How good an approximation is the 72 rule really?

Exponential growth (or exponential decay) is a mathematical model describing a phenomenon where some principal changes by a fixed proportion every given unit of time. Population, nuclear decay, and, most importantly, interest work in this fashion. In such a case, it is useful to describe the doubling time (or half-life) of the rate of change. This means -- "how long until this grows to twice the size (or shrinks to half the size) as when it originally started out?"

The rule of 72 states that the doubling time is approximately 72 divided by the interest rate times 100. For instance, if your interest rate is 7% APY, your money can be expected to double in 72/7, or 10.29 years. In other words,

Interestingly, there is also the rule of 70, which uses precisely the same technique, just a different approximation.

The REAL doubling time can be calculated by solving our exponential growth equation when the principle is doubled for the exponent (i.e. the amount of time). This is shown:

It is apparent, then, where the rule of 72 approximation is derived: ln(2) is about 0.693, and ln(1+r) is about r, when r is small.

Using this method, our doubling time for a 7% APY is about 10.24 years. Therefore, the rule of 72 approximation is 0.05 years off the mark, or an error of 0.004 or 0.4%.

Here is a chart detailing how the doubling time is affected by annualized returns with the both approximations for reference:

The red function is the actual doubling time; blue is rule of 72 approximation; and green is rule of 70.

So how accurate are the approximations? I graphed the error related to the annualized return:

Where the red function is the percentage error of the rule of 72 approximation, and the purple is the error of the rule of 70.

We can see that at very low rates of return, the approximations are pretty good, but at higher rates, misses the mark. Within this, the rule of 70 is the better approximation for APY's 0-5% (e.g. bonds, savings accounts, CD's, etc.) and the rule of 72 is better for APY's 5% and higher (e.g. equities, etc.). The rule of 70 overestimates until 2% returns, after which it underestimates. The rule of 72 overestimates until 8%, after which ir underestimates.

Overall, this error is significantly less than what can be expected in error from projected returns, so I feel that people are justified in using these approximations. However, since the correct value is already so easy to calculate, I really don't see any need for them.