2 % (c) The AQUA Project, Glasgow University, 1994-1996
5 \section[PrelNum]{Module @PrelNum@}
8 {-# OPTIONS -fno-implicit-prelude #-}
12 import {-# SOURCE #-} PrelErr
19 infixl 7 %, /, `quot`, `rem`, `div`, `mod`
25 %*********************************************************
27 \subsection{Standard numeric classes}
29 %*********************************************************
32 class (Eq a, Show a) => Num a where
33 (+), (-), (*) :: a -> a -> a
36 fromInteger :: Integer -> a
37 fromInt :: Int -> a -- partain: Glasgow extension
41 fromInt (I# i#) = fromInteger (S# i#)
42 -- Go via the standard class-op if the
43 -- non-standard one ain't provided
45 class (Num a, Ord a) => Real a where
46 toRational :: a -> Rational
48 class (Real a, Enum a) => Integral a where
49 quot, rem, div, mod :: a -> a -> a
50 quotRem, divMod :: a -> a -> (a,a)
51 toInteger :: a -> Integer
52 toInt :: a -> Int -- partain: Glasgow extension
54 n `quot` d = q where (q,_) = quotRem n d
55 n `rem` d = r where (_,r) = quotRem n d
56 n `div` d = q where (q,_) = divMod n d
57 n `mod` d = r where (_,r) = divMod n d
58 divMod n d = if signum r == negate (signum d) then (q-1, r+d) else qr
59 where qr@(q,r) = quotRem n d
61 class (Num a) => Fractional a where
64 fromRational :: Rational -> a
69 class (Fractional a) => Floating a where
71 exp, log, sqrt :: a -> a
72 (**), logBase :: a -> a -> a
73 sin, cos, tan :: a -> a
74 asin, acos, atan :: a -> a
75 sinh, cosh, tanh :: a -> a
76 asinh, acosh, atanh :: a -> a
78 x ** y = exp (log x * y)
79 logBase x y = log y / log x
82 tanh x = sinh x / cosh x
84 class (Real a, Fractional a) => RealFrac a where
85 properFraction :: (Integral b) => a -> (b,a)
86 truncate, round :: (Integral b) => a -> b
87 ceiling, floor :: (Integral b) => a -> b
89 truncate x = m where (m,_) = properFraction x
91 round x = let (n,r) = properFraction x
92 m = if r < 0 then n - 1 else n + 1
93 in case signum (abs r - 0.5) of
95 0 -> if even n then n else m
98 ceiling x = if r > 0 then n + 1 else n
99 where (n,r) = properFraction x
101 floor x = if r < 0 then n - 1 else n
102 where (n,r) = properFraction x
104 class (RealFrac a, Floating a) => RealFloat a where
105 floatRadix :: a -> Integer
106 floatDigits :: a -> Int
107 floatRange :: a -> (Int,Int)
108 decodeFloat :: a -> (Integer,Int)
109 encodeFloat :: Integer -> Int -> a
111 significand :: a -> a
112 scaleFloat :: Int -> a -> a
113 isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE
118 exponent x = if m == 0 then 0 else n + floatDigits x
119 where (m,n) = decodeFloat x
121 significand x = encodeFloat m (negate (floatDigits x))
122 where (m,_) = decodeFloat x
124 scaleFloat k x = encodeFloat m (n+k)
125 where (m,n) = decodeFloat x
129 | x == 0 && y > 0 = pi/2
130 | x < 0 && y > 0 = pi + atan (y/x)
131 |(x <= 0 && y < 0) ||
132 (x < 0 && isNegativeZero y) ||
133 (isNegativeZero x && isNegativeZero y)
135 | y == 0 && (x < 0 || isNegativeZero x)
136 = pi -- must be after the previous test on zero y
137 | x==0 && y==0 = y -- must be after the other double zero tests
138 | otherwise = x + y -- x or y is a NaN, return a NaN (via +)
142 %*********************************************************
144 \subsection{Instances for @Int@}
146 %*********************************************************
149 instance Num Int where
150 (+) x y = plusInt x y
151 (-) x y = minusInt x y
152 negate x = negateInt x
153 (*) x y = timesInt x y
154 abs n = if n `geInt` 0 then n else (negateInt n)
156 signum n | n `ltInt` 0 = negateInt 1
160 fromInteger (S# i#) = I# i#
161 fromInteger (J# s# d#)
162 = case (integer2Int# s# d#) of { i# -> I# i# }
166 instance Real Int where
167 toRational x = toInteger x % 1
169 instance Integral Int where
170 a@(I# _) `quotRem` b@(I# _) = (a `quotInt` b, a `remInt` b)
171 -- OK, so I made it a little stricter. Shoot me. (WDP 94/10)
173 -- Following chks for zero divisor are non-standard (WDP)
174 a `quot` b = if b /= 0
176 else error "Prelude.Integral.quot{Int}: divide by 0"
177 a `rem` b = if b /= 0
179 else error "Prelude.Integral.rem{Int}: divide by 0"
181 x `div` y = if x > 0 && y < 0 then quotInt (x-y-1) y
182 else if x < 0 && y > 0 then quotInt (x-y+1) y
184 x `mod` y = if x > 0 && y < 0 || x < 0 && y > 0 then
185 if r/=0 then r+y else 0
190 divMod x@(I# _) y@(I# _) = (x `div` y, x `mod` y)
191 -- Stricter. Sorry if you don't like it. (WDP 94/10)
193 --OLD: even x = eqInt (x `mod` 2) 0
194 --OLD: odd x = neInt (x `mod` 2) 0
196 toInteger (I# i) = int2Integer i -- give back a full-blown Integer
201 %*********************************************************
203 \subsection{Instances for @Integer@}
205 %*********************************************************
208 instance Ord Integer where
209 (S# i) <= (S# j) = i <=# j
210 (J# s d) <= (S# i) = cmpIntegerInt# s d i <=# 0#
211 (S# i) <= (J# s d) = cmpIntegerInt# s d i >=# 0#
212 (J# s1 d1) <= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <=# 0#
214 (S# i) > (S# j) = i ># j
215 (J# s d) > (S# i) = cmpIntegerInt# s d i ># 0#
216 (S# i) > (J# s d) = cmpIntegerInt# s d i <# 0#
217 (J# s1 d1) > (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) ># 0#
219 (S# i) < (S# j) = i <# j
220 (J# s d) < (S# i) = cmpIntegerInt# s d i <# 0#
221 (S# i) < (J# s d) = cmpIntegerInt# s d i ># 0#
222 (J# s1 d1) < (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) <# 0#
224 (S# i) >= (S# j) = i >=# j
225 (J# s d) >= (S# i) = cmpIntegerInt# s d i >=# 0#
226 (S# i) >= (J# s d) = cmpIntegerInt# s d i <=# 0#
227 (J# s1 d1) >= (J# s2 d2) = (cmpInteger# s1 d1 s2 d2) >=# 0#
229 compare (S# i) (S# j)
233 compare (J# s d) (S# i)
234 = case cmpIntegerInt# s d i of { res# ->
235 if res# <# 0# then LT else
236 if res# ># 0# then GT else EQ
238 compare (S# i) (J# s d)
239 = case cmpIntegerInt# s d i of { res# ->
240 if res# ># 0# then LT else
241 if res# <# 0# then GT else EQ
243 compare (J# s1 d1) (J# s2 d2)
244 = case cmpInteger# s1 d1 s2 d2 of { res# ->
245 if res# <# 0# then LT else
246 if res# ># 0# then GT else EQ
249 toBig (S# i) = case int2Integer# i of { (# s, d #) -> J# s d }
252 instance Num Integer where
253 (+) i1@(S# i) i2@(S# j)
254 = case addIntC# i j of { (# r, c #) ->
255 if c ==# 0# then S# r
256 else toBig i1 + toBig i2 }
257 (+) i1@(J# _ _) i2@(S# _) = i1 + toBig i2
258 (+) i1@(S# _) i2@(J# _ _) = toBig i1 + i2
259 (+) (J# s1 d1) (J# s2 d2)
260 = case plusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
262 (-) i1@(S# i) i2@(S# j)
263 = case subIntC# i j of { (# r, c #) ->
264 if c ==# 0# then S# r
265 else toBig i1 - toBig i2 }
266 (-) i1@(J# _ _) i2@(S# _) = i1 - toBig i2
267 (-) i1@(S# _) i2@(J# _ _) = toBig i1 - i2
268 (-) (J# s1 d1) (J# s2 d2)
269 = case minusInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
271 (*) i1@(S# i) i2@(S# j)
272 = case mulIntC# i j of { (# r, c #) ->
273 if c ==# 0# then S# r
274 else toBig i1 * toBig i2 }
275 (*) i1@(J# _ _) i2@(S# _) = i1 * toBig i2
276 (*) i1@(S# _) i2@(J# _ _) = toBig i1 * i2
277 (*) (J# s1 d1) (J# s2 d2)
278 = case timesInteger# s1 d1 s2 d2 of (# s, d #) -> J# s d
280 negate (S# (-2147483648#)) = 2147483648
281 negate (S# i) = S# (negateInt# i)
282 negate (J# s d) = J# (negateInt# s) d
284 -- ORIG: abs n = if n >= 0 then n else -n
286 abs (S# i) = case abs (I# i) of I# j -> S# j
288 = if (cmpIntegerInt# s d 0#) >=# 0#
290 else J# (negateInt# s) d
292 signum (S# i) = case signum (I# i) of I# j -> S# j
295 cmp = cmpIntegerInt# s d 0#
297 if cmp ># 0# then S# 1#
298 else if cmp ==# 0# then S# 0#
299 else S# (negateInt# 1#)
303 fromInt (I# i) = S# i
305 instance Real Integer where
308 instance Integral Integer where
309 -- ToDo: a `rem` b returns a small integer if b is small,
310 -- a `quot` b returns a small integer if a is small.
311 quotRem (S# i) (S# j)
312 = case quotRem (I# i) (I# j) of ( I# i, I# j ) -> ( S# i, S# j)
313 quotRem i1@(J# _ _) i2@(S# _) = quotRem i1 (toBig i2)
314 quotRem i1@(S# _) i2@(J# _ _) = quotRem (toBig i1) i2
315 quotRem (J# s1 d1) (J# s2 d2)
316 = case (quotRemInteger# s1 d1 s2 d2) of
318 -> (J# s3 d3, J# s4 d4)
320 {- USING THE UNDERLYING "GMP" CODE IS DUBIOUS FOR NOW:
322 divMod (J# a1 s1 d1) (J# a2 s2 d2)
323 = case (divModInteger# a1 s1 d1 a2 s2 d2) of
324 Return2GMPs a3 s3 d3 a4 s4 d4
325 -> (J# a3 s3 d3, J# a4 s4 d4)
329 toInt (J# s d) = case (integer2Int# s d) of { n# -> I# n# }
331 -- the rest are identical to the report default methods;
332 -- you get slightly better code if you let the compiler
333 -- see them right here:
334 (S# n) `quot` (S# d) = S# (n `quotInt#` d)
335 n `quot` d = if d /= 0 then q else
336 error "Prelude.Integral.quot{Integer}: divide by 0"
337 where (q,_) = quotRem n d
339 (S# n) `rem` (S# d) = S# (n `remInt#` d)
340 n `rem` d = if d /= 0 then r else
341 error "Prelude.Integral.rem{Integer}: divide by 0"
342 where (_,r) = quotRem n d
344 n `div` d = q where (q,_) = divMod n d
345 n `mod` d = r where (_,r) = divMod n d
347 divMod n d = case (quotRem n d) of { qr@(q,r) ->
348 if signum r == negate (signum d) then (q - 1, r+d) else qr }
349 -- Case-ified by WDP 94/10
351 ------------------------------------------------------------------------
352 instance Enum Integer where
355 toEnum n = toInteger n
358 {-# INLINE enumFrom #-}
359 {-# INLINE enumFromThen #-}
360 {-# INLINE enumFromTo #-}
361 {-# INLINE enumFromThenTo #-}
362 enumFrom x = build (\c _ -> enumDeltaIntegerFB c x 1)
363 enumFromThen x y = build (\c _ -> enumDeltaIntegerFB c x (y-x))
364 enumFromTo x lim = build (\c n -> enumDeltaToIntegerFB c n x 1 lim)
365 enumFromThenTo x y lim = build (\c n -> enumDeltaToIntegerFB c n x (y-x) lim)
367 enumDeltaIntegerFB :: (Integer -> b -> b) -> Integer -> Integer -> b
368 enumDeltaIntegerFB c x d = x `c` enumDeltaIntegerFB c (x+d) d
370 enumDeltaIntegerList :: Integer -> Integer -> [Integer]
371 enumDeltaIntegerList x d = x : enumDeltaIntegerList (x+d) d
373 enumDeltaToIntegerFB c n x delta lim
374 | delta >= 0 = up_fb c n x delta lim
375 | otherwise = dn_fb c n x delta lim
377 enumDeltaToIntegerList x delta lim
378 | delta >= 0 = up_list x delta lim
379 | otherwise = dn_list x delta lim
381 up_fb c n x delta lim = go (x::Integer)
384 | otherwise = x `c` go (x+delta)
385 dn_fb c n x delta lim = go (x::Integer)
388 | otherwise = x `c` go (x+delta)
390 up_list x delta lim = go (x::Integer)
393 | otherwise = x : go (x+delta)
394 dn_list x delta lim = go (x::Integer)
397 | otherwise = x : go (x+delta)
400 "enumDeltaInteger" enumDeltaIntegerFB (:) = enumDeltaIntegerList
401 "enumDeltaToInteger" enumDeltaToIntegerFB (:) [] = enumDeltaToIntegerList
404 ------------------------------------------------------------------------
406 instance Show Integer where
407 showsPrec x = showSignedInteger x
408 showList = showList__ (showsPrec 0)
411 showSignedInteger :: Int -> Integer -> ShowS
412 showSignedInteger p n r
413 | n < 0 && p > 6 = '(':jtos n (')':r)
414 | otherwise = jtos n r
416 jtos :: Integer -> String -> String
418 | i < 0 = '-' : jtos' (-i) rs
419 | otherwise = jtos' i rs
421 jtos' :: Integer -> String -> String
423 | n < 10 = chr (fromInteger n + (ord_0::Int)) : cs
424 | otherwise = jtos' q (chr (toInt r + (ord_0::Int)) : cs)
426 (q,r) = n `quotRem` 10
429 ord_0 = fromInt (ord '0')
432 %*********************************************************
434 \subsection{The @Ratio@ and @Rational@ types}
436 %*********************************************************
439 data (Integral a) => Ratio a = !a :% !a deriving (Eq)
440 type Rational = Ratio Integer
442 {-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
443 (%) :: (Integral a) => a -> a -> Ratio a
444 numerator, denominator :: (Integral a) => Ratio a -> a
447 \tr{reduce} is a subsidiary function used only in this module .
448 It normalises a ratio by dividing both numerator and denominator by
449 their greatest common divisor.
452 reduce :: (Integral a) => a -> a -> Ratio a
453 reduce _ 0 = error "Ratio.%: zero denominator"
454 reduce x y = (x `quot` d) :% (y `quot` d)
459 x % y = reduce (x * signum y) (abs y)
461 numerator (x :% _) = x
462 denominator (_ :% y) = y
466 %*********************************************************
468 \subsection{Overloaded numeric functions}
470 %*********************************************************
474 {-# SPECIALISE subtract :: Int -> Int -> Int #-}
475 subtract :: (Num a) => a -> a -> a
478 even, odd :: (Integral a) => a -> Bool
479 even n = n `rem` 2 == 0
482 {-# SPECIALISE gcd ::
484 Integer -> Integer -> Integer #-}
485 gcd :: (Integral a) => a -> a -> a
486 gcd 0 0 = error "Prelude.gcd: gcd 0 0 is undefined"
487 gcd x y = gcd' (abs x) (abs y)
489 gcd' a b = gcd' b (a `rem` b)
491 {-# SPECIALISE lcm ::
493 Integer -> Integer -> Integer #-}
494 lcm :: (Integral a) => a -> a -> a
497 lcm x y = abs ((x `quot` (gcd x y)) * y)
499 {-# SPECIALISE (^) ::
500 Integer -> Integer -> Integer,
501 Integer -> Int -> Integer,
502 Int -> Int -> Int #-}
503 (^) :: (Num a, Integral b) => a -> b -> a
505 x ^ n | n > 0 = f x (n-1) x
507 f a d y = g a d where
508 g b i | even i = g (b*b) (i `quot` 2)
509 | otherwise = f b (i-1) (b*y)
510 _ ^ _ = error "Prelude.^: negative exponent"
512 {- SPECIALISE (^^) ::
513 Double -> Int -> Double,
514 Rational -> Int -> Rational #-}
515 (^^) :: (Fractional a, Integral b) => a -> b -> a
516 x ^^ n = if n >= 0 then x^n else recip (x^(negate n))