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{-# LANGUAGE Safe #-}
-----------------------------------------------------------------------------
-- |
-- Module : Data.Ratio
-- Copyright : (c) The University of Glasgow 2001
-- License : BSD-style (see the file libraries/base/LICENSE)
--
-- Maintainer : libraries@haskell.org
-- Stability : stable
-- Portability : portable
--
-- Standard functions on rational numbers
--
-----------------------------------------------------------------------------
module Data.Ratio
( Ratio
, Rational
, (%)
, numerator
, denominator
, approxRational
) where
import Prelude
import GHC.Real -- The basic defns for Ratio
-- -----------------------------------------------------------------------------
-- approxRational
-- | '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
--
-- * @'abs' ('numerator' y) <= 'abs' ('numerator' y')@, and
--
-- * @'denominator' y <= 'denominator' y'@.
--
-- Any real interval contains a unique simplest rational;
-- in particular, note that @0\/1@ is the simplest rational of all.
-- Implementation details: Here, for simplicity, we assume a closed rational
-- interval. If such an interval includes at least one whole number, then
-- the simplest rational is the absolutely least whole number. Otherwise,
-- the bounds are of the form q%1 + r%d and q%1 + r'%d', where abs r < d
-- and abs r' < d', and the simplest rational is q%1 + the reciprocal of
-- the simplest rational between d'%r' and d%r.
approxRational :: (RealFrac a) => a -> a -> Rational
approxRational rat eps = simplest (rat-eps) (rat+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 = toRational x
n = numerator xr
d = denominator xr
nd' = toRational y
n' = numerator nd'
d' = denominator nd'
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'
nd'' = simplest' d' r' d r
n'' = numerator nd''
d'' = denominator nd''