Data update
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Ingy döt Net
parent
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commit
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7735 changed files with 38060 additions and 199180 deletions
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@ -1,116 +1,114 @@
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(phixonline)-->
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<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">12</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">function</span> <span style="color: #000000;">spouge_gamma</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
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<span style="color: #004080;">atom</span> <span style="color: #000000;">accm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">accm</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
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<span style="color: #000000;">accm</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">)</span>
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<span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">accm</span>
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<span style="color: #004080;">atom</span> <span style="color: #000000;">k1_factrl</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #000080;font-style:italic;">-- (k - 1)!*(-1)^k with 0!==1</span>
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<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">12</span> <span style="color: #008080;">do</span>
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<span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">exp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">13</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">13</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1.5</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">k1_factrl</span>
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<span style="color: #000000;">k1_factrl</span> <span style="color: #0000FF;">*=</span> <span style="color: #0000FF;">-(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">12</span> <span style="color: #008080;">do</span>
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<span style="color: #000000;">accm</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]/(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">+</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
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<span style="color: #000000;">accm</span> <span style="color: #0000FF;">*=</span> <span style="color: #7060A8;">exp</span><span style="color: #0000FF;">(-(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">+</span><span style="color: #000000;">12</span><span style="color: #0000FF;">))*</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">+</span><span style="color: #000000;">12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- Gamma(z+1)</span>
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<span style="color: #008080;">return</span> <span style="color: #000000;">accm</span><span style="color: #0000FF;">/</span><span style="color: #000000;">z</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
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with javascript_semantics
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sequence c = repeat(0,12)
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function spouge_gamma(atom z)
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atom accm = c[1]
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if accm=0 then
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accm = sqrt(2*PI)
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c[1] = accm
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atom k1_factrl = 1 -- (k - 1)!*(-1)^k with 0!==1
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for k=2 to 12 do
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c[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl
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k1_factrl *= -(k-1)
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end for
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end if
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for k=2 to 12 do
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accm += c[k]/(z+k-1)
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end for
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accm *= exp(-(z+12))*power(z+12,z+0.5) -- Gamma(z+1)
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return accm/z
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end function
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<span style="color: #008080;">function</span> <span style="color: #000000;">taylor_gamma</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
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<span style="color: #000080;font-style:italic;">-- (good for values between 0 and 1, apparently)</span>
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<span style="color: #008080;">constant</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">1.00000_00000_00000_00000</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.57721_56649_01532_86061</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.65587_80715_20253_88108</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.04200_26350_34095_23553</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.16653_86113_82291_48950</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.04219_77345_55544_33675</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00962_19715_27876_97356</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00721_89432_46663_09954</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00116_51675_91859_06511</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00021_52416_74114_95097</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00012_80502_82388_11619</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00002_01348_54780_78824</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_12504_93482_14267</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_11330_27231_98170</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_02056_33841_69776</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00061_16095_10448</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00050_02007_64447</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_00011_81274_57049</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00001_04342_67117</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00000_07782_26344</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_00000_03696_80562</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00000_00510_03703</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_00000_00020_58326</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_00000_00005_34812</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00000_00001_22678</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_00000_00000_11813</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00000_00000_00119</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00000_00000_00141</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.00000_00000_00000_00023</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">0.00000_00000_00000_00002</span> <span style="color: #0000FF;">}</span>
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<span style="color: #004080;">atom</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">[$]</span>
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<span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
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<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">[</span><span style="color: #000000;">n</span><span style="color: #0000FF;">]</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
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<span style="color: #008080;">return</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">s</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
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function taylor_gamma(atom x)
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-- (good for values between 0 and 1, apparently)
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constant t = { 1.00000_00000_00000_00000,
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0.57721_56649_01532_86061,
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-0.65587_80715_20253_88108,
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-0.04200_26350_34095_23553,
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0.16653_86113_82291_48950,
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-0.04219_77345_55544_33675,
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-0.00962_19715_27876_97356,
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0.00721_89432_46663_09954,
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-0.00116_51675_91859_06511,
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-0.00021_52416_74114_95097,
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0.00012_80502_82388_11619,
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-0.00002_01348_54780_78824,
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-0.00000_12504_93482_14267,
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0.00000_11330_27231_98170,
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-0.00000_02056_33841_69776,
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0.00000_00061_16095_10448,
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0.00000_00050_02007_64447,
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-0.00000_00011_81274_57049,
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0.00000_00001_04342_67117,
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0.00000_00000_07782_26344,
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-0.00000_00000_03696_80562,
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0.00000_00000_00510_03703,
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-0.00000_00000_00020_58326,
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-0.00000_00000_00005_34812,
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0.00000_00000_00001_22678,
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-0.00000_00000_00000_11813,
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0.00000_00000_00000_00119,
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0.00000_00000_00000_00141,
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-0.00000_00000_00000_00023,
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0.00000_00000_00000_00002 }
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atom y = x-1,
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s = t[$]
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for n=length(t)-1 to 1 by -1 do
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s = s*y + t[n]
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end for
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return 1/s
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end function
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<span style="color: #008080;">function</span> <span style="color: #000000;">lanczos_gamma</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">z</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0.5</span> <span style="color: #008080;">then</span>
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<span style="color: #008080;">return</span> <span style="color: #004600;">PI</span> <span style="color: #0000FF;">/</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">sin</span><span style="color: #0000FF;">(</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">lanczos_gamma</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">z</span><span style="color: #0000FF;">))</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #000080;font-style:italic;">-- use a lanczos approximation:</span>
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<span style="color: #004080;">atom</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0.99999999999980993</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">z</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">6.5</span><span style="color: #0000FF;">;</span>
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">676.5203681218851</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">1259.1392167224028</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">771.32342877765313</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">176.61502916214059</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">12.507343278686905</span><span style="color: #0000FF;">,</span>
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<span style="color: #0000FF;">-</span><span style="color: #000000;">0.13857109526572012</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">9.9843695780195716e-6</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">1.5056327351493116e-7</span> <span style="color: #0000FF;">}</span>
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<span style="color: #000000;">z</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
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<span style="color: #000000;">x</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">/</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">z</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
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<span style="color: #008080;">return</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #7060A8;">exp</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">x</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
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function lanczos_gamma(atom z)
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if z<0.5 then
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return PI / (sin(PI*z)*lanczos_gamma(1-z))
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end if
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-- use a lanczos approximation:
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atom x = 0.99999999999980993,
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t = z + 6.5;
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sequence p = { 676.5203681218851,
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-1259.1392167224028,
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771.32342877765313,
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-176.61502916214059,
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12.507343278686905,
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-0.13857109526572012,
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9.9843695780195716e-6,
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1.5056327351493116e-7 }
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z -= 1
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for i=1 to length(p) do
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x += p[i] / (z + i)
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end for
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return sqrt(2*PI) * power(t,z+0.5) * exp(-t) * x
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end function
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<span style="color: #008080;">constant</span> <span style="color: #000000;">sqPI</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">)</span>
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constant sqPI = sqrt(PI)
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<span style="color: #008080;">procedure</span> <span style="color: #000000;">sq</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">zm</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">string</span> <span style="color: #000000;">fmt</span><span style="color: #0000FF;">=</span><span style="color: #008000;">"%19.16f"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">mul</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">zm</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">mul</span>
|
||||
<span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">spouge_gamma</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">),</span>
|
||||
<span style="color: #000000;">taylor_gamma</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">),</span>
|
||||
<span style="color: #000000;">lanczos_gamma</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)},</span>
|
||||
<span style="color: #000000;">error</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_abs</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #008000;">", "</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">:=</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">)&</span><span style="color: #008000;">", "</span>
|
||||
<span style="color: #004080;">integer</span> <span style="color: #000000;">bdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">smallest</span><span style="color: #0000FF;">(</span><span style="color: #000000;">error</span><span style="color: #0000FF;">,</span><span style="color: #000000;">return_index</span><span style="color: #0000FF;">:=</span><span style="color: #004600;">true</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">best</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">bdx</span><span style="color: #0000FF;">],</span>
|
||||
<span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">bdx</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">mul</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">-- (potentially mark >1)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">best</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000000;">t</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">*</span><span style="color: #000000;">22</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..</span><span style="color: #000000;">i</span><span style="color: #0000FF;">*</span><span style="color: #000000;">22</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"*,"</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">es</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%5g: %s %s, %19.16f\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">es</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
|
||||
procedure sq(sequence zm, string fmt="%19.16f")
|
||||
atom {z, mul} = zm
|
||||
atom e = sqPI/mul
|
||||
sequence s = {spouge_gamma(z),
|
||||
taylor_gamma(z),
|
||||
lanczos_gamma(z)},
|
||||
error = sq_abs(sq_sub(s,e))
|
||||
string t = join(s,", ",fmt:=fmt)&", "
|
||||
integer bdx = smallest(error,return_index:=true)
|
||||
atom best = s[bdx],
|
||||
p = s[bdx]*mul
|
||||
for i=1 to length(s) do -- (potentially mark >1)
|
||||
if s[i]=best then
|
||||
t[i*22-2..i*22-1] = "*,"
|
||||
end if
|
||||
end for
|
||||
string es = sprintf(fmt,e)
|
||||
printf(1,"%5g: %s %s, %19.16f\n",{z,t,es,p*p})
|
||||
end procedure
|
||||
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" z ------ spouge ----- ----- taylor ------ ----- lanczos ----- ---- expected ----- %19.16f\n"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">papply</span><span style="color: #0000FF;">({{-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">/</span><span style="color: #000000;">4</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">5</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">7</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">8</span><span style="color: #0000FF;">/</span><span style="color: #000000;">15</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">6</span><span style="color: #0000FF;">}},</span><span style="color: #000000;">sq</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">sq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">0.001</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">999.4237725</span><span style="color: #0000FF;">},</span><span style="color: #008000;">"%19.15f"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">sq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">0.01</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">99.43258512</span><span style="color: #0000FF;">},</span><span style="color: #008000;">"%19.16f"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">sq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">0.1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">9.513507699</span><span style="color: #0000FF;">},</span><span style="color: #008000;">"%19.16f"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">sq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">362880</span><span style="color: #0000FF;">},</span><span style="color: #008000;">"%19.12f"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">sq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">100</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">9.332621544e155</span><span style="color: #0000FF;">},</span><span style="color: #008000;">"%19.13g"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">machine_bits</span><span style="color: #0000FF;">()=</span><span style="color: #000000;">64</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000000;">sq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">150</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sqPI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3.808922638e260</span><span style="color: #0000FF;">},</span><span style="color: #008000;">"%19.13g"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (fatal power overflow error on 32 bits)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<!--
|
||||
printf(1," z ------ spouge ----- ----- taylor ------ ----- lanczos ----- ---- expected ----- %19.16f\n",PI)
|
||||
papply({{-3/2,3/4},{-1/2,-1/2},{1/2,1},{1,sqPI},{3/2,2},{2,sqPI},{5/2,4/3},{3,sqPI/2},{7/2,8/15},{4,sqPI/6}},sq)
|
||||
sq({0.001,sqPI/999.4237725},"%19.15f")
|
||||
sq({0.01,sqPI/99.43258512},"%19.16f")
|
||||
sq({0.1,sqPI/9.513507699},"%19.16f")
|
||||
sq({10,sqPI/362880},"%19.12f")
|
||||
sq({100,sqPI/9.332621544e155},"%19.13g")
|
||||
if machine_bits()=64 then
|
||||
sq({150,sqPI/3.808922638e260},"%19.13g") -- (fatal power overflow error on 32 bits)
|
||||
end if
|
||||
|
|
|
|||
|
|
@ -1,109 +1,107 @@
|
|||
-->
|
||||
<span style="color: #008080;">without</span> <span style="color: #008080;">javascript_semantics</span> <span style="color: #000080;font-style:italic;">-- (no mpfr_exp(), mpfr_gamma() in pwa/p2js)</span>
|
||||
<span style="color: #008080;">constant</span> <span style="color: #000000;">dp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">30</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">fmt</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"%5s: %33s, %33s, %32s\n"</span>
|
||||
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.2"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (mpfr_get_fixed(maxlen), mpfr_gamma)</span>
|
||||
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
|
||||
<span style="color: #7060A8;">mpfr_set_default_precision</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">87</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- 87 decimal places. </span>
|
||||
<span style="color: #004080;">sequence</span> <span style="color: #000000;">mc</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">40</span><span style="color: #0000FF;">)</span>
|
||||
without javascript_semantics -- (no mpfr_exp(), mpfr_gamma() in pwa/p2js)
|
||||
constant dp = 30
|
||||
string fmt = "%5s: %33s, %33s, %32s\n"
|
||||
requires("1.0.2") -- (mpfr_get_fixed(maxlen), mpfr_gamma)
|
||||
include mpfr.e
|
||||
mpfr_set_default_precision(-87) -- 87 decimal places.
|
||||
sequence mc = mpfr_inits(40)
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">mpfr_spouge_gamma</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpfr</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">accm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpfr_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000080;font-style:italic;">-- mc[1] := sqrt(2*PI)</span>
|
||||
<span style="color: #7060A8;">mpfr_const_pi</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000080;font-style:italic;">-- k1_factrl = (k - 1)!*(-1)^k with 0!==1</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">k1_factrl</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span>
|
||||
<span style="color: #000000;">tmk</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(),</span>
|
||||
<span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">()</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #000080;font-style:italic;">-- mc[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl</span>
|
||||
<span style="color: #7060A8;">mpfr_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tmk</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mpfr_exp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">],</span><span style="color: #000000;">tmk</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_set_d</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1.5</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mpfr_pow</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">tmk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k1_factrl</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">],</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">],</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000080;font-style:italic;">-- k1_factrl *= -(k-1)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k1_factrl</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k1_factrl</span><span style="color: #0000FF;">,-(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #000000;">accm</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #000080;font-style:italic;">-- accm += mc[k]/(z+k-1)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">ck</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]),</span>
|
||||
<span style="color: #000000;">zk</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">zk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">zk</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ck</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ck</span><span style="color: #0000FF;">,</span><span style="color: #000000;">zk</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ck</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #000080;font-style:italic;">-- atom zc = z+length(mc)
|
||||
-- accm *= exp(-zc)*power(zc,z+0.5) -- Gamma(z+1)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">),</span>
|
||||
<span style="color: #000000;">ez</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(),</span>
|
||||
<span style="color: #000000;">zh</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mc</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #7060A8;">mpfr_neg</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ez</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mpfr_exp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ez</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ez</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">zh</span><span style="color: #0000FF;">,</span><span style="color: #000000;">zh</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mpfr_pow</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">zh</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ez</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000080;font-style:italic;">-- return accm/z</span>
|
||||
<span style="color: #7060A8;">mpfr_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">accm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">accm</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
function mpfr_spouge_gamma(mpfr z)
|
||||
mpfr accm = mc[1]
|
||||
if mpfr_cmp_si(accm,0)=0 then
|
||||
-- mc[1] := sqrt(2*PI)
|
||||
mpfr_const_pi(accm)
|
||||
mpfr_mul_si(accm,accm,2)
|
||||
mpfr_sqrt(accm,accm)
|
||||
-- k1_factrl = (k - 1)!*(-1)^k with 0!==1
|
||||
mpfr k1_factrl = mpfr_init(1),
|
||||
tmk = mpfr_init(),
|
||||
p = mpfr_init()
|
||||
for k=2 to length(mc) do
|
||||
-- mc[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl
|
||||
mpfr_set_si(tmk,length(mc)+1-k)
|
||||
mpfr_exp(mc[k],tmk)
|
||||
mpfr_set_d(p,k-1.5)
|
||||
mpfr_pow(p,tmk,p)
|
||||
mpfr_div(p,p,k1_factrl)
|
||||
mpfr_mul(mc[k],mc[k],p)
|
||||
-- k1_factrl *= -(k-1)
|
||||
mpfr_mul_si(k1_factrl,k1_factrl,-(k-1))
|
||||
end for
|
||||
end if
|
||||
accm = mpfr_init_set(accm)
|
||||
for k=2 to length(mc) do
|
||||
-- accm += mc[k]/(z+k-1)
|
||||
mpfr ck = mpfr_init_set(mc[k]),
|
||||
zk = mpfr_init_set(z)
|
||||
mpfr_add_si(zk,zk,k-1)
|
||||
mpfr_div(ck,ck,zk)
|
||||
mpfr_add(accm,accm,ck)
|
||||
end for
|
||||
-- atom zc = z+length(mc)
|
||||
-- accm *= exp(-zc)*power(zc,z+0.5) -- Gamma(z+1)
|
||||
mpfr p = mpfr_init_set(z),
|
||||
ez = mpfr_init(),
|
||||
zh = mpfr_init(0.5)
|
||||
mpfr_add_si(p,p,length(mc))
|
||||
mpfr_neg(ez,p)
|
||||
mpfr_exp(ez,ez)
|
||||
mpfr_add(zh,zh,z)
|
||||
mpfr_pow(p,p,zh)
|
||||
mpfr_mul(accm,accm,ez)
|
||||
mpfr_mul(accm,accm,p)
|
||||
-- return accm/z
|
||||
mpfr_div(accm,accm,z)
|
||||
return accm
|
||||
end function
|
||||
|
||||
<span style="color: #008080;">constant</span> <span style="color: #000000;">mPI</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(),</span>
|
||||
<span style="color: #000000;">mqPI</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">()</span>
|
||||
<span style="color: #7060A8;">mpfr_const_pi</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mPI</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">pistr</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mPI</span><span style="color: #0000FF;">,</span><span style="color: #000000;">dp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mPI</span><span style="color: #0000FF;">)</span>
|
||||
constant mPI = mpfr_init(),
|
||||
mqPI = mpfr_init()
|
||||
mpfr_const_pi(mPI)
|
||||
string pistr = mpfr_get_fixed(mPI,dp)
|
||||
mpfr_sqrt(mqPI,mPI)
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">makempfr</span><span style="color: #0000FF;">(</span><span style="color: #004080;">object</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">res</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #004080;">string</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">split</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'/'</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
|
||||
<span style="color: #7060A8;">mpfr_div_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">to_integer</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]))</span>
|
||||
<span style="color: #008080;">elsif</span> <span style="color: #004080;">sequence</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #004080;">mpfr</span> <span style="color: #000000;">x1</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">object</span> <span style="color: #000000;">x2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x</span>
|
||||
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #004080;">string</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">else</span>
|
||||
<span style="color: #7060A8;">mpfr_div_d</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">else</span>
|
||||
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
function makempfr(object x)
|
||||
mpfr res
|
||||
if string(x) then
|
||||
x = split(x,'/')
|
||||
res = mpfr_init(x[1])
|
||||
mpfr_div_si(res,res,to_integer(x[2]))
|
||||
elsif sequence(x) then
|
||||
{mpfr x1, object x2} = x
|
||||
res = mpfr_init_set(x1)
|
||||
if string(x2) then
|
||||
mpfr d = mpfr_init(x2)
|
||||
mpfr_div(res,res,d)
|
||||
else
|
||||
mpfr_div_d(res,res,x2)
|
||||
end if
|
||||
else
|
||||
res = mpfr_init(x)
|
||||
end if
|
||||
return res
|
||||
end function
|
||||
|
||||
<span style="color: #008080;">procedure</span> <span style="color: #000000;">mq</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">zm</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">=</span><span style="color: #000000;">dp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">mul</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">apply</span><span style="color: #0000FF;">(</span><span style="color: #000000;">zm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">makempfr</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mpfr_spouge_gamma</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">maxlen</span><span style="color: #0000FF;">:=</span><span style="color: #000000;">dp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init</span><span style="color: #0000FF;">()</span>
|
||||
<span style="color: #7060A8;">mpfr_gamma</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mul</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">zs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">),</span>
|
||||
<span style="color: #000000;">es</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">maxlen</span><span style="color: #0000FF;">:=</span><span style="color: #000000;">dp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">),</span>
|
||||
<span style="color: #000000;">ps</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_get_fixed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">dp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">zs</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">es</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ps</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
|
||||
procedure mq(sequence zm, integer d=dp)
|
||||
mpfr {z, mul} = apply(zm,makempfr)
|
||||
mpfr s = mpfr_spouge_gamma(z)
|
||||
string t = mpfr_get_fixed(s,d,10,maxlen:=dp+2)
|
||||
mpfr e = mpfr_init()
|
||||
mpfr_gamma(e,z)
|
||||
mpfr p = mpfr_init_set(s)
|
||||
mpfr_mul(p,p,mul)
|
||||
mpfr_mul(p,p,p)
|
||||
string zs = mpfr_get_fixed(z,3),
|
||||
es = mpfr_get_fixed(e,d,10,maxlen:=dp+2),
|
||||
ps = mpfr_get_fixed(p,dp,10)
|
||||
printf(1,fmt,{zs,t,es,ps})
|
||||
end procedure
|
||||
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" z %s %s %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">pad</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" spouge "</span><span style="color: #0000FF;">,</span><span style="color: #000000;">dp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"BOTH"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'-'</span><span style="color: #0000FF;">),</span><span style="color: #7060A8;">pad</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" expected "</span><span style="color: #0000FF;">,</span><span style="color: #000000;">dp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"BOTH"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'-'</span><span style="color: #0000FF;">),</span><span style="color: #000000;">pistr</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #7060A8;">papply</span><span style="color: #0000FF;">({{-</span><span style="color: #000000;">1.5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0.75</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}},{</span><span style="color: #000000;">1.5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}},{</span><span style="color: #000000;">2.5</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"4/3"</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">}},{</span><span style="color: #000000;">3.5</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"8/15"</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">}}},</span><span style="color: #000000;">mq</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mq</span><span style="color: #0000FF;">({</span><span style="color: #008000;">"1/1000"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"999.4237724845954661149822012996"</span><span style="color: #0000FF;">}},</span><span style="color: #000000;">28</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mq</span><span style="color: #0000FF;">({</span><span style="color: #008000;">"1/100"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"99.43258511915060371353298887051"</span><span style="color: #0000FF;">}},</span><span style="color: #000000;">29</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mq</span><span style="color: #0000FF;">({</span><span style="color: #008000;">"1/10"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"9.513507698668731836292487177265"</span><span style="color: #0000FF;">}},</span><span style="color: #000000;">30</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #000000;">362880</span><span style="color: #0000FF;">}},</span><span style="color: #000000;">25</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">100</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"9.332621544394415268169923885627e155"</span><span style="color: #0000FF;">}},</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">mq</span><span style="color: #0000FF;">({</span><span style="color: #000000;">150</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">mqPI</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"3.80892263763056972698595524350735e260"</span><span style="color: #0000FF;">}},</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
|
||||
<!--
|
||||
printf(1," z %s %s %s\n",{pad(" spouge ",dp+2,"BOTH",'-'),pad(" expected ",dp+2,"BOTH",'-'),pistr})
|
||||
papply({{-1.5,0.75},{-0.5,-0.5},{0.5,1},{1,{mqPI,1}},{1.5,2},{2,{mqPI,1}},{2.5,"4/3"},{3,{mqPI,2}},{3.5,"8/15"},{4,{mqPI,6}}},mq)
|
||||
mq({"1/1000",{mqPI,"999.4237724845954661149822012996"}},28)
|
||||
mq({"1/100",{mqPI,"99.43258511915060371353298887051"}},29)
|
||||
mq({"1/10",{mqPI,"9.513507698668731836292487177265"}},30)
|
||||
mq({10,{mqPI,362880}},25)
|
||||
mq({100,{mqPI,"9.332621544394415268169923885627e155"}},0)
|
||||
mq({150,{mqPI,"3.80892263763056972698595524350735e260"}},0)
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue