Quadruple-precision floating-point format: Difference between revisions

Revision as of 04:42, 11 July 2025

Revision as of 11:25, 11 July 2025

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The stored exponents 0000₁₆ and 7FFF₁₆ are interpreted specially.

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

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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.

<pre>

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

≈ 6.4751751194380251109244389582276465525 × 10⁻⁴⁹⁶⁶

(smallest positive subnormal number)

</pre>

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

<pre>

≈ 6.4751751194380251109244389582276465525 × 10⁻⁴⁹⁶⁶

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

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

(largest subnormal number)

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

</pre>

≈ (largest3.3621031431120935062626778173217519551 normal× number)10⁻⁴⁹³²

(smallest positive(largest subnormal number)

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

<pre>

≈ (largest3.3621031431120935062626778173217526026 number× less than one)10⁻⁴⁹³²

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

≈ 3.3621031431120935062626778173217526026 × 10⁻⁴⁹³²(smallest positive normal number)

(smallest positive normal number)

</pre>

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

<pre>

≈ (smallest1.1897314953572317650857593266280070162 number× larger than one)10⁴⁹³²

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

≈ 1.1897314953572317650857593266280070162 × 10⁴⁹³²(largest normal number)

(largest normal number)

</pre>

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

<pre>

≈ (smallest positive normal number)0.9999999999999999999999999999999999037

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

≈ 0.9999999999999999999999999999999999037 (largest number less than one)

(largest number less than one)

</pre>

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

<pre>

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

</pre>

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

<pre>

≈ 1.0000000000000000000000000000000001926

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

≈ 1.0000000000000000000000000000000001926 (smallest number larger than one)

(smallest number larger than one)

</pre>

c000 4000 0000 0000 0000 0000 0000 0000 0000₁₆ = −22

<pre>

4000 c000 0000 0000 0000 0000 0000 0000 0000₁₆ = 2−2

c000 0000 0000 0000 0000 0000 0000 0000₁₆ = −2

</pre>

8000 0000 0000 0000 0000 0000 0000 0000 0000₁₆ = −00

<pre>

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

8000 0000 0000 0000 0000 0000 0000 0000₁₆ = −0

</pre>

ffff 7fff 0000 0000 0000 0000 0000 0000 0000₁₆ = −infinityinfinity

<pre>

7fff ffff 0000 0000 0000 0000 0000 0000 0000₁₆ = infinity−infinity

ffff 0000 0000 0000 0000 0000 0000 0000₁₆ = −infinity

</pre>

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

<pre>

(largestclosest approximation subnormalto numberπ)

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

(closest approximation to π)

</pre>

3ffd 5555 5555 5555 5555 5555 5555 5555₁₆ ≈ 0.3333333333333333333333333333333333173

<pre>

(closest approximation to π1/3)

3ffd 5555 5555 5555 5555 5555 5555 5555₁₆ ≈ 0.3333333333333333333333333333333333173

(closest approximation to 1/3)

</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]].