I imagine these are the same as the old fixed function OpenGL pipeline, which I believe are as follows:

Linear: f = inverse_lerp(fog_start_distance, fog_end_distance, distance)
Exp:    f = exp(-fog_density * distance)
Exp2:   f = exp(-(fog_density * distance) ^ 2)

f = clamp(f, 0.0, 1.0)

C = f * Cr + (1.0 - f) * Cf

where:
f is fog factor
C is resulting fragment color
Cr is original fragment color
Cf is fog color