Depending on one's need for accuracy, there are any number of ways to calculate the Equation of Time. Wikipedia's EoT page and its Analemma page give a number of these. The simplest rely on the fact that both the Eccentricity and Obliquity effects are almost sine curves. Others are Fourier-based solutions. Finally there are those that use sound astronomical methods. Here you will find an example each method.
WHY BOTHER TO CALCULATE AT ALL...
On September one, trust the Sun. 
Come Halloween, subtract sixteen. 
 On Christmas Day, you're OK. 
 For your Valentine true, add a dozen and two. 
The mid of month four, add no more
At the mid of May, take four away. 
 On June fourteen, don't add a bean. 
 When August begins, add seven little mins. 
 The rest is easy: for any date, All you do is interpolate
a poem by Tad Dunne
VERY SIMPLE METHOD
A 1970 letter from my Uncle John Wigham Richardson explaining how to calculate the Equation of Time
SIMPLE METHOD
This method is adapted from the Astronomical Almanac and is accurate to +/- 6 seconds between 1950 and 2050
FOURIER METHOD
This method took many, many EoT calculations from a sophisticated astronomical package (MICA from the US Naval Observatory) and subjected those values to rigorous Fourier analysis. This gives accurate and easily calculable
ASTRONOMICAL METHOD
This method is adapted from Practical Astronomy with your Calculator by Peter Duffett-Smith
FURTHER DETAILS ON CALCULATING THE ELLIPTICITY EFFECT
MEEUS' ALGORITHMS
The most complete 'non-professional' algorithms for calculation of the Equation of Time are provided in by Jan Meeus -  Astronomical Algorithms (1998), 2nd ed - ISBN 0-943396-61-1.
You can view the author's Javascript implementation of Meeus' algorithms for the Sun here. If you would like to use these algorithms, contact the author, who would be happy to send you a text file.
OTHER SOURCES
For more information of the astronomical background of the Equation of Time, see the article below which was published in NASS Compendium : Vol 25 Nos 3 & 4, Sept & Dec 2018. (Including some corrections for the published text)
MICA
From the US Naval Observatory, this is a relatively cheap program that provides everything that an serious amateur astronomer, gnomonist or navigator might need
HORIZONS
This is the program, in whose user guide, it states that the user should consult the web-master if using the program for manned planetary landings. It is a little bit cumbersome to use, at first. But once used a few times, it is quite straight-forward. It is lightening fast, free and the best that there is.
The above image gives an indication of how input to Horizons is set up. Note the time must be UTC. The quantities, under Table Settings, will output Right Ascension & Declination, Azimuth & Altitude, and Local Solar Time (subtract the UTC value to get the gnomonical convention topographical Equation of Time plus the longitude correction)
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