Singleprecision floatingpoint format
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Singleprecision floatingpoint 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 radix point.
A floatingpoint variable can represent a wider range of numbers than a fixedpoint variable of the same bit width at the cost of precision. A signed 32bit integer variable has a maximum value of 2^{31} − 1 = 2,147,483,647, whereas an IEEE 754 32bit base2 floatingpoint variable has a maximum value of (2 − 2^{−23}) × 2^{127} ≈ 3.4028235 × 10^{38}. All integers with 7 or fewer decimal digits, and any 2^{n} for a whole number −149 ≤ n ≤ 127, can be converted exactly into an IEEE 754 singleprecision floatingpoint value.
In the IEEE 7542008 standard, the 32bit base2 format is officially referred to as binary32; it was called single in IEEE 7541985. IEEE 754 specifies additional floatingpoint types, such as 64bit base2 double precision and, more recently, base10 representations.
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 floatingpoint data types depended on the computer manufacturer and computer model, and upon decisions made by programminglanguage designers. E.g., GWBASIC's singleprecision data type was the 32bit MBF floatingpoint format.
Single precision is termed REAL in Fortran,^{[1]} SINGLEFLOAT in Common Lisp,^{[2]} float in C, C++, C#, Java,^{[3]} Float in Haskell^{[4]} and Swift,^{[5]} and Single in Object Pascal (Delphi), Visual Basic, and MATLAB. However, float in Python, Ruby, PHP, and OCaml and single in versions of Octave before 3.2 refer to doubleprecision numbers. In most implementations of PostScript, and some embedded systems, the only supported precision is single.
Floatingpoint formats 

IEEE 754 

Other 
Alternatives 
IEEE 754 standard: binary32[edit]
The IEEE 754 standard specifies a binary32 as having:
 Sign bit: 1 bit
 Exponent width: 8 bits
 Significand precision: 24 bits (23 explicitly stored)
This gives from 6 to 9 significant decimal digits precision. If a decimal string with at most 6 significant digits is converted to the IEEE 754 singleprecision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original string. If an IEEE 754 singleprecision number is converted to a decimal string with at least 9 significant digits, and then converted back to singleprecision representation, the final result must match the original number.^{[6]}
The sign bit determines the sign of the number, which is the sign of the significand as well. The exponent is an 8bit unsigned integer from 0 to 255, in biased form: an exponent value of 127 represents the actual zero. Exponents range from −126 to +127 because exponents of −127 (all 0s) and +128 (all 1s) are reserved for special numbers.
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 32bit binary32 data with a given sign, biased exponent e (the 8bit unsigned integer), and a 23bit fraction is
 ,
which yields
In this example:
 ,
 ,
 ,
 ,
 .
thus:
 .
Note:
 ,
 ,
 ,
 .
Exponent encoding[edit]
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 offsetbinary 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  fraction = 0  fraction ≠ 0  Equation 

00_{H} = 00000000_{2}  ±zero  subnormal number  
01_{H}, ..., FE_{H} = 00000001_{2}, ..., 11111110_{2}  normal value  
FF_{H} = 11111111_{2}  ±infinity  NaN (quiet, signalling) 
The minimum positive normal value is and the minimum positive (subnormal) value is .
Converting decimal to binary32[edit]
This section possibly contains original research. (February 2020) 
This section may be confusing or unclear to readers. In particular, the examples are simple particular cases (simple values exactly representable in binary, without an exponent part). This section is also probably offtopic: this is not an article about conversion, and conversion from decimal using decimal arithmetic (as opposed to conversion from a character string) is uncommon. (February 2020) 
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 base10 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 repeat with the 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.
 , the integer part represents the binary fraction digit. Remultiply 0.750 by 2 to proceed
 , fraction = 0.011, terminate
We see that can be exactly represented in binary as . Not all decimal fractions can be represented in a finite digit binary fraction. For example, decimal 0.1 cannot be represented in binary exactly, only approximated. Therefore:
Since IEEE 754 binary32 format requires real values to be represented in format (see Normalized number, Denormalized number), 1100.011 is shifted to the right by 3 digits to become
Finally we can see that:
From which we deduce:
 The exponent is 3 (and in the biased form it is therefore
 The fraction is 100011 (looking to the right of the binary point)
From these we can form the resulting 32bit IEEE 754 binary32 format representation of 12.375:
Note: consider converting 68.123 into IEEE 754 binary32 format: Using the above procedure you expect to get with the last 4 bits being 1001. However, due to the default rounding behaviour of IEEE 754 format, what you get is , whose last 4 bits are 1010.
Example 1: Consider decimal 1. We can see that:
From which we deduce:
 The exponent is 0 (and in the biased form it is therefore
 The fraction is 0 (looking to the right of the binary point in 1.0 is all )
From these we can form the resulting 32bit IEEE 754 binary32 format representation of real number 1:
Example 2: Consider a value 0.25. We can see that:
From which we deduce:
 The exponent is −2 (and in the biased form it is )
 The fraction is 0 (looking to the right of binary point in 1.0 is all zeroes)
From these we can form the resulting 32bit IEEE 754 binary32 format representation of real number 0.25:
Example 3: Consider a value of 0.375. We saw that
Hence after determining a representation of 0.375 as we can proceed as above:
 The exponent is −2 (and in the biased form it is )
 The fraction is 1 (looking to the right of binary point in 1.1 is a single )
From these we can form the resulting 32bit IEEE 754 binary32 format representation of real number 0.375:
Converting binary32 to decimal[edit]
This section possibly contains original research. (February 2020) 
This section may be confusing or unclear to readers. In particular, there is only a very simple example, without rounding. This section is also probably offtopic: this is not an article about conversion, and conversion to decimal, using decimal arithmetic, is uncommon. (February 2020) 
If the binary32 value, 41C80000 in this example, is in hexadecimal we first convert it to binary:
then we break it down into three parts: sign bit, exponent, and significand.
 Sign bit:
 Exponent:
 Significand:
We then add the implicit 24th bit to the significand:
 Significand:
and decode the exponent value by subtracting 127:
 Raw exponent:
 Decoded exponent:
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 18 = 0.03125 bit 17 = 0.015625 . . bit 6 = 0.00000762939453125 bit 5 = 0.000003814697265625 bit 4 = 0.0000019073486328125 bit 3 = 0.00000095367431640625 bit 2 = 0.000000476837158203125 bit 1 = 0.0000002384185791015625 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:
Then we need to multiply with the base, 2, to the power of the exponent, to get the final result:
Thus
This is equivalent to:
where s is the sign bit, x is the exponent, and m is the significand.
Precision limitations on decimal values (between 1 and 16777216)[edit]
 Decimals between 1 and 2: fixed interval 2^{−23} (1+2^{−23} is the next largest float after 1)
 Decimals between 2 and 4: fixed interval 2^{−22}
 Decimals between 4 and 8: fixed interval 2^{−21}
 ...
 Decimals between 2^{n} and 2^{n+1}: fixed interval 2^{n23}
 ...
 Decimals between 2^{22}=4194304 and 2^{23}=8388608: fixed interval 2^{−1}=0.5
 Decimals between 2^{23}=8388608 and 2^{24}=16777216: fixed interval 2^{0}=1
Precision limitations on integer values[edit]
 Integers between 0 and 16777216 can be exactly represented (also applies for negative integers between −16777216 and 0)
 Integers between 2^{24}=16777216 and 2^{25}=33554432 round to a multiple of 2 (even number)
 Integers between 2^{25} and 2^{26} round to a multiple of 4
 ...
 Integers between 2^{n} and 2^{n+1} round to a multiple of 2^{n23}
 ...
 Integers between 2^{127} and 2^{128} round to a multiple of 2^{104}
 Integers greater than or equal to 2^{128} are rounded to "infinity".
Notable singleprecision cases[edit]
These examples are given in bit representation, in hexadecimal and binary, of the floatingpoint value. This includes the sign, (biased) exponent, and significand.
0 00000000 00000000000000000000001_{2} = 0000 0001_{16} = 2^{−126} × 2^{−23} = 2^{−149} ≈ 1.4012984643 × 10^{−45} (smallest positive subnormal number)
0 00000000 11111111111111111111111_{2} = 007f ffff_{16} = 2^{−126} × (1 − 2^{−23}) ≈ 1.1754942107 ×10^{−38} (largest subnormal number)
0 00000001 00000000000000000000000_{2} = 0080 0000_{16} = 2^{−126} ≈ 1.1754943508 × 10^{−38} (smallest positive normal number)
0 11111110 11111111111111111111111_{2} = 7f7f ffff_{16} = 2^{127} × (2 − 2^{−23}) ≈ 3.4028234664 × 10^{38} (largest normal number)
0 01111110 11111111111111111111111_{2} = 3f7f ffff_{16} = 1 − 2^{−24} ≈ 0.999999940395355225 (largest number less than one)
0 01111111 00000000000000000000000_{2} = 3f80 0000_{16} = 1 (one)
0 01111111 00000000000000000000001_{2} = 3f80 0001_{16} = 1 + 2^{−23} ≈ 1.00000011920928955 (smallest number larger than one)
1 10000000 00000000000000000000000_{2} = c000 0000_{16} = −2 0 00000000 00000000000000000000000_{2} = 0000 0000_{16} = 0 1 00000000 00000000000000000000000_{2} = 8000 0000_{16} = −0 0 11111111 00000000000000000000000_{2} = 7f80 0000_{16} = infinity 1 11111111 00000000000000000000000_{2} = ff80 0000_{16} = −infinity 0 10000000 10010010000111111011011_{2} = 4049 0fdb_{16} ≈ 3.14159274101257324 ≈ π ( pi ) 0 01111101 01010101010101010101011_{2} = 3eaa aaab_{16} ≈ 0.333333343267440796 ≈ 1/3 x 11111111 10000000000000000000001_{2} = ffc0 0001_{16} = qNaN (on x86 and ARM processors) x 11111111 00000000000000000000001_{2} = ff80 0001_{16} = sNaN (on x86 and ARM processors)
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.
Encodings of qNaN and sNaN are not specified in IEEE 754 and implemented differently on different processors. The x86 family and the ARM family processors use the most significant bit of the significand field to indicate a quiet NaN. The PARISC processors use the bit to indicate a signalling NaN.
Optimizations[edit]
The design of floatingpoint format allows various optimisations, resulting from the easy generation of a base2 logarithm approximation from an integer view of the raw bit pattern. Integer arithmetic and bitshifting can yield an approximation to reciprocal square root (fast inverse square root), commonly required in computer graphics.
See also[edit]
 IEEE 754
 ISO/IEC 10967, language independent arithmetic
 Primitive data type
 Numerical stability
 Scientific notation
References[edit]
 ^ "REAL Statement". scc.ustc.edu.cn. Archived from the original on 20210224. Retrieved 20130228.
 ^ "CLHS: Type SHORTFLOAT, SINGLEFLOAT, DOUBLEFLOAT..."
 ^ "Primitive Data Types". Java Documentation.
 ^ "6 Predefined Types and Classes". haskell.org. 20 July 2010.
 ^ "Float". Apple Developer Documentation.
 ^ William Kahan (1 October 1997). "Lecture Notes on the Status of IEEE Standard 754 for Binary FloatingPoint Arithmetic" (PDF). p. 4.