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.

decimal64Decimal64 wasis a decimal floating-point format, formally introduced in the [[IEEE 754-2008 revision|2008 revision]]<ref name="IEEE-754_2008">{{cite book |author=IEEE Computer Society |url=https://ieeexplore.ieee.org/document/4610935 |title=IEEE Standard for Floating-Point Arithmetic |author=IEEE Computer Society |date=2008-08-29 |publisher=[[IEEE]] |isbnid=978IEEE Std 754-0-7381-5753-52008 |doi=10.1109/IEEESTD.2008.4610935 |id=IEEE Std 754-2008 |ref=CITEREFIEEE_7542008 |access-dateisbn=2016978-020-087381-5753-5 }}</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 |url=https://www.iso.org/standard/57469.html |title=ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic |dateurl=June 2011https://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 supports 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).

== Representation / encodingEncoding of decimal64 values ==

decimal64 values are represented in a 'not normalized' near to 'scientific format', with combining some bits of the exponent with the leading bits of the significand in a 'combination field'.

{| 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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If the {{val|2|u=bits}} after the sign bit are "11", then the 10-bit exponent field is shifted {{val|2|u=bits}} to the right (after both the sign bit and the "11" bits thereafter), and the represented significand is in the remaining {{val|51|u=bits}}. In this case there is an implicit (that is, not stored) leading 3-bit sequence "100" for the MSB bits of the true significand (in the remaining lower bits ''ttt...ttt'' of the significand, not all possible values are used).

Be aware that the bit numbering used in the tables for e.g. m₁₂ … m₀  is in opposite direction than that used in the paper for the IEEE 754 standard G₀ … G₁₂.

{| class="wikitable" style="text-align:left; border-width:0;"

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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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: {{math|1=(−1)^sign × 10^{exponent−398} × significand}} <!-- Remember, significand is defined as an integer: 0 <= significand < 10^16 -->

If the four bits after the sign bit are "1111" then the value is an infinity or a NaN, as described above:

0 11110 xx...x +infinity

1 11110 xx...x -infinity

x 11111 0x...x a quiet NaN

x 11111 1x...x a signalling NaN

=== Densely packed decimal significand field ===

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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.

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:<math>(-1)^\text{signbit}\times 10^{\text{exponentbits}_2-398_{10}}\times \text{truesignificand}_{10}</math>

== History ==

decimal64 was formally introduced in the [[IEEE 754-2008 revision|2008 revision]]<ref name="IEEE-754_2008">{{cite book |author=IEEE Computer Society |url=https://ieeexplore.ieee.org/document/4610935 |title=IEEE Standard for Floating-Point Arithmetic |date=2008-08-29 |publisher=[[IEEE]] |isbn=978-0-7381-5753-5 |doi=10.1109/IEEESTD.2008.4610935 |id=IEEE Std 754-2008 |ref=CITEREFIEEE_7542008 |access-date=2016-02-08}}</ref> of the [[IEEE 754]] standard, which was taken over into the ISO/IEC/IEEE 60559:2011<ref name="ISO-60559_2011">{{Cite book |last=ISO/IEC JTC 1/SC 25 |url=https://www.iso.org/standard/57469.html |title=ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic |date=June 2011 |publisher=ISO |pages=1–58}}</ref> standard.

== Side effects, more info ==

Zero has 768 possible representations (1536 accounting signed zeroes, in two different cohorts), (even many more if you account the 'illegal' significands which have to be treated as zeroes).

The gain in range and precision by the 'combination encoding' evolves because the taken 2 bits from the exponent only use three states, and the 4 MSBs of the significand stay within 0000&nbsp;…&nbsp;1001 (10 states). In total that is {{math|1=3&nbsp;×&nbsp;10&nbsp;=&nbsp;30}} possible values when combined in one encoding, which is representable in 5 bits ({{tmath|1=2^5=32}}).

== See also ==

* [[ISO/IEC 10967]], Language Independent Arithmetic

* [[Primitive data type]]

* [[Q notation (scientific notation)|D (E) notation (scientific notation)]]

== References ==