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
21 #define BASE 1000000000
22 #elif SIZEOF_HSWORD == 8
23 #define LEFTMOST_BIT 9223372036854775808
25 #define BASE 1000000000000000000
27 #error Please define LEFTMOST_BIT to be 2^(SIZEOF_HSWORD*8-1)
28 -- DIGITS should be the largest integer such that 10^DIGITS < LEFTMOST_BIT
29 -- BASE should be 10^DIGITS. Note that ^ is not available yet.
42 default () -- Double isn't available yet,
43 -- and we shouldn't be using defaults anyway
46 %*********************************************************
48 \subsection{Standard numeric class}
50 %*********************************************************
53 -- | Basic numeric class.
55 -- Minimal complete definition: all except 'negate' or @(-)@
56 class (Eq a, Show a) => Num a where
57 (+), (-), (*) :: a -> a -> a
62 -- | Sign of a number.
63 -- The functions 'abs' and 'signum' should satisfy the law:
65 -- > abs x * signum x == x
67 -- For real numbers, the 'signum' is either @-1@ (negative), @0@ (zero)
70 -- | Conversion from an 'Integer'.
71 -- An integer literal represents the application of the function
72 -- 'fromInteger' to the appropriate value of type 'Integer',
73 -- so such literals have type @('Num' a) => a@.
74 fromInteger :: Integer -> a
79 -- | the same as @'flip' ('-')@.
81 -- Because @-@ is treated specially in the Haskell grammar,
82 -- @(-@ /e/@)@ is not a section, but an application of prefix negation.
83 -- However, @('subtract'@ /exp/@)@ is equivalent to the disallowed section.
84 {-# INLINE subtract #-}
85 subtract :: (Num a) => a -> a -> a
90 %*********************************************************
92 \subsection{Instances for @Int@}
94 %*********************************************************
97 instance Num Int where
102 abs n = if n `geInt` 0 then n else negateInt n
104 signum n | n `ltInt` 0 = negateInt 1
108 fromInteger = integer2Int
110 quotRemInt :: Int -> Int -> (Int, Int)
111 quotRemInt a@(I# _) b@(I# _) = (a `quotInt` b, a `remInt` b)
112 -- OK, so I made it a little stricter. Shoot me. (WDP 94/10)
114 divModInt :: Int -> Int -> (Int, Int)
115 divModInt x@(I# _) y@(I# _) = (x `divInt` y, x `modInt` y)
116 -- Stricter. Sorry if you don't like it. (WDP 94/10)
119 %*********************************************************
121 \subsection{The @Integer@ type}
123 %*********************************************************
126 -- | Arbitrary-precision integers.
128 = S# Int# -- small integers
130 | J# Int# ByteArray# -- large integers
132 | J# Void BigInteger -- .NET big ints
134 foreign type dotnet "BigInteger" BigInteger
138 Convenient boxed Integer PrimOps.
141 zeroInteger :: Integer
144 int2Integer :: Int -> Integer
145 {-# INLINE int2Integer #-}
146 int2Integer (I# i) = S# i
148 integer2Int :: Integer -> Int
149 integer2Int (S# i) = I# i
150 integer2Int (J# s d) = case (integer2Int# s d) of { n# -> I# n# }
152 toBig (S# i) = case int2Integer# i of { (# s, d #) -> J# s d }
157 %*********************************************************
159 \subsection{Dividing @Integers@}
161 %*********************************************************
164 quotRemInteger :: Integer -> Integer -> (Integer, Integer)
165 quotRemInteger a@(S# (-LEFTMOST_BIT#)) b = quotRemInteger (toBig a) b
166 quotRemInteger (S# i) (S# j)
167 = case quotRemInt (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j )
168 quotRemInteger i1@(J# _ _) i2@(S# _) = quotRemInteger i1 (toBig i2)
169 quotRemInteger i1@(S# _) i2@(J# _ _) = quotRemInteger (toBig i1) i2
170 quotRemInteger (J# s1 d1) (J# s2 d2)
171 = case (quotRemInteger# s1 d1 s2 d2) of
173 -> (J# s3 d3, J# s4 d4)
175 divModInteger a@(S# (-LEFTMOST_BIT#)) b = divModInteger (toBig a) b
176 divModInteger (S# i) (S# j)
177 = case divModInt (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j)
178 divModInteger i1@(J# _ _) i2@(S# _) = divModInteger i1 (toBig i2)
179 divModInteger i1@(S# _) i2@(J# _ _) = divModInteger (toBig i1) i2
180 divModInteger (J# s1 d1) (J# s2 d2)
181 = case (divModInteger# s1 d1 s2 d2) of
183 -> (J# s3 d3, J# s4 d4)
185 remInteger :: Integer -> Integer -> Integer
187 | ib == 0 = error "Prelude.Integral.rem{Integer}: divide by 0"
188 remInteger a@(S# (-LEFTMOST_BIT#)) b = remInteger (toBig a) b
189 remInteger (S# a) (S# b) = S# (remInt# a b)
190 {- Special case doesn't work, because a 1-element J# has the range
191 -(2^32-1) -- 2^32-1, whereas S# has the range -2^31 -- (2^31-1)
192 remInteger ia@(S# a) (J# sb b)
193 | sb ==# 1# = S# (remInt# a (word2Int# (integer2Word# sb b)))
194 | sb ==# -1# = S# (remInt# a (0# -# (word2Int# (integer2Word# sb b))))
196 | otherwise = S# (0# -# a)
198 remInteger ia@(S# _) ib@(J# _ _) = remInteger (toBig ia) ib
199 remInteger (J# sa a) (S# b)
200 = case int2Integer# b of { (# sb, b #) ->
201 case remInteger# sa a sb b of { (# sr, r #) ->
202 S# (integer2Int# sr r) }}
203 remInteger (J# sa a) (J# sb b)
204 = case remInteger# sa a sb b of (# sr, r #) -> J# sr r
206 quotInteger :: Integer -> Integer -> Integer
208 | ib == 0 = error "Prelude.Integral.quot{Integer}: divide by 0"
209 quotInteger a@(S# (-LEFTMOST_BIT#)) b = quotInteger (toBig a) b
210 quotInteger (S# a) (S# b) = S# (quotInt# a b)
211 {- Special case disabled, see remInteger above
212 quotInteger (S# a) (J# sb b)
213 | sb ==# 1# = S# (quotInt# a (word2Int# (integer2Word# sb b)))
214 | sb ==# -1# = S# (quotInt# a (0# -# (word2Int# (integer2Word# sb b))))
215 | otherwise = zeroInteger
217 quotInteger ia@(S# _) ib@(J# _ _) = quotInteger (toBig ia) ib
218 quotInteger (J# sa a) (S# b)
219 = case int2Integer# b of { (# sb, b #) ->
220 case quotInteger# sa a sb b of (# sq, q #) -> J# sq q }
221 quotInteger (J# sa a) (J# sb b)
222 = case quotInteger# sa a sb b of (# sg, g #) -> J# sg g
228 gcdInteger :: Integer -> Integer -> Integer
229 -- SUP: Do we really need the first two cases?
230 gcdInteger a@(S# (-LEFTMOST_BIT#)) b = gcdInteger (toBig a) b
231 gcdInteger a b@(S# (-LEFTMOST_BIT#)) = gcdInteger a (toBig b)
232 gcdInteger (S# a) (S# b) = case gcdInt (I# a) (I# b) of { I# c -> S# c }
233 gcdInteger ia@(S# 0#) ib@(J# 0# _) = error "GHC.Num.gcdInteger: gcd 0 0 is undefined"
234 gcdInteger ia@(S# a) ib@(J# sb b)
237 | otherwise = S# (gcdIntegerInt# absSb b absA)
238 where absA = if a <# 0# then negateInt# a else a
239 absSb = if sb <# 0# then negateInt# sb else sb
240 gcdInteger ia@(J# _ _) ib@(S# _) = gcdInteger ib ia
241 gcdInteger (J# 0# _) (J# 0# _) = error "GHC.Num.gcdInteger: gcd 0 0 is undefined"
242 gcdInteger (J# sa a) (J# sb b)
243 = case gcdInteger# sa a sb b of (# sg, g #) -> J# sg g
245 lcmInteger :: Integer -> Integer -> Integer
251 = (divExact aa (gcdInteger aa ab)) * ab
255 divExact :: Integer -> Integer -> Integer
256 divExact a@(S# (-LEFTMOST_BIT#)) b = divExact (toBig a) b
257 divExact (S# a) (S# b) = S# (quotInt# a b)
258 divExact (S# a) (J# sb b)
259 = S# (quotInt# a (integer2Int# sb b))
260 divExact (J# sa a) (S# b)
261 = case int2Integer# b of
262 (# sb, b #) -> case divExactInteger# sa a sb b of (# sd, d #) -> J# sd d
263 divExact (J# sa a) (J# sb b)
264 = case divExactInteger# sa a sb b of (# sd, d #) -> J# sd d
268 %*********************************************************
270 \subsection{The @Integer@ instances for @Eq@, @Ord@}
272 %*********************************************************
275 instance Eq Integer where
276 (S# i) == (S# j) = i ==# j
277 (S# i) == (J# s d) = cmpIntegerInt# s d i ==# 0#
278 (J# s d) == (S# i) = cmpIntegerInt# s d i ==# 0#
279 (J# s1 d1) == (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) ==# 0#
281 (S# i) /= (S# j) = i /=# j
282 (S# i) /= (J# s d) = cmpIntegerInt# s d i /=# 0#
283 (J# s d) /= (S# i) = cmpIntegerInt# s d i /=# 0#
284 (J# s1 d1) /= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) /=# 0#
286 ------------------------------------------------------------------------
287 instance Ord Integer where
288 (S# i) <= (S# j) = i <=# j
289 (J# s d) <= (S# i) = cmpIntegerInt# s d i <=# 0#
290 (S# i) <= (J# s d) = cmpIntegerInt# s d i >=# 0#
291 (J# s1 d1) <= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <=# 0#
293 (S# i) > (S# j) = i ># j
294 (J# s d) > (S# i) = cmpIntegerInt# s d i ># 0#
295 (S# i) > (J# s d) = cmpIntegerInt# s d i <# 0#
296 (J# s1 d1) > (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) ># 0#
298 (S# i) < (S# j) = i <# j
299 (J# s d) < (S# i) = cmpIntegerInt# s d i <# 0#
300 (S# i) < (J# s d) = cmpIntegerInt# s d i ># 0#
301 (J# s1 d1) < (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <# 0#
303 (S# i) >= (S# j) = i >=# j
304 (J# s d) >= (S# i) = cmpIntegerInt# s d i >=# 0#
305 (S# i) >= (J# s d) = cmpIntegerInt# s d i <=# 0#
306 (J# s1 d1) >= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) >=# 0#
308 compare (S# i) (S# j)
312 compare (J# s d) (S# i)
313 = case cmpIntegerInt# s d i of { res# ->
314 if res# <# 0# then LT else
315 if res# ># 0# then GT else EQ
317 compare (S# i) (J# s d)
318 = case cmpIntegerInt# s d i of { res# ->
319 if res# ># 0# then LT else
320 if res# <# 0# then GT else EQ
322 compare (J# s1 d1) (J# s2 d2)
323 = case cmpInteger# s1 d1 s2 d2 of { res# ->
324 if res# <# 0# then LT else
325 if res# ># 0# then GT else EQ
330 %*********************************************************
332 \subsection{The @Integer@ instances for @Num@}
334 %*********************************************************
337 instance Num Integer where
341 negate = negateInteger
344 -- ORIG: abs n = if n >= 0 then n else -n
345 abs (S# (-LEFTMOST_BIT#)) = LEFTMOST_BIT
346 abs (S# i) = case abs (I# i) of I# j -> S# j
347 abs n@(J# s d) = if (s >=# 0#) then n else J# (negateInt# s) d
349 signum (S# i) = case signum (I# i) of I# j -> S# j
352 cmp = cmpIntegerInt# s d 0#
354 if cmp ># 0# then S# 1#
355 else if cmp ==# 0# then S# 0#
356 else S# (negateInt# 1#)
358 plusInteger i1@(S# i) i2@(S# j) = case addIntC# i j of { (# r, c #) ->
359 if c ==# 0# then S# r
360 else toBig i1 + toBig i2 }
361 plusInteger i1@(J# _ _) i2@(S# _) = i1 + toBig i2
362 plusInteger i1@(S# _) i2@(J# _ _) = toBig i1 + i2
363 plusInteger (J# s1 d1) (J# s2 d2) = case plusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
365 minusInteger i1@(S# i) i2@(S# j) = case subIntC# i j of { (# r, c #) ->
366 if c ==# 0# then S# r
367 else toBig i1 - toBig i2 }
368 minusInteger i1@(J# _ _) i2@(S# _) = i1 - toBig i2
369 minusInteger i1@(S# _) i2@(J# _ _) = toBig i1 - i2
370 minusInteger (J# s1 d1) (J# s2 d2) = case minusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
372 timesInteger i1@(S# i) i2@(S# j) = if mulIntMayOflo# i j ==# 0#
374 else toBig i1 * toBig i2
375 timesInteger i1@(J# _ _) i2@(S# _) = i1 * toBig i2
376 timesInteger i1@(S# _) i2@(J# _ _) = toBig i1 * i2
377 timesInteger (J# s1 d1) (J# s2 d2) = case timesInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
379 negateInteger (S# (-LEFTMOST_BIT#)) = LEFTMOST_BIT
380 negateInteger (S# i) = S# (negateInt# i)
381 negateInteger (J# s d) = J# (negateInt# s) d
385 %*********************************************************
387 \subsection{The @Integer@ instance for @Enum@}
389 %*********************************************************
392 instance Enum Integer where
395 toEnum n = int2Integer n
396 fromEnum n = integer2Int n
398 {-# INLINE enumFrom #-}
399 {-# INLINE enumFromThen #-}
400 {-# INLINE enumFromTo #-}
401 {-# INLINE enumFromThenTo #-}
402 enumFrom x = enumDeltaInteger x 1
403 enumFromThen x y = enumDeltaInteger x (y-x)
404 enumFromTo x lim = enumDeltaToInteger x 1 lim
405 enumFromThenTo x y lim = enumDeltaToInteger x (y-x) lim
408 "enumDeltaInteger" [~1] forall x y. enumDeltaInteger x y = build (\c _ -> enumDeltaIntegerFB c x y)
409 "efdtInteger" [~1] forall x y l.enumDeltaToInteger x y l = build (\c n -> enumDeltaToIntegerFB c n x y l)
410 "enumDeltaInteger" [1] enumDeltaIntegerFB (:) = enumDeltaInteger
411 "enumDeltaToInteger" [1] enumDeltaToIntegerFB (:) [] = enumDeltaToInteger
414 enumDeltaIntegerFB :: (Integer -> b -> b) -> Integer -> Integer -> b
415 enumDeltaIntegerFB c x d = x `c` enumDeltaIntegerFB c (x+d) d
417 enumDeltaInteger :: Integer -> Integer -> [Integer]
418 enumDeltaInteger x d = x : enumDeltaInteger (x+d) d
420 enumDeltaToIntegerFB c n x delta lim
421 | delta >= 0 = up_fb c n x delta lim
422 | otherwise = dn_fb c n x delta lim
424 enumDeltaToInteger x delta lim
425 | delta >= 0 = up_list x delta lim
426 | otherwise = dn_list x delta lim
428 up_fb c n x delta lim = go (x::Integer)
431 | otherwise = x `c` go (x+delta)
432 dn_fb c n x delta lim = go (x::Integer)
435 | otherwise = x `c` go (x+delta)
437 up_list x delta lim = go (x::Integer)
440 | otherwise = x : go (x+delta)
441 dn_list x delta lim = go (x::Integer)
444 | otherwise = x : go (x+delta)
449 %*********************************************************
451 \subsection{The @Integer@ instances for @Show@}
453 %*********************************************************
456 instance Show Integer where
458 | p > 6 && n < 0 = '(' : jtos n (')' : r)
459 -- Minor point: testing p first gives better code
460 -- in the not-uncommon case where the p argument
462 | otherwise = jtos n r
463 showList = showList__ (showsPrec 0)
465 -- Divide an conquer implementation of string conversion
466 jtos :: Integer -> String -> String
468 | n < 0 = '-' : jtos' (-n) cs
469 | otherwise = jtos' n cs
471 jtos' :: Integer -> String -> String
473 | n < BASE = jhead (fromInteger n) cs
474 | otherwise = jprinth (jsplitf (BASE*BASE) n) cs
476 -- Split n into digits in base p. We first split n into digits
477 -- in base p*p and then split each of these digits into two.
478 -- Note that the first 'digit' modulo p*p may have a leading zero
479 -- in base p that we need to drop - this is what jsplith takes care of.
480 -- jsplitb the handles the remaining digits.
481 jsplitf :: Integer -> Integer -> [Integer]
484 | otherwise = jsplith p (jsplitf (p*p) n)
486 jsplith :: Integer -> [Integer] -> [Integer]
488 if q > 0 then fromInteger q : fromInteger r : jsplitb p ns
489 else fromInteger r : jsplitb p ns
491 (q, r) = n `quotRemInteger` p
493 jsplitb :: Integer -> [Integer] -> [Integer]
495 jsplitb p (n:ns) = q : r : jsplitb p ns
497 (q, r) = n `quotRemInteger` p
499 -- Convert a number that has been split into digits in base BASE^2
500 -- this includes a last splitting step and then conversion of digits
501 -- that all fit into a machine word.
502 jprinth :: [Integer] -> String -> String
504 if q > 0 then jhead q $ jblock r $ jprintb ns cs
505 else jhead r $ jprintb ns cs
507 (q', r') = n `quotRemInteger` BASE
511 jprintb :: [Integer] -> String -> String
513 jprintb (n:ns) cs = jblock q $ jblock r $ jprintb ns cs
515 (q', r') = n `quotRemInteger` BASE
519 -- Convert an integer that fits into a machine word. Again, we have two
520 -- functions, one that drops leading zeros (jhead) and one that doesn't
522 jhead :: Int -> String -> String
524 | n < 10 = case unsafeChr (ord '0' + n) of
526 | otherwise = case unsafeChr (ord '0' + r) of
527 c@(C# _) -> jhead q (c : cs)
529 (q, r) = n `quotRemInt` 10
531 jblock = jblock' {- ' -} DIGITS
533 jblock' :: Int -> Int -> String -> String
535 | d == 1 = case unsafeChr (ord '0' + n) of
537 | otherwise = case unsafeChr (ord '0' + r) of
538 c@(C# _) -> jblock' (d - 1) q (c : cs)
540 (q, r) = n `quotRemInt` 10