Reliable Computing 8: 307–312, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
Interval Arithmetic on Multimedia Architectures
URGEN WOLFF VON GUDENBERG
ur Informatik II, Universit
urzburg, Am Hubland, D–97074 W
(Received: 16 February 2001; accepted: 4 March 2002)
Abstract. In this paper, we show how to implement interval arithmetic on multimedia architecture
extensions. Due to the difﬁculties that current architectures have with the directed rounding modes
the speed-up is rather modest. The study, nevertheless, may be helpful in future design of hardware
extensions for interval arithmetic.
Recent hardware architectures provide a potential for parallelism in evaluating an
arithmetic expression or running an algorithm. The architecture of the extension
usually follows the SIMD principle where a single operation combines multiple
operands in a pipeline. The cheapest and nearly ubiquitously available specimen
of such architectures are the multimedia extensions like Intel’s ISSE , AMD’s
3DNow! , or Motorola’s AltiVec .
In our study we implemented interval arithmetic on these architectures in order
to exploit the parallelism to improve the runtime. This goal, however, was only
reached for one architecture. This is mainly due to the lack of directed rounding
modes or the inefﬁcient switch between the rounding modes. We, hence, tested
various alternatives for directed roundings. This is similar to our portable software
2. Implementation of Interval Arithmetic on Multimedia Architectures
The extension of the architecture for multimedia applications usually provides one
or more data types and instructions to support the pipelined execution of (vector)
arithmetic operations, and optionally a set of registers.
We tested 3 different rounding modes, “native switched” by an assembler state-
ment after each operation, “one-sided” using
(x)=−(∆(−x)), and “multiplicative”
using round to nearest and multiplication with pred(1) or succ(1), respectively. The
ﬁrst two are tested only, if the architecture provides directed roundings.
For interval multiplication (and division) two different methods were checked.
The calculation of 8 products and subsequent min/max computation with one-sided
rounding is called the brute force algorithm. In nearly all software implementations
the case selection algorithm is prefered. The bounds contributing to the minimum