Are there any instances of scientific numerical problems where the choice of rounding mode matters?
There are usually a number of different rounding modes available: to $0$, away from $0$, to $\pm\infty$, nearest ties to even, nearest ties down. Apart from various currency-manipulation problems where rounding is mandated by problem details, does anyone know an example of where choice of rounding mode might actually matter?
A numerically stable algorithm would be insensitive to round-off errors and hence to choice of rounding mode, but are there any other issues that might be relevant?