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 bemin, where b is the base (radix) of the format (usually 2 or 10) and emin depends on the size and layout of the format.
Similarly, the magnitude of the largest normal number in a format is given by
- bemax × (b − b1−p),
where p is the precision of the format in digits and emax is (−emin)+1.
In the IEEE 754 binary and decimal formats, b, p, emin, and emax have the following values:[1]
| Format | b | p | emin | emax |
|---|---|---|---|---|
| binary16 | 2 | 11 | −14 | 15 |
| binary32 | 2 | 24 | −126 | 127 |
| binary64 | 2 | 53 | −1022 | 1023 |
| binary128 | 2 | 113 | −16382 | 16383 |
| decimal32 | 10 | 7 | −95 | 96 |
| decimal64 | 10 | 16 | −383 | 384 |
| decimal128 | 10 | 34 | −6143 | 6144 |
For example, in the smallest decimal format, 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 denormal (or subnormal) numbers. Zero is neither normal nor subnormal.
See also
References
- ^ IEEE Standard for Floating-Point Arithmetic, 2008-08-29, doi:10.1109/IEEESTD.2008.4610935, ISBN 978-0-7381-5752-8, retrieved 2015-04-26