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<title>Data/Number/Fixed.hs</title>
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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE
<a name="line-2"></a> EmptyDataDecls,
<a name="line-3"></a> GeneralizedNewtypeDeriving,
<a name="line-4"></a> ScopedTypeVariables,
<a name="line-5"></a> Rank2Types #-}</span>
<a name="line-6"></a>
<a name="line-7"></a><span class='hs-comment'>-- | Numbers with a fixed number of decimals.</span>
<a name="line-8"></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'>Fixed</span><span class='hs-layout'>(</span>
<a name="line-9"></a> <span class='hs-conid'>Fixed</span><span class='hs-layout'>,</span>
<a name="line-10"></a> <span class='hs-conid'>Epsilon</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eps1</span><span class='hs-layout'>,</span> <span class='hs-conid'>EpsDiv10</span><span class='hs-layout'>,</span> <span class='hs-conid'>Prec10</span><span class='hs-layout'>,</span> <span class='hs-conid'>Prec50</span><span class='hs-layout'>,</span> <span class='hs-conid'>PrecPlus20</span><span class='hs-layout'>,</span>
<a name="line-11"></a> <span class='hs-varid'>convertFixed</span><span class='hs-layout'>,</span> <span class='hs-varid'>dynamicEps</span><span class='hs-layout'>,</span> <span class='hs-varid'>precision</span><span class='hs-layout'>,</span> <span class='hs-varid'>with_added_precision</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-12"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Numeric</span>
<a name="line-13"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Char</span>
<a name="line-14"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Ratio</span>
<a name="line-15"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</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'>FixedFunctions</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>F</span>
<a name="line-16"></a>
<a name="line-17"></a><a name="Epsilon"></a><span class='hs-comment'>-- | The 'Epsilon' class contains the types that can be used to determine the</span>
<a name="line-18"></a><a name="Epsilon"></a><span class='hs-comment'>-- precision of a 'Fixed' number.</span>
<a name="line-19"></a><a name="Epsilon"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span> <span class='hs-keyword'>where</span>
<a name="line-20"></a> <span class='hs-varid'>eps</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span>
<a name="line-21"></a>
<a name="line-22"></a><a name="Eps1"></a><span class='hs-comment'>-- | An epsilon of 1, i.e., no decimals.</span>
<a name="line-23"></a><a name="Eps1"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Eps1</span>
<a name="line-24"></a><a name="instance%20Epsilon%20Eps1"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Epsilon</span> <span class='hs-conid'>Eps1</span> <span class='hs-keyword'>where</span>
<a name="line-25"></a> <span class='hs-varid'>eps</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span>
<a name="line-26"></a>
<a name="line-27"></a><a name="Epsilon"></a><span class='hs-comment'>-- | A type construct that gives one more decimals than the argument.</span>
<a name="line-28"></a><a name="Epsilon"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>EpsDiv10</span> <span class='hs-varid'>p</span>
<a name="line-29"></a><a name="instance%20Epsilon%20(EpsDiv10%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Epsilon</span> <span class='hs-layout'>(</span><span class='hs-conid'>EpsDiv10</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-30"></a> <span class='hs-varid'>eps</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>eps</span> <span class='hs-layout'>(</span><span class='hs-varid'>un</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-num'>10</span>
<a name="line-31"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>un</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>EpsDiv10</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span>
<a name="line-32"></a> <span class='hs-varid'>un</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-33"></a>
<a name="line-34"></a><a name="Prec10"></a><span class='hs-comment'>-- | Ten decimals.</span>
<a name="line-35"></a><a name="Prec10"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Prec10</span>
<a name="line-36"></a><a name="instance%20Epsilon%20Prec10"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Epsilon</span> <span class='hs-conid'>Prec10</span> <span class='hs-keyword'>where</span>
<a name="line-37"></a> <span class='hs-varid'>eps</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1e-10</span>
<a name="line-38"></a>
<a name="line-39"></a><a name="Prec50"></a><span class='hs-comment'>-- | 50 decimals.</span>
<a name="line-40"></a><a name="Prec50"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Prec50</span>
<a name="line-41"></a><a name="instance%20Epsilon%20Prec50"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Epsilon</span> <span class='hs-conid'>Prec50</span> <span class='hs-keyword'>where</span>
<a name="line-42"></a> <span class='hs-varid'>eps</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1e-50</span>
<a name="line-43"></a>
<a name="line-44"></a><a name="Prec500"></a><span class='hs-comment'>-- | 500 decimals.</span>
<a name="line-45"></a><a name="Prec500"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Prec500</span>
<a name="line-46"></a><a name="instance%20Epsilon%20Prec500"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Epsilon</span> <span class='hs-conid'>Prec500</span> <span class='hs-keyword'>where</span>
<a name="line-47"></a> <span class='hs-varid'>eps</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1e-500</span>
<a name="line-48"></a>
<a name="line-49"></a><a name="Epsilon"></a><span class='hs-comment'>-- A type that gives 20 more decimals than the argument.</span>
<a name="line-50"></a><a name="Epsilon"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>PrecPlus20</span> <span class='hs-varid'>e</span>
<a name="line-51"></a><a name="instance%20Epsilon%20(PrecPlus20%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Epsilon</span> <span class='hs-layout'>(</span><span class='hs-conid'>PrecPlus20</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-52"></a> <span class='hs-varid'>eps</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1e-20</span> <span class='hs-varop'>*</span> <span class='hs-varid'>eps</span> <span class='hs-layout'>(</span><span class='hs-varid'>un</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span>
<a name="line-53"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>un</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PrecPlus20</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span>
<a name="line-54"></a> <span class='hs-varid'>un</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-55"></a>
<a name="line-56"></a><span class='hs-comment'>-----------</span>
<a name="line-57"></a>
<a name="line-58"></a><a name="Fixed"></a><span class='hs-comment'>-- The type of fixed precision numbers. The type /e/ determines the precision.</span>
<a name="line-59"></a><a name="Fixed"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</span> <span class='hs-conid'>Rational</span> <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'>Ord</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span><span class='hs-layout'>,</span> <span class='hs-conid'>Real</span><span class='hs-layout'>,</span> <span class='hs-conid'>RealFrac</span><span class='hs-layout'>)</span>
<a name="line-60"></a>
<a name="line-61"></a><a name="precision"></a><span class='hs-comment'>-- Get the accuracy (the epsilon) of the type.</span>
<a name="line-62"></a><span class='hs-definition'>precision</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span>
<a name="line-63"></a><span class='hs-definition'>precision</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>getEps</span>
<a name="line-64"></a>
<a name="line-65"></a><a name="instance%20Num%20(Fixed%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</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'>Fixed</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-66"></a> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lift2</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span>
<a name="line-67"></a> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lift2</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-layout'>)</span>
<a name="line-68"></a> <span class='hs-layout'>(</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lift2</span> <span class='hs-layout'>(</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span>
<a name="line-69"></a> <span class='hs-varid'>negate</span> <span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</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-70"></a> <span class='hs-varid'>abs</span> <span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</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-71"></a> <span class='hs-varid'>signum</span> <span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</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-72"></a> <span class='hs-varid'>fromInteger</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fromInteger</span>
<a name="line-73"></a>
<a name="line-74"></a><a name="instance%20Fractional%20(Fixed%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</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'>Fixed</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-75"></a> <span class='hs-layout'>(</span><span class='hs-varop'>/</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lift2</span> <span class='hs-layout'>(</span><span class='hs-varop'>/</span><span class='hs-layout'>)</span>
<a name="line-76"></a> <span class='hs-varid'>fromRational</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>r</span>
<a name="line-77"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</span> <span class='hs-varop'>$</span> <span class='hs-varid'>approx</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>getEps</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-78"></a>
<a name="line-79"></a><a name="lift2"></a><span class='hs-definition'>lift2</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span>
<a name="line-80"></a><span class='hs-definition'>lift2</span> <span class='hs-varid'>op</span> <span class='hs-varid'>fx</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</span> <span class='hs-varop'>$</span> <span class='hs-varid'>approx</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>`op`</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>getEps</span> <span class='hs-varid'>fx</span><span class='hs-layout'>)</span>
<a name="line-81"></a>
<a name="line-82"></a><a name="approx"></a><span class='hs-definition'>approx</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span>
<a name="line-83"></a><span class='hs-definition'>approx</span> <span class='hs-varid'>x</span> <span class='hs-varid'>eps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>approxRational</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>eps</span><span class='hs-varop'>/</span><span class='hs-num'>2</span><span class='hs-layout'>)</span>
<a name="line-84"></a>
<a name="line-85"></a><a name="convertFixed"></a><span class='hs-comment'>-- | Convert between two arbitrary fixed precision types.</span>
<a name="line-86"></a><span class='hs-definition'>convertFixed</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>,</span> <span class='hs-conid'>Epsilon</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>f</span>
<a name="line-87"></a><span class='hs-definition'>convertFixed</span> <span class='hs-varid'>e</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span>
<a name="line-88"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</span> <span class='hs-varop'>$</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>feps</span> <span class='hs-varop'>></span> <span class='hs-varid'>eeps</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>approx</span> <span class='hs-varid'>x</span> <span class='hs-varid'>feps</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>x</span>
<a name="line-89"></a> <span class='hs-varid'>feps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>getEps</span> <span class='hs-varid'>f</span>
<a name="line-90"></a> <span class='hs-varid'>eeps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>getEps</span> <span class='hs-varid'>e</span>
<a name="line-91"></a>
<a name="line-92"></a><a name="getEps"></a><span class='hs-definition'>getEps</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span>
<a name="line-93"></a><span class='hs-definition'>getEps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>eps</span> <span class='hs-varop'>.</span> <span class='hs-varid'>un</span>
<a name="line-94"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>un</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span>
<a name="line-95"></a> <span class='hs-varid'>un</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-96"></a>
<a name="line-97"></a><a name="instance%20Show%20(Fixed%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</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'>Fixed</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-98"></a> <span class='hs-varid'>showsPrec</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>showSigned</span> <span class='hs-varid'>showFixed</span>
<a name="line-99"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>showFixed</span> <span class='hs-varid'>f</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>showString</span> <span class='hs-varop'>$</span> <span class='hs-varid'>show</span> <span class='hs-varid'>q</span> <span class='hs-varop'>++</span> <span class='hs-str'>"."</span> <span class='hs-varop'>++</span> <span class='hs-varid'>decimals</span> <span class='hs-varid'>r</span> <span class='hs-varid'>e</span>
<a name="line-100"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integer</span>
<a name="line-101"></a> <span class='hs-layout'>(</span><span class='hs-varid'>q</span><span class='hs-layout'>,</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>properFraction</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>+</span> <span class='hs-varid'>e</span><span class='hs-varop'>/</span><span class='hs-num'>2</span><span class='hs-layout'>)</span>
<a name="line-102"></a> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>getEps</span> <span class='hs-varid'>f</span>
<a name="line-103"></a> <span class='hs-varid'>decimals</span> <span class='hs-varid'>a</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>e</span> <span class='hs-varop'>>=</span> <span class='hs-num'>1</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>""</span>
<a name="line-104"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>intToDigit</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:</span> <span class='hs-varid'>decimals</span> <span class='hs-varid'>c</span> <span class='hs-layout'>(</span><span class='hs-num'>10</span> <span class='hs-varop'>*</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span>
<a name="line-105"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>c</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-num'>10</span> <span class='hs-varop'>*</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-106"></a>
<a name="line-107"></a><a name="instance%20Read%20(Fixed%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</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'>Fixed</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-108"></a> <span class='hs-varid'>readsPrec</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>readSigned</span> <span class='hs-varid'>readFixed</span>
<a name="line-109"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>readFixed</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-varid'>toFixed0</span> <span class='hs-layout'>(</span><span class='hs-varid'>approxRational</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</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'>readFloat</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>]</span>
<a name="line-110"></a>
<a name="line-111"></a><a name="instance%20Floating%20(Fixed%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</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'>Fixed</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-112"></a> <span class='hs-varid'>pi</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed0</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>pi</span>
<a name="line-113"></a> <span class='hs-varid'>sqrt</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>sqrt</span>
<a name="line-114"></a> <span class='hs-varid'>exp</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>with_added_precision</span> <span class='hs-varid'>r</span> <span class='hs-layout'>(</span><span class='hs-varid'>convertFixed</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>exp</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>where</span>
<a name="line-115"></a> <span class='hs-varid'>r</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-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0.1</span> <span class='hs-varop'>^</span> <span class='hs-layout'>(</span><span class='hs-varid'>ceiling</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>*</span> <span class='hs-num'>0.45</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-116"></a> <span class='hs-varid'>log</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>log</span>
<a name="line-117"></a> <span class='hs-varid'>sin</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>sin</span>
<a name="line-118"></a> <span class='hs-varid'>cos</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>cos</span>
<a name="line-119"></a> <span class='hs-varid'>tan</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>tan</span>
<a name="line-120"></a> <span class='hs-varid'>asin</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>asin</span>
<a name="line-121"></a> <span class='hs-varid'>acos</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>acos</span>
<a name="line-122"></a> <span class='hs-varid'>atan</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>atan</span>
<a name="line-123"></a> <span class='hs-varid'>sinh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>sinh</span>
<a name="line-124"></a> <span class='hs-varid'>cosh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>cosh</span>
<a name="line-125"></a> <span class='hs-varid'>tanh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>tanh</span>
<a name="line-126"></a> <span class='hs-varid'>asinh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>asinh</span>
<a name="line-127"></a> <span class='hs-varid'>acosh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>acosh</span>
<a name="line-128"></a> <span class='hs-varid'>atanh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toFixed1</span> <span class='hs-conid'>F</span><span class='hs-varop'>.</span><span class='hs-varid'>atanh</span>
<a name="line-129"></a>
<a name="line-130"></a><a name="toFixed0"></a><span class='hs-definition'>toFixed0</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span>
<a name="line-131"></a><span class='hs-definition'>toFixed0</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>r</span>
<a name="line-132"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</span> <span class='hs-varop'>$</span> <span class='hs-varid'>f</span> <span class='hs-varop'>$</span> <span class='hs-varid'>getEps</span> <span class='hs-varid'>r</span>
<a name="line-133"></a>
<a name="line-134"></a><a name="toFixed1"></a><span class='hs-definition'>toFixed1</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span>
<a name="line-135"></a><span class='hs-definition'>toFixed1</span> <span class='hs-varid'>f</span> <span class='hs-varid'>x</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>F</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>F</span> <span class='hs-varop'>$</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>getEps</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>r</span>
<a name="line-136"></a>
<a name="line-137"></a><a name="instance%20RealFloat%20(Fixed%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</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'>Fixed</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-138"></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-139"></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-140"></a> <span class='hs-varid'>isNaN</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-141"></a> <span class='hs-varid'>isInfinite</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-142"></a> <span class='hs-varid'>isDenormalized</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-143"></a> <span class='hs-varid'>isNegativeZero</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-144"></a> <span class='hs-varid'>isIEEE</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-145"></a> <span class='hs-comment'>-- Explicitly undefine these rather than omitting them; this</span>
<a name="line-146"></a> <span class='hs-comment'>-- prevents a compiler warning at least.</span>
<a name="line-147"></a> <span class='hs-varid'>floatRadix</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-148"></a> <span class='hs-varid'>floatDigits</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-149"></a> <span class='hs-varid'>floatRange</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-150"></a> <span class='hs-varid'>decodeFloat</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-151"></a> <span class='hs-varid'>encodeFloat</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-152"></a>
<a name="line-153"></a><span class='hs-comment'>-----------</span>
<a name="line-154"></a>
<a name="line-155"></a><a name="dynamicEps"></a><span class='hs-comment'>-- The call @dynmicEps r f v@ evaluates @f v@ to a precsion of @r@.</span>
<a name="line-156"></a><span class='hs-definition'>dynamicEps</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>e</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-157"></a><span class='hs-definition'>dynamicEps</span> <span class='hs-varid'>r</span> <span class='hs-varid'>f</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>loop</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Eps1</span><span class='hs-layout'>)</span>
<a name="line-158"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>loop</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varid'>x</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-159"></a> <span class='hs-varid'>loop</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>eps</span> <span class='hs-varid'>e</span> <span class='hs-varop'><=</span> <span class='hs-varid'>r</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromRational</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>loop</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>EpsDiv10</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-160"></a>
<a name="line-161"></a><a name="with_added_precision"></a><span class='hs-comment'>-- | The call @with_added_precision r f v@ evaluates @f v@, while</span>
<a name="line-162"></a><span class='hs-comment'>-- temporarily multiplying the precision of /v/ by /r/.</span>
<a name="line-163"></a><span class='hs-definition'>with_added_precision</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varid'>f</span><span class='hs-varop'>.</span><span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>e</span><span class='hs-varop'>.</span><span class='hs-layout'>(</span><span class='hs-conid'>Epsilon</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Fixed</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-164"></a><span class='hs-definition'>with_added_precision</span> <span class='hs-varid'>r</span> <span class='hs-varid'>f</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dynamicEps</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>*</span><span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>toRational</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-165"></a> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>precision</span> <span class='hs-varid'>v</span>
</pre></body>
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