Decimal64 floating-point format: Difference between revisions

Revision as of 18:21, 7 December 2024

Revision as of 17:31, 8 December 2024

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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).

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.

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 mostMSB bits of the true significand (in the remaining lower bits ''ttt...ttt'' of the significand, not all possible values are used).

s 00eeeeeeee (0)ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt

s 01eeeeeeee (0)ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt

s 10eeeeeeee (0)ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt

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

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 most bits of the true significand (in the remaining lower bits ''ttt...ttt'' of the significand, not all possible values are used).

s 1100eeeeeeee (100)t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt

s 1101eeeeeeee (100)t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt

s 1110eeeeeeee (100)t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt

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. The 2-bit sequences "00", "01", or "10" after the sign bit are part of the exponent field.

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

|+ BID Encoding

|-

! colspan="11" | Combination Field

!

!

! rowspan="2" |

! rowspan="2" | Exponent

! rowspan="2" |Significand

! rowspan="2" | Description

|-

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

!g1

!g0

|-

| colspan="17" |combination field not! starting with '11', bits ab = 00, 01 or 10

|-

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

| || style="background:#cedff2;" | '''abcdmmmmmm''' || style="background:#cef2e0;" | (0)'''efgtttttttttttttttttttttttttttttttttttttttttttttttttt''' || Finite number with small first digit of significand (0 … 7).

|-

| colspan="17" |combination field starting with '11', but not 1111, bits ab = 11, bits cd = 00, 01 or 10

|-

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

| || style="background:#cedff2;" | '''cdmmmmmmef''' || style="background:#cef2e0;" | '''100gtttttttttttttttttttttttttttttttttttttttttttttttttt''' || Finite number with big first digit of significand (8 or 9).

|-

| colspan="17" |combination field starting with '1111', bits abcd = 1111

|-

| 1 || 1 || 1 || 1 || 0 || colspan="6" |

|

|

| rowspan="3" | || || || ±Infinity

|-

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

| colspan="5" |

|

|

|

| || quiet NaN

|-

|1

|1

|1

|1

|1

|1

| colspan="5" |

|

|

|

|

|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).