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If you generate the rays the linear way instead, you don't even need any correction.

Generate two points which represent the left and right edges of the screen - you'd put them in at say 45 degrees left and right of the forward vector of the player. Then to generate the direction vector for each column of the screen, just interpolate linearly between those two points, and find the vector from the player to that intermediate point.

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The venerable lodev tutorial uses this method, which I also used for most of my engines. I learned an interesting tidbit while comparing the two methods though:

The old-school original methods used pretty small cos/sin/atan lookup tables to do the ray and then the correction calc. Using the linear method you end up with a couple divisions per ray that aren't there in the lookup method. Divisions were (and are, depending on the platform) pretty slow. Linear method still works with lookup tables but they're relatively huge.

Also IIRC With the linear method door-indents need a workaround.

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