<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html> <head> <!-- Generated by HsColour, http://code.haskell.org/~malcolm/hscolour/ --> <title>Data/Number/Dif.hs</title> <link type='text/css' rel='stylesheet' href='hscolour.css' /> </head> <body> <pre><a name="line-1"></a><span class='hs-comment'>-- | The 'Data.Number.Dif' module contains a data type, 'Dif', that allows for</span> <a name="line-2"></a><span class='hs-comment'>-- automatic forward differentiation.</span> <a name="line-3"></a><span class='hs-comment'>--</span> <a name="line-4"></a><span class='hs-comment'>-- All the ideas are from Jerzy Karczmarczuk\'s work,</span> <a name="line-5"></a><span class='hs-comment'>-- see <<a href="http://users.info.unicaen.fr/~karczma/arpap/diffalg.pdf">http://users.info.unicaen.fr/~karczma/arpap/diffalg.pdf</a>>.</span> <a name="line-6"></a><span class='hs-comment'>--</span> <a name="line-7"></a><span class='hs-comment'>-- A simple example, if we define</span> <a name="line-8"></a><span class='hs-comment'>--</span> <a name="line-9"></a><span class='hs-comment'>-- > foo x = x*x</span> <a name="line-10"></a><span class='hs-comment'>--</span> <a name="line-11"></a><span class='hs-comment'>-- then the function</span> <a name="line-12"></a><span class='hs-comment'>--</span> <a name="line-13"></a><span class='hs-comment'>-- > foo' = deriv foo</span> <a name="line-14"></a><span class='hs-comment'>--</span> <a name="line-15"></a><span class='hs-comment'>-- will behave as if its body was 2*x.</span> <a name="line-16"></a><span class='hs-comment'>--</span> <a name="line-17"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Number</span><span class='hs-varop'>.</span><span class='hs-conid'>Dif</span><span class='hs-layout'>(</span><span class='hs-conid'>Dif</span><span class='hs-layout'>,</span> <span class='hs-varid'>val</span><span class='hs-layout'>,</span> <span class='hs-varid'>df</span><span class='hs-layout'>,</span> <span class='hs-varid'>mkDif</span><span class='hs-layout'>,</span> <span class='hs-varid'>dCon</span><span class='hs-layout'>,</span> <span class='hs-varid'>dVar</span><span class='hs-layout'>,</span> <span class='hs-varid'>deriv</span><span class='hs-layout'>,</span> <span class='hs-varid'>unDif</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-18"></a> <a name="line-19"></a><a name="Dif"></a><span class='hs-comment'>-- |The 'Dif' type is the type of differentiable numbers.</span> <a name="line-20"></a><a name="Dif"></a><span class='hs-comment'>-- It's an instance of all the usual numeric classes.</span> <a name="line-21"></a><a name="Dif"></a><span class='hs-comment'>-- The computed derivative of a function is is correct</span> <a name="line-22"></a><a name="Dif"></a><span class='hs-comment'>-- except where the function is discontinuous, at these points</span> <a name="line-23"></a><a name="Dif"></a><span class='hs-comment'>-- the derivative should be a Dirac pulse, but it isn\'t.</span> <a name="line-24"></a><a name="Dif"></a><span class='hs-comment'>--</span> <a name="line-25"></a><a name="Dif"></a><span class='hs-comment'>-- The 'Dif' numbers are printed with a trailing ~~ to</span> <a name="line-26"></a><a name="Dif"></a><span class='hs-comment'>-- indicate that there is a \"tail\" of derivatives.</span> <a name="line-27"></a><a name="Dif"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-varop'>!</span><span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>C</span> <span class='hs-varop'>!</span><span class='hs-varid'>a</span> <a name="line-28"></a> <a name="line-29"></a><a name="dCon"></a><span class='hs-comment'>-- |The 'dCon' function turns a normal number into a 'Dif'</span> <a name="line-30"></a><span class='hs-comment'>-- number with the same value. Not that numeric literals</span> <a name="line-31"></a><span class='hs-comment'>-- do not need an explicit conversion due to the normal</span> <a name="line-32"></a><span class='hs-comment'>-- Haskell overloading of literals.</span> <a name="line-33"></a><span class='hs-definition'>dCon</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</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'>Dif</span> <span class='hs-varid'>a</span> <a name="line-34"></a><span class='hs-definition'>dCon</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-varid'>x</span> <a name="line-35"></a> <a name="line-36"></a><a name="dVar"></a><span class='hs-comment'>-- |The 'dVar' function turns a number into a variable</span> <a name="line-37"></a><span class='hs-comment'>-- number. This is the number with with respect to which</span> <a name="line-38"></a><span class='hs-comment'>-- the derivaticve is computed.</span> <a name="line-39"></a><span class='hs-definition'>dVar</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</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'>Dif</span> <span class='hs-varid'>a</span> <a name="line-40"></a><span class='hs-definition'>dVar</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-num'>1</span> <a name="line-41"></a> <a name="line-42"></a><a name="df"></a><span class='hs-comment'>-- |The 'df' takes a 'Dif' number and returns its first</span> <a name="line-43"></a><span class='hs-comment'>-- derivative. The function can be iterated to to get</span> <a name="line-44"></a><span class='hs-comment'>-- higher derivaties.</span> <a name="line-45"></a><span class='hs-definition'>df</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <a name="line-46"></a><span class='hs-definition'>df</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span> <a name="line-47"></a><span class='hs-definition'>df</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span> <a name="line-48"></a> <a name="line-49"></a><a name="val"></a><span class='hs-comment'>-- |The 'val' function takes a 'Dif' number back to a normal</span> <a name="line-50"></a><span class='hs-comment'>-- number, thus forgetting about all the derivatives.</span> <a name="line-51"></a><span class='hs-definition'>val</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <a name="line-52"></a><span class='hs-definition'>val</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</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-53"></a><span class='hs-definition'>val</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <a name="line-54"></a> <a name="line-55"></a><a name="mkDif"></a><span class='hs-comment'>-- |The 'mkDif' takes a value and 'Dif' value and makes</span> <a name="line-56"></a><span class='hs-comment'>-- a 'Dif' number that has the given value as its normal</span> <a name="line-57"></a><span class='hs-comment'>-- value, and the 'Dif' number as its derivatives.</span> <a name="line-58"></a><span class='hs-definition'>mkDif</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <a name="line-59"></a><span class='hs-definition'>mkDif</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <a name="line-60"></a> <a name="line-61"></a><a name="deriv"></a><span class='hs-comment'>-- |The 'deriv' function is a simple utility to take the</span> <a name="line-62"></a><span class='hs-comment'>-- derivative of a (single argument) function.</span> <a name="line-63"></a><span class='hs-comment'>-- It is simply defined as</span> <a name="line-64"></a><span class='hs-comment'>--</span> <a name="line-65"></a><span class='hs-comment'>-- > deriv f = val . df . f . dVar</span> <a name="line-66"></a><span class='hs-comment'>--</span> <a name="line-67"></a><span class='hs-definition'>deriv</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <a name="line-68"></a><span class='hs-definition'>deriv</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>val</span> <span class='hs-varop'>.</span> <span class='hs-varid'>df</span> <span class='hs-varop'>.</span> <span class='hs-varid'>f</span> <span class='hs-varop'>.</span> <span class='hs-varid'>dVar</span> <a name="line-69"></a> <a name="line-70"></a><a name="unDif"></a><span class='hs-comment'>-- |Convert a 'Dif' function to an ordinary function.</span> <a name="line-71"></a><span class='hs-definition'>unDif</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <a name="line-72"></a><span class='hs-definition'>unDif</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>val</span> <span class='hs-varop'>.</span> <span class='hs-varid'>f</span> <span class='hs-varop'>.</span> <span class='hs-varid'>dVar</span> <a name="line-73"></a> <a name="line-74"></a><a name="instance%20Show%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Show</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-75"></a> <span class='hs-varid'>show</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-varid'>val</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span> <span class='hs-str'>"~~"</span> <a name="line-76"></a> <a name="line-77"></a><a name="instance%20Read%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Read</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Read</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-78"></a> <span class='hs-varid'>readsPrec</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-varid'>s'</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-layout'>,</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>readsPrec</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span><span class='hs-keyglyph'>]</span> <a name="line-79"></a> <a name="line-80"></a><a name="instance%20Eq%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Eq</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-81"></a> <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'>val</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>val</span> <span class='hs-varid'>y</span> <a name="line-82"></a> <a name="line-83"></a><a name="instance%20Ord%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Ord</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-84"></a> <span class='hs-varid'>x</span> <span class='hs-varop'>`compare`</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>val</span> <span class='hs-varid'>x</span> <span class='hs-varop'>`compare`</span> <span class='hs-varid'>val</span> <span class='hs-varid'>y</span> <a name="line-85"></a> <a name="line-86"></a><a name="instance%20Num%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</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'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-87"></a> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</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-layout'>)</span> <a name="line-88"></a> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</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-layout'>)</span> <span class='hs-varid'>y'</span> <a name="line-89"></a> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</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-layout'>)</span> <span class='hs-varid'>x'</span> <a name="line-90"></a> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</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-layout'>)</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-layout'>)</span> <a name="line-91"></a> <a name="line-92"></a> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-comment'>-</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <a name="line-93"></a> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-comment'>-</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>y</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> <a name="line-94"></a> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-comment'>-</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varid'>x'</span> <a name="line-95"></a> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-comment'>-</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-comment'>-</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <a name="line-96"></a> <a name="line-97"></a> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varop'>*</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-num'>0</span> <a name="line-98"></a> <span class='hs-keyword'>_</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-num'>0</span> <a name="line-99"></a> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</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-layout'>)</span> <a name="line-100"></a> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</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-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varop'>*</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <a name="line-101"></a> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</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-conid'>C</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</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-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varop'>*</span> <span class='hs-varid'>q</span><span class='hs-layout'>)</span> <a name="line-102"></a> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</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-conid'>D</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</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-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varop'>*</span> <span class='hs-varid'>q</span> <span class='hs-varop'>+</span> <span class='hs-varid'>p</span> <span class='hs-varop'>*</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <a name="line-103"></a> <a name="line-104"></a> <span class='hs-varid'>negate</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>negate</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-105"></a> <span class='hs-varid'>negate</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>negate</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>negate</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <a name="line-106"></a> <a name="line-107"></a> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromInteger</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <a name="line-108"></a> <a name="line-109"></a> <span class='hs-varid'>abs</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>abs</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-110"></a> <span class='hs-varid'>abs</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>abs</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>signum</span> <span class='hs-varid'>p</span> <span class='hs-varop'>*</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <a name="line-111"></a> <a name="line-112"></a> <span class='hs-comment'>-- The derivative of the signum function is (2*) the Dirac impulse,</span> <a name="line-113"></a> <span class='hs-comment'>-- but there's not really any good way to encode this.</span> <a name="line-114"></a> <span class='hs-comment'>-- We could do it by +Infinity (1/0) at 0.</span> <a name="line-115"></a> <span class='hs-varid'>signum</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>signum</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-116"></a> <span class='hs-varid'>signum</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>signum</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-117"></a> <a name="line-118"></a><a name="instance%20Fractional%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</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'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-119"></a> <span class='hs-varid'>recip</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>recip</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-120"></a> <span class='hs-varid'>recip</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ip</span> <a name="line-121"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>ip</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>recip</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>x'</span> <span class='hs-varop'>*</span> <span class='hs-varid'>ip</span> <span class='hs-varop'>*</span> <span class='hs-varid'>ip</span><span class='hs-layout'>)</span> <a name="line-122"></a> <span class='hs-varid'>fromRational</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromRational</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <a name="line-123"></a> <a name="line-124"></a><a name="lift"></a><span class='hs-definition'>lift</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></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-keyglyph'>-></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span> <a name="line-125"></a><span class='hs-definition'>lift</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-conop'>:</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-126"></a><span class='hs-definition'>lift</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-conop'>:</span> <span class='hs-varid'>f'</span><span class='hs-layout'>)</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varop'>*</span> <span class='hs-varid'>lift</span> <span class='hs-varid'>f'</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <a name="line-127"></a><span class='hs-definition'>lift</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"lift"</span> <a name="line-128"></a> <a name="line-129"></a><a name="instance%20Floating%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Floating</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</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'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-130"></a> <span class='hs-varid'>pi</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-varid'>pi</span> <a name="line-131"></a> <a name="line-132"></a> <span class='hs-varid'>exp</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>exp</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-133"></a> <span class='hs-varid'>exp</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</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-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>exp</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varop'>*</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <a name="line-134"></a> <a name="line-135"></a> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>log</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-136"></a> <span class='hs-varid'>log</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>log</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varop'>/</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <a name="line-137"></a> <a name="line-138"></a> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>sqrt</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-139"></a> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</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-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>sqrt</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varop'>/</span> <span class='hs-layout'>(</span><span class='hs-num'>2</span> <span class='hs-varop'>*</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-140"></a> <a name="line-141"></a> <span class='hs-varid'>sin</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lift</span> <span class='hs-layout'>(</span><span class='hs-varid'>cycle</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>sin</span><span class='hs-layout'>,</span> <span class='hs-varid'>cos</span><span class='hs-layout'>,</span> <span class='hs-varid'>negate</span> <span class='hs-varop'>.</span> <span class='hs-varid'>sin</span><span class='hs-layout'>,</span> <span class='hs-varid'>negate</span> <span class='hs-varop'>.</span> <span class='hs-varid'>cos</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <a name="line-142"></a> <span class='hs-varid'>cos</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lift</span> <span class='hs-layout'>(</span><span class='hs-varid'>cycle</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>cos</span><span class='hs-layout'>,</span> <span class='hs-varid'>negate</span> <span class='hs-varop'>.</span> <span class='hs-varid'>sin</span><span class='hs-layout'>,</span> <span class='hs-varid'>negate</span> <span class='hs-varop'>.</span> <span class='hs-varid'>cos</span><span class='hs-layout'>,</span> <span class='hs-varid'>sin</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <a name="line-143"></a> <a name="line-144"></a> <span class='hs-varid'>acos</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>acos</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-145"></a> <span class='hs-varid'>acos</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>acos</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>x'</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'>p</span><span class='hs-varop'>*</span><span class='hs-varid'>p</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-146"></a> <span class='hs-varid'>asin</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>asin</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-147"></a> <span class='hs-varid'>asin</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>asin</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span> <span class='hs-varid'>x'</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'>p</span><span class='hs-varop'>*</span><span class='hs-varid'>p</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-148"></a> <span class='hs-varid'>atan</span> <span class='hs-layout'>(</span><span class='hs-conid'>C</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>C</span> <span class='hs-layout'>(</span><span class='hs-varid'>atan</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-149"></a> <span class='hs-varid'>atan</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>D</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>D</span> <span class='hs-layout'>(</span><span class='hs-varid'>atan</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span> <span class='hs-varid'>x'</span> <span class='hs-varop'>/</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>*</span><span class='hs-varid'>p</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-150"></a> <a name="line-151"></a> <span class='hs-varid'>sinh</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>exp</span> <span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>exp</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span> <a name="line-152"></a> <span class='hs-varid'>cosh</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>exp</span> <span class='hs-varid'>x</span> <span class='hs-varop'>+</span> <span class='hs-varid'>exp</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span> <a name="line-153"></a> <span class='hs-varid'>asinh</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sqrt</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-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-154"></a> <span class='hs-varid'>acosh</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sqrt</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-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-155"></a> <span class='hs-varid'>atanh</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-varop'>+</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-comment'>-</span> <span class='hs-varid'>log</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-comment'>-</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span> <a name="line-156"></a> <a name="line-157"></a><a name="instance%20Real%20(Dif%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Real</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Real</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-158"></a> <span class='hs-varid'>toRational</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toRational</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-159"></a> <a name="line-160"></a><a name="instance%20RealFrac%20(Dif%20a)"></a><span class='hs-keyword'>instance</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-conid'>RealFrac</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-161"></a> <span class='hs-comment'>-- Second component should have an impulse derivative.</span> <a name="line-162"></a> <span class='hs-varid'>properFraction</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>fromIntegral</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>properFraction</span> <span class='hs-layout'>(</span><span class='hs-varid'>val</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <a name="line-163"></a> <span class='hs-varid'>truncate</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>truncate</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-164"></a> <span class='hs-varid'>round</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>round</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-165"></a> <span class='hs-varid'>ceiling</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ceiling</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-166"></a> <span class='hs-varid'>floor</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>floor</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-167"></a> <a name="line-168"></a><a name="instance%20RealFloat%20(Dif%20a)"></a><span class='hs-comment'>-- Partial definition on purpose, more could be defined.</span> <a name="line-169"></a><a name="instance%20RealFloat%20(Dif%20a)"></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'>RealFloat</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dif</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <a name="line-170"></a> <span class='hs-varid'>floatRadix</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>floatRadix</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-171"></a> <span class='hs-varid'>floatDigits</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>floatDigits</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-172"></a> <span class='hs-varid'>floatRange</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>floatRange</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-173"></a> <span class='hs-varid'>exponent</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span> <a name="line-174"></a> <span class='hs-varid'>scaleFloat</span> <span class='hs-num'>0</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <a name="line-175"></a> <span class='hs-varid'>isNaN</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>isNaN</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-176"></a> <span class='hs-varid'>isInfinite</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>isInfinite</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-177"></a> <span class='hs-varid'>isDenormalized</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>isDenormalized</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-178"></a> <span class='hs-varid'>isNegativeZero</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>isNegativeZero</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-179"></a> <span class='hs-varid'>isIEEE</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>isIEEE</span> <span class='hs-varop'>.</span> <span class='hs-varid'>val</span> <a name="line-180"></a> <span class='hs-comment'>-- Set these to undefined rather than omit them to avoid compiler</span> <a name="line-181"></a> <span class='hs-comment'>-- warnings.</span> <a name="line-182"></a> <span class='hs-varid'>decodeFloat</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span> <a name="line-183"></a> <span class='hs-varid'>encodeFloat</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span> </pre></body> </html>
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