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A floating-point technique for extending the available precision

T. J. Dekker
- 01 Jun 1971 - 
- Vol. 18, Iss: 3, pp 224-242
TLDR
A technique is described for expressing multilength floating-point arithmetic in terms of singlelength floating point arithmetic, i.e. the arithmetic for an availablefloating-point number system.
Abstract: 
A technique is described for expressing multilength floating-point arithmetic in terms of singlelength floating point arithmetic, i.e. the arithmetic for an available (say: single or double precision) floating-point number system. The basic algorithms are exact addition and multiplication of two singlelength floating-point numbers, delivering the result as a doublelength floating-point number. A straight-forward application of the technique yields a set of algorithms for doublelength arithmetic which are given as ALGOL 60 procedures.

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REKENAFDEL
I
NG
T.J.
DEKKER
stichting
mathematisch
centrum
MR
118/70
.f>.
FLOATING-POINT
TECHNIQUE
FOR
EXTENDING
THE
AVAILABLE
PRECISION
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A
FLOATING-POINT
TECHNIQUE
FOR
EXTENDING
THE
AVAILABLE
PRECISION
Abstract
by
*)
T.J.
Dekker
A
technique
is
described
for
expressing
multilength
floating-point
arith-
metic
in
terms
of
singlelength
floating-point
arithmetic,
i.e.
the
arith-
metic
for
an
available
(say:
single
or
double
precision)
floating-point
number
system.
The
basic
algorithms
are
exact
addition
and
multiplication
of
two
singlelength
floating-point
numbers,
delivering
the
result
as
a
doublelength
floating-point
number. A
straightforward
application
of
the
technique
yields
a
set
of
algorithms
for
doublelength
arithmetic
which
are
given
as
ALGOL
60
procedures.
*)
Part
. . . • .
of
this
research
was done
while
the
author
was
visiting
Bell
Telephone
Laboratories,
Murray
Hill,
New
Jersey.
Contents
1
Introduction
2
Floating-point
number systems
3
Floating-point
operations
4 Exact
addition
5
Exact
multiplication
6
Splitting
into
halflength
numbers
7
Doublelength
arithmetic
8 Appendix.
ALGOL
60
procedures
9 Acknowledgements
10
References
page 1
4
6
8
16
21
24
32
36
36
Citations
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What every computer scientist should know about floating-point arithmetic

David E. Goldberg
- 01 Mar 1991 - 
TL;DR: This paper presents a tutorial on the aspects of floating-point that have a direct impact on designers of computer systems, and concludes with examples of how computer system builders can better support floating point.
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The Tera computer system

Robert Alverson, +5 more
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Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates,

Jonathan Richard Shewchuk
- 01 Oct 1997 - 
TL;DR: This article offers fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values and proposes a technique for adaptive precision arithmetic that can often speed these algorithms when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound.
Journal ArticleDOI

Accurate Sum and Dot Product

Takeshi Ogita, +2 more
- 01 Jun 2005 - 
TL;DR: Algorithms for summation and dot product of floating-point numbers are presented which are fast in terms of measured computing time and it is shown that the computed results are as accurate as if computed in twice or K-fold working precision.

Delaunay refinement mesh generation

Jonathan Richard Shewchuk, +2 more
TL;DR: This thesis aims to further this progress by cementing the foundations of two-dimensional Delaunay refinement, and by extending the technique and its analysis to three dimensions.
References
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Book

The Art of Computer Programming

Donald Ervin Knuth
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Book

Rounding Errors in Algebraic Processes

James Hardy Wilkinson
Journal ArticleDOI

Revised report on the algorithm language ALGOL 60

J. W. Backus, +12 more
- 01 Jan 1963 - 
TL;DR: A survey of the basic coustituents arid fcuturcs of the language is given, and the formal notation, by which the syntactic structure is defined, is explained.
Journal ArticleDOI

Pracniques: further remarks on reducing truncation errors

William Kahan
- 01 Jan 1965 - 
TL;DR: The AND and NOT operatimts are transf0rlned to multiplication and subtraction operations as described in (1) and (3).
Journal ArticleDOI

Quasi double-precision in floating point addition

Ole Møller
- 01 Mar 1965 - 
TL;DR: In this article, the authors proposed a method to avoid the loss of accuracy incurred in adding a small, accurate quantity to a larger one, using floating point addition, by keeping account of a small correction to the sum.