An Inconsistency in Numerical Approximation Consider the expression 
$$
10^5 - \frac{10^{10}}{1+10^5}.
$$
Using the elementary properties of fractions we can evaluate the expression as 
$$
10^5 - \frac{10^{10}}{1+10^5} = \frac{10^5 + 10^{10} - 10^{10}}{1+10^5} = \frac{10^5}{1+10^5}\approx 1.
$$
Note that the approximation $10^5+1 \approx 10^5$ is used in the last step. Now suppose we use the same approximation, but apply it before we perform the subtraction. We get
$$
10^5 - \frac{10^{10}}{1+10^5} \approx 10^5 - \frac{10^{10}}{10^5} = 0.
$$
The same logic works for 
$$
10^p - \frac{10^{2p}}{1+10^p}
$$
for arbitrary large $p$, so it cannot be simply an issue with the accuracy of the approximation. 
Is there an easy explanation of what's going on here?

It is simply an issue of accuracy of approximation.  Let me write $x = 10^p$.  Then your expression is 
$$ x - \frac{x^2}{1+x}$$
Note that $$\frac{x^2}{1+x} = \frac{x}{1/x + 1} = x (1 - 1/x + O(1/x^2)) = x - 1 + O(1/x)$$
so that
$$ x - \frac{x^2}{1+x} = x - (x - 1 + O(1/x)) = 1 + O(1/x)$$
In your second calculation you only evaluated $x^2/(1+x)$ to within $O(1)$, not $O(1/x)$, so naturally you have an error at the end that is $O(1)$.
