this one is empirical, i started from 0.25 and moved around

This is how I am simulating
16 bit integer math with this
software suite.

change const2 to 1.0 to get result in radians (I personally prefer working in degrees)

this one here just means "quarter circle" so pi/4 or 90 degrees

main formula

the sign-related stuff can be done very quickly with some bitwise and if or case statements. Note that this time there are 2 divisions as you want to multiply by const1a before dividing by const1b

tweak these to see what happens

comparison between atan2 and its approximation

error in degrees over range