Another case where writing your numerical algorithms can be faster than using the standard lib is if you know you're only working on a small subset of the possible input domain and can write a highly specialized function.
For example, a certain function may take arbitrary dimensional complex matrices as input. However if you know that you will only be passing in 3x3 Real positive semidefinite matrices you can probably write a faster function that only works on those matrices by using all that extra information you have. In Matlab I've gotten order of magnitude speedups over the standard functions by rolling my own highly specialized functions.
For example, a certain function may take arbitrary dimensional complex matrices as input. However if you know that you will only be passing in 3x3 Real positive semidefinite matrices you can probably write a faster function that only works on those matrices by using all that extra information you have. In Matlab I've gotten order of magnitude speedups over the standard functions by rolling my own highly specialized functions.