You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 3.92.0.0/16 has been blocked by ‪Blablubbs‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider or colocation provider. To prevent abuse, web hosts and colocation providers may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a colocation provider's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: ((unblock|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~)). You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 10:52, 20 August 2024. Your current IP address is 3.92.45.30.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
This IP address range has been globally blocked.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

This block affects editing on all Wikimedia wikis.
The IP address or range 3.92.0.0/16 has been globally blocked by ‪Jon Kolbert‬ for the following reason(s):

Open proxy/Webhost: See the help page if you are affected

This block will expire on 05:35, 21 February 2029. Your current IP address is 3.92.45.30.

Even while globally blocked, you will usually still be able to edit pages on Meta-Wiki.

If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

Other useful links: Global blocks · Help:I have been blocked

You can view and copy the source of this page:

((short description|256-bit computer number format))
((refimprove|date=June 2016))
((use dmy dates|date=December 2022|cs1-dates=y))
((use list-defined references|date=December 2022))
((Floating-point))
((Computer architecture bit widths))
In [[computing]], '''octuple precision''' is a binary [[floating-point]]-based [[computer number format]] that occupies 32 [[byte]]s (256 [[bit]]s) in computer memory. This 256-[[bit]] octuple precision is for applications requiring results in higher than [[quadruple precision]]. This format is rarely used and very few environments support it.((According to whom|date=June 2024))

== IEEE 754 octuple-precision binary floating-point format: binary256 ==

In its 2008 revision, the [[IEEE 754]] standard specifies a '''binary256''' format among the ''interchange formats'' (it is not a basic format), as having:
* [[Sign bit]]: 1 bit
* [[Exponent]] width: 19 bits
* [[Significand]] [[precision (arithmetic)|precision]]: 237 bits (236 explicitly stored)
<!-- "significand", with a d at the end, is a technical term, please do not confuse with "significant" -->

The format is written with an implicit lead bit with value 1 unless the exponent is all zeros. Thus only 236 bits of the [[significand]] appear in the memory format, but the total precision is 237 bits (approximately 71 decimal digits: ((nowrap|log<sub>10</sub>(2<sup>237</sup>) ≈ 71.344))).
<!-- (Commented out since the image is incorrect; it could be re-added once corrected.)-->
The bits are laid out as follows:

[[File:Octuple precision visual demonstration.svg|1000px|Layout of octuple-precision floating-point format]]

=== Exponent encoding ===

The octuple-precision binary floating-point exponent is encoded using an [[offset binary]] representation, with the zero offset being 262143; also known as exponent bias in the IEEE&nbsp;754 standard.

* E<sub>min</sub> = −262142
* E<sub>max</sub> = 262143
* [[Exponent bias]] = 3FFFF<sub>16</sub> = 262143

Thus, as defined by the offset binary representation, in order to get the true exponent the offset of 262143 has to be subtracted from the stored exponent.

The stored exponents 00000<sub>16</sub> and 7FFFF<sub>16</sub> are interpreted specially.

{|class="wikitable" style="text-align:center"
! Exponent !! Significand zero !! Significand non-zero !! Equation
|-
| 00000<sub>16</sub> || [[0 (number)|0]], [[−0]] || [[subnormal numbers]] || (-1)<sup>signbit</sup> × 2<sup>−262142</sup> × 0.significandbits<sub>2</sub>
|-
| 00001<sub>16</sub>, ..., 7FFFE<sub>16</sub> ||colspan=2| normalized value || (-1)<sup>signbit</sup> × 2<sup>exponent bits<sub>2</sub></sup> × 1.significandbits<sub>2</sub>
|-
| 7FFFF<sub>16</sub> || ±[[infinity|∞]] || [[NaN]] (quiet, signalling)
|}

The minimum strictly positive (subnormal) value is ((nowrap|2<sup>−262378</sup> ≈ 10<sup>−78984</sup>)) and has a precision of only one bit.
The minimum positive normal value is 2<sup>−262142</sup> ≈ 2.4824 × 10<sup>−78913</sup>.
The maximum representable value is 2<sup>262144</sup> − 2<sup>261907</sup> ≈ 1.6113 × 10<sup>78913</sup>.

=== Octuple-precision examples ===

These examples are given in bit ''representation'', in [[hexadecimal]],
of the floating-point value. This includes the sign, (biased) exponent, and significand.

 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub>16</sub> = +0
 8000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub>16</sub> = −0

 7fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub>16</sub> = +infinity
 ffff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub>16</sub> = −infinity

 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001<sub>16</sub>
 = 2<sup>−262142</sup> × 2<sup>−236</sup> = 2<sup>−262378</sup>
 ≈ 2.24800708647703657297018614776265182597360918266100276294348974547709294462 × 10<sup>−78984</sup>
   (smallest positive subnormal number)

 0000 0fff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff<sub>16</sub>
 = 2<sup>−262142</sup> × (1 − 2<sup>−236</sup>)
 ≈ 2.4824279514643497882993282229138717236776877060796468692709532979137875392 × 10<sup>−78913</sup>
   (largest subnormal number)

 0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub>16</sub>
 = 2<sup>−262142</sup>
 ≈ 2.48242795146434978829932822291387172367768770607964686927095329791378756168 × 10<sup>−78913</sup>
   (smallest positive normal number)

 7fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff<sub>16</sub>
 = 2<sup>262143</sup> × (2 − 2<sup>−236</sup>)
 ≈ 1.61132571748576047361957211845200501064402387454966951747637125049607182699 × 10<sup>78913</sup>
   (largest normal number)

 3fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff<sub>16</sub>
 = 1 − 2<sup>−237</sup>
 ≈ 0.999999999999999999999999999999999999999999999999999999999999999999999995472
   (largest number less than one)

 3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub>16</sub>
 = 1 (one)

 3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001<sub>16</sub>
 = 1 + 2<sup>−236</sup>
 ≈ 1.00000000000000000000000000000000000000000000000000000000000000000000000906
   (smallest number larger than one)

By default, 1/3 rounds down like [[double precision]], because of the odd number of bits in the significand.
So the bits beyond the rounding point are <code>0101...</code> which is less than 1/2 of a [[unit in the last place]].

==Implementations==
Octuple precision is rarely implemented since usage of it is extremely rare. [[Apple Inc.]] had an implementation of addition, subtraction and multiplication of octuple-precision numbers with a 224-bit [[two's complement]] significand and a 32-bit exponent.<ref name="Crandall-Papadopoulos_2002"/> One can use general [[arbitrary-precision arithmetic]] libraries to obtain octuple (or higher) precision, but specialized octuple-precision implementations may achieve higher performance.

=== Hardware support ===
There is no known hardware implementation of octuple precision.

== See also ==
* [[IEEE 754]]
* [[ISO/IEC 10967]], Language-independent arithmetic
* [[Primitive data type]]
* [[Scientific notation]]

== References ==
((reflist|refs=
<ref name="Crandall-Papadopoulos_2002">((cite web |title=Octuple-precision floating point on Apple G4 (archived copy on web.archive.org) |author-first1=Richard E. |author-last1=Crandall |author-link1=Richard E. Crandall |author-first2=Jason S. |author-last2=Papadopoulos |date=2002-05-08 |url=http://images.apple.com/ca/acg/pdf/oct3a.pdf |url-status=unfit |archive-url=https://web.archive.org/web/20060728140052/http://images.apple.com/ca/acg/pdf/oct3a.pdf |archive-date=2006-07-28)) (8 pages)</ref>
))

==Further reading==
* ((cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |date=2017-08-22 |location=Salt Lake City, UT, USA |publisher=[[Springer International Publishing AG]] |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 ))

((data types))

[[Category:Binary arithmetic]]
[[Category:Floating point types]]

Pages transcluded onto the current version of this page (help):

Octuple-precision floating-point format (edit)
Template:According to whom (view source) (template editor protected)
Template:Ambox (view source) (template editor protected)
Template:Category handler (view source) (protected)
Template:Cite book (view source) (protected)
Template:Cite web (view source) (protected)
Template:Computer architecture bit widths (edit)
Template:DMCA (view source) (template editor protected)
Template:Data types (edit)
Template:Dated maintenance category (view source) (template editor protected)
Template:Dated maintenance category (articles) (view source) (template editor protected)
Template:FULLROOTPAGENAME (view source) (template editor protected)
Template:Find sources mainspace (view source) (template editor protected)
Template:Fix-span (view source) (template editor protected)
Template:Fix/category (view source) (protected)
Template:Floating-point (edit)
Template:Hlist (view source) (protected)
Template:Hlist/styles.css (view source) (protected)
Template:Main other (view source) (protected)
Template:More citations needed (view source) (template editor protected)
Template:Navbox (view source) (template editor protected)
Template:Nowrap (view source) (protected)
Template:Ns has subpages (view source) (protected)
Template:Pagetype (view source) (protected)
Template:Plainlist/styles.css (view source) (protected)
Template:Refimprove (view source) (template editor protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Template:SDcat (view source) (protected)
Template:Short description (view source) (protected)
Template:Short description/lowercasecheck (view source) (protected)
Template:Sidebar (view source) (template editor protected)
Template:Use dmy dates (view source) (template editor protected)
Template:Use list-defined references (view source) (extended confirmed protected)
Module:Arguments (view source) (protected)
Module:Category handler (view source) (protected)
Module:Category handler/blacklist (view source) (protected)
Module:Category handler/config (view source) (protected)
Module:Category handler/data (view source) (protected)
Module:Category handler/shared (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:DecodeEncode (view source) (template editor protected)
Module:Disambiguation/templates (view source) (protected)
Module:Find sources (view source) (template editor protected)
Module:Find sources/config (view source) (template editor protected)
Module:Find sources/links (view source) (template editor protected)
Module:Find sources/templates/Find sources mainspace (view source) (template editor protected)
Module:List (view source) (protected)
Module:Message box (view source) (protected)
Module:Message box/ambox.css (view source) (protected)
Module:Message box/configuration (view source) (protected)
Module:Namespace detect/config (view source) (protected)
Module:Namespace detect/data (view source) (protected)
Module:Navbar (view source) (protected)
Module:Navbar/configuration (view source) (protected)
Module:Navbar/styles.css (view source) (protected)
Module:Navbox (view source) (template editor protected)
Module:Navbox/configuration (view source) (template editor protected)
Module:Navbox/styles.css (view source) (template editor protected)
Module:Ns has subpages (view source) (protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Pagetype/disambiguation (view source) (protected)
Module:Pagetype/rfd (view source) (template editor protected)
Module:Pagetype/setindex (view source) (protected)
Module:Pagetype/softredirect (view source) (protected)
Module:SDcat (view source) (protected)
Module:Sidebar (view source) (template editor protected)
Module:Sidebar/configuration (view source) (template editor protected)
Module:Sidebar/styles.css (view source) (template editor protected)
Module:String (view source) (protected)
Module:TableTools (view source) (protected)
Module:Unsubst (view source) (protected)
Module:Wikitext Parsing (view source) (protected)
Module:Yesno (view source) (protected)

Return to Octuple-precision floating-point format.

Retrieved from "https://en.wikipedia.org/wiki/Octuple-precision_floating-point_format"