[chongli]

Rounding error is not the only source of error with floating point. There is also loss of significance, which in the worst case is called catastrophic cancellation [1]. This occurs when subtracting two numbers which are very close in magnitude, for example:

1.23456789 - 1.23456788 = 0.00000001 = 1 * 10^-8

So here we’ve gone from 9 significant figures down to 1. This phenomenon will make a naïve Taylor series approximation of e^x be very inaccurate for negative x, due to the sign alternating between positive and negative on every term, causing a lot of catastrophic cancellation.

[1] https://en.wikipedia.org/wiki/Loss_of_significance

[saagarjha]

(Note that in your example no accuracy is lost)