Normal number (computing)

In computing, a normal number is a non-zero number in a floating-point representation which is within the balanced range supported by a given floating-point format: it is a floating point number that can be represented without leading zeros in its significand.

The magnitude of the smallest normal number in a format is given by:

$b^{E_{min}}$

where b is the base (radix) of the format (like common values 2 or 10, for binary and decimal number systems), and $E_{min}$ depends on the size and layout of the format.

Similarly, the magnitude of the largest normal number in a format is given by

$b^{E_{max}} \cdot (b - b^{1 - p})$

where p is the precision of the format in digits and $E_{min}$ is related to $E_{max}$ as:

$E_{min} \overset{Δ}{\equiv} 1 - E_{max} = (- E_{max}) + 1$

In the IEEE 754 binary and decimal formats, b, p, $E_{min}$ , and $E_{max}$ have the following values:

Smallest and Largest Normal Numbers for common numerical Formats
Format	$b$	$p$	$E_{min}$	$E_{max}$	Smallest Normal Number	Largest Normal Number
binary16	2	11	−14	15	$2^{- 14} \equiv 0.00006103515625$	$2^{15} \cdot (2 - 2^{1 - 11}) \equiv 65504$
binary32	2	24	−126	127	$2^{- 126} \equiv \frac{1}{2^{126}}$	$2^{127} \cdot (2 - 2^{1 - 24})$
binary64	2	53	−1022	1023	$2^{- 1022} \equiv \frac{1}{2^{1022}}$	$2^{1023} \cdot (2 - 2^{1 - 53})$
binary128	2	113	−16382	16383	$2^{- 16382} \equiv \frac{1}{2^{16382}}$	$2^{16383} \cdot (2 - 2^{1 - 113})$
decimal32	10	7	−95	96	$1 0^{- 95} \equiv \frac{1}{1 0^{95}}$	$1 0^{96} \cdot (10 - 1 0^{1 - 7}) \equiv 9.999999 \cdot 1 0^{96}$
decimal64	10	16	−383	384	$1 0^{- 383} \equiv \frac{1}{1 0^{383}}$	$1 0^{384} \cdot (10 - 1 0^{1 - 16})$
decimal128	10	34	−6143	6144	$1 0^{- 6143} \equiv \frac{1}{1 0^{6143}}$	$1 0^{6144} \cdot (10 - 1 0^{1 - 34})$

For example, in the smallest decimal format in the table (decimal32), the range of positive normal numbers is 10⁻⁹⁵ through 9.999999 × 10⁹⁶.

Non-zero numbers smaller in magnitude than the smallest normal number are called subnormal numbers (or denormal numbers).

Zero is considered neither normal nor subnormal.

References

↑ IEEE Standard for Floating-Point Arithmetic, 2008-08-29, doi:10.1109/IEEESTD.2008.4610935, ISBN 978-0-7381-5752-8

[1] IEEE Standard for Floating-Point Arithmetic, 2008-08-29, doi:10.1109/IEEESTD.2008.4610935, ISBN 978-0-7381-5752-8

Normal number (computing)

See also

References