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<pre><a name="line-1"></a><span class='hs-comment'>-- ERA: Exact Real Arithmetic (version 1.0)</span>
<a name="line-2"></a><span class='hs-comment'>--</span>
<a name="line-3"></a><span class='hs-comment'>-- A tolerably efficient and possibly correct implementation of the computable</span>
<a name="line-4"></a><span class='hs-comment'>-- reals using Haskell 1.2.</span>
<a name="line-5"></a><span class='hs-comment'>--</span>
<a name="line-6"></a><span class='hs-comment'>-- David Lester, Department of Computer Science, Manchester University, M13 9PL.</span>
<a name="line-7"></a><span class='hs-comment'>-- (2000-2001)</span>
<a name="line-8"></a>
<a name="line-9"></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'>CReal</span><span class='hs-layout'>(</span><span class='hs-conid'>CReal</span><span class='hs-layout'>,</span> <span class='hs-varid'>showCReal</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-10"></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-11"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Numeric</span><span class='hs-layout'>(</span><span class='hs-varid'>readFloat</span><span class='hs-layout'>,</span> <span class='hs-varid'>readSigned</span><span class='hs-layout'>)</span>
<a name="line-12"></a>
<a name="line-13"></a><a name="CReal"></a><span class='hs-comment'>-- |The 'CReal' type implements (constructive) real numbers.</span>
<a name="line-14"></a><a name="CReal"></a><span class='hs-comment'>--</span>
<a name="line-15"></a><a name="CReal"></a><span class='hs-comment'>-- Note that the comparison operations on 'CReal' may diverge</span>
<a name="line-16"></a><a name="CReal"></a><span class='hs-comment'>-- since it is (by necessity) impossible to implementent them</span>
<a name="line-17"></a><a name="CReal"></a><span class='hs-comment'>-- correctly and always terminating.</span>
<a name="line-18"></a><a name="CReal"></a><span class='hs-comment'>--</span>
<a name="line-19"></a><a name="CReal"></a><span class='hs-comment'>-- This implementation is really David Lester's ERA package.</span>
<a name="line-20"></a><a name="CReal"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Integer</span><span class='hs-layout'>)</span>
<a name="line-21"></a>
<a name="line-22"></a><a name="instance%20Eq%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Eq</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-23"></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'>s'</span> <span class='hs-layout'>(</span><span class='hs-varid'>digitsToBits</span> <span class='hs-varid'>digits</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-num'>0</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span>
<a name="line-24"></a>
<a name="line-25"></a><a name="instance%20Ord%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-26"></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'>s'</span> <span class='hs-layout'>(</span><span class='hs-varid'>digitsToBits</span> <span class='hs-varid'>digits</span><span class='hs-layout'>)</span> <span class='hs-varop'><=</span> <span class='hs-num'>0</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span>
<a name="line-27"></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'>s'</span> <span class='hs-layout'>(</span><span class='hs-varid'>digitsToBits</span> <span class='hs-varid'>digits</span><span class='hs-layout'>)</span> <span class='hs-varop'><</span> <span class='hs-num'>0</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span>
<a name="line-28"></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'>s'</span> <span class='hs-layout'>(</span><span class='hs-varid'>digitsToBits</span> <span class='hs-varid'>digits</span><span class='hs-layout'>)</span> <span class='hs-varop'>>=</span> <span class='hs-num'>0</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span>
<a name="line-29"></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'>s'</span> <span class='hs-layout'>(</span><span class='hs-varid'>digitsToBits</span> <span class='hs-varid'>digits</span><span class='hs-layout'>)</span> <span class='hs-varop'>></span> <span class='hs-num'>0</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span>
<a name="line-30"></a> <span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>y'</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-31"></a> <span class='hs-varid'>min</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>min</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>y'</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-32"></a>
<a name="line-33"></a><a name="instance%20Num%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Num</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-34"></a> <span class='hs-layout'>(</span><span class='hs-conid'>CR</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'>CR</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>+</span><span class='hs-num'>2</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-varid'>y'</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>+</span><span class='hs-num'>2</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>%</span> <span class='hs-num'>4</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-35"></a> <span class='hs-layout'>(</span><span class='hs-conid'>CR</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'>CR</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>+</span><span class='hs-varid'>sy</span><span class='hs-layout'>)</span><span class='hs-varop'>*</span><span class='hs-varid'>y'</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>+</span><span class='hs-varid'>sx</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>%</span> <span class='hs-num'>2</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'>sx</span><span class='hs-varop'>+</span><span class='hs-varid'>sy</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-36"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>x0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>abs</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span><span class='hs-varop'>+</span><span class='hs-num'>2</span><span class='hs-layout'>;</span> <span class='hs-varid'>y0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>abs</span> <span class='hs-layout'>(</span><span class='hs-varid'>y'</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span><span class='hs-varop'>+</span><span class='hs-num'>2</span>
<a name="line-37"></a> <span class='hs-varid'>sx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sizeinbase</span> <span class='hs-varid'>x0</span> <span class='hs-num'>2</span><span class='hs-varop'>+</span><span class='hs-num'>3</span><span class='hs-layout'>;</span> <span class='hs-varid'>sy</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sizeinbase</span> <span class='hs-varid'>y0</span> <span class='hs-num'>2</span><span class='hs-varop'>+</span><span class='hs-num'>3</span>
<a name="line-38"></a> <span class='hs-varid'>negate</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>negate</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-39"></a> <span class='hs-varid'>abs</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>max</span> <span class='hs-varid'>x</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-40"></a> <span class='hs-varid'>signum</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-layout'>(</span><span class='hs-varid'>signum</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-varid'>digitsToBits</span> <span class='hs-varid'>digits</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-41"></a> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>n</span><span class='hs-varop'>*</span><span class='hs-num'>2</span><span class='hs-varop'>^</span><span class='hs-varid'>p</span><span class='hs-layout'>)</span>
<a name="line-42"></a>
<a name="line-43"></a><a name="instance%20Fractional%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Fractional</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-44"></a> <span class='hs-varid'>recip</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyword'>let</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>head</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'><-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-num'>3</span> <span class='hs-varop'><=</span> <span class='hs-varid'>abs</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-45"></a> <span class='hs-keyword'>in</span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-num'>2</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'>p</span><span class='hs-varop'>+</span><span class='hs-num'>2</span><span class='hs-varop'>*</span><span class='hs-varid'>s</span><span class='hs-varop'>+</span><span class='hs-num'>2</span><span class='hs-layout'>)</span> <span class='hs-varop'>%</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>+</span><span class='hs-num'>2</span><span class='hs-varop'>*</span><span class='hs-varid'>s</span><span class='hs-varop'>+</span><span class='hs-num'>2</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-46"></a> <span class='hs-varid'>fromRational</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-layout'>(</span><span class='hs-varid'>numerator</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-varid'>fromInteger</span> <span class='hs-layout'>(</span><span class='hs-varid'>denominator</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-47"></a>
<a name="line-48"></a><span class='hs-comment'>-- two useful scaling functions:</span>
<a name="line-49"></a>
<a name="line-50"></a><a name="div2n"></a><span class='hs-definition'>div2n</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-51"></a><span class='hs-definition'>div2n</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyword'>if</span> <span class='hs-varid'>p</span> <span class='hs-varop'>>=</span> <span class='hs-varid'>n</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-comment'>-</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-varid'>p</span> <span class='hs-varop'>%</span> <span class='hs-num'>2</span><span class='hs-varop'>^</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-52"></a>
<a name="line-53"></a><a name="mul2n"></a><span class='hs-definition'>mul2n</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-54"></a><span class='hs-definition'>mul2n</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span><span class='hs-varop'>+</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-55"></a>
<a name="line-56"></a><span class='hs-comment'>-- transcendental functions (mostly range reductions):</span>
<a name="line-57"></a>
<a name="line-58"></a><a name="instance%20Floating%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Floating</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-59"></a> <span class='hs-varid'>pi</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>16</span> <span class='hs-varop'>*</span> <span class='hs-varid'>atan</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromRational</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-varop'>%</span> <span class='hs-num'>5</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-60"></a> <span class='hs-comment'>-</span> <span class='hs-num'>4</span> <span class='hs-varop'>*</span> <span class='hs-varid'>atan</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromRational</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-varop'>%</span> <span class='hs-num'>239</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-61"></a> <span class='hs-varid'>sqrt</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>floorsqrt</span> <span class='hs-layout'>(</span><span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-num'>2</span><span class='hs-varop'>*</span><span class='hs-varid'>p</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</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-62"></a>
<a name="line-63"></a> <span class='hs-varid'>log</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'><</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>error</span> <span class='hs-str'>"log of negative number\n"</span> <span class='hs-keyword'>else</span>
<a name="line-64"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'><</span> <span class='hs-num'>4</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>log</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-keyword'>else</span>
<a name="line-65"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'><</span> <span class='hs-num'>8</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>log_dr</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>else</span>
<a name="line-66"></a> <span class='hs-comment'>{- 7 < t -}</span> <span class='hs-varid'>log_dr</span> <span class='hs-layout'>(</span><span class='hs-varid'>div2n</span> <span class='hs-varid'>x</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromIntegral</span> <span class='hs-varid'>n</span> <span class='hs-varop'>*</span> <span class='hs-varid'>log2</span>
<a name="line-67"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</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-layout'>;</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span> <span class='hs-num'>2</span><span class='hs-layout'>;</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sizeinbase</span> <span class='hs-varid'>t</span> <span class='hs-num'>2</span> <span class='hs-comment'>-</span> <span class='hs-num'>3</span>
<a name="line-68"></a> <span class='hs-varid'>exp</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'><</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>div2n</span> <span class='hs-layout'>(</span><span class='hs-varid'>exp_dr</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromInteger</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-69"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>></span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>mul2n</span> <span class='hs-layout'>(</span><span class='hs-varid'>exp_dr</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromInteger</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>exp_dr</span> <span class='hs-varid'>s</span>
<a name="line-70"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>u'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>log2</span><span class='hs-layout'>;</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>u'</span> <span class='hs-num'>0</span><span class='hs-layout'>;</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>fromInteger</span> <span class='hs-varid'>n</span><span class='hs-varop'>*</span><span class='hs-varid'>log2</span>
<a name="line-71"></a> <span class='hs-varid'>sin</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-72"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>1</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-73"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>2</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-74"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>3</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-75"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>4</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-76"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>5</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-77"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>6</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-78"></a> <span class='hs-comment'>{- n == 7 -}</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-79"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>z'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>piBy4</span><span class='hs-layout'>;</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-varid'>z'</span> <span class='hs-num'>2</span> <span class='hs-varop'>%</span> <span class='hs-num'>4</span><span class='hs-layout'>)</span><span class='hs-layout'>;</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>s</span> <span class='hs-varop'>`mod`</span> <span class='hs-num'>8</span>
<a name="line-80"></a> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>piBy4</span> <span class='hs-varop'>*</span> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>s</span>
<a name="line-81"></a> <span class='hs-varid'>cos</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-82"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>1</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-83"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>2</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-84"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>3</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-85"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>4</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-86"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>5</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-comment'>-</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-87"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>==</span> <span class='hs-num'>6</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span>
<a name="line-88"></a> <span class='hs-comment'>{- n == 7 -}</span> <span class='hs-varid'>sqrt1By2</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>cos_dr</span> <span class='hs-varid'>y</span> <span class='hs-varop'>+</span> <span class='hs-varid'>sin_dr</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-89"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>z'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>piBy4</span><span class='hs-layout'>;</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-varid'>z'</span> <span class='hs-num'>2</span> <span class='hs-varop'>%</span> <span class='hs-num'>4</span><span class='hs-layout'>)</span><span class='hs-layout'>;</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>s</span> <span class='hs-varop'>`mod`</span> <span class='hs-num'>8</span>
<a name="line-90"></a> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>piBy4</span> <span class='hs-varop'>*</span> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>s</span>
<a name="line-91"></a> <span class='hs-varid'>atan</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'><</span> <span class='hs-comment'>-</span><span class='hs-num'>5</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>atan_dr</span> <span class='hs-layout'>(</span><span class='hs-varid'>negate</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'>piBy2</span> <span class='hs-keyword'>else</span>
<a name="line-92"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'>==</span> <span class='hs-comment'>-</span><span class='hs-num'>4</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-</span><span class='hs-varid'>piBy4</span> <span class='hs-comment'>-</span> <span class='hs-varid'>atan_dr</span> <span class='hs-layout'>(</span><span class='hs-varid'>xp1</span><span class='hs-varop'>/</span><span class='hs-varid'>xm1</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-93"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'><</span> <span class='hs-num'>4</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>atan_dr</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>else</span>
<a name="line-94"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'>==</span> <span class='hs-num'>4</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>piBy4</span> <span class='hs-varop'>+</span> <span class='hs-varid'>atan_dr</span> <span class='hs-layout'>(</span><span class='hs-varid'>xm1</span><span class='hs-varop'>/</span><span class='hs-varid'>xp1</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-95"></a> <span class='hs-comment'>{- t > 4 -}</span> <span class='hs-varid'>piBy2</span> <span class='hs-comment'>-</span> <span class='hs-varid'>atan_dr</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-96"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</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-layout'>;</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span> <span class='hs-num'>2</span>
<a name="line-97"></a> <span class='hs-varid'>xp1</span> <span class='hs-keyglyph'>=</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-varid'>xm1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-num'>1</span>
<a name="line-98"></a> <span class='hs-varid'>asin</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>x0</span> <span class='hs-varop'>></span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>pi</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span> <span class='hs-comment'>-</span> <span class='hs-varid'>atan</span> <span class='hs-layout'>(</span><span class='hs-varid'>s</span><span class='hs-varop'>/</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-99"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>x0</span> <span class='hs-varop'>==</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>atan</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span>
<a name="line-100"></a> <span class='hs-comment'>{- x0 < 0 -}</span> <span class='hs-comment'>-</span> <span class='hs-varid'>atan</span> <span class='hs-layout'>(</span><span class='hs-varid'>s</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'>pi</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span>
<a name="line-101"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</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-layout'>;</span> <span class='hs-varid'>x0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span> <span class='hs-num'>0</span><span class='hs-layout'>;</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-comment'>-</span> <span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-102"></a> <span class='hs-varid'>acos</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>pi</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span> <span class='hs-comment'>-</span> <span class='hs-varid'>asin</span> <span class='hs-varid'>x</span>
<a name="line-103"></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'>y</span> <span class='hs-comment'>-</span> <span class='hs-varid'>recip</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>exp</span> <span class='hs-varid'>x</span>
<a name="line-104"></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'>y</span> <span class='hs-varop'>+</span> <span class='hs-varid'>recip</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-num'>2</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>exp</span> <span class='hs-varid'>x</span>
<a name="line-105"></a> <span class='hs-varid'>tanh</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span> <span class='hs-comment'>-</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span> <span class='hs-varop'>+</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>exp</span> <span class='hs-varid'>x</span><span class='hs-layout'>;</span> <span class='hs-varid'>y'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>recip</span> <span class='hs-varid'>y</span>
<a name="line-106"></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-107"></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-108"></a> <span class='hs-varid'>atanh</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-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-varop'>/</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-109"></a>
<a name="line-110"></a>
<a name="line-111"></a><a name="acc_seq"></a><span class='hs-definition'>acc_seq</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'>Integer</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-keyglyph'>[</span><span class='hs-conid'>Rational</span><span class='hs-keyglyph'>]</span>
<a name="line-112"></a><span class='hs-definition'>acc_seq</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>scanl</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-varop'>%</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span>
<a name="line-113"></a>
<a name="line-114"></a><a name="exp_dr"></a><span class='hs-definition'>exp_dr</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-115"></a><span class='hs-definition'>exp_dr</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>power_series</span> <span class='hs-layout'>(</span><span class='hs-varid'>acc_seq</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>a</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span><span class='hs-varop'>*</span><span class='hs-layout'>(</span><span class='hs-num'>1</span> <span class='hs-varop'>%</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>id</span>
<a name="line-116"></a>
<a name="line-117"></a><a name="log_dr"></a><span class='hs-definition'>log_dr</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-118"></a><span class='hs-definition'>log_dr</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y</span> <span class='hs-varop'>*</span> <span class='hs-varid'>log_drx</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-varid'>x</span>
<a name="line-119"></a>
<a name="line-120"></a><a name="log_drx"></a><span class='hs-definition'>log_drx</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-121"></a><span class='hs-definition'>log_drx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>power_series</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span> <span class='hs-varop'>%</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'><-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span><span class='hs-keyglyph'>]</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span>
<a name="line-122"></a>
<a name="line-123"></a><a name="sin_dr"></a><span class='hs-definition'>sin_dr</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-124"></a><span class='hs-definition'>sin_dr</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>power_series</span> <span class='hs-layout'>(</span><span class='hs-varid'>acc_seq</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>a</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-comment'>-</span><span class='hs-varid'>a</span><span class='hs-varop'>*</span><span class='hs-layout'>(</span><span class='hs-num'>1</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'>n</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'>n</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>id</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-125"></a>
<a name="line-126"></a><a name="cos_dr"></a><span class='hs-definition'>cos_dr</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-127"></a><span class='hs-definition'>cos_dr</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>power_series</span> <span class='hs-layout'>(</span><span class='hs-varid'>acc_seq</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>a</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-comment'>-</span><span class='hs-varid'>a</span><span class='hs-varop'>*</span><span class='hs-layout'>(</span><span class='hs-num'>1</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'>n</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'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>id</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-128"></a>
<a name="line-129"></a><a name="atan_dr"></a><span class='hs-definition'>atan_dr</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-130"></a><span class='hs-definition'>atan_dr</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varop'>*</span> <span class='hs-varid'>atan_drx</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'>x</span><span class='hs-layout'>)</span><span class='hs-varop'>/</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>x</span><span class='hs-varop'>+</span><span class='hs-num'>1</span>
<a name="line-131"></a>
<a name="line-132"></a><a name="atan_drx"></a><span class='hs-definition'>atan_drx</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-133"></a><span class='hs-definition'>atan_drx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>power_series</span> <span class='hs-layout'>(</span><span class='hs-varid'>acc_seq</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>a</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span><span class='hs-varop'>*</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-num'>2</span><span class='hs-varop'>*</span><span class='hs-varid'>n</span><span class='hs-layout'>)</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'>n</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span>
<a name="line-134"></a>
<a name="line-135"></a><span class='hs-comment'>-- power_series takes as arguments:</span>
<a name="line-136"></a><span class='hs-comment'>-- a (rational) list of the coefficients of the power series</span>
<a name="line-137"></a><span class='hs-comment'>-- a function from the desired accuracy to the number of terms needed</span>
<a name="line-138"></a><span class='hs-comment'>-- the argument x</span>
<a name="line-139"></a>
<a name="line-140"></a><a name="power_series"></a><span class='hs-definition'>power_series</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Rational</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span>
<a name="line-141"></a><span class='hs-definition'>power_series</span> <span class='hs-varid'>ps</span> <span class='hs-varid'>terms</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span>
<a name="line-142"></a> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>CR</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyword'>let</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>terms</span> <span class='hs-varid'>p</span><span class='hs-layout'>;</span> <span class='hs-varid'>l2t</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>2</span><span class='hs-varop'>*</span><span class='hs-varid'>sizeinbase</span> <span class='hs-layout'>(</span><span class='hs-varid'>toInteger</span> <span class='hs-varid'>t</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-num'>2</span><span class='hs-varop'>+</span><span class='hs-num'>6</span><span class='hs-layout'>;</span> <span class='hs-varid'>p'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p</span> <span class='hs-varop'>+</span> <span class='hs-varid'>l2t</span>
<a name="line-143"></a> <span class='hs-varid'>xr</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span> <span class='hs-varid'>p'</span><span class='hs-layout'>;</span> <span class='hs-varid'>xn</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>2</span><span class='hs-varop'>^</span><span class='hs-varid'>p'</span><span class='hs-layout'>;</span> <span class='hs-varid'>g</span> <span class='hs-varid'>yn</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>yn</span><span class='hs-varop'>*</span><span class='hs-varid'>xr</span><span class='hs-layout'>)</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'>p'</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-144"></a> <span class='hs-keyword'>in</span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-varid'>accumulate</span> <span class='hs-layout'>(</span><span class='hs-varid'>iterate</span> <span class='hs-varid'>g</span> <span class='hs-varid'>xn</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>take</span> <span class='hs-varid'>t</span> <span class='hs-varid'>ps</span><span class='hs-layout'>)</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'>l2t</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-145"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>accumulate</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span>
<a name="line-146"></a> <span class='hs-varid'>accumulate</span> <span class='hs-conid'>[]</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"CReal.power_series.accumulate"</span>
<a name="line-147"></a> <span class='hs-varid'>accumulate</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-conop'>:</span><span class='hs-varid'>cs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-varop'>*</span><span class='hs-layout'>(</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> <span class='hs-keyword'>in</span>
<a name="line-148"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>t</span> <span class='hs-varop'>==</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-num'>0</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>t</span> <span class='hs-varop'>+</span> <span class='hs-varid'>accumulate</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>cs</span>
<a name="line-149"></a>
<a name="line-150"></a><span class='hs-comment'>-- Some useful constants:</span>
<a name="line-151"></a>
<a name="line-152"></a><a name="piBy2"></a><span class='hs-definition'>piBy2</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span>
<a name="line-153"></a><span class='hs-definition'>piBy2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>div2n</span> <span class='hs-varid'>pi</span> <span class='hs-num'>1</span>
<a name="line-154"></a>
<a name="line-155"></a><a name="piBy4"></a><span class='hs-definition'>piBy4</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span>
<a name="line-156"></a><span class='hs-definition'>piBy4</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>div2n</span> <span class='hs-varid'>pi</span> <span class='hs-num'>2</span>
<a name="line-157"></a>
<a name="line-158"></a><a name="log2"></a><span class='hs-definition'>log2</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span>
<a name="line-159"></a><span class='hs-definition'>log2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>div2n</span> <span class='hs-layout'>(</span><span class='hs-varid'>log_drx</span> <span class='hs-layout'>(</span><span class='hs-varid'>recip</span> <span class='hs-num'>2</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-num'>1</span>
<a name="line-160"></a>
<a name="line-161"></a><a name="sqrt1By2"></a><span class='hs-definition'>sqrt1By2</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>CReal</span>
<a name="line-162"></a><span class='hs-definition'>sqrt1By2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sqrt</span> <span class='hs-layout'>(</span><span class='hs-varid'>recip</span> <span class='hs-num'>2</span><span class='hs-layout'>)</span>
<a name="line-163"></a>
<a name="line-164"></a><a name="instance%20Enum%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Enum</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-165"></a> <span class='hs-varid'>toEnum</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromIntegral</span> <span class='hs-varid'>i</span>
<a name="line-166"></a> <span class='hs-varid'>fromEnum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"Cannot fromEnum CReal"</span>
<a name="line-167"></a> <span class='hs-varid'>enumFrom</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>iterate</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span>
<a name="line-168"></a> <span class='hs-varid'>enumFromTo</span> <span class='hs-varid'>n</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>takeWhile</span> <span class='hs-layout'>(</span><span class='hs-varop'><=</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>iterate</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-varid'>n</span>
<a name="line-169"></a> <span class='hs-varid'>enumFromThen</span> <span class='hs-varid'>n</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>iterate</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>(</span><span class='hs-varid'>m</span><span class='hs-comment'>-</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>n</span>
<a name="line-170"></a> <span class='hs-varid'>enumFromThenTo</span> <span class='hs-varid'>n</span> <span class='hs-varid'>m</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>m</span> <span class='hs-varop'>>=</span> <span class='hs-varid'>n</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>takeWhile</span> <span class='hs-layout'>(</span><span class='hs-varop'><=</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>iterate</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>(</span><span class='hs-varid'>m</span><span class='hs-comment'>-</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>n</span>
<a name="line-171"></a> <span class='hs-keyword'>else</span> <span class='hs-varid'>takeWhile</span> <span class='hs-layout'>(</span><span class='hs-varop'>>=</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>iterate</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>(</span><span class='hs-varid'>m</span><span class='hs-comment'>-</span><span class='hs-varid'>n</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>n</span>
<a name="line-172"></a>
<a name="line-173"></a><a name="instance%20Real%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Real</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-174"></a> <span class='hs-comment'>-- toRational x@(CR x') = x' n % 2^n where n = digitsToBits digits</span>
<a name="line-175"></a> <span class='hs-varid'>toRational</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"CReal.toRational"</span>
<a name="line-176"></a><a name="instance%20RealFrac%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>RealFrac</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-177"></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-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromInteger</span> <span class='hs-varid'>n</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-comment'>-</span> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span> <span class='hs-num'>0</span>
<a name="line-178"></a>
<a name="line-179"></a><a name="instance%20RealFloat%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>RealFloat</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-180"></a> <span class='hs-varid'>floatRadix</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"CReal.floatRadix"</span>
<a name="line-181"></a> <span class='hs-varid'>floatDigits</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"CReal.floatDigits"</span>
<a name="line-182"></a> <span class='hs-varid'>floatRange</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"CReal.floatRange"</span>
<a name="line-183"></a> <span class='hs-varid'>decodeFloat</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"CReal.decodeFloat"</span>
<a name="line-184"></a> <span class='hs-varid'>encodeFloat</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'>"CReal.encodeFloat"</span>
<a name="line-185"></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-186"></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-187"></a> <span class='hs-varid'>significand</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span>
<a name="line-188"></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-189"></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-190"></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-191"></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-192"></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-193"></a>
<a name="line-194"></a><span class='hs-comment'>-- printing and reading the reals:</span>
<a name="line-195"></a>
<a name="line-196"></a><a name="showCReal"></a><span class='hs-comment'>-- |The 'showCReal' function connverts a 'CReal' to a 'String'.</span>
<a name="line-197"></a><span class='hs-definition'>showCReal</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-comment'>-- ^ The number of decimals</span>
<a name="line-198"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>CReal</span> <span class='hs-comment'>-- ^ The real number</span>
<a name="line-199"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>String</span> <span class='hs-comment'>-- ^ The resulting string</span>
<a name="line-200"></a><span class='hs-definition'>showCReal</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-conid'>CR</span> <span class='hs-varid'>x'</span><span class='hs-layout'>)</span>
<a name="line-201"></a> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-keyword'>if</span> <span class='hs-varid'>s</span> <span class='hs-keyword'>then</span> <span class='hs-str'>"-"</span> <span class='hs-keyword'>else</span> <span class='hs-str'>""</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span> <span class='hs-varid'>zs</span> <span class='hs-varop'>++</span> <span class='hs-layout'>(</span><span class='hs-keyword'>if</span> <span class='hs-varid'>d</span> <span class='hs-varop'>/=</span> <span class='hs-num'>0</span> <span class='hs-keyword'>then</span> <span class='hs-chr'>'.'</span><span class='hs-conop'>:</span><span class='hs-varid'>fs'</span> <span class='hs-keyword'>else</span> <span class='hs-str'>""</span><span class='hs-layout'>)</span>
<a name="line-202"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>digitsToBits</span> <span class='hs-varid'>d</span>
<a name="line-203"></a> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x'</span> <span class='hs-varid'>b</span>
<a name="line-204"></a> <span class='hs-varid'>ds</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-varid'>round_uk</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-varop'>*</span><span class='hs-num'>10</span><span class='hs-varop'>^</span><span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-varop'>%</span> <span class='hs-num'>2</span><span class='hs-varop'>^</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-205"></a> <span class='hs-layout'>(</span><span class='hs-varid'>s</span><span class='hs-layout'>,</span><span class='hs-varid'>ds'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>sgn</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>head</span> <span class='hs-varid'>ds</span> <span class='hs-varop'>==</span> <span class='hs-chr'>'-'</span> <span class='hs-keyword'>in</span> <span class='hs-layout'>(</span><span class='hs-varid'>sgn</span><span class='hs-layout'>,</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>sgn</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>tail</span> <span class='hs-varid'>ds</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>ds</span><span class='hs-layout'>)</span>
<a name="line-206"></a> <span class='hs-varid'>ds''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>take</span> <span class='hs-layout'>(</span><span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-varid'>d</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-comment'>-</span><span class='hs-varid'>length</span> <span class='hs-varid'>ds'</span><span class='hs-layout'>)</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>repeat</span> <span class='hs-chr'>'0'</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span> <span class='hs-varid'>ds'</span>
<a name="line-207"></a> <span class='hs-layout'>(</span><span class='hs-varid'>zs</span><span class='hs-layout'>,</span><span class='hs-varid'>fs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>splitAt</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>ds''</span> <span class='hs-comment'>-</span><span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-varid'>ds''</span>
<a name="line-208"></a> <span class='hs-varid'>fs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>reverse</span> <span class='hs-varop'>$</span> <span class='hs-varid'>dropWhile</span> <span class='hs-layout'>(</span><span class='hs-varop'>==</span> <span class='hs-chr'>'0'</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>reverse</span> <span class='hs-varid'>fs</span> <span class='hs-keyword'>of</span>
<a name="line-209"></a> <span class='hs-str'>""</span> <span class='hs-keyglyph'>-></span> <span class='hs-str'>"0"</span>
<a name="line-210"></a> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>xs</span>
<a name="line-211"></a>
<a name="line-212"></a><a name="digitsToBits"></a><span class='hs-definition'>digitsToBits</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span>
<a name="line-213"></a><span class='hs-definition'>digitsToBits</span> <span class='hs-varid'>d</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ceiling</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromIntegral</span> <span class='hs-varid'>d</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>logBase</span> <span class='hs-num'>2.0</span> <span class='hs-num'>10.0</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Double</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-num'>4</span>
<a name="line-214"></a>
<a name="line-215"></a><a name="digits"></a><span class='hs-definition'>digits</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span>
<a name="line-216"></a><span class='hs-definition'>digits</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>40</span>
<a name="line-217"></a>
<a name="line-218"></a><a name="instance%20Read%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Read</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-219"></a> <span class='hs-varid'>readsPrec</span> <span class='hs-sel'>_p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>readSigned</span> <span class='hs-varid'>readFloat</span>
<a name="line-220"></a>
<a name="line-221"></a><a name="instance%20Show%20CReal"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>CReal</span> <span class='hs-keyword'>where</span>
<a name="line-222"></a> <span class='hs-varid'>showsPrec</span> <span class='hs-varid'>p</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>showCReal</span> <span class='hs-varid'>digits</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>in</span>
<a name="line-223"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>head</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>==</span> <span class='hs-chr'>'-'</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>showParen</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varop'>></span> <span class='hs-num'>6</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>showString</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-224"></a> <span class='hs-keyword'>else</span> <span class='hs-varid'>showString</span> <span class='hs-varid'>xs</span>
<a name="line-225"></a>
<a name="line-226"></a><span class='hs-comment'>-- GMP functions not provided by Haskell</span>
<a name="line-227"></a>
<a name="line-228"></a><a name="sizeinbase"></a><span class='hs-definition'>sizeinbase</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span>
<a name="line-229"></a><span class='hs-definition'>sizeinbase</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>abs</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span>
<a name="line-230"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>f</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'><=</span> <span class='hs-num'>1</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>1</span> <span class='hs-varop'>+</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span> <span class='hs-varop'>`div`</span> <span class='hs-varid'>toInteger</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-231"></a>
<a name="line-232"></a><a name="floorsqrt"></a><span class='hs-definition'>floorsqrt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Integer</span>
<a name="line-233"></a><span class='hs-definition'>floorsqrt</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>until</span> <span class='hs-varid'>satisfy</span> <span class='hs-varid'>improve</span> <span class='hs-varid'>x</span>
<a name="line-234"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>improve</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>floor</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-varop'>*</span><span class='hs-varid'>y</span><span class='hs-varop'>+</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-num'>2</span><span class='hs-varop'>*</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-235"></a> <span class='hs-varid'>satisfy</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y</span><span class='hs-varop'>*</span><span class='hs-varid'>y</span> <span class='hs-varop'><=</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-layout'>(</span><span class='hs-varid'>y</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-varop'>*</span><span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span>
<a name="line-236"></a>
<a name="line-237"></a><a name="round_uk"></a><span class='hs-definition'>round_uk</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Integer</span>
<a name="line-238"></a><span class='hs-definition'>round_uk</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>floor</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>+</span><span class='hs-num'>1</span> <span class='hs-varop'>%</span> <span class='hs-num'>2</span><span class='hs-layout'>)</span>
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