haskell2010-1.1.1.1: Compatibility with Haskell 2010

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LanguageHaskell2010

Data.Ratio

Contents

Synopsis

Documentation

data Ratio a Source

Rational numbers, with numerator and denominator of some Integral type.

Instances

Integral a => Enum (Ratio a) 
Eq a => Eq (Ratio a) 
Integral a => Fractional (Ratio a) 
Integral a => Num (Ratio a) 
Integral a => Ord (Ratio a) 
(Integral a, Read a) => Read (Ratio a) 
Integral a => Real (Ratio a) 
Integral a => RealFrac (Ratio a) 
(Integral a, Show a) => Show (Ratio a) 

type Rational = Ratio Integer Source

Arbitrary-precision rational numbers, represented as a ratio of two Integer values. A rational number may be constructed using the % operator.

(%) :: Integral a => a -> a -> Ratio a Source

Forms the ratio of two integral numbers.

numerator :: Integral a => Ratio a -> a Source

Extract the numerator of the ratio in reduced form: the numerator and denominator have no common factor and the denominator is positive.

denominator :: Integral a => Ratio a -> a Source

Extract the denominator of the ratio in reduced form: the numerator and denominator have no common factor and the denominator is positive.

approxRational :: RealFrac a => a -> a -> Rational Source

approxRational, applied to two real fractional numbers x and epsilon, returns the simplest rational number within epsilon of x. A rational number y is said to be simpler than another y' if

Any real interval contains a unique simplest rational; in particular, note that 0/1 is the simplest rational of all.

Specification

 module  Data.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 "Data.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