Single-precision floating-point format

Single-precision floating-point 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 754-2008 the 32-bit base-2 format is officially referred to as binary32. It was called single in IEEE 754-1985. In older computers, different floating-point formats of 4 bytes were used, e.g., GW-BASIC's single-precision data type was the 32-bit MBF floating-point format.

One of the first programming languages 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 the double float data type depended on the computer manufacturer and computer model.

Single-precision binary floating-point is used due to its wider range over fixed point (of the same bit-width), even if at the cost of precision. A signed 32-bit integer can have a maximum value of 2³¹ − 1 = 2,147,483,647, whereas the maximum representable IEEE 754 floating-point value is (2 − 2⁻²³) × 2¹²⁷ ≈ 3.402823 × 10³⁸. All integers with 6 or fewer significant decimal digits can be converted to an IEEE 754 floating-point value without loss of precision, some integers up to 9 significant decimal digits can be converted to an IEEE 754 floating-point value without loss of precision, but no more than 9 significant decimal digits can be stored. As an example, the 32-bit integer 2,147,483,647 converts to 2,147,483,650 in IEEE 754 form.

Single precision is termed REAL in Fortran,^[1] float in C, C++, C#, Java,^[2] Float in Haskell,^[3] 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 double-precision numbers. In most implementations of PostScript, the only real precision is single.

Floating point precisions
IEEE 754
16-bit: Half (binary16) 32-bit: Single (binary32), decimal32 64-bit: Double (binary64), decimal64 128-bit: Quadruple (binary128), decimal128 256-bit: Octuple (binary256) Extended precision formats
Other
Minifloat Microsoft Binary Format IBM Floating Point Architecture Arbitrary precision

IEEE 754 single-precision binary floating-point format: binary32

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 decimal digits is converted to IEEE 754 single-precision form and then converted back to the same number of significant decimal digits, then the final string should match the original; and if an IEEE 754 single-precision number is converted to a decimal string with at least 9 significant decimal digits and then converted back, 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). 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 log₁₀(2²⁴) ≈ 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 sign, exponent e (the 8-bit unsigned integer), and a 23-bit fraction is

(-1)^{b_{31}}\times (1.b_{22}b_{21}\dots b_{0})_{2}\times 2^{(b_{30}b_{29}\dots b_{23})_{2}-127},

which in decimal yields

{\text{value}}=(-1)^{\text{sign}}\times \left(1+\sum _{i=1}^{23}b_{23-i}2^{-i}\right)\times 2^{(e-127)}.

In this example:

${\text{sign}}=b_{31}=0$ ,
$(-1)^{\text{sign}}=(-1)^{0}=+1\in \{-1,+1\}$ ,
$e=b_{30}b_{29}\dots b_{23}=\sum _{i=0}^{7}b_{23+i}2^{+i}=124\in \{1,\ldots ,(2^{8}-1)-1\}=\{1,\ldots ,254\}$ ,
$2^{(e-127)}=2^{124-127}=2^{-3}\in \{2^{-126},\ldots ,2^{127}\}$ ,
$1.b_{22}b_{21}...b_{0}=1+\sum _{i=1}^{23}b_{23-i}2^{-i}=1+1\cdot 2^{-2}=1.25\in \{1,1+2^{-23},\ldots ,2-2^{-23}\}\subset [1;2-2^{-23}]\subset [1;2)$ .

thus:

${\text{value}}=(+1)\times 1.25\times 2^{-3}=+0.15625$ .

Note:

$1+2^{-23}\approx 1.000\,000\,119$ ,
$2-2^{-23}\approx 1.999\,999\,881$ ,
$2^{-126}\approx 1.175\,494\,35\times 10^{-38}$ ,
$2^{+127}\approx 1.701\,411\,83\times 10^{+38}$ .

Exponent encoding

The single-precision binary floating-point exponent is encoded using an offset-binary 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 non-zero	Equation
00_H	zero, −0	denormal numbers	(−1)^signbit×2⁻¹²⁶× 0.significandbits
01_H, ..., FE_H	normalized value		(−1)^signbit×2^{exponentbits−127}× 1.significandbits
FF_H	±infinity	NaN (quiet, signalling)

The minimum positive normal value is 2⁻¹²⁶ ≈ 1.18 × 10⁻³⁸ and the minimum positive (denormal) value is 2⁻¹⁴⁹ ≈ 1.4 × 10⁻⁴⁵.

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 re-multiply 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. Re-multiply 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

Since IEEE 754 binary32 format requires real values to be represented in $(1.x_{1}x_{2}...x_{23})_{2}\times 2^{e}$ 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: 0-10000010-10001100000000000000000 = 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: 0-01111111-00000000000000000000000 = 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: 0-01111101-00000000000000000000000 = 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 = x₁)

From these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 0.375 as: 0-01111101-10000000000000000000000 = 3ec00000_H

Single-precision examples

These examples are given in bit representation, in hexadecimal, of the floating-point value. This includes the sign, (biased) exponent, and significand.

3f80 0000  = 1
c000 0000  = −2

7f7f ffff  = (1 − 2⁻²⁴) × 2¹²⁸  ≈ 3.402823466 × 10³⁸  (max finite positive value in single precision)
0080 0000  = 2⁻¹²⁶ ≈ 1.175494351 × 10⁻³⁸ (min normalized positive value in 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 single-precision binary to decimal

We start with the hexadecimal representation of the value, 41c80000, in this example, and convert it to binary:

41c8 0000₁₆ = 0100 0001 1100 1000 0000 0000 0000 0000₂

then we break it down into three parts: sign bit, exponent, and significand.

Sign bit: 0
Exponent: 1000 0011₂ = 83₁₆ = 131
Significand: 100 1000 0000 0000 0000 0000₂ = 480000₁₆

We then add the implicit 24th bit to the significand:

Significand: 1100 1000 0000 0000 0000 0000₂ = C80000₁₆

and decode the exponent value by subtracting 127:

Raw exponent: 83₁₆ = 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 24 = 1
bit 23 = 0.5
bit 22 = 0.25
bit 21 = 0.125
bit 20 = 0.0625
bit 19 = 0.03125
.
.
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 = C80000/2²³

Then we need to multiply with the base, 2, to the power of the exponent, to get the final result:

1.5625 × 2⁴ = 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.

Precision limits on integer values

Integers in $[-16777216,16777216]$ can be exactly represented
Integers in $[-33554432,-16777217]$ or in $[16777217,33554432]$ round to a multiple of 2
Integers in $[-2^{26},-2^{25}-1]$ or in $[2^{25}+1,2^{26}]$ round to a multiple of 4
....
Integers in $[-2^{127},-2^{126}-1]$ or in $[2^{126}+1,2^{127}]$ round to a multiple of $2^{103}$
Integers in $[-2^{128}+2^{104},-2^{127}-1]$ or in $[2^{127}+1,2^{128}-2^{104}]$ round to a multiple of $2^{127-23}$
Integers larger or equal than $2^{128}$ or smaller or equal than $-2^{128}$ are rounded to "infinity".

Optimizations

The design of floating-point format allows various optimisations, resulting from the easy generation of a base-2 logarithm approximation from an integer view of the raw bit pattern. Integer arithmetic and bit-shifting can yield an approximation to reciprocal square root (fast inverse square root), commonly required in computer graphics).

References

↑ "REAL Statement". scc.ustc.edu.cn.
↑ "Primitive Data Types". Java Documentation.
↑ "6 Predefined Types and Classes". haskell.org. 20 July 2010.
↑ William Kahan (1 October 1987). "Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic" (PDF).

External links

Data types

Uninterpreted	Bit Byte Trit Tryte Word Bit array

Numeric	Arbitrary-precision or bignum Complex Decimal Fixed point Floating point Double precision Extended precision Half precision Long double Minifloat Octuple precision Quadruple precision Single precision Integer signedness Interval Rational

Text	Character String null-terminated

Pointer	Address physical virtual Reference

Composite	Algebraic data type generalized Array Associative array Class Dependent Equality Inductive List Object metaobject Option type Product Record Set Union tagged

Other	Boolean Bottom type Collection Enumerated type Exception Function type Opaque data type Recursive data type Semaphore Stream Top type Type class Unit type Void

Related topics	Abstract data type Data structure Generic Kind metaclass Parametric polymorphism Primitive data type Protocol interface Subtyping Type constructor Type conversion Type system

See also platform-dependent and independent units of information

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Single-precision floating-point format

IEEE 754 single-precision binary floating-point format: binary32

Exponent encoding

Converting from decimal representation to binary32 format

Single-precision examples

Converting from single-precision binary to decimal

Precision limits on integer values

Optimizations

See also

References

External links