The Haskell 98 Report
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module Ratio ( Ratio, Rational, (%), numerator, denominator, approxRational ) where infixl 7 % data (Integral a) => Ratio a = ... type Rational = Ratio Integer (%) :: (Integral a) => a -> a -> Ratio a numerator, denominator :: (Integral a) => Ratio a -> a approxRational :: (RealFrac a) => a -> a -> Rational instance (Integral a) => Eq (Ratio a) where ... instance (Integral a) => Ord (Ratio a) where ... instance (Integral a) => Num (Ratio a) where ... instance (Integral a) => Real (Ratio a) where ... instance (Integral a) => Fractional (Ratio a) where ... instance (Integral a) => RealFrac (Ratio a) where ... instance (Integral a) => Enum (Ratio a) where ... instance (Read a,Integral a) => Read (Ratio a) where ... instance (Integral a) => Show (Ratio a) where ... |
それぞれの Integral 型 t に対して、一つの型 Ratio t が存在して、その構成要素は一組の t 型である。型名 Rational は Ratio Integer の型の同義名である。
Ratio は Eq、Ord、Num、 Real、Fractional、RealFrac、Enum、 Read および Show クラスのインスタンスである。 それぞれの場合について、Ratio t に対するインスタンスは単に、 対応する t 上の演算を「リフト」するだけである。もし、t が有限の 型であれば、結果は前もっては予測不可能である。たとえば、 Ratio Int は絶対値が小さい有理数の場合にも整数の 桁あふれを起す可能性がある。
演算子 (%) は 2 つの整数 (integral number) の比を形成し、 分母が正数の既約分数に約分する。関数 numerator と denominator は分母が正数の既約分数の場合の比の構成要素を 取り出す。Ratio は抽象データ型である。たとえば、 12 % 8 は約分されて 3/2 となり、 12 % (-8) は約分されて (-3)/2 となる。
approxRational 関数は2つの実小数 x と epsilon に適用されて、開区間 (x-epsilon, x+epsilon) 内にある最も単純な 有理数を返す。既約有理数 n/d が 別の既約有理数 n'/d' よりも単純であると言えるのは、|n| <=|n'| かつ d <=d' の場合である。あらゆる実数区間は唯一の最も単純な 有理数を含むということが証明可能であることに注意せよ。
-- Standard functions on rational numbers
module Ratio (
Ratio, Rational, (%), numerator, denominator, approxRational ) where
infixl 7 %
ratPrec = 7 :: Int
data (Integral a) => Ratio a = !a :% !a deriving (Eq)
type Rational = Ratio Integer
(%) :: (Integral a) => a -> a -> Ratio a
numerator, denominator :: (Integral a) => Ratio a -> a
approxRational :: (RealFrac a) => a -> a -> Rational
-- "reduce" is a subsidiary function used only in this module.
-- It normalises a ratio by dividing both numerator
-- and denominator by their greatest common divisor.
--
-- E.g., 12 `reduce` 8 == 3 :% 2
-- 12 `reduce` (-8) == 3 :% (-2)
reduce _ 0 = error "Ratio.% : zero denominator"
reduce x y = (x `quot` d) :% (y `quot` d)
where d = gcd x y
x % y = reduce (x * signum y) (abs y)
numerator (x :% _) = x
denominator (_ :% y) = y
instance (Integral a) => Ord (Ratio a) where
(x:%y) <= (x':%y') = x * y' <= x' * y
(x:%y) < (x':%y') = x * y' < x' * y
instance (Integral a) => Num (Ratio a) where
(x:%y) + (x':%y') = reduce (x*y' + x'*y) (y*y')
(x:%y) * (x':%y') = reduce (x * x') (y * y')
negate (x:%y) = (-x) :% y
abs (x:%y) = abs x :% y
signum (x:%y) = signum x :% 1
fromInteger x = fromInteger x :% 1
instance (Integral a) => Real (Ratio a) where
toRational (x:%y) = toInteger x :% toInteger y
instance (Integral a) => Fractional (Ratio a) where
(x:%y) / (x':%y') = (x*y') % (y*x')
recip (x:%y) = y % x
fromRational (x:%y) = fromInteger x :% fromInteger y
instance (Integral a) => RealFrac (Ratio a) where
properFraction (x:%y) = (fromIntegral q, r:%y)
where (q,r) = quotRem x y
instance (Integral a) => Enum (Ratio a) where
succ x = x+1
pred x = x-1
toEnum = fromIntegral
fromEnum = fromInteger . truncate
-- May overflow
enumFrom = numericEnumFrom
-- These numericEnumXXX functions
enumFromThen = numericEnumFromThen
-- are as defined in Prelude.hs
enumFromTo = numericEnumFromTo
-- but not exported from it!
enumFromThenTo = numericEnumFromThenTo
instance (Read a, Integral a) => Read (Ratio a) where
readsPrec p = readParen (p > ratPrec)
(\r -> [(x%y,u) | (x,s) <- readsPrec (ratPrec+1) r,
("%",t) <- lex s,
(y,u) <- readsPrec (ratPrec+1) t ])
instance (Integral a) => Show (Ratio a) where
showsPrec p (x:%y) = showParen (p > ratPrec)
(showsPrec (ratPrec+1) x .
showString " % " .
showsPrec (ratPrec+1) y)
approxRational x eps = simplest (x-eps) (x+eps)
where simplest x y | y < x = simplest y x
| x == y = xr
| x > 0 = simplest' n d n' d'
| y < 0 = - simplest' (-n') d' (-n) d
| otherwise = 0 :% 1
where xr@(n:%d) = toRational x
(n':%d') = toRational y
simplest' n d n' d' -- assumes 0 < n%d < n'%d'
| r == 0 = q :% 1
| q /= q' = (q+1) :% 1
| otherwise = (q*n''+d'') :% n''
where (q,r) = quotRem n d
(q',r') = quotRem n' d'
(n'':%d'') = simplest' d' r' d r
The Haskell 98 Report
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December 2002