Decimal64 floating-point format: Difference between revisions

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

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

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

It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations.

Decimal64 is 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 }}</ref> of the [[IEEE 754]] standard, also known as 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>

Decimal64 supports 16 [[decimal digit]]s of [[significand]] and an [[exponent]] range of −383 to +384, i.e. {{gaps|±0.000|000|000|000|000|e=-383}} to {{gaps|±9.999|999|999|999|999|e=384}}. (Equivalently, {{gaps|±0|000|000|000|000|000|e=-398}} to {{gaps|±9|999|999|999|999|999|e=369}}.) In contrast, the corresponding binary format, which is the most commonly used type, has an approximate range of {{gaps|±0.000|000|000|000|001|e=-308}} to {{gaps|±1.797|693|134|862|315|e=308}}. Because the significand is not normalized, most values with less than 16 [[significant digits]] have multiple possible representations; {{gaps|1×10²|&#61;|0.1×10³|&#61;|0.01×10⁴}}, etc. Zero has 768 possible representations (1536 if you include both [[signed zero]]s).

== Format ==

Decimal64 floating point is a relatively new decimal floating-point format, formally introduced in the 2008 version of [[IEEE 754]].

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.×10⁻³⁹⁸, [[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}}.

== Representation of decimal64 values ==

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=100000 × 10⁻¹=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).

IEEE 754 allows two alternative representation methods for decimal64 values. The standard does not specify how to signify which representation is used, for instance in a situation where decimal64 values are communicated between systems:

== Encoding of decimal64 values ==

* In the [[#Binary integer significand field|binary representation method]], the 16-digit significand is represented as a binary coded positive integer, based on [[binary integer decimal]].

* In the [[#Densely packed decimal significand field|decimal representation method]], the 16-digit significand is represented as a decimal coded positive integer, based on [[densely packed decimal]] with 5 groups of 3 digits (except the most significant digit encoded specially) are each represented in declets (10-bit sequences).

{| class="wikitable"

Both alternatives provide exactly the same range of representable numbers: 16 digits of significand and 3×2⁸ = 768 possible decimal exponent values. (All the possible decimal exponent values storable in a [[binary64]] number are representable in decimal64, and most bits of the significand of a binary64]] are stored keeping roughly the same number of decimal digits in the significand.)

|-

! Sign !! Combination !! Significand continuation

|-

! 1 bit !! 13 bits !! 50 bits

|-

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

|}

IEEE 754 allows two alternative encodings for decimal64 values. The standard does not specify how to signify which representation is used, for instance in a situation where decimal64 values are communicated between systems:

In both cases, the most significant 4 bits of the significand (which actually only have 10 possible values) are combined with the most significant 2 bits of the exponent (3 possible values) to use 30 of the 32 possible values of a 5-bit field. The remaining combinations encode [[infinity|infinities]] and [[NaN]]s.

* In the [[#Binary integer significand field|binary encoding]], the 16-digit significand is represented as a binary coded positive integer, based on [[binary integer decimal]] (BID).

If the most significant digit of the significand is between 0 and 7 (encodable on three bits ''mmm''), the number begins as follows

* In the [[#Densely packed decimal significand field|decimal encoding]], the 16-digit significand is represented as a decimal coded positive integer, based on [[densely packed decimal]] (DPD) with 5 groups of 3 digits (except the most significant digit encoded specially) are each represented in declets (10-bit sequences). This is pretty efficient, because 2¹⁰ = 1024, is only little more than needed to still contain all numbers from 0 to 999.

Both alternatives provide exactly the same set of representable numbers: 16 digits of significand and {{math|size=100%|1=3 × 2⁸ = 768}} possible decimal exponent values. (All the possible decimal exponent values storable in a [[binary64]] number are representable in decimal64, and most bits of the significand of a binary64 are stored keeping roughly the same number of decimal digits in the significand.)

s 00 xxx xxx..xxx Exponent begins with 00, significand with 0xxx

s 01 xxx xxx..xxx Exponent begins with 01, significand with 0xxx

s 10 xxx xxx..xxx Exponent begins with 10, significand with 0xxx

IfIn both cases, the most significant digit4 bits of the significand us(which 8actually oronly 9have (representable10 aspossible anvalues) binaryare numbercombined ''100m''with wheretwo onlybits of the exponent (3 possible values) to use 30 of the 32 possible values of a 5-bit ''m''field. isThe storedremaining combinations encode [[infinity|infinities]] and bits[[NaN]]s. ''100''BID areand implicit),DPD use different bits of the numbercombination beginsfield asfor follows:that.

In the cases of Infinity and NaN, all other bits of the encoding are ignored. Thus, it is possible to initialize an array to Infinities or NaNs by filling it with a single byte value.

s 11 00x xxx...xxx Exponent begins with 00, significand with 100x

s 11 01x xxx...xxx Exponent begins with 01, significand with 100x

s 11 10x xxx...xxx Exponent begins with 10, significand with 100x

The following bits (''xxx...xxx'' in the above) encode the remaining 8 bits of the exponent and the remaining 15 significant digits, but the details vary depending on the encoding alternative used (either the digits of the significand, or bits of the exponent may also come first). There is no particular reason for this difference, other than historical reasons in the eight-year long development of IEEE 754-2008.

The final combinations are used for infinities and NaNs, and are the same for both alternative encodings:

s 11 110 xxx...xxx ±Infinity (see [[Extended real number line]])

s 11 111 0xx...xxx quiet NaN (sign bit ignored)

s 11 111 1xx...xxx signaling NaN (sign bit ignored)

In the latter cases, all other ''xxx...xxx'' bits of the encoding are ignored. Thus, it is possible to initialize an array to NaNs by filling it with a single byte value.

=== Binary integer significand field ===

This format uses a binary significand from 0 to {{math|size=100%|1=10¹⁶−1 − 1 = {{gaps|9|999|999|999|999|999}} = 2386F26FC0FFFF₁₆ = {{gaps|1000|1110000110|1111001001|1011111100|0000111111|1111111111₂}}.}}The encoding, completely stored on 64 bits, can represent binary significands up to {{math|size=100%|1=10 × 2⁵⁰ − 1 = {{gaps|11|258|999|068|426|239}} = 27FFFFFFFFFFFF₁₆,}} but values larger than {{math|size=100%|1=10¹⁶ − 1}} are illegal (and the standard requires implementations to treat them as 0, if encountered on input).

As described above, the encoding varies depending on whether the most significant {{val|4|u=bits}} of the significand are in the range 0 to 7 (0000₂ to 0111₂), or higher (1000₂ or 1001₂).

The encoding, completely stored on 64 bits, can represent binary significands up to 10×2⁵⁰−1 = {{gaps|11|258|999|068|426|239}} = 27FFFFFFFFFFFF₁₆, but values larger than 10¹⁶−1 are illegal (and the standard requires implementations to treat them as 0, if encountered on input).

If the 2 after the sign bit are "00", "01", or "10", then the exponent field consists of the {{val|10|u=bits}} following the sign bit, and the significand is the remaining {{val|53|u=bits}}, with an implicit leading {{val|0|u=bit}}. This includes [[subnormal numbers]] where the leading significand digit is 0.

As described above, the encoding varies depending on whether the most significant 4 bits of the significand are in the range 0 to 7 (0000₂ to 0111₂), or higher (1000₂ or 1001₂).

If the {{val|2 |u=bits}} after the sign bit are "00", "01", or "1011", then the 10-bit exponent field consistsis ofshifted {{val|2|u=bits}} to the 10right bits(after followingboth the sign bit and the "11" bits thereafter), and the represented significand is in the remaining 53 {{val|51|u=bits,}}. withIn this case there is an implicit leading(that 0is, 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).

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

s 00 eeeeeeee [(0)ttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt]

|+ BID Encoding

s 01 eeeeeeee [(0)ttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt]

|-

s 10 eeeeeeee [(0)ttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt]

! colspan="13" | Combination Field

! rowspan="2" |

This includes [[subnormal numbers]] where the leading significand digit is 0.

! rowspan="2" | Exponent

! rowspan="2" |Significand / Description

If the 4 bits after the sign bit are "1100", "1101", or "1110", then the 10-bit exponent field is shifted 2 bits to the right (after both the sign bit and the "11" bits thereafter), and the represented significand is in the remaining 51 bits. In this case there is an implicit (that is, not stored) leading 3-bit sequence "100" for the most bits of the true significand.

|-

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

s 11 00eeeeeeee [(100)ttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt]

!g1

s 11 01eeeeeeee [(100)ttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt]

!g0

s 11 10eeeeeeee [(100)ttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt][tttttttt]

|-

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

The 2-bit sequence "11" after the sign bit indicates that there is an ''implicit'' 3-bit prefix "100" to the significand. Compare having an implicit 1-bit prefix "1" in the significand of normal values for the binary formats. Note also that the 2-bit sequences "00", "01", or "10" after the sgn bit are part of the exponent field.

|-

| 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'''

Note that the leading bits of the significand field do ''not'' encode the most significant decimal digit; they are simply part of a larger pure-binary number. For example, a significand of {{gaps|8|000|000|000|000|000}} is encoded as binary {{gaps|0111|0001101011|1111010100|1001100011|0100000000|0000000000}}, with the leading 4 bits encoding 7; the first significand which requires a 54th bit is 2⁵³ = {{gaps|9|007|199|254|740|992}}.

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

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:#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 big first digit of significand (8 or 9).

|-

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

|-

| 1 || 1 || 1 || 1 || 0 || colspan="8" |

| rowspan="3" | || || ±Infinity

|-

| 1 || 1 || 1 || 1 || 1 ||0

| colspan="7" |

|

| quiet NaN

|-

|1

|1

|1

|1

|1

|1

| colspan="7" |

|

|signaling NaN (with payload in significand)

|}

The leading bits of the significand field do ''not'' encode the most significant decimal digit; they are simply part of a larger pure-binary number. For example, a significand of {{gaps|8|000|000|000|000|000}} is encoded as binary {{gaps|0111|0001101011|1111010100|1001100011|0100000000|0000000000}}₂, with the leading {{val|4|u=bits}} encoding 7; the first significand which requires a 54th bit is {{math|size=100%|1=2⁵³ = {{gaps|9|007|199|254|740|992}}.}} The highest valid significant is {{gaps|9|999|999|999|999|999}} whose binary encoding is

{{gaps|(100)0|1110000110|1111001001|1011111100|0000111111|1111111111}}₂ (with the 3 most significant bits (100) not stored but implicit as shown above; and the next bit is always zero in valid encodings).

In the above cases, the value represented is

: {{math|1=(−1)<sup>sign</subsup> × 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:

s0 11110 xx...x ±+infinity

s1 1111111110 0xxx...x a quiet NaN-infinity

sx 11111 1x0x...x a signallingquiet NaN

x 11111 1x...x a signalling NaN

=== Densely packed decimal significand field ===

In this version, the significand is stored as a series of decimal digits. The leading digit is between 0 and 9 (3 or 4 binary bits), and the rest of the significand uses the [[densely packed decimal]] (DPD) encoding.

Unlike the binary integer significand version, where the exponent changed position and came before the significand, this encoding, combines theThe leading {{val|2 |u=bits}} of the exponent and the leading digit (3 or {{val|4 |u=bits}}) of the significand are combined into the five bits that follow the sign bit.

This eight bits after that are the exponent continuation field, providing the less-significant bits of the exponent.

The last {{val|50|u=bits}} are the significand continuation field, consisting of five 10-bit ''[[declet (computing)|declet]]s''.<ref name="Muller_2010">{{cite book |author-last1=Muller |author-first1=Jean-Michel |author-last2=Brisebarre |author-first2=Nicolas |author-last3=de Dinechin |author-first3=Florent |author-last4=Jeannerod |author-first4=Claude-Pierre |author-last5=Lefèvre |author-first5=Vincent |author-last6=Melquiond |author-first6=Guillaume |author-last7=Revol |author-first7=Nathalie|author7-link=Nathalie Revol |author-last8=Stehlé |author-first8=Damien |author-last9=Torres |author-first9=Serge |title=Handbook of Floating-Point Arithmetic |year=2010 |publisher=[[Birkhäuser]] |edition=1 |isbn=978-0-8176-4704-9<!-- print --> |doi=10.1007/978-0-8176-4705-6 |lccn=2009939668<!-- |isbn=978-0-8176-4705-6 (online), ISBN 0-8176-4704-X (print) -->|url=https://cds.cern.ch/record/1315760 }}</ref> Each declet encodes three decimal digits<ref name="Muller_2010"/> using the DPD encoding.

The last 50 bits are the significand continuation field, consisting of 5 "declets" (10-bit each). Each declet encodes three decimal digits using the DPD encoding.

If the first two bits after the sign bit are "00", "01", or "10", then those are the leading bits of the exponent, and the three bits "TTTcde" after that are interpreted as the leading decimal digit (0 to 7):

If the first two bits after the sign bit are "11", then the second 2-bits are the leading bits of the exponent, and the next bit "e" is prefixed with implicit bits "100" to form the leading decimal digit (8 or 9):

s 00 TTT eeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]

s 01 TTT eeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]

s 10 TTT eeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]

The remaining two combinations (11 110 and 11 111) of the 5-bit field after the sign bit are used to represent ±infinity and NaNs, respectively.

If the 4 bits after the sign bit are "1100", "1101", or "1110", then the second 2-bits are the leading bits of the exponent, and the next bit "T" is prefixed with implicit bits "100" to form the leading decimal digit (8 or 9):

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

s 11 00T eeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]

|+ DPD Encoding

s 11 01T eeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]

|-

s 11 10T eeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]

! colspan="13" | Combination Field

! rowspan="2" |

The remaining two combinations (11 110 and 11 111) of the 5-bit field after the sign bit are used to represent ±infinity and NaNs, respectively.

! rowspan="2" | Exponent

! rowspan="2" |Significand / Description

|-

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

!g1

!g0

|-

| 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).

|-

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

|-

| 1 || 1 || 1 || 1 || 0 || colspan="8" |

| rowspan="3" | || || ±Infinity

|-

| 1 || 1 || 1 || 1 || 1 ||0

| colspan="7" |

|

| quiet NaN

|-

|1

|1

|1

|1

|1

|1

| colspan="7" |

|

|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.

Line 104 ⟶ 168:

The 8 decimal values whose digits are all 8s or 9s have four codings each.

The bits marked x in the table above are [[don't care|ignored]] on input, but will always be 0 in computed results.

(The 8×3{{math|size=100%|1=8 × 3 = 24}} non-standard encodings fill in the gap between {{math|size=100%|1=10³ = 1000 and 2¹⁰ = 1024.}})

In the above cases, with the ''true significand'' as the sequence of decimal digits decoded, the value represented is

Line 112 ⟶ 176:

== See also ==

* [[IEEE 754-2008|IEEE Standard for Floating-Point Arithmetic (IEEE 754)]]

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

* [[Primitive data type]]

* [[D notation (scientific notation)]]

== References ==

Line 121 ⟶ 185:

[[Category:Computer arithmetic]]

[[Category:Data types]]

[[Category:Floating point types]]