Octuple-precision floating-point format: Difference between revisions

Revision as of 20:53, 15 June 2025

Revision as of 19:33, 10 July 2025

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== 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

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=== 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 754 standard.

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The stored exponents 00000₁₆ and 7FFFF₁₆ are interpreted specially.

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

|-

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

|-

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=== 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.

<pre<includeonly></includeonly>

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = +0

80000000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = −0+0

00008000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = +0−0

</pre>

<pre<includeonly></includeonly>

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

ffff7fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = −infinity+infinity

7fffffff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = +infinity−infinity

</pre>

<pre<includeonly></includeonly>

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

= 2^−262142 × 2⁻²³⁶ = 2^−262378

= 2^−262142 × 2⁻²³⁶ = 2^−262378
≈ 2.24800708647703657297018614776265182597360918266100276294348974547709294462 × 10⁻⁷⁸⁹⁸⁴

(smallest positive subnormal number)

</pre>

<pre<includeonly></includeonly>

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

= 2^−262142 × (1 − 2⁻²³⁶)

= 2^−262142 × (1 − 2⁻²³⁶)
≈ 2.4824279514643497882993282229138717236776877060796468692709532979137875392 × 10⁻⁷⁸⁹¹³

(largest subnormal number)

</pre>

<pre<includeonly></includeonly>

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

= 2^−262142

= 2^−262142
≈ 2.48242795146434978829932822291387172367768770607964686927095329791378756168 × 10⁻⁷⁸⁹¹³

(smallest positive normal number)

</pre>

<pre<includeonly></includeonly>

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

= 2²⁶²¹⁴³ × (2 − 2⁻²³⁶)

= 2²⁶²¹⁴³ × (2 − 2⁻²³⁶)
≈ 1.61132571748576047361957211845200501064402387454966951747637125049607182699 × 10⁷⁸⁹¹³

(largest normal number)

</pre>

<pre<includeonly></includeonly>

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

= 1 − 2⁻²³⁷

= 1 − 2^{−262142−237}

≈ 0.999999999999999999999999999999999999999999999999999999999999999999999995472

(largest number less than one)

</pre>

<pre<includeonly></includeonly>

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

= 1 (one)

</pre>

<pre<includeonly></includeonly>

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

= 1 + 2⁻²³⁶

= 2^−262142 × (1 −+ 2⁻²³⁶)

≈ 1.00000000000000000000000000000000000000000000000000000000000000000000000906

(smallest number larger than one)

</pre>

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 implementationwith ofnative support for octuple precision.

== See also ==

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}}

== 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 }}