Editing IEEE 754-1985 (section)

== Range and precision ==
[[File:IEEE 754 relative precision.svg|thumb|Relative precision of single (binary32) and double precision (binary64) numbers, compared with decimal representations using a fixed number of [[significant digits]]. Relative precision is defined here as ulp(''x'')/''x'', where ulp(''x'') is the [[unit in the last place]] in the representation of ''x'', i.e. the gap between ''x'' and the next representable number.]]
Precision is defined as the minimum difference between two successive mantissa representations; thus it is a function only in the mantissa; while the gap is defined as the difference between two successive numbers.<ref>{{citation |title=Computer Arithmetic |author1=Hossam A. H. Fahmy |author2=Shlomo Waser |author3=Michael J. Flynn |url=http://arith.stanford.edu/~hfahmy/webpages/arith_class/arith.pdf |access-date=2011-01-02 |url-status=dead |archive-url=https://web.archive.org/web/20101008203307/http://arith.stanford.edu/~hfahmy/webpages/arith_class/arith.pdf |archive-date=2010-10-08}}</ref>

=== Single precision ===
[[Single-precision]] numbers occupy 32 bits. In single precision:
* The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the exponent field and the binary value 1 in the fraction field) are
*: ±2<sup>&minus;23</sup> &times; 2<sup>&minus;126</sup> ≈ ±1.40130{{e|−45}}
* The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the exponent field and 0 in the fraction field) are
*: ±1 &times; 2<sup>&minus;126</sup> ≈ ±1.17549{{e|−38}}
* The finite positive and finite negative numbers furthest from zero (represented by the value with 254 in the exponent field and all 1s in the fraction field) are
*: ±(2−2<sup>−23</sup>) &times; 2<sup>127</sup><ref name="Kahan">{{Cite web
  | author = William Kahan |author-link=William Kahan
  | title = Lecture Notes on the Status of IEEE 754
  | date =  October 1, 1997
  | publisher = University of California, Berkeley
  | url = http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF
  | access-date = 2007-04-12 }}</ref> ≈ ±3.40282{{e|38}}

Some example range and gap values for given exponents in single precision:

{| class="wikitable" style="text-align:right;"
|- style="text-align:center;"
! Actual Exponent (unbiased)
! Exp (biased)
! Minimum
! Maximum
! Gap
|-
| −1
| 126
| 0.5
| ≈ 0.999999940395
| ≈ 5.96046e-8
|-
| 0
| 127
| 1
| ≈ 1.999999880791
| ≈ 1.19209e-7
|-
| 1
| 128
| 2
| ≈ 3.999999761581
| ≈ 2.38419e-7
|-
| 2
| 129
| 4
| ≈ 7.999999523163
| ≈ 4.76837e-7
|-
| 10
| 137
| 1024
| ≈ 2047.999877930
| ≈ 1.22070e-4
|-
| 11
| 138
| 2048
| ≈ 4095.999755859
| ≈ 2.44141e-4
|-
| 23
| 150
| 8388608
| 16777215
| 1
|-
| 24
| 151
| 16777216
| 33554430
| 2
|-
| 127
| 254
| ≈ 1.70141e38
| ≈ 3.40282e38
| ≈ 2.02824e31
|}

As an example, 16,777,217 cannot be encoded as a 32-bit float as it will be rounded to 16,777,216. However, all integers within the representable range that are a power of 2 can be stored in a 32-bit float without rounding.

=== Double precision ===

[[Double-precision]] numbers occupy 64 bits. In double precision:
* The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the Exp field and the binary value 1 in the Fraction field) are
*: ±2<sup>&minus;52</sup> &times; 2<sup>&minus;1022</sup> ≈ ±4.94066{{e|−324}}
* The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the Exp field and 0 in the fraction field) are
*: ±1 &times; 2<sup>&minus;1022</sup> ≈ ±2.22507{{e|−308}}
* The finite positive and finite negative numbers furthest from zero (represented by the value with 2046 in the Exp field and all 1s in the fraction field) are
*: ±(2−2<sup>−52</sup>) &times; 2<sup>1023</sup><ref name="Kahan" /> ≈ ±1.79769{{e|308}}

Some example range and gap values for given exponents in double precision:

{| class="wikitable" style="text-align:right;"
|- style="text-align:center;"
! Actual Exponent (unbiased)
! Exp (biased)
! Minimum
! Maximum
! Gap
|-
| −1
| 1022
| 0.5
| ≈ 0.999999999999999888978
| ≈ 1.11022e-16
|-
| 0
| 1023
| 1
| ≈ 1.999999999999999777955
| ≈ 2.22045e-16
|-
| 1
| 1024
| 2
| ≈ 3.999999999999999555911
| ≈ 4.44089e-16
|-
| 2
| 1025
| 4
| ≈ 7.999999999999999111822
| ≈ 8.88178e-16
|-
| 10
| 1033
| 1024
| ≈ 2047.999999999999772626
| ≈ 2.27374e-13
|-
| 11
| 1034
| 2048
| ≈ 4095.999999999999545253
| ≈ 4.54747e-13
|-
| 52
| 1075
| 4503599627370496	
| 9007199254740991
| 1
|-
| 53
| 1076
| 9007199254740992	
| 18014398509481982
| 2
|-
| 1023
| 2046
| ≈ 8.98847e307
| ≈ 1.79769e308
| ≈ 1.99584e292
|}

=== Extended formats ===

The standard also recommends extended format(s) to be used to perform internal computations at a higher precision than that required for the final result, to minimise round-off errors: the standard only specifies minimum precision and exponent requirements for such formats. The [[x87]] [[extended precision|80-bit extended format]] is the most commonly implemented extended format that meets these requirements.