Why WolframAlpha Result is Different?

Question

http://www.wolframalpha.com/input/?i=x%3D10%5E-15%3B+%28e%5Ex-1%29%2Fx

When i click Approximate Form in result, the result is:

1.11022

but when i click More Digit:

1.000000000000000500000000000000166666666666666708333333333333341666666666666668055555555555555753968

Why the number is different? (instead of showing more digit) and what is the actual solution?

Answer 1

The first one is evaluated using WolframAlpha's exponential function. Now I'm not sure how $\exp$ is implemented in WA, but in Java, for instance, if $r$ is small, $\exp(r)$ will be approximated as: \begin{align*} \exp(r) &= 1+r+\frac{rR_1}{2-R_1},\\ R_1 &= r-\left(\frac{r^2}6 + P_2\,r^4 + P_3\,r^6 + P_4\,r^8 + P_5\,r^{10}\right) \end{align*} where $|P_2|,\ldots,|P_5|\ll1$. So, if $r=10^{-15}$ and double-precision floating point arithmetic is used, $\frac{rR_1}{2-R_1}$ will be dropped and hence $\exp(r)$ will be equal to the result of storing $1+r$ in double precision, i.e. $1+2^{-50}+2^{-52}\approx1+1.11022\times10^{-15}$. Hence $\dfrac{e^r-1}{r}=\dfrac{2^{-50}+2^{-52}}{10^{-15}}\approx1.11022$. (Matlab also gives the same number.) This explains the first result. I have no idea how the second result was calculated.

Answer 2

I am not totally sure but i guess this is a problem of the double floating point precision. If you only calculate with a precision of $10^{-16}$ (so called machine precision) you always get errors in the result. If you chose more digits it will calculate with a higher precision and the final result will be nearer at the exact solution.

Answer 3

The approximations are incorrect and artifact of the rounding and precision errors. Better start with the expansion of $e^x = 1 + x + \frac{x^2}2 + \frac{x^3}{3!} \dots$ Your function $\frac{e^x-1}x = 1 + \frac{x}2 + \frac{x^2}{3!}+\dots$ Substitute x and truncate the series based on your desired precision.

ps. In WolframAlpha if you substitute $x=10^{-14}$ you'll get a result less than $1$, which is nonsense.

Answer 4

You are seeing exactly the exponential expansion in both cases, limited to the available precision, which is computed with a 64 bit double in the default case and a hundred digits in the "More Digits" case.

Let's try it by hand: $$e^x \approx 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \dots$$

With a double: $1 + 10^{-15} + 10^{-30}/2 + \dots$ and we have 53 bits $\approx$ 17.46 decimal digits of accuracy, so we can drop the $x^2$ and higher terms.

Now, what is $1+10^{-15}$ in 64 bit floating point representation? Well, it certainly can't represent it exactly. Rather, we have a 53 bit binary number which tries to get as close as possible. Note that $2^{-50}$ + $2^{-52}$ = $1.110223\dots \cdot 10^{-15}$, so that's the best we can do.

Subtract off $1$ and you're left with this as an error term.

The hundred-digit one is correct. You can see the $10^{-15}$ term, the half-$10^{-30}$ term (5 and then a bunch of zeros), the cubic term--it's $1 over 6$ so you get 16666 in the decimal expansion, and then the quartic term comes in on top of it $6666666... + 0416666... \rightarrow 70833333$ etc..