  = case gcdInt# a b of g -> S# g
+gcdInteger ia@(S# a) ib@(J# sb b)
+  | a  ==# 0#  = abs ib
+  | sb ==# 0#  = abs ia
+  | otherwise  = case gcdIntegerInt# sb b a of g -> S# g
+gcdInteger ia@(J# sa a) ib@(S# b)
+  | sa ==# 0#  = abs ib
+  | b ==# 0#   = abs ia
+  | otherwise  = case gcdIntegerInt# sa a b of g -> S# g
+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 (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 #) ->
@@ -258,90 +354,21 @@ instance  Num Integer  where
     fromInteger	x	=  x
 
     fromInt (I# i)	=  S# i
+\end{code}
 
-instance  Real Integer  where
-    toRational x	=  x % 1
-
-instance  Integral Integer where
-	-- ToDo:  a `rem`  b returns a small integer if b is small,
-	--	  a `quot` b returns a small integer if a is small.
-    quotRem (S# i) (S# j)         
-      = case quotRem (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j) 
-    quotRem i1@(J# _ _) i2@(S# _) = quotRem i1 (toBig i2)
-    quotRem i1@(S# _) i2@(J# _ _) = quotRem (toBig i1) i2
-    quotRem (J# s1 d1) (J# s2 d2)
-      = case (quotRemInteger# s1 d1 s2 d2) of
-	  (# s3, d3, s4, d4 #)
-	    -> (J# s3 d3, J# s4 d4)
-
-    toInteger n	     = n
-    toInt (S# i)     = I# i
-    toInt (J# s d)   = case (integer2Int# s d) of { n# -> I# n# }
-
-	-- we've got specialised quot/rem methods for Integer (see below)
-    n `quot` d = n `quotInteger` d
-    n `rem`  d = n `remInteger`  d
-
-    n `div` d	=  q  where (q,_) = divMod n d
-    n `mod` d	=  r  where (_,r) = divMod n d
-
-    divMod (S# i) (S# j)         
-      = case divMod (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j) 
-    divMod i1@(J# _ _) i2@(S# _) = divMod i1 (toBig i2)
-    divMod i1@(S# _) i2@(J# _ _) = divMod (toBig i1) i2
-    divMod (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 (S# a) (S# b) = S# (remInt# a b)
-remInteger ia@(S# a) (J# sb b)
-  = if sb ==# 1#
-    then
-      S# (remInt# a (word2Int# (integer2Word# sb b)))
-    else if sb ==# -1# then
-      S# (remInt# a (0# -# (word2Int# (integer2Word# sb b))))
-    else if 0# <# sb then
-      ia
-    else
-      S# (0# -# a)
-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 (S# a) (S# b) = S# (quotInt# a b)
-quotInteger (S# a) (J# sb b)
-  = if sb ==# 1#
-    then
-      S# (quotInt# a (word2Int# (integer2Word# sb b)))
-    else if sb ==# -1# then
-      S# (quotInt# a (0# -# (word2Int# (integer2Word# sb b))))
-    else
-      zeroInteger
-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
 
-zeroInteger :: Integer
-zeroInteger = S# 0#
+%*********************************************************
+%*							*
+\subsection{The @Integer@ instance for @Enum@}
+%*							*
+%*********************************************************
 
-------------------------------------------------------------------------
+\begin{code}
 instance  Enum Integer  where
     succ x		 = x + 1
     pred x		 = x - 1
-    toEnum n		 =  toInteger n
-    fromEnum n		 =  toInt n
+    toEnum n		 = int2Integer n
+    fromEnum n		 = integer2Int n
 
     {-# INLINE enumFrom #-}
     {-# INLINE enumFromThen #-}
@@ -390,9 +417,10 @@ dn_list x delta lim = go (x::Integer)
  #-}
 \end{code}
 
+
 %*********************************************************
 %*							*
-\subsection{Show code for Integers}
+\subsection{The @Integer@ instances for @Show@}
 %*							*
 %*********************************************************
 
@@ -414,147 +442,7 @@ jtos i rs
   jtos' :: Integer -> String -> String
   jtos' n cs
    | n < 10    = chr (fromInteger n + (ord_0::Int)) : cs
-   | otherwise = jtos' q (chr (toInt r + (ord_0::Int)) : cs)
+   | otherwise = jtos' q (chr (integer2Int r + (ord_0::Int)) : cs)
     where
-     (q,r) = n `quotRem` 10
-
-ord_0 :: Num a => a
-ord_0 = fromInt (ord '0')
-\end{code}
-
-%*********************************************************
-%*							*
-\subsection{The @Ratio@ and @Rational@ types}
-%*							*
-%*********************************************************
-
-\begin{code}
-data  (Integral a)	=> Ratio a = !a :% !a  deriving (Eq)
-type  Rational		=  Ratio Integer
-
-{-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
-(%)			:: (Integral a) => a -> a -> Ratio a
-numerator, denominator	:: (Integral a) => Ratio a -> a
-\end{code}
-
-\tr{reduce} is a subsidiary function used only in this module .
-It normalises a ratio by dividing both numerator and denominator by
-their greatest common divisor.
-
-\begin{code}
-reduce ::  (Integral a) => a -> a -> Ratio a
-reduce _ 0		=  error "Ratio.%: zero denominator"
-reduce x y		=  (x `quot` d) :% (y `quot` d)
-			   where d = gcd x y
-\end{code}
-
-\begin{code}
-x % y			=  reduce (x * signum y) (abs y)
-
-numerator   (x :% _)	=  x
-denominator (_ :% y)	=  y
-
-\end{code}
-
-%*********************************************************
-%*							*
-\subsection{Overloaded numeric functions}
-%*							*
-%*********************************************************
-
-\begin{code}
-
-{-# SPECIALISE subtract :: Int -> Int -> Int #-}
-subtract	:: (Num a) => a -> a -> a
-subtract x y	=  y - x
-
-even, odd	:: (Integral a) => a -> Bool
-even n		=  n `rem` 2 == 0
-odd		=  not . even
-
-gcd		:: (Integral a) => a -> a -> a
-gcd 0 0		=  error "Prelude.gcd: gcd 0 0 is undefined"
-gcd x y		=  gcd' (abs x) (abs y)
-		   where gcd' a 0  =  a
-			 gcd' a b  =  gcd' b (a `rem` b)
-
-{-# SPECIALISE lcm ::
-	Int -> Int -> Int,
-	Integer -> Integer -> Integer #-}
-lcm		:: (Integral a) => a -> a -> a
-lcm _ 0		=  0
-lcm 0 _		=  0
-lcm x y		=  abs ((x `quot` (gcd x y)) * y)
-
-{-# SPECIALISE (^) ::
-	Integer -> Integer -> Integer,
-	Integer -> Int -> Integer,
-	Int -> Int -> Int #-}
-(^)		:: (Num a, Integral b) => a -> b -> a
-_ ^ 0		=  1
-x ^ n | n > 0	=  f x (n-1) x
-		   where f _ 0 y = y
-		         f a d y = g a d  where
-			           g b i | even i  = g (b*b) (i `quot` 2)
-				         | otherwise = f b (i-1) (b*y)
-_ ^ _		= error "Prelude.^: negative exponent"
-
-{- SPECIALISE (^^) ::
-	Double -> Int -> Double,
-	Rational -> Int -> Rational #-}
-(^^)		:: (Fractional a, Integral b) => a -> b -> a
-x ^^ n		=  if n >= 0 then x^n else recip (x^(negate n))
-\end{code}
-
-%*********************************************************
-%*							*
-\subsection{Specialized versions of gcd/lcm for Int/Integer}
-%*							*
-%*********************************************************
-
-\begin{code}
-{-# RULES
-"Int.gcd"      forall a b . gcd  a b = gcdInt a b
-"Integer.gcd"  forall a b . gcd  a b = gcdInteger  a b
-"Integer.lcm"  forall a b . lcm  a b = lcmInteger  a b
- #-}
-
-gcdInt :: Int -> Int -> Int
-gcdInt (I# a)  (I# b)
-  = I# (gcdInt# a b)
-
-gcdInteger :: Integer -> Integer -> Integer
-gcdInteger (S# a) (S# b)
-  = case gcdInt# a b of g -> S# g
-gcdInteger ia@(S# a) ib@(J# sb b)
-  | a  ==# 0#  = abs ib
-  | sb ==# 0#  = abs ia
-  | otherwise  = case gcdIntegerInt# sb b a of g -> S# g
-gcdInteger ia@(J# sa a) ib@(S# b)
-  | sa ==# 0#  = abs ib
-  | b ==# 0#   = abs ia
-  | otherwise  = case gcdIntegerInt# sa a b of g -> S# g
-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 (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
+     (q,r) = n `quotRemInteger` 10
 \end{code}