This paper presents the design of a high-speed processor which performs staggered interval arithmetic. Each staggered interval is represented as the sum of a set of floating point numbers plus an interval, which consists of two floating point endpoints. Staggered interval arithmetic allows the precision of the computation to be specified and the accuracy of the result to be determined. Efficient arithmetic algorithms, which reduce the number of floating point operations needed to perform staggered interval arithmetic, are introduced. To achieve high performance, the processor employs an array of pipelined floating point arithmetic units and two long accumulators. The processor provides direct hardware support for accurate and numerically reliable vector and matrix computations.