# Decimal64 floating-point format

In computing, decimal64 is a decimal floating-point computer number format that occupies 8 bytes (64 bits) in computer memory.
Decimal64 is a decimal floating-point format, formally introduced in the 2008 revision of the IEEE 754 standard, also known as ISO/IEC/IEEE 60559:2011.

## Format

Decimal64 supports 'normal' values that can have 16 digit precision from ±1.000000000000000×10^$^{−383}$ to ±9.999999999999999×10^$^{384}$, plus 'denormal' values with ramp-down relative precision down to ±1.×10$^{−398}$, signed zeros, 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 ±5×10^$^{−324}$, over normal-min with full 53-bit precision ±2.2250738585072014×10^$^{−308}$ to max ±1.7976931348623157×10^$^{+308}$.
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 × 10$^0$=1000 × 10$^1$ all have the value 10000. These sets of representations for a 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).

## Encoding of decimal64 values

| Sign | Combination | Significand continuation |
| 1 bit | 13 bits | 50 bits |
| s | mmmmmmmmmmmmm | 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 the binary encoding, the 16-digit significand is represented as a binary coded positive integer, based on binary integer decimal (BID).
* In the 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 2$^{10}$ = 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 3 × 2$^8$ = 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.)
In both cases, the most significant 4 bits of the significand (which actually 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 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.

### Binary integer significand field

This format uses a binary significand from 0 to 10$^{16}$ − 1 = 9999999999999999 = 2386F26FC0FFFF$_{16}$ = 100011100001101111001001101111110000001111111111111111$_2$.The encoding, completely stored on 64 bits, can represent binary significands up to 10 × 2$^{50}$ − 1 = 11258999068426239 = 27FFFFFFFFFFFF$_{16}$, but values larger than 10$^{16}$ − 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 4 bits of the significand are in the range 0 to 7 (0000$_2$ to 0111$_2$), or higher (1000$_2$ or 1001$_2$).
If the 2 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. 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 (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 MSB bits of the true significand (in the remaining lower bits ttt...ttt of the significand, not all possible values are used).
BID Encoding :
| Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Exponent | Significand / Description |
| g12 | g11 | g10 | g9 | g8 | g7 | g6 | g5 | g4 | g3 | g2 | g1 | g0 | Exponent | Significand / Description |
| combination field not! starting with '11', bits ab = 00, 01 or 10 ||||||||||||||||
| a | b | c | d | m | m | m | m | m | m | e | f | g | abcdmmmmmm | (0)efgtttttttttttttttttttttttttttttttttttttttttttttttttt Finite number with small first digit of significand (0 .. 7). |
| combination field starting with '11', but not 1111, bits ab = 11, bits cd = 00, 01 or 10 ||||||||||||||||
| 1 | 1 | c | d | m | m | m | m | m | m | e | f | g | cdmmmmmmef | 100gtttttttttttttttttttttttttttttttttttttttttttttttttt Finite number with big first digit of significand (8 or 9). |
| combination field starting with '1111', bits abcd = 1111 ||||||||||||||||
| 1 | 1 | 1 | 1 | 0 | | | | | | | | | | ±Infinity |
| 1 | 1 | 1 | 1 | 1 | 0 | | | | | | | | | quiet NaN |
| 1 | 1 | 1 | 1 | 1 | 1 | | | | | | | | | 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 8000000000000000 is encoded as binary 011100011010111111010100100110001101000000000000000000$_2$, with the leading 4 bits encoding 7; the first significand which requires a 54th bit is 2$^{53}$ = 9007199254740992. The highest valid significant is 9999999999999999 whose binary encoding is (100)011100001101111001001101111110000001111111111111111$_2$ (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
    (−1)$^{sign}$ × 10$^{exponent−398}$ × significand
If the four bits after the sign bit are "1111" then the value is an infinity or a NaN, as described above:

### 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.
The leading 2 bits of the exponent and the leading digit (3 or 4 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 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 "cde" 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):
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.
DPD Encoding :
| Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Combination Field | Exponent | Significand / Description |
| g12 | g11 | g10 | g9 | g8 | g7 | g6 | g5 | g4 | g3 | g2 | g1 | g0 | Exponent | Significand / Description |
| combination field not! starting with '11', bits ab = 00, 01 or 10 ||||||||||||||||
| a | b | c | d | e | m | m | m | m | m | m | m | m | abmmmmmmmm | (0)cde tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt Finite number with small first digit of significand (0 … 7). |
| combination field starting with '11', but not 1111, bits ab = 11, bits cd = 00, 01 or 10 ||||||||||||||||
| 1 | 1 | c | d | e | m | m | m | m | m | m | m | m | cdmmmmmmmm | 100e tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt Finite number with big first digit of significand (8 or 9). |
| combination field starting with '1111', bits abcd = 1111 ||||||||||||||||
| 1 | 1 | 1 | 1 | 0 | | | | | | | | | | ±Infinity |
| 1 | 1 | 1 | 1 | 1 | 0 | | | | | | | | | quiet NaN |
| 1 | 1 | 1 | 1 | 1 | 1 | | | | | | | | | 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.
Densely packed decimal encoding rules :
| DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | DPD encoded value | Decimal digits | Decimal digits | Decimal digits | Decimal digits | Decimal digits | Decimal digits |
| Code space (1024 states) | b9 | b8 | b7 | b6 | b5 | b4 | b3 | b2 | b1 | b0 | d2 | d1 | d0 | Values encoded | Description | Occurrences (1000 states) |
| 50.0% (512 states) | a | b | c | d | e | f | 0 | g | h | i | 0abc | 0def | 0ghi | (0–7) (0–7) (0–7) | 3 small digits | 51.2% (512 states) |
| 37.5% (384 states) | a | b | c | d | e | f | 1 | 0 | 0 | i | 0abc | 0def | 100i | (0–7) (0–7) (8–9) | 2 small digits, 1 large digit | 38.4% (384 states) |
| 37.5% (384 states) | a | b | c | g | h | f | 1 | 0 | 1 | i | 0abc | 100f | 0ghi | (0–7) (8–9) (0–7) | 2 small digits, 1 large digit | 38.4% (384 states) |
| 37.5% (384 states) | g | h | c | d | e | f | 1 | 1 | 0 | i | 100c | 0def | 0ghi | (8–9) (0–7) (0–7) | 2 small digits, 1 large digit | 38.4% (384 states) |
| 9.375% (96 states) | g | h | c | 0 | 0 | f | 1 | 1 | 1 | i | 100c | 100f | 0ghi | (8–9) (8–9) (0–7) | 1 small digit, 2 large digits | 9.6% (96 states) |
| 9.375% (96 states) | d | e | c | 0 | 1 | f | 1 | 1 | 1 | i | 100c | 0def | 100i | (8–9) (0–7) (8–9) | 1 small digit, 2 large digits | 9.6% (96 states) |
| 9.375% (96 states) | a | b | c | 1 | 0 | f | 1 | 1 | 1 | i | 0abc | 100f | 100i | (0–7) (8–9) (8–9) | 1 small digit, 2 large digits | 9.6% (96 states) |
| 3.125% (32 states, 8 used) | x | x | c | 1 | 1 | f | 1 | 1 | 1 | i | 100c | 100f | 100i | (8–9) (8–9) (8–9) | 3 large digits, b9, b8: don't care | 0.8% (8 states) |
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 8 × 3 = 24 non-standard encodings fill in the gap between 10$^3$ = 1000 and 2$^{10}$ = 1024.)
In the above cases, with the true significand as the sequence of decimal digits decoded, the value represented is
    $(-1)^{\text{signbit}}\times 10^{{\text{exponentbits}}_{2}-398_{10}}\times {\text{truesignificand}}_{10}$