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<pre><a name="line-1"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-2"></a><span class='hs-comment'>-- |</span>
<a name="line-3"></a><span class='hs-comment'>-- Module : Data.Ratio</span>
<a name="line-4"></a><span class='hs-comment'>-- Copyright : (c) The University of Glasgow 2001</span>
<a name="line-5"></a><span class='hs-comment'>-- License : BSD-style (see the file libraries/base/LICENSE)</span>
<a name="line-6"></a><span class='hs-comment'>-- </span>
<a name="line-7"></a><span class='hs-comment'>-- Maintainer : libraries@haskell.org</span>
<a name="line-8"></a><span class='hs-comment'>-- Stability : stable</span>
<a name="line-9"></a><span class='hs-comment'>-- Portability : portable</span>
<a name="line-10"></a><span class='hs-comment'>--</span>
<a name="line-11"></a><span class='hs-comment'>-- Standard functions on rational numbers</span>
<a name="line-12"></a><span class='hs-comment'>--</span>
<a name="line-13"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-14"></a>
<a name="line-15"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Ratio</span>
<a name="line-16"></a> <span class='hs-layout'>(</span> <span class='hs-conid'>Ratio</span>
<a name="line-17"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Rational</span>
<a name="line-18"></a> <span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>%</span><span class='hs-layout'>)</span> <span class='hs-comment'>-- :: (Integral a) => a -> a -> Ratio a</span>
<a name="line-19"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>numerator</span> <span class='hs-comment'>-- :: (Integral a) => Ratio a -> a</span>
<a name="line-20"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>denominator</span> <span class='hs-comment'>-- :: (Integral a) => Ratio a -> a</span>
<a name="line-21"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>approxRational</span> <span class='hs-comment'>-- :: (RealFrac a) => a -> a -> Rational</span>
<a name="line-22"></a>
<a name="line-23"></a> <span class='hs-comment'>-- Ratio instances: </span>
<a name="line-24"></a> <span class='hs-comment'>-- (Integral a) => Eq (Ratio a)</span>
<a name="line-25"></a> <span class='hs-comment'>-- (Integral a) => Ord (Ratio a)</span>
<a name="line-26"></a> <span class='hs-comment'>-- (Integral a) => Num (Ratio a)</span>
<a name="line-27"></a> <span class='hs-comment'>-- (Integral a) => Real (Ratio a)</span>
<a name="line-28"></a> <span class='hs-comment'>-- (Integral a) => Fractional (Ratio a)</span>
<a name="line-29"></a> <span class='hs-comment'>-- (Integral a) => RealFrac (Ratio a)</span>
<a name="line-30"></a> <span class='hs-comment'>-- (Integral a) => Enum (Ratio a)</span>
<a name="line-31"></a> <span class='hs-comment'>-- (Read a, Integral a) => Read (Ratio a)</span>
<a name="line-32"></a> <span class='hs-comment'>-- (Integral a) => Show (Ratio a)</span>
<a name="line-33"></a>
<a name="line-34"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-35"></a>
<a name="line-36"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Prelude</span>
<a name="line-37"></a>
<a name="line-38"></a><span class='hs-cpp'>#ifdef __GLASGOW_HASKELL__</span>
<a name="line-39"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>GHC</span><span class='hs-varop'>.</span><span class='hs-conid'>Real</span> <span class='hs-comment'>-- The basic defns for Ratio</span>
<a name="line-40"></a><span class='hs-cpp'>#endif</span>
<a name="line-41"></a>
<a name="line-42"></a><span class='hs-cpp'>#ifdef __HUGS__</span>
<a name="line-43"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Hugs</span><span class='hs-varop'>.</span><span class='hs-conid'>Prelude</span><span class='hs-layout'>(</span><span class='hs-conid'>Ratio</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>%</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>numerator</span><span class='hs-layout'>,</span> <span class='hs-varid'>denominator</span><span class='hs-layout'>)</span>
<a name="line-44"></a><span class='hs-cpp'>#endif</span>
<a name="line-45"></a>
<a name="line-46"></a><span class='hs-cpp'>#ifdef __NHC__</span>
<a name="line-47"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Ratio</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ratio</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>%</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>numerator</span><span class='hs-layout'>,</span> <span class='hs-varid'>denominator</span><span class='hs-layout'>,</span> <span class='hs-varid'>approxRational</span><span class='hs-layout'>)</span>
<a name="line-48"></a><span class='hs-cpp'>#else</span>
<a name="line-49"></a>
<a name="line-50"></a><span class='hs-comment'>-- -----------------------------------------------------------------------------</span>
<a name="line-51"></a><span class='hs-comment'>-- approxRational</span>
<a name="line-52"></a>
<a name="line-53"></a><span class='hs-comment'>-- | 'approxRational', applied to two real fractional numbers @x@ and @epsilon@,</span>
<a name="line-54"></a><span class='hs-comment'>-- returns the simplest rational number within @epsilon@ of @x@.</span>
<a name="line-55"></a><span class='hs-comment'>-- A rational number @y@ is said to be /simpler/ than another @y'@ if</span>
<a name="line-56"></a><span class='hs-comment'>--</span>
<a name="line-57"></a><span class='hs-comment'>-- * @'abs' ('numerator' y) <= 'abs' ('numerator' y')@, and</span>
<a name="line-58"></a><span class='hs-comment'>--</span>
<a name="line-59"></a><span class='hs-comment'>-- * @'denominator' y <= 'denominator' y'@.</span>
<a name="line-60"></a><span class='hs-comment'>--</span>
<a name="line-61"></a><span class='hs-comment'>-- Any real interval contains a unique simplest rational;</span>
<a name="line-62"></a><span class='hs-comment'>-- in particular, note that @0\/1@ is the simplest rational of all.</span>
<a name="line-63"></a>
<a name="line-64"></a><span class='hs-comment'>-- Implementation details: Here, for simplicity, we assume a closed rational</span>
<a name="line-65"></a><span class='hs-comment'>-- interval. If such an interval includes at least one whole number, then</span>
<a name="line-66"></a><span class='hs-comment'>-- the simplest rational is the absolutely least whole number. Otherwise,</span>
<a name="line-67"></a><span class='hs-comment'>-- the bounds are of the form q%1 + r%d and q%1 + r'%d', where abs r < d</span>
<a name="line-68"></a><span class='hs-comment'>-- and abs r' < d', and the simplest rational is q%1 + the reciprocal of</span>
<a name="line-69"></a><span class='hs-comment'>-- the simplest rational between d'%r' and d%r.</span>
<a name="line-70"></a>
<a name="line-71"></a><a name="approxRational"></a><span class='hs-definition'>approxRational</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFrac</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span>
<a name="line-72"></a><span class='hs-definition'>approxRational</span> <span class='hs-varid'>rat</span> <span class='hs-varid'>eps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>simplest</span> <span class='hs-layout'>(</span><span class='hs-varid'>rat</span><span class='hs-comment'>-</span><span class='hs-varid'>eps</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>rat</span><span class='hs-varop'>+</span><span class='hs-varid'>eps</span><span class='hs-layout'>)</span>
<a name="line-73"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>simplest</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-varop'><</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>simplest</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span>
<a name="line-74"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>xr</span>
<a name="line-75"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varop'>></span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>simplest'</span> <span class='hs-varid'>n</span> <span class='hs-varid'>d</span> <span class='hs-varid'>n'</span> <span class='hs-varid'>d'</span>
<a name="line-76"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-varop'><</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-comment'>-</span> <span class='hs-varid'>simplest'</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>n'</span><span class='hs-layout'>)</span> <span class='hs-varid'>d'</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-varid'>d</span>
<a name="line-77"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span> <span class='hs-conop'>:%</span> <span class='hs-num'>1</span>
<a name="line-78"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>xr</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toRational</span> <span class='hs-varid'>x</span>
<a name="line-79"></a> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>numerator</span> <span class='hs-varid'>xr</span>
<a name="line-80"></a> <span class='hs-varid'>d</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>denominator</span> <span class='hs-varid'>xr</span>
<a name="line-81"></a> <span class='hs-varid'>nd'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toRational</span> <span class='hs-varid'>y</span>
<a name="line-82"></a> <span class='hs-varid'>n'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>numerator</span> <span class='hs-varid'>nd'</span>
<a name="line-83"></a> <span class='hs-varid'>d'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>denominator</span> <span class='hs-varid'>nd'</span>
<a name="line-84"></a>
<a name="line-85"></a> <span class='hs-varid'>simplest'</span> <span class='hs-varid'>n</span> <span class='hs-varid'>d</span> <span class='hs-varid'>n'</span> <span class='hs-varid'>d'</span> <span class='hs-comment'>-- assumes 0 < n%d < n'%d'</span>
<a name="line-86"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>r</span> <span class='hs-varop'>==</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>q</span> <span class='hs-conop'>:%</span> <span class='hs-num'>1</span>
<a name="line-87"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>q</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>q'</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>q</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-conop'>:%</span> <span class='hs-num'>1</span>
<a name="line-88"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>q</span><span class='hs-varop'>*</span><span class='hs-varid'>n''</span><span class='hs-varop'>+</span><span class='hs-varid'>d''</span><span class='hs-layout'>)</span> <span class='hs-conop'>:%</span> <span class='hs-varid'>n''</span>
<a name="line-89"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>q</span><span class='hs-layout'>,</span><span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>quotRem</span> <span class='hs-varid'>n</span> <span class='hs-varid'>d</span>
<a name="line-90"></a> <span class='hs-layout'>(</span><span class='hs-varid'>q'</span><span class='hs-layout'>,</span><span class='hs-varid'>r'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>quotRem</span> <span class='hs-varid'>n'</span> <span class='hs-varid'>d'</span>
<a name="line-91"></a> <span class='hs-varid'>nd''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>simplest'</span> <span class='hs-varid'>d'</span> <span class='hs-varid'>r'</span> <span class='hs-varid'>d</span> <span class='hs-varid'>r</span>
<a name="line-92"></a> <span class='hs-varid'>n''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>numerator</span> <span class='hs-varid'>nd''</span>
<a name="line-93"></a> <span class='hs-varid'>d''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>denominator</span> <span class='hs-varid'>nd''</span>
<a name="line-94"></a><span class='hs-cpp'>#endif</span>
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