Octuple-precision floating-point format: Difference between revisions

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{{short description|256-bit computer number format}}

#REDIRECT [[IEEE floating point]]

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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]].

The range greatly exceeds what is needed to describe all known physical limitations within the observable universe or precisions better than [[Planck units]].

== 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|log₁₀(2²³⁷) ≈ 71.344}}).

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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_min = −262142

* E_max = 262143

* [[Exponent bias]] = 3FFFF₁₆ = 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 00000₁₆ and 7FFFF₁₆ are interpreted specially.

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

|-

! Exponent !! Significand zero !! Significand non-zero !! Equation

|-

| 00000₁₆ || [[0 (number)|0]], [[−0]] || [[subnormal numbers]] || (−1)^signbit × 2^−262142 × 0.significandbits₂

|-

| 00001₁₆, ..., 7FFFE₁₆ ||colspan=2| normalized value || (−1)^signbit × 2^{exponent bits₂} × 1.significandbits₂

|-

| 7FFFF₁₆ || ±[[infinity|∞]] || [[NaN]] (quiet, signaling)

|}

The minimum strictly positive (subnormal) value is {{nowrap|2^−262378 ≈ 10⁻⁷⁸⁹⁸⁴}} and has a precision of only one bit.

The minimum positive normal value is 2^−262142 ≈ 2.4824 × 10⁻⁷⁸⁹¹³.

The maximum representable value is 2²⁶²¹⁴⁴ − 2²⁶¹⁹⁰⁷ ≈ 1.6113 × 10⁷⁸⁹¹³.

=== 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 0000₁₆ = +0

8000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = −0

7fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = +infinity

ffff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = −infinity

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001₁₆

= 2^−262142 × 2⁻²³⁶ = 2^−262378

≈ 2.24800708647703657297018614776265182597360918266100276294348974547709294462 × 10⁻⁷⁸⁹⁸⁴

(smallest positive subnormal number)

0000 0fff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff₁₆

= 2^−262142 × (1 − 2⁻²³⁶)

≈ 2.4824279514643497882993282229138717236776877060796468692709532979137875392 × 10⁻⁷⁸⁹¹³

(largest subnormal number)

0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆

= 2^−262142

≈ 2.48242795146434978829932822291387172367768770607964686927095329791378756168 × 10⁻⁷⁸⁹¹³

(smallest positive normal number)

7fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff₁₆

= 2²⁶²¹⁴³ × (2 − 2⁻²³⁶)

≈ 1.61132571748576047361957211845200501064402387454966951747637125049607182699 × 10⁷⁸⁹¹³

(largest normal number)

3fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff₁₆

= 1 − 2⁻²³⁷

≈ 0.999999999999999999999999999999999999999999999999999999999999999999999995472

(largest number less than one)

3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆

= 1 (one)

3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001₁₆

= 1 + 2⁻²³⁶

≈ 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 with native support for 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]]