Single-precision floating-point format explained

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 radix point.

A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2³¹ − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2⁻²³) × 2¹²⁷ ≈ 3.4028235 × 10³⁸. All integers with 7 or fewer decimal digits, and any 2ⁿ for a whole number −149 ≤ n ≤ 127, can be converted exactly into an IEEE 754 single-precision floating-point value.

In the IEEE 754 standard, the 32-bit base-2 format is officially referred to as binary32; it was called single in IEEE 754-1985. IEEE 754 specifies additional floating-point types, such as 64-bit base-2 double precision and, more recently, base-10 representations.

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 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 in Fortran;^[1] SINGLE-FLOAT in Common Lisp;^[2] float in C, C++, C# and 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 double-precision numbers. In most implementations of PostScript, and some embedded systems, the only supported precision is single.

IEEE 754 standard: 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 digits is converted to the IEEE 754 single-precision 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 single-precision number is converted to a decimal string with at least 9 significant digits, and then converted back to single-precision 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 field is an 8-bit unsigned integer from 0 to 255, in biased form: a value of 127 represents the actual exponent zero. Exponents range from −126 to +127 (thus 1 to 254 in the exponent field), because the biased exponent values 0 (all 0s) and 255 (all 1s) are reserved for special numbers (subnormal numbers, signed zeros, infinities, and NaNs).

The true significand of normal numbers 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. Subnormal numbers and zeros (which are the floating-point numbers smaller in magnitude than the least positive normal number) are represented with the biased exponent value 0, giving the implicit leading bit the value 0. 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 sign, biased exponent e (the 8-bit unsigned integer), and a 23-bit fraction is

( -1 ) ^{b _{31}} x 2 ^{(b _{30} b _{29} ... b _{23} ) _{2} -127} x ( 1 .b _{22} b _{21} ... b _{0)} _{2} ,which yieldsvalue = ( -1 ) ^{sign} x 2 ^{(E-127)} x \left( 1 + \sum ^{23} _{i =1} b _{23-i} 2 ^{-i} \right).

In this example:

sign = b _{31} = 0 ,( -1 ) ^{sign} = ( -1 ) ^{0} = +1 \in \{ -1 , +1 \} ,E = (b _{30} b _{29} ... b _{23} ) _{2} = \sum ^{7} _{i =0} 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 ^{23} _{i =1} b _{23-i} 2 ^{-i} = 1 + 1 ⋅ 2 ^{-2} = 1 . 25 \in \{ 1 , 1+2 ^{-23} , \ldots, 2-2 ^{-23} \} \subset [ 1 ; 2 - 2 ^{-23} ] \subset [ 1 ; 2 ) .thus:value = ( +1 ) x 2 ^{-3} x 1 . 25 = +0 . 15625 .

Note:

1+2 ^{-23} ≈ 1 . 000000119 ,2-2 ^{-23} ≈ 1 . 999999881 ,2 ^{-126} ≈ 1 . 17549435 x 10 ^{-38} ,2 ^{+127} ≈ 1 . 70141183 x 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	fraction = 0	fraction ≠ 0	Equation
00H = 000000002	±zero		( -1 ) ^{sign x 2} ^{-126} x 0 . fraction
01H, ..., FEH = 000000012, ..., 111111102	normal value		( -1 ) ^{sign x 2} ^{exponent-127} x 1 . fraction
FFH = 111111112		NaN (quiet, signalling)

The minimum positive normal value is

2 ^{-126} ≈ 1 . 18 x 10 ^{-38} and the minimum positive (subnormal) value is 2 ^{-149} ≈ 1 . 4 x 10 ^{-45} .

Converting decimal to binary32

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 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.

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 proceed0 . 750 x 2 = 1 . 500 = 1 + 0 . 500 ⇒ b _{-2} = 1 0 . 500 x 2 = 1 . 000 = 1 + 0 . 000 ⇒ b _{-3} = 1 , fraction = 0.011, 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, 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 _{1x} _{2...x} _{23} ) _{2} x 2 ^{e} format (see Normalized number, Denormalized number), 1100.011 is shifted to the right by 3 digits to become ( 1 . 100011 ) _{2} x 2 ^{3}

Finally we can see that:

( 12 . 375 ) _{10} = ( 1 . 100011 ) _{2} x 2 ^{3}

From which we deduce:

• The exponent is 3 (and in the biased form it is therefore

( 127+3 ) _{10} = ( 130 ) _{10} = ( 1000 0010 ) _{2} )

• 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:

( 12 . 375 ) _{10} = ( 0 10000010 10001100000000000000000 ) _{2} = ( 41460000 ) _{16}

Note: consider converting 68.123 into IEEE 754 binary32 format: Using the above procedure you expect to get

( 42883EF9 ) _{16} with the last 4 bits being 1001. However, due to the default rounding behaviour of IEEE 754 format, what you get is ( 42883EFA ) _{16} , whose last 4 bits are 1010.

Example 1:Consider decimal 1. We can see that:

( 1 ) _{10} = ( 1 . 0 ) _{2} x 2 ^{0}

From which we deduce:

• The exponent is 0 (and in the biased form it is therefore

( 127+0 ) _{10} = ( 127 ) _{10} = ( 0111 1111 ) _{2}

• 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:

( 1 ) _{10} = ( 0 01111111 00000000000000000000000 ) _{2} = ( 3F800000 ) _{16}

Example 2:Consider a value 0.25. We can see that:

( 0 . 25 ) _{10} = ( 1 . 0 ) _{2} x 2 ^{-2}

From which we deduce:

• The exponent is −2 (and in the biased form it is

( 127+ ( -2 )) _{10} = ( 125 ) _{10} = ( 0111 1101 ) _{2} )

• The fraction is 0 (looking to the right of binary point in 1.0 is all zeroes)

From these we can form the resulting 32-bit IEEE 754 binary32 format representation of real number 0.25:

( 0 . 25 ) _{10} = ( 0 01111101 00000000000000000000000 ) _{2} = ( 3E800000 ) _{16}

Example 3:Consider a value of 0.375. We saw that

0 . 375 = {( 0 . 011 ) _{2}} = {( 1 . 1 ) _{2} x} 2 ^{-2}

Hence after determining a representation of 0.375 as

{( 1 . 1 ) _{2} x} 2 ^{-2} we can proceed as above:

• The exponent is −2 (and in the biased form it is

( 127+ ( -2 )) _{10} = ( 125 ) _{10} = ( 0111 1101 ) _{2} )

• 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:

( 0 . 375 ) _{10} = ( 0 01111101 10000000000000000000000 ) _{2} = ( 3EC00000 ) _{16}

Converting binary32 to decimal

If the binary32 value, in this example, is in hexadecimal we first convert it to binary:

41C80000 _{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 _{2}

• Exponent:

1000 0011 _{2} = 83 _{16} = 131 _{10}

• Significand:

100 1000 0000 0000 0000 0000 _{2} = 480000 _{16}

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

• Significand:

1 100 1000 0000 0000 0000 0000 _{2} = C80000 _{16}

and decode the exponent value by subtracting 127:

• Raw exponent:

83 _{16} = 131 _{10}

• 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 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:

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 x 2 ^{4} = 25

Thus

41C80000 = 25

This is equivalent to:

n = ( -1 ) ^{s} x ( 1+ m* 2 ^{-23} ) x 2 ^{x} where is the sign bit, is the exponent, and is the significand.

Precision limitations on decimal values (between 1 and 16777216)

• 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 2ⁿ and 2ⁿ⁺¹: fixed interval 2^n-23
• ...
• Decimals between 2²²=4194304 and 2²³=8388608: fixed interval 2⁻¹=0.5
• Decimals between 2²³=8388608 and 2²⁴=16777216: fixed interval 2⁰=1

Precision limitations on integer values

• Integers between 0 and 16777216 can be exactly represented (also applies for negative integers between −16777216 and 0)
• Integers between 2²⁴=16777216 and 2²⁵=33554432 round to a multiple of 2 (even number)
• Integers between 2²⁵ and 2²⁶ round to a multiple of 4
• ...
• Integers between 2ⁿ and 2ⁿ⁺¹ round to a multiple of 2^n-23
• ...
• Integers between 2¹²⁷ and 2¹²⁸ round to a multiple of 2¹⁰⁴
• Integers greater than or equal to 2¹²⁸ are rounded to "infinity".

Notable single-precision cases

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

0 00000000 00000000000000000000001₂ = 0000 0001₁₆ = 2⁻¹²⁶ × 2⁻²³ = 2⁻¹⁴⁹ ≈ 1.4012984643 × 10⁻⁴⁵ (smallest positive subnormal number)

0 00000000 11111111111111111111111₂ = 007f ffff₁₆ = 2⁻¹²⁶ × (1 − 2⁻²³) ≈ 1.1754942107 ×10⁻³⁸ (largest subnormal number)

0 00000001 00000000000000000000000₂ = 0080 0000₁₆ = 2⁻¹²⁶ ≈ 1.1754943508 × 10⁻³⁸ (smallest positive normal number)

0 11111110 11111111111111111111111₂ = 7f7f ffff₁₆ = 2¹²⁷ × (2 − 2⁻²³) ≈ 3.4028234664 × 10³⁸ (largest normal number)

0 01111110 11111111111111111111111₂ = 3f7f ffff₁₆ = 1 − 2⁻²⁴ ≈ 0.999999940395355225 (largest number less than one)

0 01111111 00000000000000000000000₂ = 3f80 0000₁₆ = 1 (one)

0 01111111 00000000000000000000001₂ = 3f80 0001₁₆ = 1 + 2⁻²³ ≈ 1.00000011920928955 (smallest number larger than one)

1 10000000 00000000000000000000000₂ = c000 0000₁₆ = −2 0 00000000 00000000000000000000000₂ = 0000 0000₁₆ = 0 1 00000000 00000000000000000000000₂ = 8000 0000₁₆ = −0 0 11111111 00000000000000000000000₂ = 7f80 0000₁₆ = infinity 1 11111111 00000000000000000000000₂ = ff80 0000₁₆ = −infinity 0 10000000 10010010000111111011011₂ = 4049 0fdb₁₆ ≈ 3.14159274101257324 ≈ π (pi) 0 01111101 01010101010101010101011₂ = 3eaa aaab₁₆ ≈ 0.333333343267440796 ≈ 1/3 x 11111111 10000000000000000000001₂ = ffc0 0001₁₆ = qNaN (on x86 and ARM processors) x 11111111 00000000000000000000001₂ = ff80 0001₁₆ = 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 PA-RISC processors use the bit to indicate a signalling NaN.

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.

See also

• IEEE 754
• ISO/IEC 10967, language independent arithmetic
• Primitive data type
• Numerical stability
• Scientific notation

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

Notes and References

1. Web site: REAL Statement. scc.ustc.edu.cn. 2013-02-28. 2021-02-24. https://web.archive.org/web/20210224045812/http://scc.ustc.edu.cn/zlsc/sugon/intel/compiler_f/main_for/lref_for/source_files/rfreals.htm. dead.
2. Web site: CLHS: Type SHORT-FLOAT, SINGLE-FLOAT, DOUBLE-FLOAT.... www.lispworks.com.
3. Web site: Primitive Data Types. Java Documentation.
4. Web site: 6 Predefined Types and Classes. 20 July 2010. haskell.org.
5. Web site: Float. Apple Developer Documentation.
6. Web site: Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic . William Kahan . 1 October 1997 . 4. https://web.archive.org/web/20120208075518/http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF. 8 February 2012.