Quadruple-precision floating-point format: Difference between revisions

Revision as of 12:36, 11 July 2025

Revision as of 20:29, 13 July 2025

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=== Quadruple precision examples ===

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

{| style="font-family: monospace, monospace;"

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0000 0000 0000 0000 0000 0000 0000 0001₁₆ = 2⁻¹⁶³⁸² × 2⁻¹¹² = 2⁻¹⁶⁴⁹⁴<br>

{{spaces|42}}≈ 6.4751751194380251109244389582276465525 × 10⁻⁴⁹⁶⁶<br>

{{spaces|42}}(smallest positive subnormal number)

0000 0000ffff 0000ffff 0000ffff 0000ffff 0000ffff 0000ffff 0001ffff₁₆ = 2⁻¹⁶³⁸² × (1 − 2⁻¹¹² = 2)<sup>−16494</supbr>

{{spaces|42}}≈ 3.3621031431120935062626778173217519551 × 10⁻⁴⁹³²<br>

≈ 6.4751751194380251109244389582276465525 × 10⁻⁴⁹⁶⁶

{{spaces|42}}(smallest positivelargest subnormal number)

0001 0000 ffff0000 ffff0000 ffff0000 ffff0000 ffff0000 ffff ffff0000₁₆ = 2⁻¹⁶³⁸² × (1 − 2<sup>−112</supbr>)

{{spaces|42}}≈ 3.36210314311209350626267781732175195513621031431120935062626778173217526026 × 10⁻⁴⁹³²<br>

{{spaces|42}}(smallest positive normal number)

(largest subnormal number)

00017ffe 0000ffff 0000ffff 0000ffff 0000ffff 0000ffff 0000ffff 0000ffff₁₆ = 2^{−1638216383} × (2 − 2⁻¹¹²)<br>

{{spaces|42}}≈ 1.1897314953572317650857593266280070162 × 10⁴⁹³²<br>

≈ 3.3621031431120935062626778173217526026 × 10⁻⁴⁹³²

{{spaces|42}}(smallest positivelargest normal number)

7ffe3ffe ffff ffff ffff ffff ffff ffff ffff₁₆ = 2¹⁶³⁸³ × (21 − 2^−112−113)<br>

{{spaces|42}}≈ 0.9999999999999999999999999999999999037<br>

≈ 1.1897314953572317650857593266280070162 × 10⁴⁹³²

{{spaces|42}}(largest number less than one)

(largest normal number)

3ffe3fff ffff0000 ffff0000 ffff0000 ffff0000 ffff0000 ffff0000 ffff0000₁₆ = 1 − 2⁻¹¹³(one)

≈ 0.9999999999999999999999999999999999037

(largest number less than one)

3fff 0000 0000 0000 0000 0000 0000 00000001₁₆ = 1 (one)+ 2⁻¹¹²<br>

{{spaces|42}}≈ 1.0000000000000000000000000000000001926<br>

{{spaces|42}}(smallest number larger than one)

3fff4000 0000 0000 0000 0000 0000 0000 00010000₁₆ = 1 + 2<sup>−112</supbr>

ffffc000 0000 0000 0000 0000 0000 0000 0000₁₆ = −infinity−2

≈ 1.0000000000000000000000000000000001926

(smallest number larger than one)

40000000 0000 0000 0000 0000 0000 0000 0000₁₆ = 20<br>

c0008000 0000 0000 0000 0000 0000 0000 0000₁₆ = −2−0

00007fff 0000 0000 0000 0000 0000 0000 0000₁₆ = 0infinity<br>

8000ffff 0000 0000 0000 0000 0000 0000 0000₁₆ = −0−infinity

7fff4000 0000921f 0000b544 000042d1 00008469 0000898c 0000c517 000001b8₁₆ =≈ infinity3.1415926535897932384626433832795027975<br>

{{spaces|42}}(closest approximation to π)

ffff 0000 0000 0000 0000 0000 0000 0000₁₆ = −infinity

40003ffd 921f5555 b5445555 42d15555 84695555 898c5555 c5175555 01b85555₁₆ ≈ 30.14159265358979323846264338327950279753333333333333333333333333333333333173<br>

{{spaces|42}}(closest approximation to π1/3)

|}

3ffd 5555 5555 5555 5555 5555 5555 5555₁₆ ≈ 0.3333333333333333333333333333333333173

(closest approximation to 1/3)

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]].