Octuple-precision floating-point format: Difference between revisions

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

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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 or [[64]] [[nibble]]s) in computer memory. This [[256]] [[bit]] octuple precision is for applications requiring results in higher than [[quadruple precision]]. Disclaimer: This format is rarely (if ever) used and very few things support it.

{{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 or [[64]] [[nibble]]s) in computer memory. This [[256]] -[[bit]] octuple precision is for applications requiring results in higher than [[quadruple precision]]. Disclaimer: This format is rarely (if ever) used and very few things support it.

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: binary128 ==

== IEEE 754 octuple-precision binary floating-point format: binary128binary256 ==

The IEEE 754 standard specifies a '''binary128''' as having:

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: 1819 bits

* [[Significand]] [[precision (arithmetic)|precision]]: 238237 bits (237236 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 stored with all zeros. Thus only 112236 bits of the [[significand]] appear in the memory format, but the total precision is 113237 bits (approximately 3471 decimal digits,: {{nowrap|log<mathsub>\log_{10}</sub>(2^{113}²³⁷) \approx≈ 3471.016</math>344}}). The bits are laid out as follows:

This gives from 33 - 36 significant decimal digits precision (if a decimal string with at most 33 significant decimal is converted to IEEE 754 octuple precision and then converted back to the same number of significant decimal, then the final string should match the original; and if an IEEE 754 octuple precision is converted to a decimal string with at least 36 significant decimal and then converted back to octuple, then the final number must match the original <ref name=whyieee>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF|title=Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic| author=William Kahan |date=1 October 1987}}</ref>).

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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]]

The format is written with an implicit lead bit with value 1 unless the exponent is stored with all zeros. Thus only 112 bits of the [[significand]] appear in the memory format, but the total precision is 113 bits (approximately 34 decimal digits, <math>\log_{10}(2^{113}) \approx 34.016</math>). The bits are laid out as follows:

[[|thumbnail|center]]

[[File:Octuple persision visual demontration.png|Octuple persision visual demontration]]

=== 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 = −262143−262142

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 754 standard.

* E_min = −262143

* E_max = 262143

* [[Exponent bias]] = 3FFF3FFFF₁₆ = 16383262143

Thus, as defined by the offset binary representation, in order to get the true exponent the offset of 16383262143 has to be subtracted from the stored exponent.

The stored exponents 000000000₁₆ and 7FFF7FFFF₁₆ are interpreted specially.

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

|-

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

|-

| 000000000₁₆ || [[0 (number)|0]], [[−0]] || [[subnormal numbers]] || <math>(-1−1)^{\text{^signbit}} \times× 2^{-16382}^−262142 \times× 0.\text{significandbits}_2<sub>2</mathsub>

|-

| 000100001₁₆, ..., 7FFE7FFFE₁₆ ||colspan=2| normalized value || <math>(-1−1)^{\text{^signbit}} \times× 2^{{\text{exponentbits}_2}^{exponent -bits₂} 16383} \times× 1.\text{significandbits}_2<sub>2</mathsub>

|-

| 7FFF7FFFF₁₆ || ±[[infinity|∞]] || [[NaN]] (quiet, signallingsignaling)

|}

The minimum strictly positive (subnormal) value is {{nowrap|2^{−16494−262378} ≈ 10^{−4965−78984}}} and has a precision of only one bit.

The minimum positive normal value is 2^{−16382−262142} ≈ 32.36214824 × 10^{−4932−78913} and has a precision of 112&nbsp;bits, i.e. ±2&thinsp;⁻¹⁶⁴⁹⁴ as well.

The maximum representable value is 2^16384262144 -− 2^16272261907 ≈ 1.18976113 × 10^493278913.

=== 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 ₁₆ = +0

8000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 ₁₆ = -0−0

7fff 0000f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 ₁₆ = +infinity

ffff 0000f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 ₁₆ = -infinity−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 if ever implemented in to software 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 octrupleoctuple-precision implementations may achieve higher performance.

===Computer-language support===

In C++, It is possible to make a library to handle Octuple-precision floating-point arithmetic. Controversially, in theory it is possible to do Octuple-precision floating-point arithmic in binary (but it would be incredibly hard, painful torture).

=== Hardware support ===

There is little to no known hardware with native support for octuple precision arithmetic.

== See also ==

* [[IEEE 754]]

* [[IEEE 754-2008|IEEE Standard for Floating-Point Arithmetic (IEEE 754)]]

* [[ISO/IEC 10967]], Language-independent Independent Arithmeticarithmetic

* [[Extended precision]] (80-bit)

* [[ISO/IEC 10967]], Language Independent Arithmetic

* [[Primitive data type]]

* [[longScientific doublenotation]]

== References ==

{{reflist|colwidthrefs=30em}}

<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 ==

==Source For The Layout Of This Entire Page==

* {{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 }}

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

{{data types}}

[[Category:Binary arithmetic]]

[[Category:Data types]]

[[Category:Binary arithmetic]]

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[[Category:DataFloating point types]]