Here are a few websites in which the combination "formula to + gerund" has been used.
The only link I am able to follow is the first, and it leads to an example that I find unidiomatic. If you continue to search for examples of "formula to + gerund," you will find some, and you can safely assume that I will find all of them unidiomatic; so there is no need for you to present them here or to give links to the examples.
Should you continue your search, I recommend that you compare the relative frequency of "formula for + gerund," using the same gerund. You will thus be able to compile some statistics showing you that the overwhelming preference is for "for."
The Oxford English Dictionary (OED), in its definitions of "formula," does not contain a single example of "formula to + gerund." It does, however, contain an example of "formula for + gerund": "1850 C. G. B. Daubeny (ed. 2) v. 156 A general formula for calculating the specific heat of each class of compounds."
On COCA (The Corpus of Contemporary American English), there are 319 examples of "formula to," and only 2 of them are instances of "formula to + gerund." I find both of the sentences to be unidiomatic, and would have revised "to" to "for" if I had been able to edit the sentences:
- "Joe Linta may have come up with a two-question formula to evaluating it."
- "You might think that you had discerned the secret formula to becoming one of our favorite ski areas."
COCA displays 1225 results for "formula for," and I do not have time to go through all of them for you, but you're welcome to join the corpus yourself and have a look. Suffice it to say that the results for "formula for + gerund" dwarf the pitiful two results for "formula to + gerund." Here are a handful of examples:
- "Some post the formula for calculating pay."
- "In an era of constant disruption, clinging to purpose and recruiting for resilience is the formula for building the workforce of the future."
- "it did suggest a sharp departure from the usual post-1989 formula for exercising global leadership."
- "the use of total population is supported by the Constitution's formula for allocating seats in the House of Representatives among the States."
- "He must have a magic formula for reaching the Final Four, which his teams have done seven times."
I don't have time to keep providing you with more and more evidence right now.