Robert Matthews did all of the creative work in deriving the algebraic expression. He has described them in this paper, as well as the one I linked to above. Both are worth reading. Further work in this area has been done by Leonhard Held.
Conventional reports dichotomize results into “significant” and “not significant”. Given only the CI, plug in the limits into this formula to derive the “advocacy bound” when given a “not significant” report.
For ratios, the advocacy limit (AL) is:
AL = exp(-\frac{ ln(UL) \times ln^2(\frac{U}{L}) } {2\times ln(U) \times ln(L)})
The Skeptical Limit (SL) for ratios (when given a “significant” report) is:
SL = exp(\frac{ln^2(\frac{U}{L})} {4\sqrt{ln(U)ln(L)}})
In both formulas, U = upper limit and L = Lower Limit
My spreadsheet should really be an R script, but I haven’t spent the time to write one up with validated test cases yet. The spreadsheet is close (up to 3 significant figures) to reproducing the examples given in his articles.