''Why is the interval package slow?''
All arithmetic interval operations are simulated in high-level octave language using C99
floating-point routines or multi-precision floating-point routines, which is a lot slower than hardware implementations [https://books.google.de/books?id=JTc4XdXFnQIC&pg=PA61]. Building interval arithmetic operations from floating-point routines is easy for simple monotonic functions, e. g., addition and subtraction, but is complex for others, e. g., [http://exp.ln0.de/heimlich-power-2011.htm interval power function], atan2, or [[#Reverse_arithmetic_operations|reverse functions]]. For some interval operations it is not even possible to rely on floating-point routines, since not all required routines are available in C99 or BLAS.
Great algorithms and optimizations exist for matrix arithmetic in GNU
octave. Good interval versions of these still have to be found and implemented.
''Why is the interval package accurate?''
Some basic operations are provided by the C library on common platforms with directed rounding and correctly rounded result: plus, minus, division, multiplication and square root. All other GNU Octave arithmetic functions are not guaranteed to produce accurate results, because they are based on C99 floating-point routines [http: //www.gnu.org/software/libc/manual/html_node/Errors-in-Math-Functions.html#Errors-in-Math-Functions]. Their results depend on hardware, system libraries and compilation options.
The interval package handles all arithmetic functions with the help of the [http://www.mpfr.org/ GNU MPFR library]. With MPFR it is possible to compute system-independent, valid and tight enclosures of the correct results.