Normal number (computing): Difference between revisions

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

<math display="block">b^{E_{\text{eminmin}}}</math>

where ''b'' is the base (radix) of the format (like common values 2 or 10, for binary and decimal number systems), and ''emin<math display="inline">E_{\text{min}}</math>'' depends on the size and layout of the format.

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

:<math display="block">b^{E_{\text{emaxmax}}}\cdot\left(b - b^{1-p}\right)</math>

where ''p'' is the precision of the format in [[numerical digit|digit]]s and ''emax<math display="inline">E_{\text{min}}</math>'' is (&minus;related to ''emin<math display="inline">E_{\text{max}}</math>'')+1. as:

<math display="block">E_{\text{min}}\, \overset{\Delta}{\equiv}\, 1 - E_{\text{max}} = \left(-E_{\text{max}}\right) + 1</math>

In the [[IEEE 754]] binary and decimal formats, ''b'', ''p'', ''emin'', and ''emax'' have the following values:<ref>{{Citation

In the [[IEEE 754]] binary and decimal formats, ''b'', ''p'', ''emin''<math display="inline">E_{\text{min}}</math>, and ''emax<math display="inline">E_{\text{max}}</math>'' have the following values:<ref>{{Citation

| title = IEEE Standard for Floating-Point Arithmetic

| date = 2008-08-29

Line 25 ⟶ 27:

| accessdate = 2015-04-26| isbn = 978-0-7381-5752-8

}}</ref>

{| class="wikitable" | style="text-align: right;" |

|+Smallest and Largest Normal Numbers for common numerical Formats

!Format!!<math>b</math>!!<math>p</math>!!<math>E_{\text{min}}</math>!!<math>E_{\text{max}}</math>

!Format!!b!!p!!emin!!emax

!Smallest Normal Number

!Largest Normal Number

|-

|[[Half-precision floating-point format|binary16]]||2||11||−14||15

|<math>2^{-14} \equiv 0.00006103515625</math>

|<math>2^{15}\cdot\left(2 - 2^{1-11}\right) \equiv 65504</math>

|-

|[[Single-precision floating-point format|binary32]]||2||24||−126||127

|<math>2^{-126} \equiv \frac{1}{2^{126}}</math>

|<math>2^{127}\cdot\left(2 - 2^{1-24}\right)</math>

|-

|[[Double-precision floating-point format|binary64]]||2||53||−1022||1023

|<math>2^{-1022} \equiv \frac{1}{2^{1022}}</math>

|<math>2^{1023}\cdot\left(2 - 2^{1-53}\right)</math>

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== See also ==

* [[Normalized number]]

* [[Half-precision floating-point format]]

* [[Single-precision floating-point format]]

* [[Double-precision floating-point format]]

== References ==