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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-------------------------------------------------------------------------------moduleData.Ratio(Ratio,Rational,(%),numerator,denominator,approxRational)whereimportGHC.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::(RealFraca)=>a->a->RationalapproxRationalrateps=-- We convert rat and eps to rational *before* subtracting/adding since-- otherwise we may overflow. This was the cause of #14425.simplest(toRationalrat-toRationaleps)(toRationalrat+toRationaleps)wheresimplestxy|y<x=simplestyx|x==y=xr|x>0=simplest'ndn'd'|y<0=-simplest'(-n')d'(-n)d|otherwise=0:%1wherexr=toRationalxn=numeratorxrd=denominatorxrnd'=toRationalyn'=numeratornd'd'=denominatornd'simplest'ndn'd'-- assumes 0 < n%d < n'%d'|r==0=q:%1|q/=q'=(q+1):%1|otherwise=(q*n''+d''):%n''where(q,r)=quotRemnd(q',r')=quotRemn'd'nd''=simplest'd'r'drn''=numeratornd''d''=denominatornd''