Normal number (computing)
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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 {\displaystyle b^{E_{\text{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 {\textstyle E_{\text{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 ⋅ ( b − b 1 − p ) {\displaystyle b^{E_{\text{max}}}\cdot \left(b-b^{1-p}\right)}
where p is the precision of the format in digits and E min {\textstyle E_{\text{min}}} is related to E max {\textstyle E_{\text{max}}} as:
E min ≡ Δ 1 − E max = ( − E max ) + 1 {\displaystyle E_{\text{min}}\,{\overset {\Delta }{\equiv }}\,1-E_{\text{max}}=\left(-E_{\text{max}}\right)+1}
In the IEEE 754 binary and decimal formats, b, p, E min {\textstyle E_{\text{min}}}, and E max {\textstyle E_{\text{max}}} have the following values:
| Format | b {\displaystyle b} | p {\displaystyle p} | E min {\displaystyle E_{\text{min}}} | E max {\displaystyle E_{\text{max}}} | Smallest Normal Number | Largest Normal Number |
|---|---|---|---|---|---|---|
| binary16 | 2 | 11 | −14 | 15 | 2 − 14 ≡ 0.00006103515625 {\displaystyle 2^{-14}\equiv 0.00006103515625} | 2 15 ⋅ ( 2 − 2 1 − 11 ) ≡ 65504 {\displaystyle 2^{15}\cdot \left(2-2^{1-11}\right)\equiv 65504} |
| binary32 | 2 | 24 | −126 | 127 | 2 − 126 ≡ 1 2 126 {\displaystyle 2^{-126}\equiv {\frac {1}{2^{126}}}} | 2 127 ⋅ ( 2 − 2 1 − 24 ) {\displaystyle 2^{127}\cdot \left(2-2^{1-24}\right)} |
| binary64 | 2 | 53 | −1022 | 1023 | 2 − 1022 ≡ 1 2 1022 {\displaystyle 2^{-1022}\equiv {\frac {1}{2^{1022}}}} | 2 1023 ⋅ ( 2 − 2 1 − 53 ) {\displaystyle 2^{1023}\cdot \left(2-2^{1-53}\right)} |
| binary128 | 2 | 113 | −16382 | 16383 | 2 − 16382 ≡ 1 2 16382 {\displaystyle 2^{-16382}\equiv {\frac {1}{2^{16382}}}} | 2 16383 ⋅ ( 2 − 2 1 − 113 ) {\displaystyle 2^{16383}\cdot \left(2-2^{1-113}\right)} |
| decimal32 | 10 | 7 | −95 | 96 | 10 − 95 ≡ 1 10 95 {\displaystyle 10^{-95}\equiv {\frac {1}{10^{95}}}} | 10 96 ⋅ ( 10 − 10 1 − 7 ) ≡ 9.999999 ⋅ 10 96 {\displaystyle 10^{96}\cdot \left(10-10^{1-7}\right)\equiv 9.999999\cdot 10^{96}} |
| decimal64 | 10 | 16 | −383 | 384 | 10 − 383 ≡ 1 10 383 {\displaystyle 10^{-383}\equiv {\frac {1}{10^{383}}}} | 10 384 ⋅ ( 10 − 10 1 − 16 ) {\displaystyle 10^{384}\cdot \left(10-10^{1-16}\right)} |
| decimal128 | 10 | 34 | −6143 | 6144 | 10 − 6143 ≡ 1 10 6143 {\displaystyle 10^{-6143}\equiv {\frac {1}{10^{6143}}}} | 10 6144 ⋅ ( 10 − 10 1 − 34 ) {\displaystyle 10^{6144}\cdot \left(10-10^{1-34}\right)} |
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.