As detailed here and elsewhere, Feynman and others at Los Alamos could calculate many problems to 10% accuracy in minutes:
When I was at Los Alamos I found out that Hans Bethe was absolutely topnotch at calculating… A few minutes later we need to take the cube root of $2 \frac 1 2$… and he says, "It's about $1.35$."… I had a lot of fun trying to do arithmetic fast, by tricks, with Hans… I announced, "I can work out in sixty seconds the answer to any problem that anybody can state in ten seconds, to 10 percent!"… People started giving me problems they thought were difficult, such as integrating a function like $\frac 1 {1 + x^4}$, which hardly changed over the range they gave me. The hardest
one somebody gave me was the binomial coefficient of $x^{10}$ in $(1 + x)^{20}$Íž I got that just in time…. [Paul Olum] says, $\tan 10^{100}$. I was sunk: you have to divide by $\pi$ to $100$ decimal places!
What methods do Feynman, Bethe, Olum, use to do these? Or, since we can't really know the answer to that: What methods can we use to easily approximate calculations within 10% error?
Now, one might simply respond: Wolfram Alpha. But we do this not for lack of a calculator! For example, Sanjay Mahajan requires his students to show "number sense" by
Without a calculator, estimate $\sqrt{1.3}, \sqrt[3]{1.6}, \sin 7,$ and $1.01^{100}$.
What methods can we use to do this? I'll post my collection as an answer below. Surprisingly, while I've found good methods for estimating logs, exponents, and trig, approximate long division (in less steps than the real thing) proves to be the hardest!
Best Answer
Log
$$\log (1 + x) = x - \frac {x^2}2 + \frac{x^3} 3 +... \\ \log 2 \approx 0.69 \ \log 3 \approx 1.1 \log 10 \approx 2.3$$ allows easily computing any log to within 1%.
Exponential
$$e^x = 1 + x + \frac {x^2}{2!}+...$$ along with the constants above allows easily computing any exp to within 1%.
Trig
$$\sin x = x - \frac {x^3}{3!} + ... \\ \cos x = 1 - \frac {x^2} {2!} + ...$$ provide sufficient estimates of all trig functions. The cubic term is only needed near $\frac \pi4$. Near $\frac \pi 2$, use cofunctions near $0$. For inverse functions, use the above and algebra.
Factorial
Stirling's approximation $$n! \approx \sqrt{2 \pi n} \left[ \frac n e \right]^n$$ along with the exp estimators allows computing good estimates. It may be easier to take the log: $$\log (n!) \approx n \log n - n + \frac 1 2 \log n + \frac 1 2 \log 2 \pi.$$
Integrals
Numeric estimates via sampling are surprisingly accurate. For example, for Feynman's example $\int \frac {dx} {x^4 + 1}$ (with bounds $[0,1]$) it suffices to take three easy points $0 \leadsto 1, 1 \leadsto \frac 1 2, \frac 1 2 \leadsto \frac {16} {17}$, which average to $\frac {83} {102}$, within range of the true value $0.867...$.
Sometimes thought is required where and how to sample. For example, to solve Arnold's problem of $\int_0^{\frac \pi 2}\cos^{100}x$, notice that $\cos^{100}x$ is nearly always either close to $1$ or very close to $0$. We can therefore treat it as a square pulse which begins at $0$ and ends when $\cos^{100} x$ crosses $\frac 1 2$.
To find this threshold $x$, first solve $t^{100} = \frac 1 2$, $$100 \log t \approx -0.69 \\ t \approx \exp (-0.0069) \approx 0.9931$$ and then $$\arccos x = t \approx 0.9931 \\ x^2/2 \approx 0.0069 \\ x \approx 0.12$$ which is within about 4% of the true value $0.12502...$.
Polynomials and Rational Functions
$(1+x)^n \approx 1 + nx$ for small $x$
$\frac 1 {1+x} \approx 1 - x + x^2$ for small $x$
Division
Division within 10% is easy by rounding (and perhaps correcting), but often division errors are magnified, and therefore require closer to 2% accuracy.
No good method found!
Is there a method that can e.g. determine $\frac {562} {2.3}$ to within 2% error with less steps than long division?!