Quadruple-precision floating-point format: Difference between revisions

Revision as of 11:07, 3 July 2025

Revision as of 14:35, 10 July 2025

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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 0001₁₆ = 2⁻¹⁶³⁸² × 2⁻¹¹² = 2⁻¹⁶⁴⁹⁴

≈ 6.4751751194380251109244389582276465525 × 10⁻⁴⁹⁶⁶

(smallest positive subnormal number)

<pre>

0000 ffff ffff ffff ffff ffff ffff ffff₁₆ = 2⁻¹⁶³⁸² × (1 − 2⁻¹¹²)

0000 0000 0000 0000 0000 0000 0000 0001₁₆ = 2⁻¹⁶³⁸² × 2⁻¹¹² = 2⁻¹⁶⁴⁹⁴

≈ 3.3621031431120935062626778173217519551 × 10⁻⁴⁹³²

≈ 6.4751751194380251109244389582276465525 × (largest subnormal number)10⁻⁴⁹⁶⁶

(smallest positive subnormal number)

</pre>

<pre>

0001 0000 0000 0000 0000 0000 0000 0000₁₆ = 2⁻¹⁶³⁸²

0000 ffff ffff ffff ffff ffff ffff ffff₁₆ = 2⁻¹⁶³⁸² × (1 − 2⁻¹¹²)

≈ 3.3621031431120935062626778173217526026 × 10⁻⁴⁹³²

≈ 3.3621031431120935062626778173217519551 × (smallest positive normal number)10⁻⁴⁹³²

(smallestlargest subnormal number larger than one)

</pre>

<pre>

7ffe ffff ffff ffff ffff ffff ffff ffff₁₆ = 2¹⁶³⁸³ × (2 − 2⁻¹¹²)

0001 0000 0000 0000 0000 0000 0000 0000₁₆ = 2⁻¹⁶³⁸²

≈ 1.1897314953572317650857593266280070162 × 10⁴⁹³²

≈ 3.3621031431120935062626778173217526026 × (largest normal number)10⁻⁴⁹³²

(closestsmallest approximationpositive tonormal πnumber)

</pre>

<pre>

3ffe ffff ffff ffff ffff ffff ffff ffff₁₆ = 1 − 2⁻¹¹³

7ffe ffff ffff ffff ffff ffff ffff ffff₁₆ = 2¹⁶³⁸³ × (2 − 2⁻¹¹²)

≈ 0.9999999999999999999999999999999999037

≈ 1.1897314953572317650857593266280070162 × (largest number less than one)10⁴⁹³²

≈ 3.3621031431120935062626778173217526026(largest ×normal 10⁻⁴⁹³²number)

</pre>

<pre>

3fff 0000 0000 0000 0000 0000 0000 0000₁₆ = 1 (one)

3ffe ffff ffff ffff ffff ffff ffff ffff₁₆ = 1 − 2⁻¹¹³

≈ 60.4751751194380251109244389582276465525 × 10⁻⁴⁹⁶⁶9999999999999999999999999999999999037

≈ 1.1897314953572317650857593266280070162(largest ×number 10⁴⁹³²less than one)

</pre>

<pre>

3fff 0000 0000 0000 0000 0000 0000 0001₁₆ = 1 + 2⁻¹¹²

3fff 0000 0000 0000 0000 0000 0000 0000₁₆ = 1 (one)

≈ 1.0000000000000000000000000000000001926

</pre>

(smallest number larger than one)

<pre>

4000 0000 0000 0000 0000 0000 0000 0000₁₆ = 2

c0003fff 0000 0000 0000 0000 0000 0000 00000001₁₆ = −21 + 2⁻¹¹²

≈ 31.3621031431120935062626778173217519551 × 10⁻⁴⁹³²0000000000000000000000000000000001926

(closestsmallest number approximationlarger tothan 1/3one)

</pre>

<pre>

0000 0000 0000 0000 0000 0000 0000 0000₁₆ = 0

80004000 0000 0000 0000 0000 0000 0000 0000₁₆ = −02

3fffc000 0000 0000 0000 0000 0000 0000 00010000₁₆ = 1 + 2⁻¹¹²−2

</pre>

<pre>

7fff 0000 0000 0000 0000 0000 0000 0000₁₆ = infinity

ffff0000 0000 0000 0000 0000 0000 0000 0000₁₆ = −infinity0

40008000 0000 0000 0000 0000 0000 0000 0000₁₆ = 2−0

</pre>

<pre>

4000 921f b544 42d1 8469 898c c517 01b8₁₆ ≈ 3.1415926535897932384626433832795027975

00007fff 0000 0000 0000 0000 0000 0000 0000₁₆ = 0infinity

(closest approximation to π)

7fffffff 0000 0000 0000 0000 0000 0000 0000₁₆ = infinity−infinity

</pre>

<pre>

3ffd 5555 5555 5555 5555 5555 5555 5555₁₆ ≈ 0.3333333333333333333333333333333333173

4000 921f b544 42d1 8469 898c c517 01b8₁₆ ≈ 3.1415926535897932384626433832795027975

(closest approximation to 1/3)

≈ 0.9999999999999999999999999999999999037(closest approximation to π)

</pre>

<pre>

3ffd 5555 5555 5555 5555 5555 5555 5555₁₆ ≈ 0.3333333333333333333333333333333333173

≈ (closest approximation to 1.0000000000000000000000000000000001926/3)

</pre>

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

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A similar technique can be used to produce a '''double-quad arithmetic''', which is represented as a sum of two quadruple-precision values. They can represent operations with at least 226 (or 227) bits.<ref>sourceware.org [http://sourceware.org/ml/libc-alpha/2012-03/msg01024.html Re: The state of glibc libm]</ref>

== Implementations ==

Quadruple precision is often implemented in software by a variety of techniques (such as the double-double technique above, although that technique does not implement IEEE quadruple precision), since direct hardware support for quadruple precision is, {{as of|2016|lc=on}}, less common (see "[[#Hardware support|Hardware support]]" below). One can use general [[arbitrary-precision arithmetic]] libraries to obtain quadruple (or higher) precision, but specialized quadruple-precision implementations may achieve higher performance.

=== Computer-language support ===

A separate question is the extent to which quadruple-precision types are directly incorporated into computer [[programming language]]s.

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As of 2024, [[Rust (programming language)|Rust]] is currently working on adding a new <code>f128</code> type for IEEE quadruple-precision 128-bit floats.<ref>{{cite web |last1=Cross |first1=Travis |title=Tracking Issue for f16 and f128 float types |url=https://github.com/rust-lang/rust/issues/116909 |website=GitHub |access-date=2024-07-05}}</ref>

=== Libraries and toolboxes ===

* The [[GNU Compiler Collection|GCC]] quad-precision math library, [https://gcc.gnu.org/onlinedocs/libquadmath libquadmath], provides <code>__float128</code> and <code>__complex128</code> operations.

* The [[Boost (C++ libraries)|Boost]] multiprecision library Boost.Multiprecision provides unified cross-platform C++ interface for <code>__float128</code> and <code>_Quad</code> types, and includes a custom implementation of the standard math library.<ref>{{cite web |title=Boost.Multiprecision – float128 |url=http://www.boost.org/doc/libs/1_58_0/libs/multiprecision/doc/html/boost_multiprecision/tut/floats/float128.html |access-date=2015-06-22}}</ref>