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...

a poem by Tad Dunne

VERY SIMPLE METHOD

ASTRONOMICAL ALMANAC METHOD

This method is adapted from Section C5 of the Astronomical Almanac and EoT is accurate to +/- 3.5 seconds during this Century.

FOURIER METHOD

This method provides the quickest way to find the Longitude Corrected Equation of Time. The routine below can be pasted directly into Microsoft Excel by opening a Visual Basic module,

This was developed by many EoT calculations (every 6 hours every day over this Century) from NASA/JPL Horizons application (see below for details of this application). Amongst the many variables that Horizons can calculate is the Hour Angle of the Sun, from which it is easy to deduce the EoT. The EoT values thus provided were subjected to rigorous Fourier analysis. This gives results that are within +/- 13 seconds of the value predicted by Horizons over this Century.

Function EoT(Year, Month, Day, Hour, Zone, DST, Longitude):

UTC_Hour = Hour - Zone - DST

Long_Corr = 15 * Zone - Longitude

aaa = 367 * Year - 730531.5

bbb = Int((7 * Int(Year + (Month + 9) / 12)) / 4)

ccc = Int(275 * Month / 9) + Day

Theta = 0.004301 * (4 * (aaa - bbb + ccc + UTC_Hour / 24#))

EoT1 = 7.353 * Sin(1 * Theta + 6.208)

EoT2 = 9.927 * Sin(2 * Theta + 0.370)

EoT3 = 0.337 * Sin(3 * Theta + 0.304)

EoT4 = 0.231 * Sin(4 * Theta + 0.716)

EoT = 0.017 + EoT1 + EoT2 + EoT3 + EoT4 + Long_Corr

End Function

UTC_Hour = Hour - Zone - DST

Long_Corr = 15 * Zone - Longitude

aaa = 367 * Year - 730531.5

bbb = Int((7 * Int(Year + (Month + 9) / 12)) / 4)

ccc = Int(275 * Month / 9) + Day

Theta = 0.004301 * (4 * (aaa - bbb + ccc + UTC_Hour / 24#))

EoT1 = 7.353 * Sin(1 * Theta + 6.208)

EoT2 = 9.927 * Sin(2 * Theta + 0.370)

EoT3 = 0.337 * Sin(3 * Theta + 0.304)

EoT4 = 0.231 * Sin(4 * Theta + 0.716)

EoT = 0.017 + EoT1 + EoT2 + EoT3 + EoT4 + Long_Corr

End Function

In the above, the date & time are your local civil time. Longitude is in degrees +ve East of Greenwich. Zone is is in Hours +ve East of Greenwich. DST is in hours.

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. N,B. The Mac version of MICA does NOT work under recent Apple Mac OSX.

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 east to use and quite straight-forward. It is lightening fast, free and astronomically the best that there is.