Data/Ratio.hs

    {-# 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 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''