Singleprecision floatingpoint format is a computer number format that occupies 4 bytes (32 bits) in computer memory and represents a wide dynamic range of values by using a floating point.
In IEEE 7542008 the 32bit base 2 format is officially referred to as binary32. It was called single in IEEE 7541985. In older computers, other floatingpoint formats of 4 bytes were used.
One of the first programming languages to provide single and doubleprecision floatingpoint data types was Fortran. Before the widespread adoption of IEEE 7541985, the representation and properties of the double float data type depended on the computer manufacturer and computer model.
Singleprecision binary floatingpoint is used due to its wider range over fixed point (of the same bitwidth), even if at the cost of precision.
Single precision is known as REAL in Fortran,^{[1]} as float in C, C++, C#, Java,^{[2]} as Float in Haskell,^{[3]} and as single in Delphi (Pascal), Visual Basic, and MATLAB. However, float in Python, Ruby, PHP, and OCaml and single in versions of Octave prior to 3.2 refer to doubleprecision numbers. In PostScript the only floatingpoint precision is single.
Contents

IEEE 754 singleprecision binary floatingpoint format: binary32 1

Exponent encoding 1.1

Converting from decimal representation to binary32 format 1.2

Singleprecision examples 1.3

Converting from singleprecision binary to decimal 1.4

Trivia 1.5

See also 2

External links 3

References 4
IEEE 754 singleprecision binary floatingpoint format: binary32
The IEEE 754 standard specifies a binary32 as having:
This gives from 6 to 9 significant decimal digits precision (if a decimal string with at most 6 significant decimal is converted to IEEE 754 single precision and then converted back to the same number of significant decimal, then the final string should match the original; and if an IEEE 754 single precision is converted to a decimal string with at least 9 significant decimal and then converted back to single, then the final number must match the original ^{[4]}).
Sign bit determines the sign of the number, which is the sign of the significand as well. Exponent is either an 8 bit signed integer from −128 to 127 (2's Complement) or an 8 bit unsigned integer from 0 to 255 which is the accepted biased form in IEEE 754 binary32 definition. If the unsigned integer format is used, the exponent value used in the arithmetic is the exponent shifted by a bias – for the IEEE 754 binary32 case, an exponent value of 127 represents the actual zero (i.e. for 2^{e − 127} to be one, e must be 127).
The true significand includes 23 fraction bits to the right of the binary point and an implicit leading bit (to the left of the binary point) with value 1 unless the exponent is stored with all zeros. Thus only 23 fraction bits of the significand appear in the memory format but the total precision is 24 bits (equivalent to log_{10}(2^{24}) ≈ 7.225 decimal digits). The bits are laid out as follows:
The real value assumed by a given 32 bit binary32 data with a given biased exponent e (the 8 bit unsigned integer) and a 23 bit fraction is = (1)^\text{sign}(1.b_{22}b_{21}...b_{0})_2 \times 2^{e127} where more precisely we have \text{value} = (1)^\text{sign}\left(1 + \sum_{i=1}^{23} b_{23i} 2^{i} \right)\times 2^{(e127)}.
In this example:

\text{sign} = 0

1 + \sum_{i=1}^{23} b_{23i} 2^{i} = 1 + 2^{2} = 1.25

2^{(e127)} = 2^{124127} = 2^{3}
thus:

\text{value} = 1.25 \times 2^{3} = 0.15625
Exponent encoding
The singleprecision binary floatingpoint exponent is encoded using an offsetbinary representation, with the zero offset being 127; also known as exponent bias in the IEEE 754 standard.

E_{min} = 01_{H}−7F_{H} = −126

E_{max} = FE_{H}−7F_{H} = 127

Exponent bias = 7F_{H} = 127
Thus, in order to get the true exponent as defined by the offset binary representation, the offset of 127 has to be subtracted from the stored exponent.
The stored exponents 00_{H} and FF_{H} are interpreted specially.
Exponent

Significand zero

Significand nonzero

Equation

00_{H}

zero, −0

denormal numbers

(−1)^{signbits}×2^{−126}× 0.significandbits

01_{H}, ..., FE_{H}

normalized value

(−1)^{signbits}×2^{exponentbits−127}× 1.significandbits

FF_{H}

±infinity

NaN (quiet, signalling)


The minimum positive (denormal) value is 2^{−149} ≈ 1.4 × 10^{−45}. The minimum positive normal value is 2^{−126} ≈ 1.18 × 10^{−38}. The maximum representable value is (2−2^{−23}) × 2^{127} ≈ 3.4 × 10^{38}.
Converting from decimal representation to binary32 format
In general refer to the IEEE 754 standard itself for the strict conversion (including the rounding behaviour) of a real number into its equivalent binary32 format.
Here we can show how to convert a base 10 real number into an IEEE 754 binary32 format using the following outline:

consider a real number with an integer and a fraction part such as 12.375

convert and normalize the integer part into binary

convert the fraction part using the following technique as shown here

add the two results and adjust them to produce a proper final conversion
Conversion of the fractional part:
consider 0.375, the fractional part of 12.375. To convert it into a binary fraction, multiply the fraction by 2, take the integer part and remultiply new fraction by 2 until a fraction of zero is found or until the precision limit is reached which is 23 fraction digits for IEEE 754 binary32 format.
0.375 x 2 = 0.750 = 0 + 0.750 => b_{−1} = 0, the integer part represents the binary fraction digit. Remultiply 0.750 by 2 to proceed
0.750 x 2 = 1.500 = 1 + 0.500 => b_{−2} = 1
0.500 x 2 = 1.000 = 1 + 0.000 => b_{−3} = 1, fraction = 0.000, terminate
We see that (0.375)_{10} can be exactly represented in binary as (0.011)_{2}. Not all decimal fractions can be represented in a finite digit binary fraction. For example decimal 0.1 cannot be represented in binary exactly. So it is only approximated.
Therefore (12.375)_{10} = (12)_{10} + (0.375)_{10} = (1100)_{2} + (0.011)_{2} = (1100.011)_{2}
Also IEEE 754 binary32 format requires that you represent real values in (1.x_1x_2...x_{23})_2 \times 2^{e} format, (see Normalized number, Denormalized number) so that 1100.011 is shifted to the right by 3 digits to become (1.100011)_2 \times 2^{3}
Finally we can see that: (12.375)_{10} =(1.100011)_2 \times 2^{3}
From which we deduce:

The exponent is 3 (and in the biased form it is therefore 130 = 1000 0010)

The fraction is 100011 (looking to the right of the binary point)
From these we can form the resulting 32 bit IEEE 754 binary32 format representation of 12.375 as: 01000001010001100000000000000000 = 41460000_{H}
Note: consider converting 68.123 into IEEE 754 binary32 format: Using the above procedure you expect to get 42883EF9_{H} with the last 4 bits being 1001 However due to the default rounding behaviour of IEEE 754 format what you get is 42883EFA_{H} whose last 4 bits are 1010 .
Ex 1: Consider decimal 1 We can see that: (1)_{10} =(1.0)_2 \times 2^{0}
From which we deduce:

The exponent is 0 (and in the biased form it is therefore 127 = 0111 1111 )

The fraction is 0 (looking to the right of the binary point in 1.0 is all 0 = 000...0)
From these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 1 as: 00111111100000000000000000000000 = 3f800000_{H}
Ex 2: Consider a value 0.25 . We can see that: (0.25)_{10} =(1.0)_2 \times 2^{2}
From which we deduce:

The exponent is −2 (and in the biased form it is 127+(−2)= 125 = 0111 1101 )

The fraction is 0 (looking to the right of binary point in 1.0 is all zeros)
From these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 0.25 as: 00111110100000000000000000000000 = 3e800000_{H}
Ex 3: Consider a value of 0.375 . We saw that 0.375 = {(1.1)_2}\times 2^{2}
Hence after determining a representation of 0.375 as {(1.1)_2}\times 2^{2} we can proceed as above:

The exponent is −2 (and in the biased form it is 127+(−2)= 125 = 0111 1101 )

The fraction is 1 (looking to the right of binary point in 1.1 is a single 1 = x_{1})
From these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 0.375 as: 00111110110000000000000000000000 = 3ec00000_{H}
Singleprecision examples
These examples are given in bit representation, in hexadecimal, of the floatingpoint value. This includes the sign, (biased) exponent, and significand.
3f80 0000 = 1
c000 0000 = −2
7f7f ffff ≈ 3.4028234 × 10^{38} (max single precision)
0000 0000 = 0
8000 0000 = −0
7f80 0000 = infinity
ff80 0000 = −infinity
3eaa aaab ≈ 1/3
By default, 1/3 rounds up, instead of down like double precision, because of the even number of bits in the significand. The bits of 1/3 beyond the rounding point are 1010...
which is more than 1/2 of a unit in the last place.
Converting from singleprecision binary to decimal
We start with the hexadecimal representation of the value, 41c80000, in this example, and convert it to binary
41c8 0000_{16} = 0100 0001 1100 1000 0000 0000 0000 0000_{2}
then we break it down into three parts; sign bit, exponent and significand.
Sign bit: 0
Exponent: 1000 0011_{2} = 83_{16} = 131
Significand: 100 1000 0000 0000 0000 0000_{2} = 480000_{16}
We then add the implicit 24th bit to the significand
Significand: 1100 1000 0000 0000 0000 0000_{2} = C80000_{16}
and decode the exponent value by subtracting 127
Raw exponent: 83_{16} = 131
Decoded exponent: 131 − 127 = 4
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
bit 23 = 1
bit 22 = 0.5
bit 21 = 0.25
bit 20 = 0.125
bit 19 = 0.0625
.
.
bit 0 = 0.00000011920928955078125
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.
Decoded significand: 1 + 0.5 + 0.0625 = 1.5625 = C80000/2^{23}
Then we need to multiply with the base, 2, to the power of the exponent to get the final result
1.5625 × 2^{4} = 25
Thus
41c8 0000 = 25
This is equivalent to:
n = (1)^s \times
(1+m*2^{23})\times
2^{x  127}
where s is the sign bit, x is the exponent, and m is the significand.
Trivia
A fascinating example of how the floatingpoint format can be misused in a good way is shown in the Fast inverse square root implementation, where the complex calculation of square root and inversion are replaced (approximately) by a bitshift and subtraction operated on the 32bits of the floating point encoding as if it were an integer.
See also
External links

Online calculator

Online converter for IEEE 754 numbers with single precision

C source code to convert between IEEE double, single, and half precision
References

^ http://scc.ustc.edu.cn/zlsc/sugon/intel/compiler_f/main_for/lref_for/source_files/rfreals.htm

^ http://java.sun.com/docs/books/tutorial/java/nutsandbolts/datatypes.html

^ https://www.haskell.org/onlinereport/haskell2010/haskellch6.html#x131350006.4

^ William Kahan (1 October 1987). "Lecture Notes on the Status of IEEE Standard 754 for Binary FloatingPoint Arithmetic".
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.