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New Developments in Interval Arithmetic and Their Implications for Floating-Point Standardization

M. H. van Emden

TL;DR

This paper argues that interval arithmetic can be implemented at speeds comparable to conventional floating-point units, enabling all arithmetic to carry interval guarantees with no exceptions. It formalizes a modern theory of approximation using finite floating-point intervals, and develops relational, closed interval operations that bound all real-valued possibilities. By employing set extensions and careful endpoint rounding, the authors show addition, subtraction, multiplication, and division can be performed without generating NaNs and with efficient case analysis. The work suggests a path toward an interval-friendly floating-point standard that preserves useful directed roundings and signed-zero behaviors while removing the need for exception signaling, with implications for hardware design and numerical reliability.

Abstract

We consider the prospect of a processor that can perform interval arithmetic at the same speed as conventional floating-point arithmetic. This makes it possible for all arithmetic to be performed with the superior security of interval methods without any penalty in speed. In such a situation the IEEE floating-point standard needs to be compared with a version of floating-point arithmetic that is ideal for the purpose of interval arithmetic. Such a comparison requires a succinct and complete exposition of interval arithmetic according to its recent developments. We present such an exposition in this paper. We conclude that the directed roundings toward the infinities and the definition of division by the signed zeros are valuable features of the standard. Because the operations of interval arithmetic are always defined, exceptions do not arise. As a result neither Nans nor exceptions are needed. Of the status flags, only the inexact flag may be useful. Denormalized numbers seem to have no use for interval arithmetic; in the use of interval constraints, they are a handicap.

New Developments in Interval Arithmetic and Their Implications for Floating-Point Standardization

TL;DR

This paper argues that interval arithmetic can be implemented at speeds comparable to conventional floating-point units, enabling all arithmetic to carry interval guarantees with no exceptions. It formalizes a modern theory of approximation using finite floating-point intervals, and develops relational, closed interval operations that bound all real-valued possibilities. By employing set extensions and careful endpoint rounding, the authors show addition, subtraction, multiplication, and division can be performed without generating NaNs and with efficient case analysis. The work suggests a path toward an interval-friendly floating-point standard that preserves useful directed roundings and signed-zero behaviors while removing the need for exception signaling, with implications for hardware design and numerical reliability.

Abstract

We consider the prospect of a processor that can perform interval arithmetic at the same speed as conventional floating-point arithmetic. This makes it possible for all arithmetic to be performed with the superior security of interval methods without any penalty in speed. In such a situation the IEEE floating-point standard needs to be compared with a version of floating-point arithmetic that is ideal for the purpose of interval arithmetic. Such a comparison requires a succinct and complete exposition of interval arithmetic according to its recent developments. We present such an exposition in this paper. We conclude that the directed roundings toward the infinities and the definition of division by the signed zeros are valuable features of the standard. Because the operations of interval arithmetic are always defined, exceptions do not arise. As a result neither Nans nor exceptions are needed. Of the status flags, only the inexact flag may be useful. Denormalized numbers seem to have no use for interval arithmetic; in the use of interval constraints, they are a handicap.

Paper Structure

This paper contains 15 sections, 6 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Why interval arithmetic?
  3. A theory of approximation
  4. Approximation structures
  5. An approximation structure for the reals
  6. Floating-point intervals: a finite approximation structure for the reals
  7. Interval Arithmetic
  8. Set extensions of functions
  9. Relational definitions
  10. The algorithm for interval addition and subtraction
  11. The algorithm for interval multiplication
  12. Division
  13. Related work
  14. Conclusions
  15. Acknowledgments

Key Result

theorem 1

If $\mathcal{A}$ is an approximation structure for $\mathcal{T}$, then for every $S \subset \mathcal{T}$ there exists a unique least (in the sense of the set-inclusion partial order) element $S'$ of $\mathcal{A}$ such that $S \subset S'$.

Figures (3)

  • Figure 1: Classification of nonempty intervals according to whether they contain at least one real of the sign indicated at the top of the second and third columns. Classes $P$ and $N$ are further decomposed according to whether they have a zero bound. As only non-empty intervals are classified, we have $u \leq v$.
  • Figure 2: Case analysis for multiplication of real intervals, $\langle a,b\rangle *\langle c,d\rangle$.
  • Figure 3: Case analysis for relational division of real intervals, $\langle a,b\rangle /\langle c,d\rangle$ when $a \leq b$, $c \leq d$. The last column refers to how the formula has been proved ("$D$" for a direct proof, "$S_1$" and "$S_2$" refer to a symmetry used to reduce it to an earlier case.) The "class" labels, $N,N_1,N_0,M,P_0,P_1,P$ are as in Figure \ref{['fig:class']}.

Theorems & Definitions (13)

  • definition 1
  • theorem 1
  • definition 2
  • theorem 2
  • definition 3
  • definition 4
  • theorem 3
  • definition 5
  • definition 6
  • definition 7
  • ...and 3 more