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| <!DOCTYPE html><html><head><meta charset='utf-8'><title>Plot</title><script src='https://cdn.plot.ly/plotly-latest.min.js'></script><script type='text/javascript' src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'></script></head><body> | |
| <style>body{margin:auto;padding:15px;text-align:justify;line-height:140%; -webkit-font-smoothing:antialiased;-moz-osx-font-smoothing:grayscale;font-size:14px;font-smoothing:antialiased;color:#222;font-family:Roboto, sans-serif;}</style><style>h1{border-bottom: 2px solid #666; color: #333; padding: 20px 0 8px }</style><style>h2{border-bottom: 1px solid #888; color: #555; padding: 20px 0 4px }</style></style><style>div.funclist{font-size: 12px}</style><h1>Example report using arctangent</h1> | |
| This is overkill 'reporting' as demo. Actual intended use is to quickly visualize a small set of approximations over the domain. The script builds the HTML and can automatically open the results in a brower. A fair number of plot traces are so close they overlay one another | |
| Generated from: <a href=https://github.com/Marc-B-Reynolds/Stand-alone-junk/blob/master/src/Sollya/examples/atan.sollya>this script</a><p>The \( P_n \) polynomials are using the library (sollya results).<br>The \( H_n \) are from Hasting 1955.<br>The \( R_n \) <i>Efficient Approximations for the Arctangent Function</i> Rajan, Wang, Inkol, Joyal, 2006.<br>The \( E_n \) David Eberly (via Lagarde)<br>The \( S_n \) <i><a href=http://seblagarde.wordpress.com/2014/12/01/inverse-trigonometric-functions-gpu-optimization-for-amd-gcn-architecture/>Inverse trigonometric functions GPU optimization for AMD GCN architecture</a></i>, Sebastien Lagarde, 2014<br><br>All are binary32 input and working precision. Currently the reported numbers are not exact as they do not properly account for rounding errors.<p>Plots with many traces will have a scroll bar on the left side of the legend.<p> NOTE: the range of this example is not reasonable, see <a href=https://basesandframes.wordpress.com/2016/05/17/faster-math-functions>Faster Math Functions, Robin Green</a> <h2>polynomial: absolute error. \( x \in [-1,1] \)</h2> | |
| <p> | |
| <div class='funclist'>\( \begin{align} | |
| P_{0} (x) & = x-1898203 \times 2^{-23} x^{3}\\ | |
| P_{1} (x) & =8157033 \times 2^{-23} x-25159 \times 2^{-17} x^{3}\\ | |
| S_{2ea} (x) & =1.05863 * x - 0.269408 * x * abs(x)\\ | |
| R_8 (x) & =13176795\times 2^{-24}x+x(6274075\times 2^{-25}-6440505\times 2^{-25}x^2)\\ | |
| S_{3ea} (x) & =1.0307 * x - 0.17239 * x * abs(x) - 7.4563099999999999999999999999999999999999999999998e-2 * x^3\\ | |
| S_{3a} (x) & =2035479 \times 2^{-21} x-6395609 \times 2^{-25} x^{3}\\ | |
| P_{6} (x) & = x-638597 \times 2^{-21} x^{3}+12219049 \times 2^{-27} x^{5}\\ | |
| H_3 (x) & =16699269 \times 2^{-24} x-2421615 \times 2^{-23} x^{3}+10647627 \times 2^{-27} x^{5}\\ | |
| P_{8} (x) & = x-638597 \times 2^{-21} x^{3}+12219049 \times 2^{-27} x^{5}\\ | |
| P_{9} (x) & =16699335 \times 2^{-24} x-4843417 \times 2^{-24} x^{3}+10648697 \times 2^{-27} x^{5}\\ | |
| S_{5a} (x) & =1043115 \times 2^{-20} x-1207733 \times 2^{-22} x^{3}+5319787 \times 2^{-26} x^{5}\\ | |
| H_4 (x) & =8382023 \times 2^{-23} x-2694269 \times 2^{-23} x^{3}+1227057 \times 2^{-23} x^{5}-10467077 \times 2^{-28} x^{7}\\ | |
| P_{12} (x) & = x-5473369 \times 2^{-24} x^{3}+10423095 \times 2^{-26} x^{5}-11760953 \times 2^{-28} x^{7}\\ | |
| P_{13} (x) & =8382013 \times 2^{-23} x-2694211 \times 2^{-23} x^{3}+9815643 \times 2^{-26} x^{5}-10465365 \times 2^{-28} x^{7}\\ | |
| H_5 (x) & =2096871 \times 2^{-21} x-2770753 \times 2^{-23} x^{3}+6044529 \times 2^{-25} x^{5}-5713179 \times 2^{-26} x^{7}+11185759 \times 2^{-29} x^{9}\\ | |
| P_{15} (x) & = x-11129511 \times 2^{-25} x^{3}+12380977 \times 2^{-26} x^{5}-6070033 \times 2^{-26} x^{7}+12380379 \times 2^{-29} x^{9}\\ | |
| P_{16} (x) & =16774973 \times 2^{-24} x-5541591 \times 2^{-24} x^{3}+12090241 \times 2^{-26} x^{5}-11429387 \times 2^{-27} x^{7}+11190965 \times 2^{-29} x^{9}\\ | |
| H_6 (x) & =8388417 \times 2^{-23} x-348781 \times 2^{-20} x^{3}+6494241 \times 2^{-25} x^{5}-15627355 \times 2^{-27} x^{7}+7067009 \times 2^{-27} x^{9}-12585543 \times 2^{-30} x^{11}\\ | |
| P_{18} (x) & = x-2793121 \times 2^{-23} x^{3}+13098501 \times 2^{-26} x^{5}-4020455 \times 2^{-25} x^{7}+14980357 \times 2^{-28} x^{9}-13752895 \times 2^{-30} x^{11}\\ | |
| P_{19} (x) & =8388417 \times 2^{-23} x-697561 \times 2^{-21} x^{3}+12988331 \times 2^{-26} x^{5}-15626729 \times 2^{-27} x^{7}+7066423 \times 2^{-27} x^{9}-12583921 \times 2^{-30} x^{11}\\ | |
| \end{align} \) | |
| </div> | |
| <div id='fig0' style='width:80%'></div> | |
| <h2>polynomial: relative error \( x \in [-1,1] \)</h2> | |
| <p> | |
| <div class='funclist'>\( \begin{align} | |
| S_{2er} (x) & =1.01990999999999999999999999999999999999999999999998 * x - 0.218891 * x * abs(x)\\ | |
| P_{21} (x) & = x-3846473 \times 2^{-24} x^{3}\\ | |
| P_{22} (x) & =8282181 \times 2^{-23} x-1777371 \times 2^{-23} x^{3}\\ | |
| S_{3r} (x) & =16564229 \times 2^{-24} x-14218221 \times 2^{-26} x^{3}\\ | |
| S_{3er} (x) & =1.00536 * x - 8.89206e-2 * x * abs(x) - 0.135249 * x^3\\ | |
| P_{25} (x) & = x-2581139 \times 2^{-23} x^{3}+12710473 \times 2^{-27} x^{5}\\ | |
| P_{26} (x) & =16750777 \times 2^{-24} x-5050593 \times 2^{-24} x^{3}+5989505 \times 2^{-26} x^{5}\\ | |
| S_{5r} (x) & =8375371 \times 2^{-23} x-10100857 \times 2^{-25} x^{3}+11977899 \times 2^{-27} x^{5}\\ | |
| P_{28} (x) & = x-2748299 \times 2^{-23} x^{3}+10691397 \times 2^{-26} x^{5}-12481303 \times 2^{-28} x^{7}\\ | |
| P_{29} (x) & =16773657 \times 2^{-24} x-5466161 \times 2^{-24} x^{3}+5220365 \times 2^{-25} x^{5}-5949435 \times 2^{-27} x^{7}\\ | |
| P_{30} (x) & = x-5572229 \times 2^{-24} x^{3}+6268445 \times 2^{-25} x^{5}-12629619 \times 2^{-27} x^{7}+3334009 \times 2^{-27} x^{9}\\ | |
| P_{31} (x) & =16776713 \times 2^{-24} x-11130027 \times 2^{-25} x^{3}+6214785 \times 2^{-25} x^{5}-12338063 \times 2^{-27} x^{7}+12811371 \times 2^{-29} x^{9}\\ | |
| P_{32} (x) & = x-2794153 \times 2^{-23} x^{3}+13165621 \times 2^{-26} x^{5}-16443667 \times 2^{-27} x^{7}+1971997 \times 2^{-25} x^{9}-7492053 \times 2^{-29} x^{11}\\ | |
| P_{33} (x) & =16777143 \times 2^{-24} x-11173457 \times 2^{-25} x^{3}+13129041 \times 2^{-26} x^{5}-16273047 \times 2^{-27} x^{7}+7715083 \times 2^{-27} x^{9}-14476303 \times 2^{-30} x^{11}\\ | |
| \end{align} \) | |
| </div> | |
| <div id='fig1' style='width:80%'></div> | |
| <h2>absolute error: exact endpoints. \( x \in [-1,1] \)</h2> | |
| The library versions are using a hack ATM: Fix up final coefficient | |
| <p> | |
| <div class='funclist'>\( \begin{align} | |
| E_3 (x) & = x-450053 \times 2^{-21} x^{3}\\ | |
| R_5 (x) & =0.785398185253143310546875 * x + 0.284999999999999999999999999999999999999999999999997 * x * (1 - abs(x))\\ | |
| R_7 (x) & =0.785398185253143310546875 * x + 0.273 * x * (1 - abs(x))\\ | |
| P_{37} (x) & = x-14401685 \times 2^{-26} x^{3}\\ | |
| P_{38} (x) & =8157033 \times 2^{-23} x-12549085 \times 2^{-26} x^{3}\\ | |
| R_9 (x) & =0.785398185253143310546875 * x - x * (abs(x) - 1) * (0.2447 + 6.6299999999999999999999999999999999999999999999999e-2 * abs(x))\\ | |
| E_5 (x) & = x-10129915 \times 2^{-25} x^{3}+11716255 \times 2^{-27} x^{5}\\ | |
| P_{41} (x) & = x-638597 \times 2^{-21} x^{3}+12066837 \times 2^{-27} x^{5}\\ | |
| P_{42} (x) & =16699335 \times 2^{-24} x-4843417 \times 2^{-24} x^{3}+10567013 \times 2^{-27} x^{5}\\ | |
| P_{43} (x) & = x-5473369 \times 2^{-24} x^{3}+10423095 \times 2^{-26} x^{5}-5862609 \times 2^{-27} x^{7}\\ | |
| P_{44} (x) & =8382013 \times 2^{-23} x-2694211 \times 2^{-23} x^{3}+9815643 \times 2^{-26} x^{5}-5221761 \times 2^{-27} x^{7}\\ | |
| P_{45} (x) & = x-11129511 \times 2^{-25} x^{3}+12380977 \times 2^{-26} x^{5}-6070033 \times 2^{-26} x^{7}+3092785 \times 2^{-27} x^{9}\\ | |
| P_{46} (x) & =16774973 \times 2^{-24} x-5541591 \times 2^{-24} x^{3}+12090241 \times 2^{-26} x^{5}-11429387 \times 2^{-27} x^{7}+1398103 \times 2^{-26} x^{9}\\ | |
| P_{47} (x) & = x-2793121 \times 2^{-23} x^{3}+13098501 \times 2^{-26} x^{5}-4020455 \times 2^{-25} x^{7}+14980357 \times 2^{-28} x^{9}-13750363 \times 2^{-30} x^{11}\\ | |
| P_{48} (x) & =8388417 \times 2^{-23} x-697561 \times 2^{-21} x^{3}+12988331 \times 2^{-26} x^{5}-15626729 \times 2^{-27} x^{7}+7066423 \times 2^{-27} x^{9}-12582135 \times 2^{-30} x^{11}\\ | |
| \end{align} \) | |
| </div> | |
| <div id='fig2' style='width:80%'></div> | |
| <h2>relative error: with exact endpoints. \( x \in [-1,1] \)</h2> | |
| <p> | |
| <div class='funclist'>\( \begin{align} | |
| P_{49} (x) & = x-14401685 \times 2^{-26} x^{3}\\ | |
| P_{50} (x) & =8282181 \times 2^{-23} x-13550269 \times 2^{-26} x^{3}\\ | |
| M_3 (x) & =x * (-0.1784 * abs(x) - 6.6299999999999999999999999999999999999999999999999e-2 * x^2 + 1.03009999999999999999999999999999999999999999999999)\\ | |
| P_{52} (x) & = x-2581139 \times 2^{-23} x^{3}+12494853 \times 2^{-27} x^{5}\\ | |
| P_{53} (x) & =16750777 \times 2^{-24} x-5050593 \times 2^{-24} x^{3}+11812885 \times 2^{-27} x^{5}\\ | |
| P_{54} (x) & = x-2748299 \times 2^{-23} x^{3}+10691397 \times 2^{-26} x^{5}-6213381 \times 2^{-27} x^{7}\\ | |
| P_{55} (x) & =16773657 \times 2^{-24} x-5466161 \times 2^{-24} x^{3}+5220365 \times 2^{-25} x^{5}-5927071 \times 2^{-27} x^{7}\\ | |
| P_{56} (x) & = x-5572229 \times 2^{-24} x^{3}+6268445 \times 2^{-25} x^{5}-12629619 \times 2^{-27} x^{7}+832575 \times 2^{-25} x^{9}\\ | |
| P_{57} (x) & =16776713 \times 2^{-24} x-11130027 \times 2^{-25} x^{3}+6214785 \times 2^{-25} x^{5}-12338063 \times 2^{-27} x^{7}+799921 \times 2^{-25} x^{9}\\ | |
| P_{58} (x) & = x-2794153 \times 2^{-23} x^{3}+13165621 \times 2^{-26} x^{5}-16443667 \times 2^{-27} x^{7}+1971997 \times 2^{-25} x^{9}-14979887 \times 2^{-30} x^{11}\\ | |
| P_{59} (x) & =16777143 \times 2^{-24} x-11173457 \times 2^{-25} x^{3}+13129041 \times 2^{-26} x^{5}-16273047 \times 2^{-27} x^{7}+7715083 \times 2^{-27} x^{9}-14472615 \times 2^{-30} x^{11}\\ | |
| \end{align} \) | |
| </div> | |
| <div id='fig3' style='width:80%'></div> | |
| <script> | |
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| }; | |
| var data0= [trace0,trace1,trace2,trace3,trace4,trace5,trace6,trace7,trace8,trace9,trace10,trace11,trace12,trace13,trace14,trace15,trace16,trace17,trace18,trace19]; | |
| var data1= [trace20,trace21,trace22,trace23,trace24,trace25,trace26,trace27,trace28,trace29,trace30,trace31,trace32,trace33]; | |
| var data2= [trace34,trace35,trace36,trace37,trace38,trace39,trace40,trace41,trace42,trace43,trace44,trace45,trace46,trace47,trace48]; | |
| var data3= [trace49,trace50,trace51,trace52,trace53,trace54,trace55,trace56,trace57,trace58,trace59]; | |
| var layout0= {title: 'figure 1'}; | |
| Plotly.newPlot('fig0',data0, layout0, {displaylogo: false, autosizable: true}); | |
| var layout1= {title: 'figure 2'}; | |
| Plotly.newPlot('fig1',data1, layout1, {displaylogo: false, autosizable: true}); | |
| var layout2= {title: 'figure 3'}; | |
| Plotly.newPlot('fig2',data2, layout2, {displaylogo: false, autosizable: true}); | |
| var layout3= {title: 'figure 4'}; | |
| Plotly.newPlot('fig3',data3, layout3, {displaylogo: false, autosizable: true}); | |
| </script></body></html> |
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