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 (1 − 2⁻²⁴) × 2¹²⁸ ≈ 3.402823 × 10³⁸. All integers with six or fewer significant decimal digits can be converted to an IEEE 754 floating point value without loss of precision, some integers up to nine significant decimal digits can be converted to an IEEE 754 floating point value without loss of precision, but no more than nine 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] as float in C, C++, C#, Java,^[2] as Float in Haskell,^[3] and as 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.

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 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 is converted to a decimal string with at least 9 significant decimal digits 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 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 $(-1)^{b_{31}}\times (1.b_{22}b_{21}...b_{0})_2 \times 2^{b_{30}b_{29}...b_{23}-127}$ 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:

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

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
00H	zero, −0	denormal numbers	(−1)^signbit×2^−126× 0.significandbits
01H, ..., FEH	normalized value		(−1)^signbit×2^exponentbits−127× 1.significandbits
FFH	±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

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: 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 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 = C80000/2²³

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".

Trivia

A fascinating example of how the floating-point 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 bit-shift and subtraction operated on the 32-bits of the floating point encoding as if it were an integer.

See also

• IEEE Standard for Floating-Point Arithmetic (IEEE 754)
• ISO/IEC 10967, language independent arithmetic
• Primitive data type
• Numerical stability

References

1. "REAL Statement". scc.ustc.edu.cn.
2. "Primitive Data Types". Java Documentation.
3. "6 Predefined Types and Classes". haskell.org. 20 July 2010.
4. William Kahan (1 October 1987). "Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic" (PDF).

External links

• Live floating-point bit pattern editor
• Online calculator
• Online converter for IEEE 754 numbers with single precision
• C source code to convert between IEEE double, single, and half precision