Interval arithmetic and arbitrary precision arithmetic are two approaches that are useful to users whose applications require a guaranteed accuracy and who are willing to pay the price of slower execution to achieve their goals. A working of Exercises 10.17, 10.18, 13.5, and 13.11 demonstrates both the power and the limitations of interval arithmetic. The power is that guaranteed lower and upper bounds on the desired solution are computed, but the limitation is that these bounds may not be close together, especially if the problem being solved is ill conditioned. It should be clear from a working of the exercises that carrying out a complicated calculation with interval arithmetic would be very clumsy with only the rounding modes as tools. However, software packages are available that carry out interval computations automatically. INTLAB [Rum] is an efficient and powerful MATLAB toolbox for interval arithmetic.

Arbitrary precision systems make computations to arbitrarily high precision. These come in several flavors. The traditional approach is to carry out all computations using a prescribed precision, i.e., a prescribed number of significand bits or digits. Since no computer provides hardware support for this, such systems must be implemented in software, typically using an array of integers to represent the consecutive significand bits of a "big" floating point number. Operations on such numbers would traditionally be implemented using the integer arithmetic hardware operations. Systems of this kind are provided, for example, by the symbolic packages Mathematica and Maple