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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.Complex</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 : provisional</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'>-- Complex 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'>Complex</span>
<a name="line-16"></a> <span class='hs-layout'>(</span>
<a name="line-17"></a> <span class='hs-comment'>-- * Rectangular form</span>
<a name="line-18"></a> <span class='hs-conid'>Complex</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conop'>:+</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-19"></a>
<a name="line-20"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>realPart</span> <span class='hs-comment'>-- :: (RealFloat a) => Complex a -> a</span>
<a name="line-21"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>imagPart</span> <span class='hs-comment'>-- :: (RealFloat a) => Complex a -> a</span>
<a name="line-22"></a> <span class='hs-comment'>-- * Polar form</span>
<a name="line-23"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>mkPolar</span> <span class='hs-comment'>-- :: (RealFloat a) => a -> a -> Complex a</span>
<a name="line-24"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>cis</span> <span class='hs-comment'>-- :: (RealFloat a) => a -> Complex a</span>
<a name="line-25"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>polar</span> <span class='hs-comment'>-- :: (RealFloat a) => Complex a -> (a,a)</span>
<a name="line-26"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>magnitude</span> <span class='hs-comment'>-- :: (RealFloat a) => Complex a -> a</span>
<a name="line-27"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>phase</span> <span class='hs-comment'>-- :: (RealFloat a) => Complex a -> a</span>
<a name="line-28"></a> <span class='hs-comment'>-- * Conjugate</span>
<a name="line-29"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>conjugate</span> <span class='hs-comment'>-- :: (RealFloat a) => Complex a -> Complex a</span>
<a name="line-30"></a>
<a name="line-31"></a> <span class='hs-comment'>-- Complex instances:</span>
<a name="line-32"></a> <span class='hs-comment'>--</span>
<a name="line-33"></a> <span class='hs-comment'>-- (RealFloat a) => Eq (Complex a)</span>
<a name="line-34"></a> <span class='hs-comment'>-- (RealFloat a) => Read (Complex a)</span>
<a name="line-35"></a> <span class='hs-comment'>-- (RealFloat a) => Show (Complex a)</span>
<a name="line-36"></a> <span class='hs-comment'>-- (RealFloat a) => Num (Complex a)</span>
<a name="line-37"></a> <span class='hs-comment'>-- (RealFloat a) => Fractional (Complex a)</span>
<a name="line-38"></a> <span class='hs-comment'>-- (RealFloat a) => Floating (Complex a)</span>
<a name="line-39"></a> <span class='hs-comment'>-- </span>
<a name="line-40"></a> <span class='hs-comment'>-- Implementation checked wrt. Haskell 98 lib report, 1/99.</span>
<a name="line-41"></a>
<a name="line-42"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-43"></a>
<a name="line-44"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Prelude</span>
<a name="line-45"></a>
<a name="line-46"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Typeable</span>
<a name="line-47"></a><span class='hs-cpp'>#ifdef __GLASGOW_HASKELL__</span>
<a name="line-48"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Data</span> <span class='hs-layout'>(</span><span class='hs-conid'>Data</span><span class='hs-layout'>)</span>
<a name="line-49"></a><span class='hs-cpp'>#endif</span>
<a name="line-50"></a>
<a name="line-51"></a><span class='hs-cpp'>#ifdef __HUGS__</span>
<a name="line-52"></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'>Num</span><span class='hs-layout'>(</span><span class='hs-varid'>fromInt</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span><span class='hs-layout'>(</span><span class='hs-varid'>fromDouble</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-53"></a><span class='hs-cpp'>#endif</span>
<a name="line-54"></a>
<a name="line-55"></a><span class='hs-keyword'>infix</span> <span class='hs-num'>6</span> <span class='hs-conop'>:+</span>
<a name="line-56"></a>
<a name="line-57"></a><span class='hs-comment'>-- -----------------------------------------------------------------------------</span>
<a name="line-58"></a><span class='hs-comment'>-- The Complex type</span>
<a name="line-59"></a>
<a name="line-60"></a><a name="Complex"></a><span class='hs-comment'>-- | Complex numbers are an algebraic type.</span>
<a name="line-61"></a><a name="Complex"></a><span class='hs-comment'>--</span>
<a name="line-62"></a><a name="Complex"></a><span class='hs-comment'>-- For a complex number @z@, @'abs' z@ is a number with the magnitude of @z@,</span>
<a name="line-63"></a><a name="Complex"></a><span class='hs-comment'>-- but oriented in the positive real direction, whereas @'signum' z@</span>
<a name="line-64"></a><a name="Complex"></a><span class='hs-comment'>-- has the phase of @z@, but unit magnitude.</span>
<a name="line-65"></a><a name="Complex"></a><span class='hs-keyword'>data</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span>
<a name="line-66"></a> <span class='hs-keyglyph'>=</span> <span class='hs-varop'>!</span><span class='hs-varid'>a</span> <span class='hs-conop'>:+</span> <span class='hs-varop'>!</span><span class='hs-varid'>a</span> <span class='hs-comment'>-- ^ forms a complex number from its real and imaginary</span>
<a name="line-67"></a> <span class='hs-comment'>-- rectangular components.</span>
<a name="line-68"></a><span class='hs-cpp'># if __GLASGOW_HASKELL__</span>
<a name="line-69"></a> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span><span class='hs-layout'>,</span> <span class='hs-conid'>Read</span><span class='hs-layout'>,</span> <span class='hs-conid'>Data</span><span class='hs-layout'>)</span>
<a name="line-70"></a><span class='hs-cpp'># else</span>
<a name="line-71"></a> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span><span class='hs-layout'>,</span> <span class='hs-conid'>Read</span><span class='hs-layout'>)</span>
<a name="line-72"></a><span class='hs-cpp'># endif</span>
<a name="line-73"></a>
<a name="line-74"></a><span class='hs-comment'>-- -----------------------------------------------------------------------------</span>
<a name="line-75"></a><span class='hs-comment'>-- Functions over Complex</span>
<a name="line-76"></a>
<a name="line-77"></a><a name="realPart"></a><span class='hs-comment'>-- | Extracts the real part of a complex number.</span>
<a name="line-78"></a><span class='hs-definition'>realPart</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-79"></a><span class='hs-definition'>realPart</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:+</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span>
<a name="line-80"></a>
<a name="line-81"></a><a name="imagPart"></a><span class='hs-comment'>-- | Extracts the imaginary part of a complex number.</span>
<a name="line-82"></a><span class='hs-definition'>imagPart</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-83"></a><span class='hs-definition'>imagPart</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y</span>
<a name="line-84"></a>
<a name="line-85"></a><a name="conjugate"></a><span class='hs-comment'>-- | The conjugate of a complex number.</span>
<a name="line-86"></a><span class='hs-comment'>{-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}</span>
<a name="line-87"></a><span class='hs-definition'>conjugate</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span>
<a name="line-88"></a><span class='hs-definition'>conjugate</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-89"></a>
<a name="line-90"></a><a name="mkPolar"></a><span class='hs-comment'>-- | Form a complex number from polar components of magnitude and phase.</span>
<a name="line-91"></a><span class='hs-comment'>{-# SPECIALISE mkPolar :: Double -> Double -> Complex Double #-}</span>
<a name="line-92"></a><span class='hs-definition'>mkPolar</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</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'>Complex</span> <span class='hs-varid'>a</span>
<a name="line-93"></a><span class='hs-definition'>mkPolar</span> <span class='hs-varid'>r</span> <span class='hs-varid'>theta</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>r</span> <span class='hs-varop'>*</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>theta</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>r</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>theta</span>
<a name="line-94"></a>
<a name="line-95"></a><a name="cis"></a><span class='hs-comment'>-- | @'cis' t@ is a complex value with magnitude @1@</span>
<a name="line-96"></a><span class='hs-comment'>-- and phase @t@ (modulo @2*'pi'@).</span>
<a name="line-97"></a><span class='hs-comment'>{-# SPECIALISE cis :: Double -> Complex Double #-}</span>
<a name="line-98"></a><span class='hs-definition'>cis</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</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-conid'>Complex</span> <span class='hs-varid'>a</span>
<a name="line-99"></a><span class='hs-definition'>cis</span> <span class='hs-varid'>theta</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>theta</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>theta</span>
<a name="line-100"></a>
<a name="line-101"></a><a name="polar"></a><span class='hs-comment'>-- | The function 'polar' takes a complex number and</span>
<a name="line-102"></a><span class='hs-comment'>-- returns a (magnitude, phase) pair in canonical form:</span>
<a name="line-103"></a><span class='hs-comment'>-- the magnitude is nonnegative, and the phase in the range @(-'pi', 'pi']@;</span>
<a name="line-104"></a><span class='hs-comment'>-- if the magnitude is zero, then so is the phase.</span>
<a name="line-105"></a><span class='hs-comment'>{-# SPECIALISE polar :: Complex Double -> (Double,Double) #-}</span>
<a name="line-106"></a><span class='hs-definition'>polar</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-107"></a><span class='hs-definition'>polar</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>magnitude</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-varid'>phase</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span>
<a name="line-108"></a>
<a name="line-109"></a><a name="magnitude"></a><span class='hs-comment'>-- | The nonnegative magnitude of a complex number.</span>
<a name="line-110"></a><span class='hs-comment'>{-# SPECIALISE magnitude :: Complex Double -> Double #-}</span>
<a name="line-111"></a><span class='hs-definition'>magnitude</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-112"></a><span class='hs-definition'>magnitude</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>scaleFloat</span> <span class='hs-varid'>k</span>
<a name="line-113"></a> <span class='hs-layout'>(</span><span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-varid'>sqr</span> <span class='hs-layout'>(</span><span class='hs-varid'>scaleFloat</span> <span class='hs-varid'>mk</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sqr</span> <span class='hs-layout'>(</span><span class='hs-varid'>scaleFloat</span> <span class='hs-varid'>mk</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-114"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-varid'>exponent</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>exponent</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-115"></a> <span class='hs-varid'>mk</span> <span class='hs-keyglyph'>=</span> <span class='hs-comment'>-</span> <span class='hs-varid'>k</span>
<a name="line-116"></a> <span class='hs-varid'>sqr</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>z</span> <span class='hs-varop'>*</span> <span class='hs-varid'>z</span>
<a name="line-117"></a>
<a name="line-118"></a><a name="phase"></a><span class='hs-comment'>-- | The phase of a complex number, in the range @(-'pi', 'pi']@.</span>
<a name="line-119"></a><span class='hs-comment'>-- If the magnitude is zero, then so is the phase.</span>
<a name="line-120"></a><span class='hs-comment'>{-# SPECIALISE phase :: Complex Double -> Double #-}</span>
<a name="line-121"></a><span class='hs-definition'>phase</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-122"></a><span class='hs-definition'>phase</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span> <span class='hs-conop'>:+</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span> <span class='hs-comment'>-- SLPJ July 97 from John Peterson</span>
<a name="line-123"></a><span class='hs-definition'>phase</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>atan2</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span>
<a name="line-124"></a>
<a name="line-125"></a>
<a name="line-126"></a><span class='hs-comment'>-- -----------------------------------------------------------------------------</span>
<a name="line-127"></a><span class='hs-comment'>-- Instances of Complex</span>
<a name="line-128"></a>
<a name="line-129"></a><span class='hs-cpp'>#include "Typeable.h"</span>
<a name="line-130"></a><span class='hs-conid'>INSTANCE_TYPEABLE1</span><span class='hs-layout'>(</span><span class='hs-conid'>Complex</span><span class='hs-layout'>,</span><span class='hs-varid'>complexTc</span><span class='hs-layout'>,</span><span class='hs-str'>"Complex"</span><span class='hs-layout'>)</span>
<a name="line-131"></a>
<a name="line-132"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Num</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-133"></a> <span class='hs-comment'>{-# SPECIALISE instance Num (Complex Float) #-}</span>
<a name="line-134"></a> <span class='hs-comment'>{-# SPECIALISE instance Num (Complex Double) #-}</span>
<a name="line-135"></a> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-conop'>:+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>+</span><span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-varop'>+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span>
<a name="line-136"></a> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-comment'>-</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-conop'>:+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-comment'>-</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span>
<a name="line-137"></a> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-conop'>:+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>x'</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span><span class='hs-varop'>*</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>y'</span><span class='hs-varop'>+</span><span class='hs-varid'>y</span><span class='hs-varop'>*</span><span class='hs-varid'>x'</span><span class='hs-layout'>)</span>
<a name="line-138"></a> <span class='hs-varid'>negate</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>negate</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>negate</span> <span class='hs-varid'>y</span>
<a name="line-139"></a> <span class='hs-varid'>abs</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>magnitude</span> <span class='hs-varid'>z</span> <span class='hs-conop'>:+</span> <span class='hs-num'>0</span>
<a name="line-140"></a> <span class='hs-varid'>signum</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-conop'>:+</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span>
<a name="line-141"></a> <span class='hs-varid'>signum</span> <span class='hs-varid'>z</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>r</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>y</span><span class='hs-varop'>/</span><span class='hs-varid'>r</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>magnitude</span> <span class='hs-varid'>z</span>
<a name="line-142"></a> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>n</span> <span class='hs-conop'>:+</span> <span class='hs-num'>0</span>
<a name="line-143"></a><span class='hs-cpp'>#ifdef __HUGS__</span>
<a name="line-144"></a> <span class='hs-varid'>fromInt</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInt</span> <span class='hs-varid'>n</span> <span class='hs-conop'>:+</span> <span class='hs-num'>0</span>
<a name="line-145"></a><span class='hs-cpp'>#endif</span>
<a name="line-146"></a>
<a name="line-147"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Fractional</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-148"></a> <span class='hs-comment'>{-# SPECIALISE instance Fractional (Complex Float) #-}</span>
<a name="line-149"></a> <span class='hs-comment'>{-# SPECIALISE instance Fractional (Complex Double) #-}</span>
<a name="line-150"></a> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-conop'>:+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>x''</span><span class='hs-varop'>+</span><span class='hs-varid'>y</span><span class='hs-varop'>*</span><span class='hs-varid'>y''</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-varid'>d</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-varop'>*</span><span class='hs-varid'>x''</span><span class='hs-comment'>-</span><span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>y''</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-varid'>d</span>
<a name="line-151"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>x''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>scaleFloat</span> <span class='hs-varid'>k</span> <span class='hs-varid'>x'</span>
<a name="line-152"></a> <span class='hs-varid'>y''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>scaleFloat</span> <span class='hs-varid'>k</span> <span class='hs-varid'>y'</span>
<a name="line-153"></a> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-comment'>-</span> <span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-varid'>exponent</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>exponent</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span>
<a name="line-154"></a> <span class='hs-varid'>d</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span><span class='hs-varop'>*</span><span class='hs-varid'>x''</span> <span class='hs-varop'>+</span> <span class='hs-varid'>y'</span><span class='hs-varop'>*</span><span class='hs-varid'>y''</span>
<a name="line-155"></a>
<a name="line-156"></a> <span class='hs-varid'>fromRational</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromRational</span> <span class='hs-varid'>a</span> <span class='hs-conop'>:+</span> <span class='hs-num'>0</span>
<a name="line-157"></a><span class='hs-cpp'>#ifdef __HUGS__</span>
<a name="line-158"></a> <span class='hs-varid'>fromDouble</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromDouble</span> <span class='hs-varid'>a</span> <span class='hs-conop'>:+</span> <span class='hs-num'>0</span>
<a name="line-159"></a><span class='hs-cpp'>#endif</span>
<a name="line-160"></a>
<a name="line-161"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Floating</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-162"></a> <span class='hs-comment'>{-# SPECIALISE instance Floating (Complex Float) #-}</span>
<a name="line-163"></a> <span class='hs-comment'>{-# SPECIALISE instance Floating (Complex Double) #-}</span>
<a name="line-164"></a> <span class='hs-varid'>pi</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>pi</span> <span class='hs-conop'>:+</span> <span class='hs-num'>0</span>
<a name="line-165"></a> <span class='hs-varid'>exp</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>expx</span> <span class='hs-varop'>*</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>y</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>expx</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>y</span>
<a name="line-166"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>expx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>exp</span> <span class='hs-varid'>x</span>
<a name="line-167"></a> <span class='hs-varid'>log</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-varid'>magnitude</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>phase</span> <span class='hs-varid'>z</span>
<a name="line-168"></a>
<a name="line-169"></a> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-conop'>:+</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span>
<a name="line-170"></a> <span class='hs-varid'>sqrt</span> <span class='hs-varid'>z</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>u</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-keyword'>if</span> <span class='hs-varid'>y</span> <span class='hs-varop'><</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span><span class='hs-varid'>v</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-171"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>x</span> <span class='hs-varop'><</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-layout'>(</span><span class='hs-varid'>v'</span><span class='hs-layout'>,</span><span class='hs-varid'>u'</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span> <span class='hs-layout'>(</span><span class='hs-varid'>u'</span><span class='hs-layout'>,</span><span class='hs-varid'>v'</span><span class='hs-layout'>)</span>
<a name="line-172"></a> <span class='hs-varid'>v'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>abs</span> <span class='hs-varid'>y</span> <span class='hs-varop'>/</span> <span class='hs-layout'>(</span><span class='hs-varid'>u'</span><span class='hs-varop'>*</span><span class='hs-num'>2</span><span class='hs-layout'>)</span>
<a name="line-173"></a> <span class='hs-varid'>u'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>magnitude</span> <span class='hs-varid'>z</span> <span class='hs-varop'>+</span> <span class='hs-varid'>abs</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span><span class='hs-layout'>)</span>
<a name="line-174"></a>
<a name="line-175"></a> <span class='hs-varid'>sin</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*</span> <span class='hs-varid'>cosh</span> <span class='hs-varid'>y</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sinh</span> <span class='hs-varid'>y</span>
<a name="line-176"></a> <span class='hs-varid'>cos</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*</span> <span class='hs-varid'>cosh</span> <span class='hs-varid'>y</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sinh</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-177"></a> <span class='hs-varid'>tan</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>sinx</span><span class='hs-varop'>*</span><span class='hs-varid'>coshy</span><span class='hs-conop'>:+</span><span class='hs-varid'>cosx</span><span class='hs-varop'>*</span><span class='hs-varid'>sinhy</span><span class='hs-layout'>)</span><span class='hs-varop'>/</span><span class='hs-layout'>(</span><span class='hs-varid'>cosx</span><span class='hs-varop'>*</span><span class='hs-varid'>coshy</span><span class='hs-conop'>:+</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>sinx</span><span class='hs-varop'>*</span><span class='hs-varid'>sinhy</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-178"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>sinx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>x</span>
<a name="line-179"></a> <span class='hs-varid'>cosx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>x</span>
<a name="line-180"></a> <span class='hs-varid'>sinhy</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sinh</span> <span class='hs-varid'>y</span>
<a name="line-181"></a> <span class='hs-varid'>coshy</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cosh</span> <span class='hs-varid'>y</span>
<a name="line-182"></a>
<a name="line-183"></a> <span class='hs-varid'>sinh</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>y</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sinh</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>y</span> <span class='hs-varop'>*</span> <span class='hs-varid'>cosh</span> <span class='hs-varid'>x</span>
<a name="line-184"></a> <span class='hs-varid'>cosh</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>y</span> <span class='hs-varop'>*</span> <span class='hs-varid'>cosh</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>y</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sinh</span> <span class='hs-varid'>x</span>
<a name="line-185"></a> <span class='hs-varid'>tanh</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>cosy</span><span class='hs-varop'>*</span><span class='hs-varid'>sinhx</span><span class='hs-conop'>:+</span><span class='hs-varid'>siny</span><span class='hs-varop'>*</span><span class='hs-varid'>coshx</span><span class='hs-layout'>)</span><span class='hs-varop'>/</span><span class='hs-layout'>(</span><span class='hs-varid'>cosy</span><span class='hs-varop'>*</span><span class='hs-varid'>coshx</span><span class='hs-conop'>:+</span><span class='hs-varid'>siny</span><span class='hs-varop'>*</span><span class='hs-varid'>sinhx</span><span class='hs-layout'>)</span>
<a name="line-186"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>siny</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sin</span> <span class='hs-varid'>y</span>
<a name="line-187"></a> <span class='hs-varid'>cosy</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cos</span> <span class='hs-varid'>y</span>
<a name="line-188"></a> <span class='hs-varid'>sinhx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sinh</span> <span class='hs-varid'>x</span>
<a name="line-189"></a> <span class='hs-varid'>coshx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cosh</span> <span class='hs-varid'>x</span>
<a name="line-190"></a>
<a name="line-191"></a> <span class='hs-varid'>asin</span> <span class='hs-varid'>z</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y'</span><span class='hs-conop'>:+</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>x'</span><span class='hs-layout'>)</span>
<a name="line-192"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-conop'>:+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:+</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-comment'>-</span> <span class='hs-varid'>z</span><span class='hs-varop'>*</span><span class='hs-varid'>z</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-193"></a> <span class='hs-varid'>acos</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y''</span><span class='hs-conop'>:+</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>x''</span><span class='hs-layout'>)</span>
<a name="line-194"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>x''</span><span class='hs-conop'>:+</span><span class='hs-varid'>y''</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-varid'>z</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span><span class='hs-conop'>:+</span><span class='hs-varid'>x'</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-195"></a> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-conop'>:+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-comment'>-</span> <span class='hs-varid'>z</span><span class='hs-varop'>*</span><span class='hs-varid'>z</span><span class='hs-layout'>)</span>
<a name="line-196"></a> <span class='hs-varid'>atan</span> <span class='hs-varid'>z</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y'</span><span class='hs-conop'>:+</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>x'</span><span class='hs-layout'>)</span>
<a name="line-197"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span><span class='hs-conop'>:+</span><span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-num'>1</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:+</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span><span class='hs-varop'>+</span><span class='hs-varid'>z</span><span class='hs-varop'>*</span><span class='hs-varid'>z</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-198"></a>
<a name="line-199"></a> <span class='hs-varid'>asinh</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-varid'>z</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span><span class='hs-varop'>+</span><span class='hs-varid'>z</span><span class='hs-varop'>*</span><span class='hs-varid'>z</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-200"></a> <span class='hs-varid'>acosh</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-varid'>z</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-varid'>z</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>z</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-varop'>/</span><span class='hs-layout'>(</span><span class='hs-varid'>z</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-201"></a> <span class='hs-varid'>atanh</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0.5</span> <span class='hs-varop'>*</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-num'>1.0</span><span class='hs-varop'>+</span><span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-layout'>(</span><span class='hs-num'>1.0</span><span class='hs-comment'>-</span><span class='hs-varid'>z</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
</pre></body>
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