Decimal64 floating-point format


In computing, decimal64 is a decimal floating-point computer numbering format that occupies 8 bytes in computer memory.
It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations.
Decimal64 supports 16 decimal digits of significand and an exponent range of −383 to +384, i.e. to. In contrast, the corresponding binary format, which is the most commonly used type, has an approximate range of to. Because the significand is not normalized, most values with less than 16 significant digits have multiple possible representations;, etc. Zero has 768 possible representations.
Decimal64 floating point is a relatively new decimal floating-point format, formally introduced in the 2008 version of IEEE 754 as well as with.

Representation of decimal64 values

SignCombinationExponent continuationCoefficient continuation
1 bit5 bits8 bits50 bits
smmmmmxxxxxxxxcccccccccccccccccccccccccccccccccccccccccccccccccc

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:
Both alternatives provide exactly the same range of representable numbers: 16 digits of significand and 3 × 28 = 768 possible decimal exponent values.
In both cases, the most significant 4 bits of the significand are combined with the most significant 2 bits of the exponent to use 30 of the 32 possible values of a 5-bit field. The remaining combinations encode infinities and NaNs.
Combination fieldExponent MsbitsSignificand MsbitsOther
00mmm000xxx—
01mmm010xxx—
10mmm100xxx—
1100m00100x—
1101m01100x—
1110m10100x—
11110——±Infinity
11111——NaN. Sign bit ignored. First bit of exponent continuation field determines if NaN is signaling.

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.

Binary integer significand field

This format uses a binary significand from 0 to 1016 − 1 = = 2386F26FC0FFFF16 =.
The encoding, completely stored on 64 bits, can represent binary significands up to 10 × 250 − 1 = = 27FFFFFFFFFFFF16, but values larger than 1016 − 1 are illegal.
As described above, the encoding varies depending on whether the most significant 4 bits of the significand are in the range 0 to 7, or higher.
If the 2 bits after the sign bit are "00", "01", or "10", then the exponent field consists of the 10 bits following the sign bit, and the significand is the remaining 53 bits, with an implicit leading 0 bit:
s 00eeeeeeee ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt
s 01eeeeeeee ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt
s 10eeeeeeee ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt
This includes subnormal numbers where the leading significand digit is 0.
If the 2 bits after the sign bit are "11", then the 10-bit exponent field is shifted 2 bits to the right, and the represented significand is in the remaining 51 bits. In this case there is an implicit leading 3-bit sequence "100" for the most bits of the true significand.
s 1100eeeeeeee t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt
s 1101eeeeeeee t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt
s 1110eeeeeeee 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.
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 is encoded as binary 2, with the leading 4 bits encoding 7; the first significand which requires a 54th bit is 253 =. The highest valid significant is whose binary encoding is
2.
In the above cases, the value represented is
If the four bits after the sign bit are "1111" then the value is an infinity or a NaN, as described above:
s 11110 xx...x ±infinity
s 11111 0x...x a quiet NaN
s 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, and the rest of the significand uses the densely packed decimal encoding.
The leading 2 bits of the exponent and the leading digit 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 50 bits are the significand continuation field, consisting of five 10-bit declets. 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 "TTT" after that are interpreted as the leading decimal digit :
s 00 TTT eeeeeeee
s 01 TTT eeeeeeee
s 10 TTT eeeeeeee
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 "T" is prefixed with implicit bits "100" to form the leading decimal digit :
s 1100 T eeeeeeee
s 1101 T eeeeeeee
s 1110 T eeeeeeee
The remaining two combinations of the 5-bit field after the sign bit are used to represent ±infinity and NaNs, respectively.
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.
The 8 decimal values whose digits are all 8s or 9s have four codings each.
The bits marked x in the table above are ignored on input, but will always be 0 in computed results.
In the above cases, with the true significand as the sequence of decimal digits decoded, the value represented is