# DOF formula vs the simplified DOF formula

I was calculating the theoretical difference in the change of the DOF (depth of field), between a frame achieved on a 25mm signature prime and an "identical frame" achieved by using a 75mm signature prime (not identical in perspective but rather in subject framing).

Lens stats:

focal length (mm) horizontal AOV(°)
25 73
75 27.5

By using right angle triangles, I calculated that to achieve the same horizonal framing on the 75mm as I did with the 25mm lens at a distance of 3m from the subject, I had to be at a distance of 9.07m (roughly 3 times larger).

Now when calculating the DOF with the normal Equations I got these two values:

f-stop Circle of confusion (mm)
4 0.029

25mm lens DoF = 4.77m

75mm lens DoF = 3.48m

These are roughly the same, but then again in my opinion they do also seem relatively large apart.

Now to my knowledge when changing the focal length by the same factor as we change the distance (in this case a factor of 3), the Depth of field should stay relatively constant and from practice this is what I often see.

Furthermore, when using the simplified DOF formula, DoF = 2Nc(of)^2, it would suggest that this would be the case.

So my question is: if it is fair to assume that the values of 4.77m and 3.48m obtained from the non-simplified DOF formula, are similar enough to be assumed equivalent in practice?

I would also love to know the explanation behind how the simplified DOF formula is derived from the non-simplified formula.