Single-precision floating-point format

In computing, single precision is a computer numbering format that occupies one storage location in computer memory at a given address. A single-precision number, sometimes simply a single, may be defined to be an integer, fixed point, or floating point.

Modern computers with 32-bit words (single precision) provide 64-bit double precision. Single precision floating point is an IEEE 754 standard for encoding binary or decimal floating point numbers in 4 bytes.

Single precision binary floating-point format

Single precision binary floating-point is a commonly used format on PCs, due to it's wider range over fixed point, even if at the cost of precision on integer numbers. It's commonly known simply as float. The IEEE 754 standard defines a float as:

Sign bit: 1
Exponent width: 8
Significant precision: 23 (24 implicit)

The format is written with an implicit most-significant bit with value 1 unless the written exponent is all zeros. Thus only 23 bits of the fraction mantissa appear in the memory format but the total precision is 24 bits (better than 7 decimal digits, $\log _{10}(2^{24})\approx 7.225$ ).

Exponent encoding

The single precision binary floating-point exponent is encoded using an Excess-N representation, to be more exact, Excess-127, also known as exponent bias on the IEEE 754 standard. Examples of such representations would be:

E_min (1) = −126
E (40) = −137
E_max (255) = 127

Thus, as defined by Excess-N representation, in order to get the true exponent, the exponent bias (127) has to be subtracted from the written exponent.

All bit patterns are valid encoding, although exponents of 0x00 and 0xff are interpreted specially.

Exponent	Mantissa zero	Mantissa non-zero	Equation
0x00	zero	denormal	$(-1)^{\text{sign}}\times 2^{-126}\times 0.{\text{mantissa}}$
0x01–0xfe	normalized value		$(-1)^{\text{sign}}\times 2^{{\text{exponent}}-127}\times 1.{\text{mantissa}}$
0xff	infinity	NaN

Single precision examples in hexadecimal

3f80 0000   = 1

c000 0000   = −2

7f7f ffff   ≈ 3.4028234 x 10³⁸  (Max Single)
				
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 significant. So the bits beyond the rounding point are 1010... which is more than 1/2 of a unit in the last place.

0000 0000   = 0
8000 0000   = −0

7f80 0000   = Infinity
ff80 0000   = -Infinity

Converting from single precision binary to human readable form (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 mantissa.

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

We then add the implicit 24th bit to the mantissa

Mantissa: 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 mantissa, 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
.
.

The mantissa in this example has three bits set, bit 23, bit 22 and bit 19. We can now decode the mantissa by adding the values represented by these bits.

Decoded mantissa: 1 + 0.5 + 0.0625 = 1.5625

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:

 $n=(1-2s)(1+m2^{-23})2^{x-127}$

where s is the sign bit, x is the exponent, and m is the mantissa.

Single precision binary floating-point format

Exponent encoding

Single precision examples in hexadecimal

Converting from single precision binary to human readable form (decimal)

See also