I suspect that using the expm1 function would give you a better result.

Computing $e^x -1$ with the trivial formula in machine precision gives you only limited accuracy for small inputs: the fundamental reason is that there are only "few" floating point numbers around 1, and when you first compute $e^x$ the machine has to approximate your result to the closest floating point number. For instance, the next number after $1$ is $1+2^{-52}$, so every number in $[1, 1+2^{-52}]$ has to be replaced with one of the two extremes. So computing, for instance $e^x-1$ for $x=2^{-100}$ with good accuracy is a lost cause.

Hence there is the need for a separate library function that computes $e^x-1$ natively and is accurate also for small inputs (it can be implemented using a Taylor expansion for small inputs, for instance). Hence expm1, which is in most programming languages and is an optional part of the IEEE floating point format.

(Similarly, there is log1p to compute $\log(1+x)$. These functions are very handy when working with very small probabilities in log space, for instance.)

In any case, if you know that your function has a multiple zero, there is no reason not to use a Newton variant designed for multiple zeros.

I suspect that using the expm1 function would give you a better result.

Computing $e^x -1$ with the trivial formula in machine precision gives you only limited accuracy for small inputs: the fundamental reason is that there are only "few" floating point numbers around 1, and when you first compute $e^x$ the machine has to approximate your result to the closest floating point number. For instance, the next number after $1$ is $1+2^{-52}$, so every number in $[1, 1+2^{-52}]$ has to be replaced with one of the two extremes. So computing, for instance, $e^{2^{-100}} \approx 1 + 2^{-100}$ and then subtracting 1 results in a huge (relative) error.

Hence there is the need for a separate library function that computes $e^x-1$ natively and is accurate also for small inputs (it can be implemented using a Taylor expansion for small inputs, for instance). Hence expm1, which is in most programming languages. (The IEEE floating point standard "recommends" to include it in its implementations.)

(Similarly, there is log1p to compute $\log(1+x)$. These functions are very handy when working with very small probabilities in log space, for instance.)

In any case, if you know that your function has a multiple zero, there is no reason not to use a Newton variant designed for multiple zeros.