Decimal64 floating-point format: Difference between revisions

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{{Short description|64-bit computer number format}}

{{lowercase title}}

{{Use dmy dates|date=July 2020|cs1-dates=y}}

{{floating-point}}

In [[computing]], '''decimal64''' is a [[decimal floating point|decimal floating-point]] [[computer number format]] that occupies 8 bytes (64 bits) in computer memory.

ItDecimal64 wasis a decimal floating-point format, formally introduced in the [[IEEE 754-2008 revision|2008 revision]]<ref name="IEEE-754_2008">{{cite book |title=IEEE Standard for Floating-Point Arithmetic |author=IEEE Computer Society |date=2008-08-29 |publisher=[[IEEE]] |id=IEEE Std 754-2008 |doi=10.1109/IEEESTD.2008.4610935 |ref=CITEREFIEEE_7542008 |isbn=978-0-7381-5753-5 |url=https://ieeexplore.ieee.org/document/4610935 |access-date=2016-02-08}}</ref> of the [[IEEE 754]] standard, whichalso wasknown taken over into theas ISO/IEC/IEEE 60559:2011.<ref name="ISO-60559_2011">{{Cite book |last=ISO/IEC JTC 1/SC 25|title=ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic |url=https://www.iso.org/standard/57469.html |publisher=ISO |pages=1–58 |date=June 2011}}</ref> standard.

== Format ==

Decimal64 supports 'normal' values that can have 16 digit precision from {{gaps|±1.000|000|000|000|000|e=-383}} to {{gaps|±9.999|999|999|999|999|e=384}}, plus 'denormal' values with ramp-down relative precision down to ±1 × 10.×10⁻³⁹⁸, [[signed zero]]s, signed infinities and [[NaN]] (Not a Number). This format allowssupports two different encodings.

The binary format of the same size supports a range from denormal-min {{gaps|±5|||||e=-324|}}, over normal-min with full 53-bit precision {{gaps|±2.225|073|858|507|201|e=-308|4}} to max {{gaps|±1.797|693|134|862|315|e=+308|7}}.

Because the significand for the [[IEEE 754]] decimal formats is not normalized, most values with less than 16 [[significant digits]] have multiple possible representations; 1000000 × 10^-2−2=100000 × 10^-1−1=10000 × 10⁰=1000 × 10¹ all have the value 10000. These sets of representations for a same value are called ''[[Cohort (floating point)|cohorts]]'', the different members can be used to denote how many digits of the value are known precisely. Each signed zero has 768 possible representations (1536 for all zeros, in two different cohorts).

== Encoding of decimal64 values ==

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{| class="wikitable"

|-

! Sign !! Combination !! Significand continuation

! <u>S</u>ign !! Co<u>m</u>bination !! <u>T</u>railing significand bits

|-

! 1 bit !! 13 bits !! 50 bits

|-

| {{mono|s}} || {{mono|mmmmmmmmmmmmm}} || tttttttttttttttttttttttttttttttttttttttttttttttttt{{mono|cccccccccccccccccccccccccccccccccccccccccccccccccc}}

|}

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! rowspan="2" |Significand / Description

|-

! g12 !! g11 !! g10 !! g9 !! g8 !! g7 !! g6 !! g5 !! g4 !! g3 !! g2

! m₁₂!! m₁₁!! m₁₀!! m₉!! m₈!! m₇!! m₆!! m₅!! m₄!! m₃!! m₂

!g1

!m₁

!g0

!m₀

|-

| colspan="16" |combination field not! starting with '11', bits ab = 00, 01 or 10

|-

| style="font-family:monospace; background:#cedff2;" | '''a''' || style="font-family:monospace; background:#cedff2;" | '''b''' || style="font-family:monospace; background:#cedff2;" | '''c''' || style="font-family:monospace; background:#cedff2;" | '''d''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cef2e0;" | '''e''' || style="font-family:monospace; background:#cef2e0;" |'''f''' || style="font-family:monospace; background:#cef2e0;" |'''g'''

| || style="font-family:monospace; background:#cedff2;" | '''abcdmmmmmm''' || style="background:#cef2e0;" | {{mono|(0)'''efgtttttttttttttttttttttttttttttttttttttttttttttttttt''' }}

Finite number with significandsmall <first 9007199254740992,digit fitsof into 53significand bits(0&nbsp;..&nbsp;7).

|-

| colspan="16" |combination field starting with '11', but not 1111, bits ab = 11, bits cd = 00, 01 or 10

|-

| 1 || 1 || style="font-family:monospace; background:#cedff2;" | '''c'''|| style="font-family:monospace; background:#cedff2;" | '''d''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''e''' || style="font-family:monospace; background:#cedff2;" | '''f''' || style="font-family:monospace; background:#cef2e0;" | '''g'''

| || style="font-family:monospace; background:#cedff2;" | '''cdmmmmmmef''' || style="background:#cef2e0;" | {{mono|'''100gtttttttttttttttttttttttttttttttttttttttttttttttttt''' }}

Finite number with significandbig <first 9007199254740991,digit of significand needs(8 54or bits9).

|-

| colspan="16" |combination field starting with '1111', bits abcd = 1111

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! rowspan="2" |Significand / Description

|-

! g12 !! g11 !! g10 !! g9 !! g8 !! g7 !! g6 !! g5 !! g4 !! g3 !! g2

! m₁₂!! m₁₁!! m₁₀!! m₉!! m₈!! m₇!! m₆!! m₅!! m₄!! m₃!! m₂

!g1

!m₁

!g0

!m₀

|-

| colspan="16" |combination field not! starting with '11', bits ab = 00, 01 or 10

|-

| style="font-family:monospace; background:#cedff2;" | '''a''' || style="font-family:monospace; background:#cedff2;" | '''b''' || style="font-family:monospace; background:#cef2e0;" | '''c''' || style="font-family:monospace; background:#cef2e0;" | '''d''' || style="font-family:monospace; background:#cef2e0;" | '''e''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m'''

| || style="font-family:monospace; background:#cedff2;" | '''abmmmmmmmm'''|| style="background:#cef2e0;" | {{nowrap|{{mono|(0)'''cde tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt''' }}}}

Finite number with small first digit of significand (0&nbsp;…&nbsp;7).

|-

| colspan="16" |combination field starting with '11', but not 1111, bits ab = 11, bits cd = 00, 01 or 10

|-

| 1 || 1 || style="font-family:monospace; background:#cedff2;" | '''c''' || style="font-family:monospace; background:#cedff2;" | '''d''' || style="font-family:monospace; background:#cef2e0;" | '''e''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m''' || style="font-family:monospace; background:#cedff2;" | '''m'''

| || style="font-family:monospace; background:#cedff2;" | '''cdmmmmmmmm'''|| style="background:#cef2e0;" | {{nowrap|{{mono|'''100e tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt''' }}}}

Finite number with big first digit of significand (8 or 9).

|-

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|signaling NaN (with payload in significand)

|}

The DPD/3BCD transcoding for the declets is given by the following table. b9...b0 are the bits of the DPD, and d2...d0 are the three BCD digits.