whereas the rule is normally only accurate at the lowest end of interest rates, from 0% to about 5%. To compute the E-M approximation, multiply the rule of 69.3 result by 200/(200−r) as follows: . For example, if the interest rate is 18%, the rule of 69.3 gives t = 3.85 years, which the E-M rule multiplies by (i.e. 200/ (200−18)) to give a doubling time of 4.23 years. As the actual doubling time at this rate is 4.19 years, the E-M rule thus gives a closer approximation than the rule of 72. To obtain a similar correction for the rule of 70 or 72, one of the numerators can be set and the other adjusted to keep their product approximately the same. The E-M rule could thus be written also as or In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate. Padé approximant The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula: which simplifies to: Derivation Periodic compounding For periodic compounding, future value is given by: where is the present value, is the number of time periods, and stands for the interest rate per time period. The future value is double the present value when: which is the following condition: This equation is easily solved for : A simple rearrangement shows: If r is small, then ln(1 + r) approximately equals r (this is the first term in the Taylor series). That is, the latter factor grows slowly when is close to zero. Call this latter factor . The function is shown to be accurate in the approximation of for a small, positive interest rate when (see derivation below). , and we therefore approximate time as: Written as a percentage: This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). is written as a percentage. In order to derive the more precise adjustments presented above, it is noted that is more closely approximated by (using the second term in the Taylor series). can then be further simplified by Taylor approximations: Replacing the "R" in R/200 on the third line with 7.79 gives 72 on the numerator. This shows that the rule of 72 is most accurate for periodically compounded interests around 8%.