The problem with the shockingly simple math post is that your retirement budget != your current budget. Mine will be cheaper in some areas (kids will be out of house unless thing speed up considerably) but we also have a large travel budget.

Guesstimate a retirement budget, multiply by 25-33x depending on your preferred SWR. Then analyze savings in terms of what % of your FIRE number (including investment returns over x years) it represents.

As MDM and warfreak pointed out, the goal doesn't matter for this calculation assuming the increase in savings is a one time occurance. Since the goal is the only value affected by what your expenses in retirement will be, this concern can be overlooked.

I prefer annuity formulas for this sort of thing which clearly delineates regularly occurring savings, one-time immediate savings, and future expenses, but the fomulas aren't as pretty as the above. Using the earlier introduced variables, S for ongoing savings amount, P for current investment balance, r for constant growth rate, and t for time to retirement and adding my own variables swr for safe withdrawal rate and E for expenses in retirement, then the formula

t = ln((E / swr * r + S)/(P * r + S)) / ln(1 + r)

gives the time to retirement, t. Note that you can use any time period at all for these variables as long as you're consistant. For instance, you can express your savings, S, in amount per month as long as the rest of your variables (including interest rates, which are all effective intereset rates) are expressed per month. 3% interest per year, for instance, would be 1.03^(1/12) - 1 = .2466% per month. Also, you can use E, S, and P amounts that are nominal, like $20,000, or percents like a savings rate of 70% as long as the percents are all given with respect to the same variable such as current income.

To see the effect of saving an amount of money today, n, simply compare the value of t using P and the value of t using P + n as your current investment balances. If the savings can be repeated every period, then also subsitute S + n for S or, possibly, S + m where m is n times the number of times the savings can be repeated for the period your using (i.e. if you're using annual numbers and the savings is repeated monthly, then compare the value of t using P and S to the value of t using P + n and S + 12*n). If the savings will continue, not just during the duration of the acculation period, but also during retirement, then also substitute E - n or E - m for E also.

Using the numbers supplied above and assuming a 4% SWR, I can approximate warfreak's estimate using S = 25000, E = 40000, P = 100000, r = .03, and swr = .04 to get t = 22.84019053 years. If I increase P to P = 100001, then t = 22.84015428 years which is a difference of dt = 3.62473e-5 which translates to about 19 minutes/$. The difference is due to the difference in compounding period. If I compound my numbers daily instead of yearly, I get 18.8 minutes/$ just like warfreak.

If I assume that I can save that dollar every month, then I substitute P = 100001 and S = 25012 yeilding dt = .008702079 which is a difference of about 2.6 days per dollar saved every month until retirement.

If I further assume that I will save that dollar every month in retirement as well, then I also substitute E = 39988 to get dt = .014237285 or about 4.6 days per dollar saved every month permanently.

Note that dt/dS and dt/dE are both dependent on your expenses in retirment, unlike dt/dP, so your goal does matter when computing time saved when reducing regularly occurring expenses.