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Normal number (computing)

Number type in floating-point arithmetic

Summary

Number type in floating-point arithmetic

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_{\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_{\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_{\text{max}}}\cdot\left(b - b^{1-p}\right)

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

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_{\text{min}}, and E_{\text{max}} have the following values:{{Citation

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

References

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

computer-arithmetic

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