Exponent bias

2025-04-16T22:06:00Z

2A02:A313:2B7:2B80:351E:33C5:61F7:2582: 524387 corrected to 524287

{{Short description|Offset of the exponent field of floating-point numbers}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
In [[IEEE 754]] [[Floating-point arithmetic|floating-point]] numbers, the [[exponent]] is biased in the [[biasing|engineering sense of the word]] – the value stored is offset from the actual value by the '''exponent bias''', also called a '''biased exponent'''.<ref name="Gosling_1980"/>
Biasing is done because exponents have to be signed values in order to be able to represent both tiny and huge values, but [[two's complement]], the usual representation for signed values, would make comparison harder.

To solve this problem the exponent is stored as an unsigned value which is suitable for comparison, and when being interpreted it is converted into an exponent within a signed range by subtracting the bias.

By arranging the fields such that the sign bit takes the most significant bit position, the biased exponent takes the middle position, then the [[significand]] will be the least significant bits and the resulting value will be ordered properly. This is the case whether or not it is interpreted as a floating-point or integer value. The purpose of this is to enable high speed comparisons between floating-point numbers using fixed-point hardware.

If there are <math>E</math> bits in the exponent, the bias
is typically set as <math>b = 2^{E-1}-1</math>.<ref name="Bryant_2010"/>

Therefore, the possible integer values that the biased exponent can express lie in the range <math>[1-b, b]</math>.
To understand this range, with <math>E</math> bits in the exponent, the possible unsigned integers lie in the range <math>[0, 2^{E}-1]</math>.
However, the strings containing all zeros and all ones are reserved for special values, so the expressible integers lie in the range <math>[1, 2^{E}-2]</math>.
It follows that:

* The maximum biased value is <math>(2^{E}-2) - b = 2b - b = b</math>.
* The minimum biased value is <math>1 - b</math>.

When interpreting the floating-point number, the bias is subtracted to retrieve the actual exponent.

* For a [[Half-precision_floating-point_format|half-precision]] number, the exponent is stored in the range {{math|[1, 30]}} (0 and 31 have special meanings), and is interpreted by subtracting the bias for a 5-bit exponent (15) to get an exponent value in the range {{math|[−14, 15]}}.
* For a [[Single-precision floating-point format|single-precision]] number, the exponent is stored in the range {{math|[1, 254]}} (0 and 255 have special meanings), and is interpreted by subtracting the bias for an 8-bit exponent (127) to get an exponent value in the range {{math|[−126, 127]}}.
* For a [[Double-precision_floating-point_format|double-precision]] number, the exponent is stored in the range {{math|[1, 2046]}} (0 and 2047 have special meanings), and is interpreted by subtracting the bias for an 11-bit exponent (1023) to get an exponent value in the range {{math|[−1022, 1023]}}.
* For a [[Quadruple-precision_floating-point_format|quadruple-precision]] number, the exponent is stored in the range {{math|[1, 32766]}} (0 and 32767 have special meanings), and is interpreted by subtracting the bias for a 15-bit exponent (16383) to get an exponent value in the range {{math|[−16382, 16383]}}.
* For an [[Octuple-precision floating-point format|octuple-precision]] number, the exponent is stored in the range {{math|[1, 524286]}} (0 and 524287 have special meanings), and is interpreted by subtracting the bias for a 19-bit exponent (262143) to get an exponent value in the range {{math|[−262142, 262143]}}.

== History ==
The floating-point format of the [[IBM 704]] introduced the use of a biased exponent in 1954.

==See also==
* [[Offset binary]] (also referred to as excess-K)

==References==
{{Reflist|refs=
<ref name="Gosling_1980">{{cite book |title=Design of Arithmetic Units for Digital Computers |author-first=John B. |author-last=Gosling |editor-first=Frank H. |editor-last=Sumner |date=1980 |edition=1 |publisher=[[The Macmillan Press Ltd]] |location=Department of Computer Science, [[University of Manchester]], Manchester, UK |isbn=0-333-26397-9 |chapter=6.1 Floating-Point Notation / 6.8.5 Exponent Representation |series=Macmillan Computer Science Series |pages=74, 91, 137 |quote=[…] In [[floating-point representation]], a number ''x'' is represented by two signed numbers ''m'' and ''e'' such that ''x'' = ''m'' · ''b''<sup>''e''</sup> where ''m'' is the [[significand|mantissa]], ''e'' the [[exponent]] and ''b'' the [[base (exponentiation)|base]]. […] The mantissa is sometimes termed the characteristic and a version of the exponent also has this title from some authors. It is hoped that the terms here will be unambiguous. […] [w]e use a[n exponent] value which is shifted by half the binary range of the number. […] This special form is sometimes referred to as a biased exponent, since it is the conventional value plus a constant. Some authors have called it a characteristic, but this term should not be used, since [[Control Data Corporation|CDC]] and others use this term for the mantissa. It is also referred to as an '[[excess-K|excess -]]' representation, where, for example, - is 64 for a 7-bit exponent (2<sup>7−1</sup> = 64). […]}}</ref>
<ref name="Bryant_2010">{{cite book |title=Computer systems: a programmer's perspective |author-last1=O'Hallaron |author-first1=David R. |author-first2=Randal E. |author-last2=Bryant |author-link2=Randal E. Bryant |date=2010 |publisher=[[Prentice Hall]] |location=Boston, Massachusetts, USA |isbn=978-0-13-610804-7 |edition=2}}</ref>
}}

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[[Category:Computer arithmetic]]

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Exponent bias