In computing, tapered floating point (TFP) is a format similar to
floating point
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base.
Numbers of this form ...
, but with variable-sized entries for the
significand
The significand (also coefficient, sometimes argument, or more ambiguously mantissa, fraction, or characteristic) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its s ...
and
exponent
In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
instead of the fixed-length entries found in normal floating-point formats. In addition to this, tapered floating-point formats provide a fixed-size pointer entry indicating the number of digits in the exponent entry. The number of digits of the significand entry (including the sign) results from the difference of the fixed total length minus the length of the exponent and pointer entries.
Thus numbers with a small exponent, i.e. whose
order of magnitude
In a ratio scale based on powers of ten, the order of magnitude is a measure of the nearness of two figures. Two numbers are "within an order of magnitude" of each other if their ratio is between 1/10 and 10. In other words, the two numbers are ...
is close to the one of 1, have a higher
relative precision than those with a large exponent.
History
The tapered floating-point scheme was first proposed by
Robert Morris of
Bell Laboratories
Nokia Bell Labs, commonly referred to as ''Bell Labs'', is an American industrial research and development company owned by Finnish technology company Nokia. With headquarters located in Murray Hill, New Jersey, the company operates several lab ...
in 1971,
and refined with ''leveling'' by Masao Iri and Shouichi Matsui of
University of Tokyo
The University of Tokyo (, abbreviated as in Japanese and UTokyo in English) is a public research university in Bunkyō, Tokyo, Japan. Founded in 1877 as the nation's first modern university by the merger of several pre-westernisation era ins ...
in 1981,
and by Hozumi Hamada of
Hitachi, Ltd.
Alan Feldstein of
Arizona State University
Arizona State University (Arizona State or ASU) is a public university, public research university in Tempe, Arizona, United States. Founded in 1885 as Territorial Normal School by the 13th Arizona Territorial Legislature, the university is o ...
and Peter Turner
of
Clarkson University
Clarkson University is a private research university with its main campus in Potsdam, New York. Clarkson has additional graduate programs and research facilities in the New York Capital District. It was established in 1896 and enrolled over 4 ...
described a tapered scheme resembling a conventional floating-point system except for the overflow or underflow conditions.
In 2013,
John Gustafson proposed the
Unum
Unum Group is an American insurance company headquartered in Chattanooga, Tennessee. Founded as Union Mutual in 1848 and known as UnumProvident from 1999-2007, the company is part of the Fortune 500. Unum Group was created by the 1999 merger of ...
number system, a variant of tapered floating-point arithmetic with an ''exact'' bit added to the representation and some
interval interpretation to the non-exact values.
See also
*
Logarithmic number system (LNS)
*
Symmetric level-index arithmetic (SLI)
References
Further reading
*
*
* . Previously published in:
*
*
* {{cite book , author-first=Nelson H. F. , author-last=Beebe , title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library , chapter=Chapter H.8 - Unusual floating-point systems , date=2017-08-22 , location=Salt Lake City, Utah, USA , publisher=
Springer International Publishing AG
Springer Nature or the Springer Nature Group is a German-British academic publishing company created by the May 2015 merger of Springer Science+Business Media and Holtzbrinck Publishing Group's Nature Publishing Group, Palgrave Macmillan, and Mac ...
, edition=1 , lccn=2017947446 , isbn=978-3-319-64109-6 , doi=10.1007/978-3-319-64110-2 , page=966 , s2cid=30244721 , quote=
��representation with a moveable boundary between exponent and significand, sacrificing precision only when a larger range is needed (sometimes called ''tapered arithmetic'')
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Computer arithmetic