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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="index.html">Contents</a></li><li><a href="doc-index.html">Index</a></li></ul><p class="caption">haskell2010-1.1.0.1: Compatibility with Haskell 2010</p></div><div id="content"><div id="module-header"><table class="info"><tr><th>Safe Haskell</th><td>Safe</td></tr></table><p class="caption">Data.Ratio</p></div><div id="table-of-contents"><p class="caption">Contents</p><ul><li><a href="#g:1">Specification
</a></li></ul></div><div id="synopsis"><p id="control.syn" class="caption expander" onclick="toggleSection('syn')">Synopsis</p><ul id="section.syn" class="hide" onclick="toggleSection('syn')"><li class="src short"><span class="keyword">data</span> <a href="#t:Ratio">Ratio</a> a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Rational">Rational</a> = <a href="Data-Ratio.html#t:Ratio">Ratio</a> <a href="Prelude.html#t:Integer">Integer</a></li><li class="src short"><a href="#v:-37-">(%)</a> :: <a href="Prelude.html#t:Integral">Integral</a> a => a -> a -> <a href="Data-Ratio.html#t:Ratio">Ratio</a> a</li><li class="src short"><a href="#v:numerator">numerator</a> :: <a href="Prelude.html#t:Integral">Integral</a> a => <a href="Data-Ratio.html#t:Ratio">Ratio</a> a -> a</li><li class="src short"><a href="#v:denominator">denominator</a> :: <a href="Prelude.html#t:Integral">Integral</a> a => <a href="Data-Ratio.html#t:Ratio">Ratio</a> a -> a</li><li class="src short"><a href="#v:approxRational">approxRational</a> :: <a href="Prelude.html#t:RealFrac">RealFrac</a> a => a -> a -> <a href="Data-Ratio.html#t:Rational">Rational</a></li></ul></div><div id="interface"><h1>Documentation</h1><div class="top"><p class="src"><span class="keyword">data</span> <a name="t:Ratio" class="def">Ratio</a> a <a href="../base-4.5.1.0/src/GHC-Real.html#Ratio" class="link">Source</a></p><div class="doc"><p>Rational numbers, with numerator and denominator of some <code><a href="Prelude.html#t:Integral">Integral</a></code> type.
</p></div><div class="subs instances"><p id="control.i:Ratio" class="caption collapser" onclick="toggleSection('i:Ratio')">Instances</p><div id="section.i:Ratio" class="show"><table><tr><td class="src"><a href="../base-4.5.1.0/Data-Typeable-Internal.html#t:Typeable1">Typeable1</a> <a href="Data-Ratio.html#t:Ratio">Ratio</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Integral">Integral</a> a => <a href="Prelude.html#t:Enum">Enum</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Eq">Eq</a> a => <a href="Prelude.html#t:Eq">Eq</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Integral">Integral</a> a => <a href="Prelude.html#t:Fractional">Fractional</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Integral">Integral</a> a => <a href="Prelude.html#t:Num">Num</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Integral">Integral</a> a => <a href="Prelude.html#t:Ord">Ord</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Prelude.html#t:Integral">Integral</a> a, <a href="Prelude.html#t:Read">Read</a> a) => <a href="Prelude.html#t:Read">Read</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Integral">Integral</a> a => <a href="Prelude.html#t:Real">Real</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Prelude.html#t:Integral">Integral</a> a => <a href="Prelude.html#t:RealFrac">RealFrac</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Prelude.html#t:Integral">Integral</a> a, <a href="Prelude.html#t:Show">Show</a> a) => <a href="Prelude.html#t:Show">Show</a> (<a href="Data-Ratio.html#t:Ratio">Ratio</a> a)</td><td class="doc empty"> </td></tr></table></div></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a name="t:Rational" class="def">Rational</a> = <a href="Data-Ratio.html#t:Ratio">Ratio</a> <a href="Prelude.html#t:Integer">Integer</a><a href="../base-4.5.1.0/src/GHC-Real.html#Rational" class="link">Source</a></p><div class="doc"><p>Arbitrary-precision rational numbers, represented as a ratio of
two <code><a href="Prelude.html#t:Integer">Integer</a></code> values. A rational number may be constructed using
the <code><a href="Data-Ratio.html#v:-37-">%</a></code> operator.
</p></div></div><div class="top"><p class="src"><a name="v:-37-" class="def">(%)</a> :: <a href="Prelude.html#t:Integral">Integral</a> a => a -> a -> <a href="Data-Ratio.html#t:Ratio">Ratio</a> a<a href="../base-4.5.1.0/src/GHC-Real.html#%25" class="link">Source</a></p><div class="doc"><p>Forms the ratio of two integral numbers.
</p></div></div><div class="top"><p class="src"><a name="v:numerator" class="def">numerator</a> :: <a href="Prelude.html#t:Integral">Integral</a> a => <a href="Data-Ratio.html#t:Ratio">Ratio</a> a -> a<a href="../base-4.5.1.0/src/GHC-Real.html#numerator" class="link">Source</a></p><div class="doc"><p>Extract the numerator of the ratio in reduced form:
the numerator and denominator have no common factor and the denominator
is positive.
</p></div></div><div class="top"><p class="src"><a name="v:denominator" class="def">denominator</a> :: <a href="Prelude.html#t:Integral">Integral</a> a => <a href="Data-Ratio.html#t:Ratio">Ratio</a> a -> a<a href="../base-4.5.1.0/src/GHC-Real.html#denominator" class="link">Source</a></p><div class="doc"><p>Extract the denominator of the ratio in reduced form:
the numerator and denominator have no common factor and the denominator
is positive.
</p></div></div><div class="top"><p class="src"><a name="v:approxRational" class="def">approxRational</a> :: <a href="Prelude.html#t:RealFrac">RealFrac</a> a => a -> a -> <a href="Data-Ratio.html#t:Rational">Rational</a><a href="../base-4.5.1.0/src/Data-Ratio.html#approxRational" class="link">Source</a></p><div class="doc"><p><code><a href="Data-Ratio.html#v:approxRational">approxRational</a></code>, applied to two real fractional numbers <code>x</code> and <code>epsilon</code>,
returns the simplest rational number within <code>epsilon</code> of <code>x</code>.
A rational number <code>y</code> is said to be <em>simpler</em> than another <code>y'</code> if
</p><ul><li> <code><code><a href="Prelude.html#v:abs">abs</a></code> (<code><a href="Data-Ratio.html#v:numerator">numerator</a></code> y) <= <code><a href="Prelude.html#v:abs">abs</a></code> (<code><a href="Data-Ratio.html#v:numerator">numerator</a></code> y')</code>, and
</li><li> <code><code><a href="Data-Ratio.html#v:denominator">denominator</a></code> y <= <code><a href="Data-Ratio.html#v:denominator">denominator</a></code> y'</code>.
</li></ul><p>Any real interval contains a unique simplest rational;
in particular, note that <code>0/1</code> is the simplest rational of all.
</p></div></div><h1 id="g:1">Specification
</h1><div class="doc"><pre> 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
</pre></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.11.0</p></div></body></html>
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