Thom Hogan stated (http://www.panohelp.com/images/bythomformula.jpg):

  "You can calculate the focal length of a lens at its closest
  focus distance by using the formula minimum_focus /
  ((1/reproduction_ratio) + reproduction_ratio + 2)."

Theory: This is just the "thin lens formula" rewritten.  So, take
the "thin lens formula" and see if it can be transformed into
Thom's equation:

  1/f = 1/v + 1/b

Convert right hand side to common denominator form:

  1/f = 1/v*b/b + 1/b*v/v
  1/f = b/vb + v/vb
  1/f = (v+b)/(vb)

Divide both sides into one (reciprocal):

  1/(1/f) = 1/((v+b)/(vb))
  f = (vb)/(v+b)

Multiply numerator/denominator by same "v+b" value:

  f = (vb)*(v+b) / ((v+b)*(v+b))
  f = (vb)*(v+b) / (v+b)^2

Divide numerator/denominator by same "vb" value:

  f = ((vb)*(v+b)/(vb)) / ((v+b)^2/(vb))
  f = (v+b) / ((v+b)^2/(vb))

Multiply out:

  f = (v+b) / (v^2/vb + 2vb/vb + b^2/vb)

Simplify:

  f = (v+b) / (v/b + 2 + b/v)

Change v/b into b/v form:

  f = (v+b) / (1/(b/v) + 2 + b/v)

In the thin lens diagram, the 'reproduction ratio' is image height
divided by object height, and by "corresponding sides of similar
triangles are proportional", the 'reproduction ratio' also equals
b/v. Substitute variable:

  f = (v+b) / (1/reproduction_ratio + 2 + reproduction_ratio)

(v+b) in the thin lens formula is the distance from the object to the
image.  Substitute variable:

  f = minimum_focus / (1/(b/v) + 2 + b/v)

Therefore, the formula is just the thin lens formula, albeit
just reworked.

But for a thick lens, (v+b) does NOT equals to distance from the
object to the image.  Rather, both 'v' and 'b' are measurements
from their respective principal planes. Therefore, the formula
produces the wrong answer.  For full details, review:

  http://www.panohelp.com/thinlensformula.html