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{{short description|32-bit computer number format}}
{{Cleanup|reason=<br/>{{*}} This article doesn't provide a good structure to lead users from easy to deeper understanding
'''Single-precision floating-point format''' (sometimes called '''FP32''' or '''float32''') is a [[computer number format]], usually occupying [[32 bits]] in [[computer memory]]; it represents a wide [[dynamic range]] of numeric values by using a [[floating point|floating radix point]].
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One of the first [[programming language]]s to provide single- and double-precision floating-point data types was [[Fortran]]. Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the [[computer manufacturer]] and computer model, and upon decisions made by programming-language designers. E.g., [[GW-BASIC]]'s single-precision data type was the [[32-bit MBF]] floating-point format.
Single precision is termed ''REAL(4)'' or ''REAL*4'' in [[Fortran]];<ref>{{cite web|url=http://scc.ustc.edu.cn/zlsc/sugon/intel/compiler_f/main_for/lref_for/source_files/rfreals.htm|title=REAL Statement|website=scc.ustc.edu.cn|access-date=2013-02-28|archive-date=2021-02-24|archive-url=https://web.archive.org/web/20210224045812/http://scc.ustc.edu.cn/zlsc/sugon/intel/compiler_f/main_for/lref_for/source_files/rfreals.htm|url-status=dead}}</ref> ''SINGLE-FLOAT'' in [[Common Lisp]];<ref>{{Cite web|url=https://www.lispworks.com/documentation/HyperSpec/Body/t_short_.htm|title=CLHS: Type SHORT-FLOAT, SINGLE-FLOAT, DOUBLE-FLOAT...|website=www.lispworks.com}}</ref> ''float binary(p)'' with p≤21, ''float decimal(p)'' with the maximum value of p depending on whether the DFP (IEEE 754 DFP) attribute applies, in PL/I; ''float'' in [[C (programming language)|C]] with IEEE 754 support, [[C++]] (if it is in C), [[C Sharp (programming language)|C#]] and [[Java (programming language)|Java]];<ref>{{cite web|url=https://docs.oracle.com/javase/tutorial/java/nutsandbolts/datatypes.html|title=Primitive Data Types|website=Java Documentation}}</ref> ''Float'' in [[Haskell (programming language)|Haskell]]<ref>{{cite web|url=https://www.haskell.org/onlinereport/haskell2010/haskellch6.html#x13-1350006.4|title=6 Predefined Types and Classes|date=20 July 2010|website=haskell.org}}</ref> and [[Swift (programming language)|Swift]];<ref>{{cite web|url=https://developer.apple.com/documentation/swift/float|title=Float|website=Apple Developer Documentation}}</ref> and ''Single'' in [[Object Pascal]] ([[Delphi (programming language)|Delphi]]), [[Visual Basic]], and [[MATLAB]]. However, ''float'' in [[Python (programming language)|Python]], [[Ruby (programming language)|Ruby]], [[PHP]], and [[OCaml]] and ''single'' in versions of [[GNU Octave|Octave]] before 3.2 refer to [[double-precision floating-point format|double-precision]] numbers. In most implementations of [[PostScript]], and some [[embedded systems]], the only supported precision is single.
{{Floating-point}}
== IEEE 754 standard: binary32<span class="anchor" id="IEEE 754 single-precision binary floating-point format: binary32"></span> ==
The IEEE 754 standard specifies a ''binary32'' as having:
* [[Sign bit]]: 1 bit
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[[Image:Float example.svg]]
The real value assumed by a given 32-bit ''binary32'' data with a given ''sign'', biased exponent ''
: <math>(-1)^{b_{31}} \times 2^{(b_{30}b_{29} \dots b_{23})_2 - 127} \times (1.b_{22}b_{21} \dots b_0)_2</math>,
which yields
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The stored exponents 00<sub>H</sub> and FF<sub>H</sub> are interpreted specially.
{|class="wikitable" style="text-align: center;"
|-
! Exponent !! fraction = 0 !! fraction ≠ 0 !! Equation
|-
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| 01<sub>H</sub>, ..., FE<sub>H</sub> = 00000001<sub>2</sub>, ..., 11111110<sub>2</sub>||colspan=2| normal value || <math>(-1)^\text{sign}\times2^{\text{exponent}-127}\times1.\text{fraction}</math>
|-
| FF<sub>H</sub> = 11111111<sub>2</sub>|| ±[[infinity]] || [[NaN]] (quiet,
|}
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Each of the 24 bits of the significand (including the implicit 24th bit), bit 23 to bit 0, represents a value, starting at 1 and halves for each bit, as follows:
<pre>
bit 17 = 0.015625
.▼
bit 0 = 0.00000011920928955078125
</pre>
The significand in this example has three bits set: bit 23, bit 22, and bit 19. We can now decode the significand by adding the values represented by these bits.
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* Decimals between 4 and 8: fixed interval 2<sup>−21</sup>
* ...
* Decimals between 2<sup>n</sup> and 2<sup>n+1</sup>: fixed interval 2<sup>
* ...
* Decimals between 2<sup>22</sup>=4194304 and 2<sup>23</sup>=8388608: fixed interval 2<sup>−1</sup>=0.5
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* Integers between 2<sup>25</sup> and 2<sup>26</sup> round to a multiple of 4
* ...
* Integers between 2<sup>n</sup> and 2<sup>n+1</sup> round to a multiple of 2<sup>
* ...
* Integers between 2<sup>127</sup> and 2<sup>128</sup> round to a multiple of 2<sup>104</sup>
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These examples are given in bit ''representation'', in [[hexadecimal]] and [[Binary number|binary]], of the floating-point value. This includes the sign, (biased) exponent, and significand.
{| style="font-family: monospace, monospace;"
0 00000000 00000000000000000000001<sub>2</sub> = 0000 0001<sub>16</sub> = 2<sup>−126</sup> × 2<sup>−23</sup> = 2<sup>−149</sup> ≈ 1.4012984643 × 10<sup>−45</sup>▼
|-
(smallest positive subnormal number)▼
|
▲
|}
By default, 1/3 rounds up, instead of down like [[
Encodings of qNaN and sNaN are not specified in [[
=== Optimizations ===
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