In computing, decimal64 is a decimal floating-point computer number format that occupies 8 bytes (64 bits) in computer memory. The format was formally introduced in the 2008 revision of the IEEE 754 standard, also known as ISO/IEC/IEEE 60559:2011.
Decimal64 values are categorized as normal or subnormal (denormal) numbers and can be encoded in either binary integer decimal (BID) or densely packed decimal (DPD) formats. Normal values can possess 16-digit precision ranging from ñ1.000000000000000àto ñ9.999999999999999ÃÂ. In addition to normal and subnormal numbers, the format also includes signed zeros, infinities, and NaNs.
The binary format of identical size accommodates a spectrum from denormal-min ñ5ÃÂ, through normal-min with complete 53-bit precision ñ2.2250738585072014ÃÂ, to maximum ñ1.7976931348623157ÃÂ.
Because the significand for the IEEE 754, decimal formats are not normalized and most values with less than 16 significant digits have multiple possible representations; 1000000 à10<sup>âÂÂ2</sup> = 100000 à10<sup>âÂÂ1</sup> = 10000 à10<sup>0</sup> = 1000 à10<sup>1</sup> all have the value 10000. These sets of representations for the same value are called 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 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:
Both alternatives provide exactly the same set of representable numbers: 16 digits of significand and 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.
In both cases, the most significant 4 bits of the significand, which only have 10 possible values, are combined with two bits of the exponent (3 possible values) to use 30 of the 32 possible values of a 5-bit field. The remaining combinations encode infinities and NaNs. BID and DPD use different bits of the combination field.
For Infinity and NaN, all other bits of the encoding are not used. Thus, an array can be filled with a single byte value to set it to Infinities or NaNs.
This format uses a binary significand from 0 to The encoding, completely stored on 64 bits, can represent binary significands up to but values larger than 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 of the significand are in the range 0 to 7 (0000<sub>2</sub> to 0111<sub>2</sub>), or higher (1000<sub>2</sub> or 1001<sub>2</sub>).
If the two bits after the sign bit are "00", "01", or "10", then the exponent field consists of the following the sign bit and the significand is the remaining with an implicit leading . This includes subnormal numbers where the leading significand digit is 0.
If the after the sign bit are "11", then the 10-bit exponent field is shifted to the right (after both the sign bit and the "11" bits thereafter) and the represented significand is in the remaining . 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).
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 <sub>2</sub> with the leading encoding 7; the first significand which requires a 54th bit is The highest valid significant is whose binary encoding is <sub>2</sub> (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
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
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.
The leading of the exponent and the leading digit (3 or ) of the significand are combined into the five bits that follow the sign bit. The eight bits after that are the exponent continuation field, providing the less-significant bits of the exponent. The last are the significand continuation field, consisting of five 10-bit declets. Each declet encodes three decimal digits using the DPD encoding.
If the initial two bits following the sign bit are "00", "01", or "10", they represent the leading bits of the exponent, whereas the subsequent three bits "cde" are regarded as the leading decimal digit (ranging from 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):
The remaining two combinations (11 110 and 11 111) of the 5-bit field after the sign bit are used to represent ñinfinity 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. The non-standard encodings fill in the gap between
In the above cases, with the true significand as the sequence of decimal digits decoded, the value represented is
There are multiple libraries available for calculations with decimal data types. Some examples are listed below: