-% ------------------------------------------------------------------------------
-% $Id: Num.lhs,v 1.1 2001/06/28 14:15:03 simonmar Exp $
-%
-% (c) The University of Glasgow, 1994-2000
-%
-
-\section[GHC.Num]{Module @GHC.Num@}
-
-The class
-
- Num
-
-and the type
-
- Integer
-
-
\begin{code}
-{-# OPTIONS -fno-implicit-prelude #-}
-
+{-# OPTIONS_GHC -fno-implicit-prelude #-}
+-----------------------------------------------------------------------------
+-- |
+-- Module : GHC.Num
+-- Copyright : (c) The University of Glasgow 1994-2002
+-- License : see libraries/base/LICENSE
+--
+-- Maintainer : cvs-ghc@haskell.org
+-- Stability : internal
+-- Portability : non-portable (GHC Extensions)
+--
+-- The 'Num' class and the 'Integer' type.
+--
+-----------------------------------------------------------------------------
+
+#include "MachDeps.h"
+#if SIZEOF_HSWORD == 4
+#define LEFTMOST_BIT 2147483648
+#define DIGITS 9
+#define BASE 1000000000
+#elif SIZEOF_HSWORD == 8
+#define LEFTMOST_BIT 9223372036854775808
+#define DIGITS 18
+#define BASE 1000000000000000000
+#else
+#error Please define LEFTMOST_BIT to be 2^(SIZEOF_HSWORD*8-1)
+-- DIGITS should be the largest integer such that 10^DIGITS < LEFTMOST_BIT
+-- BASE should be 10^DIGITS. Note that ^ is not available yet.
+#endif
+
+-- #hide
module GHC.Num where
import {-# SOURCE #-} GHC.Err
import GHC.Base
-import GHC.List
import GHC.Enum
import GHC.Show
%*********************************************************
\begin{code}
+-- | Basic numeric class.
+--
+-- Minimal complete definition: all except 'negate' or @(-)@
class (Eq a, Show a) => Num a where
(+), (-), (*) :: a -> a -> a
+ -- | Unary negation.
negate :: a -> a
- abs, signum :: a -> a
+ -- | Absolute value.
+ abs :: a -> a
+ -- | Sign of a number.
+ -- The functions 'abs' and 'signum' should satisfy the law:
+ --
+ -- > abs x * signum x == x
+ --
+ -- For real numbers, the 'signum' is either @-1@ (negative), @0@ (zero)
+ -- or @1@ (positive).
+ signum :: a -> a
+ -- | Conversion from an 'Integer'.
+ -- An integer literal represents the application of the function
+ -- 'fromInteger' to the appropriate value of type 'Integer',
+ -- so such literals have type @('Num' a) => a@.
fromInteger :: Integer -> a
x - y = x + negate y
negate x = 0 - x
+-- | the same as @'flip' ('-')@.
+--
+-- Because @-@ is treated specially in the Haskell grammar,
+-- @(-@ /e/@)@ is not a section, but an application of prefix negation.
+-- However, @('subtract'@ /exp/@)@ is equivalent to the disallowed section.
{-# INLINE subtract #-}
subtract :: (Num a) => a -> a -> a
subtract x y = y - x
| otherwise = 1
fromInteger = integer2Int
-\end{code}
-
-
-\begin{code}
--- These can't go in GHC.Base 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)
+quotRemInt a@(I# _) 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)
-- Stricter. Sorry if you don't like it. (WDP 94/10)
\end{code}
-
%*********************************************************
%* *
\subsection{The @Integer@ type}
%*********************************************************
\begin{code}
+-- | Arbitrary-precision integers.
data Integer
= S# Int# -- small integers
+#ifndef ILX
| J# Int# ByteArray# -- large integers
+#else
+ | J# Void BigInteger -- .NET big ints
+
+foreign type dotnet "BigInteger" BigInteger
+#endif
\end{code}
Convenient boxed Integer PrimOps.
\begin{code}
quotRemInteger :: Integer -> Integer -> (Integer, Integer)
-quotRemInteger a@(S# (-2147483648#)) b = quotRemInteger (toBig a) b
+quotRemInteger a@(S# (-LEFTMOST_BIT#)) 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)
(# s3, d3, s4, d4 #)
-> (J# s3 d3, J# s4 d4)
-divModInteger a@(S# (-2147483648#)) b = divModInteger (toBig a) b
+divModInteger a@(S# (-LEFTMOST_BIT#)) 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)
-> (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 ia ib
+ | ib == 0 = error "Prelude.Integral.rem{Integer}: divide by 0"
+remInteger a@(S# (-LEFTMOST_BIT#)) 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)
= 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 ia ib
+ | ib == 0 = error "Prelude.Integral.quot{Integer}: divide by 0"
+quotInteger a@(S# (-LEFTMOST_BIT#)) 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)
\begin{code}
gcdInteger :: Integer -> Integer -> Integer
-- SUP: Do we really need the first two cases?
-gcdInteger a@(S# (-2147483648#)) b = gcdInteger (toBig a) b
-gcdInteger a b@(S# (-2147483648#)) = gcdInteger a (toBig b)
+gcdInteger a@(S# (-LEFTMOST_BIT#)) b = gcdInteger (toBig a) b
+gcdInteger a b@(S# (-LEFTMOST_BIT#)) = gcdInteger a (toBig b)
gcdInteger (S# a) (S# b) = case gcdInt (I# a) (I# b) of { I# c -> S# c }
gcdInteger ia@(S# 0#) ib@(J# 0# _) = error "GHC.Num.gcdInteger: gcd 0 0 is undefined"
gcdInteger ia@(S# a) ib@(J# sb b)
ab = abs b
divExact :: Integer -> Integer -> Integer
-divExact a@(S# (-2147483648#)) b = divExact (toBig a) b
+divExact a@(S# (-LEFTMOST_BIT#)) b = divExact (toBig a) b
divExact (S# a) (S# b) = S# (quotInt# a b)
divExact (S# a) (J# sb b)
= S# (quotInt# a (integer2Int# sb b))
fromInteger x = x
-- ORIG: abs n = if n >= 0 then n else -n
- abs (S# (-2147483648#)) = 2147483648
+ abs (S# (-LEFTMOST_BIT#)) = LEFTMOST_BIT
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
minusInteger i1@(S# _) i2@(J# _ _) = toBig i1 - i2
minusInteger (J# s1 d1) (J# s2 d2) = case minusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
-timesInteger i1@(S# i) i2@(S# j) = case mulIntC# i j of { (# r, c #) ->
- if c ==# 0# then S# r
- else toBig i1 * toBig i2 }
+timesInteger i1@(S# i) i2@(S# j) = if mulIntMayOflo# i j ==# 0#
+ then S# (i *# j)
+ else toBig i1 * toBig i2
timesInteger i1@(J# _ _) i2@(S# _) = i1 * toBig i2
timesInteger i1@(S# _) i2@(J# _ _) = toBig i1 * i2
timesInteger (J# s1 d1) (J# s2 d2) = case timesInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
-negateInteger (S# (-2147483648#)) = 2147483648
+negateInteger (S# (-LEFTMOST_BIT#)) = LEFTMOST_BIT
negateInteger (S# i) = S# (negateInt# i)
negateInteger (J# s d) = J# (negateInt# s) d
\end{code}
{-# INLINE enumFromThen #-}
{-# INLINE enumFromTo #-}
{-# INLINE enumFromThenTo #-}
- enumFrom x = efdInteger x 1
- enumFromThen x y = efdInteger x (y-x)
- enumFromTo x lim = efdtInteger x 1 lim
- enumFromThenTo x y lim = efdtInteger x (y-x) lim
-
-
-efdInteger = enumDeltaIntegerList
-efdtInteger = enumDeltaToIntegerList
+ enumFrom x = enumDeltaInteger x 1
+ enumFromThen x y = enumDeltaInteger x (y-x)
+ enumFromTo x lim = enumDeltaToInteger x 1 lim
+ enumFromThenTo x y lim = enumDeltaToInteger x (y-x) lim
{-# RULES
-"efdInteger" forall x y. efdInteger x y = build (\c _ -> enumDeltaIntegerFB c x y)
-"efdtInteger" forall x y l.efdtInteger x y l = build (\c n -> enumDeltaToIntegerFB c n x y l)
-"enumDeltaInteger" enumDeltaIntegerFB (:) = enumDeltaIntegerList
-"enumDeltaToInteger" enumDeltaToIntegerFB (:) [] = enumDeltaToIntegerList
+"enumDeltaInteger" [~1] forall x y. enumDeltaInteger x y = build (\c _ -> enumDeltaIntegerFB c x y)
+"efdtInteger" [~1] forall x y l.enumDeltaToInteger x y l = build (\c n -> enumDeltaToIntegerFB c n x y l)
+"enumDeltaInteger" [1] enumDeltaIntegerFB (:) = enumDeltaInteger
+"enumDeltaToInteger" [1] enumDeltaToIntegerFB (:) [] = enumDeltaToInteger
#-}
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
+enumDeltaInteger :: Integer -> Integer -> [Integer]
+enumDeltaInteger x d = x : enumDeltaInteger (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
+enumDeltaToInteger x delta lim
| delta >= 0 = up_list x delta lim
| otherwise = dn_list x delta lim
\begin{code}
instance Show Integer where
showsPrec p n r
- | n < 0 && p > 6 = '(' : jtos n (')' : r)
+ | p > 6 && n < 0 = '(' : jtos n (')' : r)
+ -- Minor point: testing p first gives better code
+ -- in the not-uncommon case where the p argument
+ -- is a constant
| otherwise = jtos n r
showList = showList__ (showsPrec 0)
+-- Divide an conquer implementation of string conversion
jtos :: Integer -> String -> String
jtos n cs
| n < 0 = '-' : jtos' (-n) cs
| otherwise = jtos' n cs
where
jtos' :: Integer -> String -> String
- jtos' n' cs'
- | n' < 10 = case unsafeChr (ord '0' + fromInteger n') of
- c@(C# _) -> c:cs'
- | otherwise = case unsafeChr (ord '0' + fromInteger r) of
- c@(C# _) -> jtos' q (c:cs')
+ jtos' n cs
+ | n < BASE = jhead (fromInteger n) cs
+ | otherwise = jprinth (jsplitf (BASE*BASE) n) cs
+
+ -- Split n into digits in base p. We first split n into digits
+ -- in base p*p and then split each of these digits into two.
+ -- Note that the first 'digit' modulo p*p may have a leading zero
+ -- in base p that we need to drop - this is what jsplith takes care of.
+ -- jsplitb the handles the remaining digits.
+ jsplitf :: Integer -> Integer -> [Integer]
+ jsplitf p n
+ | p > n = [n]
+ | otherwise = jsplith p (jsplitf (p*p) n)
+
+ jsplith :: Integer -> [Integer] -> [Integer]
+ jsplith p (n:ns) =
+ if q > 0 then fromInteger q : fromInteger r : jsplitb p ns
+ else fromInteger r : jsplitb p ns
where
- (q,r) = n' `quotRemInteger` 10
+ (q, r) = n `quotRemInteger` p
+
+ jsplitb :: Integer -> [Integer] -> [Integer]
+ jsplitb p [] = []
+ jsplitb p (n:ns) = q : r : jsplitb p ns
+ where
+ (q, r) = n `quotRemInteger` p
+
+ -- Convert a number that has been split into digits in base BASE^2
+ -- this includes a last splitting step and then conversion of digits
+ -- that all fit into a machine word.
+ jprinth :: [Integer] -> String -> String
+ jprinth (n:ns) cs =
+ if q > 0 then jhead q $ jblock r $ jprintb ns cs
+ else jhead r $ jprintb ns cs
+ where
+ (q', r') = n `quotRemInteger` BASE
+ q = fromInteger q'
+ r = fromInteger r'
+
+ jprintb :: [Integer] -> String -> String
+ jprintb [] cs = cs
+ jprintb (n:ns) cs = jblock q $ jblock r $ jprintb ns cs
+ where
+ (q', r') = n `quotRemInteger` BASE
+ q = fromInteger q'
+ r = fromInteger r'
+
+ -- Convert an integer that fits into a machine word. Again, we have two
+ -- functions, one that drops leading zeros (jhead) and one that doesn't
+ -- (jblock)
+ jhead :: Int -> String -> String
+ jhead n cs
+ | n < 10 = case unsafeChr (ord '0' + n) of
+ c@(C# _) -> c : cs
+ | otherwise = case unsafeChr (ord '0' + r) of
+ c@(C# _) -> jhead q (c : cs)
+ where
+ (q, r) = n `quotRemInt` 10
+
+ jblock = jblock' {- ' -} DIGITS
+
+ jblock' :: Int -> Int -> String -> String
+ jblock' d n cs
+ | d == 1 = case unsafeChr (ord '0' + n) of
+ c@(C# _) -> c : cs
+ | otherwise = case unsafeChr (ord '0' + r) of
+ c@(C# _) -> jblock' (d - 1) q (c : cs)
+ where
+ (q, r) = n `quotRemInt` 10
+
\end{code}
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