Calculate the effective focal length of the lens combination. The effective focal length (EFL) is given by:
EFL = f1 x f2 / (f1 + f2 -D),
where f1 and f2 are the two focal lengths and D is the separation between the two.
For example, if f1 is 160 mm and f2 is 60 mm and D is 20 mm, then the EFL is given by
EFL = 160 x 60 / (160 + 60 - 20) = 9600 / 200 = 48 mm.
Calculate the derivative of the EFL with respect to the first focal length. The result is:
d/df1 (EFL) = (f2^2 - d x f2) / (f1 + f2 - D)^2.
For the example, this is 2400/40000 = 0.06.
Calculate the derivative of the EFL with respect to the second focal length. The result is:
d/df2 (EFL) = (f1^2 - d x f1) / (f1 + f2 - D)^2.
For the example, this is 22400/40000 = 0.56.
Look up the precision of each of the lenses. This information is usually on the datasheet.
For the example problem, assume the datasheets show that the precision, or error, of each lens is 5 mm.
Add the precisions in quadrature to get the overall precision. The precision can be added in quadrature --- that is, as the sum of the squares --- because they are completely independent errors. The expression is
Net precision = sqrt([d/df1 (EFL)]^2 x delta1^2 + [d/df2 (EFL)]^2 x delta2^2),
where delta1 and delta2 are the quoted precisions of each lens.
For the example problem, this expression is
Net precision = sqrt ([0.06 x 5]^2 + [0.56 x 5]^2) which is about 3 mm.