I know there are tools that can do the conversion like pyproj, but I'm actually interested in the formula for computing it. Or if it's done with a big lookup table. I would love some information on how it's performed?

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If I am reading your question correctly, you may find Formula For State Plane to Lat/Lon Conversion at GeoNet useful. Posters here outlined resources for the math behind State Plane to Geographic Coordinates.

I'd pay particular attention to a manual from EPSG entitled "Geomatics Guidance Note 7, part 2 Coordinate Conversions & Transformations including Formulas" and to "John P. Snyder's Map Projections: A Working Manual" (PDF).

There is some good discussion in that first link culminating in two SQL queries for running through the individual conversions for lat/lon, I'd recommend checking those out, as well.

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