2 {-# OPTIONS_GHC -fno-implicit-prelude #-}
3 -----------------------------------------------------------------------------
6 -- Copyright : (c) The University of Glasgow 1994-2002
7 -- License : see libraries/base/LICENSE
9 -- Maintainer : cvs-ghc@haskell.org
10 -- Stability : internal
11 -- Portability : non-portable (GHC Extensions)
13 -- The 'Num' class and the 'Integer' type.
15 -----------------------------------------------------------------------------
18 #if SIZEOF_HSWORD == 4
19 #define LEFTMOST_BIT 2147483648
20 #elif SIZEOF_HSWORD == 8
21 #define LEFTMOST_BIT 9223372036854775808
23 #error Please define LEFTMOST_BIT to be 2^(SIZEOF_HSWORD*8-1)
28 import {-# SOURCE #-} GHC.Err
36 default () -- Double isn't available yet,
37 -- and we shouldn't be using defaults anyway
40 %*********************************************************
42 \subsection{Standard numeric class}
44 %*********************************************************
47 -- | Basic numeric class.
49 -- Minimal complete definition: all except 'negate' or @(-)@
50 class (Eq a, Show a) => Num a where
51 (+), (-), (*) :: a -> a -> a
56 -- | Sign of a number.
57 -- The functions 'abs' and 'signum' should satisfy the law:
59 -- > abs x * signum x == x
61 -- For real numbers, the 'signum' is either @-1@ (negative), @0@ (zero)
64 -- | Conversion from an 'Integer'.
65 -- An integer literal represents the application of the function
66 -- 'fromInteger' to the appropriate value of type 'Integer',
67 -- so such literals have type @('Num' a) => a@.
68 fromInteger :: Integer -> a
73 -- | the same as @'flip' ('-')@.
75 -- Because @-@ is treated specially in the Haskell grammar,
76 -- @(-@ /e/@)@ is not a section, but an application of prefix negation.
77 -- However, @('subtract'@ /exp/@)@ is equivalent to the disallowed section.
78 {-# INLINE subtract #-}
79 subtract :: (Num a) => a -> a -> a
84 %*********************************************************
86 \subsection{Instances for @Int@}
88 %*********************************************************
91 instance Num Int where
96 abs n = if n `geInt` 0 then n else negateInt n
98 signum n | n `ltInt` 0 = negateInt 1
102 fromInteger = integer2Int
104 quotRemInt :: Int -> Int -> (Int, Int)
105 quotRemInt a@(I# _) b@(I# _) = (a `quotInt` b, a `remInt` b)
106 -- OK, so I made it a little stricter. Shoot me. (WDP 94/10)
108 divModInt :: Int -> Int -> (Int, Int)
109 divModInt x@(I# _) y@(I# _) = (x `divInt` y, x `modInt` y)
110 -- Stricter. Sorry if you don't like it. (WDP 94/10)
113 %*********************************************************
115 \subsection{The @Integer@ type}
117 %*********************************************************
120 -- | Arbitrary-precision integers.
122 = S# Int# -- small integers
124 | J# Int# ByteArray# -- large integers
126 | J# Void BigInteger -- .NET big ints
128 foreign type dotnet "BigInteger" BigInteger
132 Convenient boxed Integer PrimOps.
135 zeroInteger :: Integer
138 int2Integer :: Int -> Integer
139 {-# INLINE int2Integer #-}
140 int2Integer (I# i) = S# i
142 integer2Int :: Integer -> Int
143 integer2Int (S# i) = I# i
144 integer2Int (J# s d) = case (integer2Int# s d) of { n# -> I# n# }
146 toBig (S# i) = case int2Integer# i of { (# s, d #) -> J# s d }
151 %*********************************************************
153 \subsection{Dividing @Integers@}
155 %*********************************************************
158 quotRemInteger :: Integer -> Integer -> (Integer, Integer)
159 quotRemInteger a@(S# (-LEFTMOST_BIT#)) b = quotRemInteger (toBig a) b
160 quotRemInteger (S# i) (S# j)
161 = case quotRemInt (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j )
162 quotRemInteger i1@(J# _ _) i2@(S# _) = quotRemInteger i1 (toBig i2)
163 quotRemInteger i1@(S# _) i2@(J# _ _) = quotRemInteger (toBig i1) i2
164 quotRemInteger (J# s1 d1) (J# s2 d2)
165 = case (quotRemInteger# s1 d1 s2 d2) of
167 -> (J# s3 d3, J# s4 d4)
169 divModInteger a@(S# (-LEFTMOST_BIT#)) b = divModInteger (toBig a) b
170 divModInteger (S# i) (S# j)
171 = case divModInt (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j)
172 divModInteger i1@(J# _ _) i2@(S# _) = divModInteger i1 (toBig i2)
173 divModInteger i1@(S# _) i2@(J# _ _) = divModInteger (toBig i1) i2
174 divModInteger (J# s1 d1) (J# s2 d2)
175 = case (divModInteger# s1 d1 s2 d2) of
177 -> (J# s3 d3, J# s4 d4)
179 remInteger :: Integer -> Integer -> Integer
181 = error "Prelude.Integral.rem{Integer}: divide by 0"
182 remInteger a@(S# (-LEFTMOST_BIT#)) b = remInteger (toBig a) b
183 remInteger (S# a) (S# b) = S# (remInt# a b)
184 {- Special case doesn't work, because a 1-element J# has the range
185 -(2^32-1) -- 2^32-1, whereas S# has the range -2^31 -- (2^31-1)
186 remInteger ia@(S# a) (J# sb b)
187 | sb ==# 1# = S# (remInt# a (word2Int# (integer2Word# sb b)))
188 | sb ==# -1# = S# (remInt# a (0# -# (word2Int# (integer2Word# sb b))))
190 | otherwise = S# (0# -# a)
192 remInteger ia@(S# _) ib@(J# _ _) = remInteger (toBig ia) ib
193 remInteger (J# sa a) (S# b)
194 = case int2Integer# b of { (# sb, b #) ->
195 case remInteger# sa a sb b of { (# sr, r #) ->
196 S# (integer2Int# sr r) }}
197 remInteger (J# sa a) (J# sb b)
198 = case remInteger# sa a sb b of (# sr, r #) -> J# sr r
200 quotInteger :: Integer -> Integer -> Integer
202 = error "Prelude.Integral.quot{Integer}: divide by 0"
203 quotInteger a@(S# (-LEFTMOST_BIT#)) b = quotInteger (toBig a) b
204 quotInteger (S# a) (S# b) = S# (quotInt# a b)
205 {- Special case disabled, see remInteger above
206 quotInteger (S# a) (J# sb b)
207 | sb ==# 1# = S# (quotInt# a (word2Int# (integer2Word# sb b)))
208 | sb ==# -1# = S# (quotInt# a (0# -# (word2Int# (integer2Word# sb b))))
209 | otherwise = zeroInteger
211 quotInteger ia@(S# _) ib@(J# _ _) = quotInteger (toBig ia) ib
212 quotInteger (J# sa a) (S# b)
213 = case int2Integer# b of { (# sb, b #) ->
214 case quotInteger# sa a sb b of (# sq, q #) -> J# sq q }
215 quotInteger (J# sa a) (J# sb b)
216 = case quotInteger# sa a sb b of (# sg, g #) -> J# sg g
222 gcdInteger :: Integer -> Integer -> Integer
223 -- SUP: Do we really need the first two cases?
224 gcdInteger a@(S# (-LEFTMOST_BIT#)) b = gcdInteger (toBig a) b
225 gcdInteger a b@(S# (-LEFTMOST_BIT#)) = gcdInteger a (toBig b)
226 gcdInteger (S# a) (S# b) = case gcdInt (I# a) (I# b) of { I# c -> S# c }
227 gcdInteger ia@(S# 0#) ib@(J# 0# _) = error "GHC.Num.gcdInteger: gcd 0 0 is undefined"
228 gcdInteger ia@(S# a) ib@(J# sb b)
231 | otherwise = S# (gcdIntegerInt# absSb b absA)
232 where absA = if a <# 0# then negateInt# a else a
233 absSb = if sb <# 0# then negateInt# sb else sb
234 gcdInteger ia@(J# _ _) ib@(S# _) = gcdInteger ib ia
235 gcdInteger (J# 0# _) (J# 0# _) = error "GHC.Num.gcdInteger: gcd 0 0 is undefined"
236 gcdInteger (J# sa a) (J# sb b)
237 = case gcdInteger# sa a sb b of (# sg, g #) -> J# sg g
239 lcmInteger :: Integer -> Integer -> Integer
245 = (divExact aa (gcdInteger aa ab)) * ab
249 divExact :: Integer -> Integer -> Integer
250 divExact a@(S# (-LEFTMOST_BIT#)) b = divExact (toBig a) b
251 divExact (S# a) (S# b) = S# (quotInt# a b)
252 divExact (S# a) (J# sb b)
253 = S# (quotInt# a (integer2Int# sb b))
254 divExact (J# sa a) (S# b)
255 = case int2Integer# b of
256 (# sb, b #) -> case divExactInteger# sa a sb b of (# sd, d #) -> J# sd d
257 divExact (J# sa a) (J# sb b)
258 = case divExactInteger# sa a sb b of (# sd, d #) -> J# sd d
262 %*********************************************************
264 \subsection{The @Integer@ instances for @Eq@, @Ord@}
266 %*********************************************************
269 instance Eq Integer where
270 (S# i) == (S# j) = i ==# j
271 (S# i) == (J# s d) = cmpIntegerInt# s d i ==# 0#
272 (J# s d) == (S# i) = cmpIntegerInt# s d i ==# 0#
273 (J# s1 d1) == (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) ==# 0#
275 (S# i) /= (S# j) = i /=# j
276 (S# i) /= (J# s d) = cmpIntegerInt# s d i /=# 0#
277 (J# s d) /= (S# i) = cmpIntegerInt# s d i /=# 0#
278 (J# s1 d1) /= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) /=# 0#
280 ------------------------------------------------------------------------
281 instance Ord Integer where
282 (S# i) <= (S# j) = i <=# j
283 (J# s d) <= (S# i) = cmpIntegerInt# s d i <=# 0#
284 (S# i) <= (J# s d) = cmpIntegerInt# s d i >=# 0#
285 (J# s1 d1) <= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <=# 0#
287 (S# i) > (S# j) = i ># j
288 (J# s d) > (S# i) = cmpIntegerInt# s d i ># 0#
289 (S# i) > (J# s d) = cmpIntegerInt# s d i <# 0#
290 (J# s1 d1) > (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) ># 0#
292 (S# i) < (S# j) = i <# j
293 (J# s d) < (S# i) = cmpIntegerInt# s d i <# 0#
294 (S# i) < (J# s d) = cmpIntegerInt# s d i ># 0#
295 (J# s1 d1) < (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <# 0#
297 (S# i) >= (S# j) = i >=# j
298 (J# s d) >= (S# i) = cmpIntegerInt# s d i >=# 0#
299 (S# i) >= (J# s d) = cmpIntegerInt# s d i <=# 0#
300 (J# s1 d1) >= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) >=# 0#
302 compare (S# i) (S# j)
306 compare (J# s d) (S# i)
307 = case cmpIntegerInt# s d i of { res# ->
308 if res# <# 0# then LT else
309 if res# ># 0# then GT else EQ
311 compare (S# i) (J# s d)
312 = case cmpIntegerInt# s d i of { res# ->
313 if res# ># 0# then LT else
314 if res# <# 0# then GT else EQ
316 compare (J# s1 d1) (J# s2 d2)
317 = case cmpInteger# s1 d1 s2 d2 of { res# ->
318 if res# <# 0# then LT else
319 if res# ># 0# then GT else EQ
324 %*********************************************************
326 \subsection{The @Integer@ instances for @Num@}
328 %*********************************************************
331 instance Num Integer where
335 negate = negateInteger
338 -- ORIG: abs n = if n >= 0 then n else -n
339 abs (S# (-LEFTMOST_BIT#)) = LEFTMOST_BIT
340 abs (S# i) = case abs (I# i) of I# j -> S# j
341 abs n@(J# s d) = if (s >=# 0#) then n else J# (negateInt# s) d
343 signum (S# i) = case signum (I# i) of I# j -> S# j
346 cmp = cmpIntegerInt# s d 0#
348 if cmp ># 0# then S# 1#
349 else if cmp ==# 0# then S# 0#
350 else S# (negateInt# 1#)
352 plusInteger i1@(S# i) i2@(S# j) = case addIntC# i j of { (# r, c #) ->
353 if c ==# 0# then S# r
354 else toBig i1 + toBig i2 }
355 plusInteger i1@(J# _ _) i2@(S# _) = i1 + toBig i2
356 plusInteger i1@(S# _) i2@(J# _ _) = toBig i1 + i2
357 plusInteger (J# s1 d1) (J# s2 d2) = case plusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
359 minusInteger i1@(S# i) i2@(S# j) = case subIntC# i j of { (# r, c #) ->
360 if c ==# 0# then S# r
361 else toBig i1 - toBig i2 }
362 minusInteger i1@(J# _ _) i2@(S# _) = i1 - toBig i2
363 minusInteger i1@(S# _) i2@(J# _ _) = toBig i1 - i2
364 minusInteger (J# s1 d1) (J# s2 d2) = case minusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
366 timesInteger i1@(S# i) i2@(S# j) = if mulIntMayOflo# i j ==# 0#
368 else toBig i1 * toBig i2
369 timesInteger i1@(J# _ _) i2@(S# _) = i1 * toBig i2
370 timesInteger i1@(S# _) i2@(J# _ _) = toBig i1 * i2
371 timesInteger (J# s1 d1) (J# s2 d2) = case timesInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
373 negateInteger (S# (-LEFTMOST_BIT#)) = LEFTMOST_BIT
374 negateInteger (S# i) = S# (negateInt# i)
375 negateInteger (J# s d) = J# (negateInt# s) d
379 %*********************************************************
381 \subsection{The @Integer@ instance for @Enum@}
383 %*********************************************************
386 instance Enum Integer where
389 toEnum n = int2Integer n
390 fromEnum n = integer2Int n
392 {-# INLINE enumFrom #-}
393 {-# INLINE enumFromThen #-}
394 {-# INLINE enumFromTo #-}
395 {-# INLINE enumFromThenTo #-}
396 enumFrom x = enumDeltaInteger x 1
397 enumFromThen x y = enumDeltaInteger x (y-x)
398 enumFromTo x lim = enumDeltaToInteger x 1 lim
399 enumFromThenTo x y lim = enumDeltaToInteger x (y-x) lim
402 "enumDeltaInteger" [~1] forall x y. enumDeltaInteger x y = build (\c _ -> enumDeltaIntegerFB c x y)
403 "efdtInteger" [~1] forall x y l.enumDeltaToInteger x y l = build (\c n -> enumDeltaToIntegerFB c n x y l)
404 "enumDeltaInteger" [1] enumDeltaIntegerFB (:) = enumDeltaInteger
405 "enumDeltaToInteger" [1] enumDeltaToIntegerFB (:) [] = enumDeltaToInteger
408 enumDeltaIntegerFB :: (Integer -> b -> b) -> Integer -> Integer -> b
409 enumDeltaIntegerFB c x d = x `c` enumDeltaIntegerFB c (x+d) d
411 enumDeltaInteger :: Integer -> Integer -> [Integer]
412 enumDeltaInteger x d = x : enumDeltaInteger (x+d) d
414 enumDeltaToIntegerFB c n x delta lim
415 | delta >= 0 = up_fb c n x delta lim
416 | otherwise = dn_fb c n x delta lim
418 enumDeltaToInteger x delta lim
419 | delta >= 0 = up_list x delta lim
420 | otherwise = dn_list x delta lim
422 up_fb c n x delta lim = go (x::Integer)
425 | otherwise = x `c` go (x+delta)
426 dn_fb c n x delta lim = go (x::Integer)
429 | otherwise = x `c` go (x+delta)
431 up_list x delta lim = go (x::Integer)
434 | otherwise = x : go (x+delta)
435 dn_list x delta lim = go (x::Integer)
438 | otherwise = x : go (x+delta)
443 %*********************************************************
445 \subsection{The @Integer@ instances for @Show@}
447 %*********************************************************
450 instance Show Integer where
452 | p > 6 && n < 0 = '(' : jtos n (')' : r)
453 -- Minor point: testing p first gives better code
454 -- in the not-uncommon case where the p argument
456 | otherwise = jtos n r
457 showList = showList__ (showsPrec 0)
459 jtos :: Integer -> String -> String
461 | n < 0 = '-' : jtos' (-n) cs
462 | otherwise = jtos' n cs
464 jtos' :: Integer -> String -> String
466 | n' < 10 = case unsafeChr (ord '0' + fromInteger n') of
468 | otherwise = case unsafeChr (ord '0' + fromInteger r) of
469 c@(C# _) -> jtos' q (c:cs')
471 (q,r) = n' `quotRemInteger` 10