Decimal32 floating point format

Representation of decimal32 values

IEEE 754 allows two alternative representation methods for decimal32 values. The standard does not specify how to signify which representation is used, for instance in a situation where decimal32 values are communicated between systems.

In one representation method, based on binary integer decimal, the significand is represented as binary coded positive integer.

The other, alternative, representation method is based on densely packed decimal for most of the significand (except the most significant digit).

Both alternatives provide exactly the same range of representable numbers: 7 digits of significand and 3×2⁶=192 possible exponent values.

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 called the combination field. The remaining combinations encode infinities and NaNs.

Binary integer significand field

This format uses a binary significand from 0 to 10⁷−1 = 9999999 = 98967F₁₆ = 100110001001011001111111₂. The encoding can represent binary significands up to 10×2²⁰−1 = 10485759 = 9FFFFF₁₆ = 100111111111111111111111₂, but values larger than 10⁷−1 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 (0000₂ to 0111₂), or higher (1000₂ or 1001₂).

If the 2 bits after the sign bit are "00", "01", or "10", then the exponent field consists of the 8 bits following the sign bit, and the significand is the remaining 23 bits, with an implicit leading 0 bit:

s 00eeeeee (0)ttt tttttttttt tttttttttt s 01eeeeee (0)ttt tttttttttt tttttttttt s 10eeeeee (0)ttt 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 8-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 21 bits. In this case there is an implicit (that is, not stored) leading 3-bit sequence "100" in the true significand.

s 1100eeeeee (100)t tttttttttt tttttttttt s 1101eeeeee (100)t tttttttttt tttttttttt s 1110eeeeee (100)t tttttttttt tttttttttt

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

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 8000000 is encoded as binary 011110100001001000000000, with the leading 4 bits encoding 7; the first significand which requires a 24th bit is 2²³ = 8388608

In the above cases, the value represented is

(−1)^sign × 10^{exponent−101} × significand

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 (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 the leading 2 bits of the exponent and the leading digit (3 or 4 bits) of the significand into the five bits that follow the sign bit.

These six bits after that are the exponent continuation field, providing the less-significant bits of the exponent.

The last 20 bits are the significand continuation field, consisting of two 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 after that are interpreted as the leading decimal digit (0 to 7):

s 00 TTT (00)eeeeee (0TTT)[tttttttttt][tttttttttt] s 01 TTT (01)eeeeee (0TTT)[tttttttttt][tttttttttt] s 10 TTT (10)eeeeee (0TTT)[tttttttttt][tttttttttt]

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

s 1100 T (00)eeeeee (100T)[tttttttttt][tttttttttt] s 1101 T (01)eeeeee (100T)[tttttttttt][tttttttttt] s 1110 T (10)eeeeee (100T)[tttttttttt][tttttttttt]

The remaining two combinations (11110 and 11111) of the 5-bit field 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. (The 8×3 = 24 non-standard encodings fill in the gap between 10³=1000 and 2¹⁰=1024.)

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

( − 1 ) signbit × 10 exponentbits 2 − 101 10 × truesignificand 10