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. Both alternatives provide exactly the same range of representable numbers: 7 digits of significand and possible 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 called the combination field. The remaining combinations encode infinities and NaNs. The sign bit of NaNs is ignored. The first bit of the remaining exponent determines whether the NaN is quiet or signaling.
Binary integer significand field
This format uses a binary significand from 0 to 107 − 1 = = 98967F16 =. The encoding can represent binary significands up to 10 × 220 − 1 = = 9FFFFF16 =, but values larger than 107 − 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 8 bits following the sign bit, and the significand is the remaining 23 bits, with an implicit leading 0 bit: s 00eeeeee ttt tttttttttt tttttttttt s 01eeeeee ttt tttttttttt tttttttttt s 10eeeeee 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, and the represented significand is in the remaining 21 bits. In this case there is an implicit leading 3-bit sequence "100" in the true significand. s 1100eeeeee t tttttttttt tttttttttt s 1101eeeeee t tttttttttt tttttttttt s 1110eeeeee 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. The "00", "01", or "10" bits 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, with the leading 4 bits encoding 7; the first significand which requires a 24th bit is 223 = 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
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. 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 : s 00 TTT eeeeee s 01 TTT eeeeee s 10 TTT eeeeee 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 : s 1100 T eeeeee s 1101 T eeeeee s 1110 T eeeeee The remaining two combinations 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. In the above cases, with the true significand as the sequence of decimal digits decoded, the value represented is
