Approximation of $e^{-x}$

If $x$ is "small enough", I like using the $(2,2)$ Padé approximant, which is easily cast into a memorable form:

$$\exp\,x\approx \frac{(x+3)^2+3}{(x-3)^2+3}$$

For $|x| < \frac12$, the absolute difference between the approximant and the true function is $< 8\times 10^{-5}$. Pretty good approximation for a mere rational function... of course, as with all such approximants, the further you go from $0$, the less accurate it becomes.

Dilip was asking in the comments about "Feynman's method"; since this is too long for a comment, I shall be quoting the relevant paragraphs here:

Lucky Numbers

One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for $e^x$, which is $1+x+x^2/2!+x^3/3!$ Each term you get by multiplying the preceding term by $x$ and dividing by the next number. For example, to get the next term after $x^4/4!$ you multiply that term by $x$ and divide by $5$. It's very simple.

When I was a kid I was excited by series, and had played with this thing. I had computed $e$ using that series, and had seen how quickly the new terms became very small.

I mumbled something about how it was easy to calculate $e$ to any power using that series (you just substitute that power for $x$).

"Oh yeah?" they said. "Well, then what's $e$ to the $3.3$?" said some joker—I think it was Tukey.

I say, "That's easy. It's $27.11$."

Tukey knows it isn't so easy to compute all that in your head. "Hey! How'd you do that?"

Another guy says, "You know Feynman, he's just faking it. It's not really right."

They go to get a table, and while they're doing that, I put on a few more figures. "$27.1126$," I say.

They find it in the table. "It's right! But how'd you do it!"

"I just summed the series."

"Nobody can sum the series that fast. You must just happen to know that one. How about $e$ to the $3$?"

"Look," I say. "It's hard work! Only one a day!"

"Hah! It's a fake!" they say, happily.

"All right," I say, "It's $20.085$."

They look in the book as I put a few more figures on. They're all excited now, because I got another one right.

Here are these great mathematicians of the day, puzzled at how I can compute $e$ to any power! One of them says, "He just can't be substituting and summing—it's too hard. There's some trick. You couldn't do just any old number like $e$ to the $1.4$."

I say, "It's hard work, but for you, OK. It's $4.05$."

As they're looking it up, I put on a few more digits and say, "And that's the last one for the day!" and walk out.

What happened was this: I happened to know three numbers—the logarithm of $10$ to the base $e$ (needed to convert numbers from base $10$ to base $e$), which is $2.3026$ (so I knew that $e$ to the $2.3$ is very close to $10$), and because of radioactivity (mean-life and half-life), I knew the $\log$ of $2$ to the base $e$, which is $.69315$ (so I also knew that $e$ to the $.7$ is nearly equal to $2$). I also knew $e$ (to the $1$), which is $2.71828$.

The first number they gave me was $e$ to the $3.3$, which is $e$ to the $2.3$—ten—times $e$, or $27.18$. While they were sweating about how I was doing it, I was correcting for the extra $.0026$—$2.3026$ is a little high.

I knew I couldn't do another one; that was sheer luck. But then the guy said $e$ to the $3$: that's $e$ to the $2.3$ times $e$ to the $.7$, or ten times two. So I knew that it was $20.$something, and while they were worrying about how I did it, I adjusted for the $.693$.

Now I was sure I couldn't do another one, because the last one was again by sheer luck. But the guy said $e$ to the $1.4$, which is $e$ to the $.7$ times itself. So all I had to do is fix up $4$ a little bit!

They never did figure out how I did it.

...

There's less to it than meets the eye. ;)