\begin{code}
{-# OPTIONS_GHC -fno-implicit-prelude #-}
-----------------------------------------------------------------------------
-- |
-- Module : GHC.Num
-- Copyright : (c) The University of Glasgow 1994-2002
-- License : see libraries/base/LICENSE
--
-- Maintainer : [email protected]
-- 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.Enum
import GHC.Show
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}
-- | 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
-- | 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
\end{code}
%*********************************************************
%* *
\subsection{Instances for @Int@}
%* *
%*********************************************************
\begin{code}
instance Num Int where
(+) = plusInt
(-) = minusInt
negate = negateInt
(*) = timesInt
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 = integer2Int
quotRemInt :: Int -> Int -> (Int, Int)
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)
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}
-- | 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}
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# }
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# (-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)
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# (-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)
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 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)
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# (integer2Int# 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 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)
| 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
-- SUP: Do we really need the first two cases?
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)
| a ==# 0# = abs ib
| sb ==# 0# = abs ia
| otherwise = S# (gcdIntegerInt# absSb b absA)
where absA = if a <# 0# then negateInt# a else a
absSb = if sb <# 0# then negateInt# sb else sb
gcdInteger ia@(J# _ _) ib@(S# _) = gcdInteger ib ia
gcdInteger (J# 0# _) (J# 0# _) = error "GHC.Num.gcdInteger: gcd 0 0 is undefined"
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# (-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))
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
(+) = plusInteger
(-) = minusInteger
(*) = timesInteger
negate = negateInteger
fromInteger x = x
-- ORIG: abs n = if n >= 0 then n else -n
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
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#)
plusInteger i1@(S# i) i2@(S# j) = case addIntC# i j of { (# r, c #) ->
if c ==# 0# then S# r
else toBig i1 + toBig i2 }
plusInteger i1@(J# _ _) i2@(S# _) = i1 + toBig i2
plusInteger i1@(S# _) i2@(J# _ _) = toBig i1 + i2
plusInteger (J# s1 d1) (J# s2 d2) = case plusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
minusInteger i1@(S# i) i2@(S# j) = case subIntC# i j of { (# r, c #) ->
if c ==# 0# then S# r
else toBig i1 - toBig i2 }
minusInteger i1@(J# _ _) i2@(S# _) = i1 - toBig i2
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) = 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# (-LEFTMOST_BIT#)) = LEFTMOST_BIT
negateInteger (S# i) = S# (negateInt# i)
negateInteger (J# s d) = J# (negateInt# s) d
\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 = 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
"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
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
enumDeltaToInteger 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)
\end{code}
%*********************************************************
%* *
\subsection{The @Integer@ instances for @Show@}
%* *
%*********************************************************
\begin{code}
instance Show Integer where
showsPrec p 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 < 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` 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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