(As I did ..) ... it occurred to me that others might be doing a back of a fag packet derivation of OS' formula. If so a couple of points might make it a bit easier to derive .. (if you are playing with TODA, then replace STODA by TODA) ..
(a) draw a sketch of the runway slope with the STODA gradients. (Apologies for no sketch to make it easier but I don't have the facility on this machine)
(b) the presumption is that the second segment gradient is driven to equal the STODA gradient, so draw the first segment from the runway to intersect the STODA gradient. OS has suggested that you use the critical difference between the second and first segments .. that's fine as the difference is pretty constant, however, as you need to check the AFM for the first segment distance, you might just as well use the actual first segment gradient for the calculation .. keep in mind that this is all going on the background as part of generating a table of some sort for the pilot to use ..
(c) in the triangle bounded by
(i) the distance delta (D) to be found
(ii) STODA surface from the runway to the intersection with first segment gradient
(iii) first segment slant distance (d)
the angles are
(i) first segment gradient - runway slope (= G1-GR)
(ii) STODA slope - first segment slope (= GS-G1)
(iii) 180 - (G1-GR) - (GS-G1) .... (sum of angles is 180 degrees)
Sine rule gives ..
D = d/cos(G1) * sin(GS-G1)/sin(180 - (G1-GR) - (GS-G1))
putting cos(G1) = 1 (error in OS' example is a whole 4 cm) gives OS' formula.
The divisor term can be simplified by expanding bracket expressions to get
D = d * sin(GS-G1)/sin(180+GR-GS)
(keep in mind, if you are playing with Excel or similar to do the sums, that you need to convert between radians and degrees for calculations ..)
The preceding presumes that I have counted on all my fingers correctly along the way ....