decimal32 floatingpoint format
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Floatingpoint formats 

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In computing, decimal32 is a decimal floatingpoint computer numbering format that occupies 4 bytes (32 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Like the binary16 format, it is intended for memory saving storage.
Decimal32 supports 7 decimal digits of significand and an exponent range of −95 to +96, i.e. ±0.000000×10 ^{−95} to ±9.999999×10 ^{96}. (Equivalently, ±0000000×10 ^{−101} to ±9999999×10 ^{90}.) Because the significand is not normalized (there is no implicit leading "1"), most values with less than 7 significant digits have multiple possible representations; 1 × 10^{2}=0.1 × 10^{3}=0.01 × 10^{4}, etc. Zero has 192 possible representations (384 when both signed zeros are included).
Decimal32 floating point is a relatively new decimal floatingpoint format, formally introduced in the 2008 version^{[1]} of IEEE 754 as well as with ISO/IEC/IEEE 60559:2011.^{[2]}
Representation of decimal32 values[edit]
Sign  Combination  Trailing significand field 

1 bit  11 bits  20 bits 
s  ggggggggggg  tttttttttttttttttttt 
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 (BID), the significand is represented as binary coded positive integer.
The other, alternative, representation method is based on densely packed decimal (DPD) 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 × 2^{6} = 192 possible exponent values.
In both encodings, BID and DPD, the 2 most significant exponent bits, and the 4 most significant bits of the significand, are combined to 5 bits. The position of the 5 bits in the combination field varies, but otherwise the encoding is identical. 5 bits suffice instead of 6, because the 2 MSBs from the exponent only encode values from 0 to 2 (3 possible values), and the 4 MSBs of the significand represent a decimal digit between 0 and 9 (10 possible values). In total we have 3*10 = 30 possible values when combined in one encoding, which is representable in 5 bits ().
Combination Field  MSBs of  LSBs of Exponent  Description  

g10  g9  g8  g7  g6  g5  g4  g3  g2  g1  g0  Exponent  Significand  
0  0  d  e  f  g  h  i  a  b  c  00  0abc  defghi  Finite number full binary significand = 0abctttttttttttttttttttt  
0  1  01  
1  0  10  
1  1  0  0  d  e  f  g  h  i  c  00  100c  Finite number full binary significand = 100ctttttttttttttttttttt  
1  1  0  1  01  
1  1  1  0  10  
1  1  1  1  0  ±Infinity  
1  1  1  1  1  NaN (with payload in Significand) 
Combination Field  MSBs of  LSBs of Exponent  Significand's Leading Decimal Digit  Description  

g10  g9  g8  g7  g6  g5  g4  g3  g2  g1  g0  Exponent  Significand  
0  0  a  b  c  d  e  f  g  h  i  00  0abc  defghi  4*a + 2*b + c  Finite number with  
0  1  01  
1  0  10  
1  1  0  0  c  00  100c  8 + c  Finite number with  
1  1  0  1  01  
1  1  1  0  10  
1  1  1  1  0  ±Infinity  
1  1  1  1  1  NaN (with payload in Significand) 
For the BID encoding, the full binary significand is obtained by appending the bits from the trailing significand field to the significand's MSBs, as shown in the BID table above. The resulting significand is a positive binary integer of 24 bits, that has to be divided by 10 repeatedly, to obtain the individual decimal digits.
For the DPD encoding, the DPD table above shows how to obtain the significand's leading decimal digit from the significand's MSBs. To obtain the trailing significand decimal digits, the significant trailing field has to be decoded according to the DPD rules (see below). The full decimal significand is then obtained by concatenating the leading and trailing decimal digits.
For ±Infinity, besides the sign bit, all the remaining bits are ignored (i.e., both the exponent and significand fields have no effect). For NaNs the sign bit has no meaning in the standard, and is ignored. Therefore, signed and unsigned NaNs are equivalent, even though some programs will show NaNs as signed. The bit g5 determines whether the NaN is quiet (0) or signaling (1). The bits of the significand are the NaN's payload and can hold user defined data (e.g., to distinguish how NaNs were generated). Like for normal significands, the payload of NaNs can be either in BID or DPD encoding.
Binary integer significand field[edit]
This format uses a binary significand from 0 to 10^{7} − 1 = 9999999 = 98967F_{16} = 100110001001011001111111_{2}. The encoding can represent binary significands up to 10 × 2^{20} − 1 = 10485759 = 9FFFFF_{16} = 100111111111111111111111_{2}, but values larger than 10^{7} − 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 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 8bit 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 3bit 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" 2bit sequence after the sign bit indicates that there is an implicit "100" 3bit 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 purebinary 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 2^{23} = 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[edit]
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.
These six bits after that are the exponent continuation field, providing the lesssignificant bits of the exponent.
The last 20 bits are the significand continuation field, consisting of two 10bit declets.^{[3]} Each declet encodes three decimal digits^{[3]} 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 5bit 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.
DPD encoded value  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)  
a  b  c  g  h  f  1  0  1  i  0abc  100f  0ghi  (0–7) (8–9) (0–7)  
g  h  c  d  e  f  1  1  0  i  100c  0def  0ghi  (8–9) (0–7) (0–7)  
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)  
d  e  c  0  1  f  1  1  1  i  100c  0def  100i  (8–9) (0–7) (8–9)  
a  b  c  1  0  f  1  1  1  i  0abc  100f  100i  (0–7) (8–9) (8–9)  
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 nonstandard 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
See also[edit]
References[edit]
 ^ IEEE Computer Society (20080829). IEEE Standard for FloatingPoint Arithmetic. IEEE. doi:10.1109/IEEESTD.2008.4610935. ISBN 9780738157535. IEEE Std 7542008. Retrieved 20160208.
 ^ "ISO/IEC/IEEE 60559:2011". 2011. Retrieved 20160208.
{{cite journal}}
: Cite journal requiresjournal=
(help)  ^ ^{a} ^{b} Muller, JeanMichel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, ClaudePierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Handbook of FloatingPoint Arithmetic (1 ed.). Birkhäuser. doi:10.1007/9780817647056. ISBN 9780817647049. LCCN 2009939668.
 ^ Cowlishaw, Michael Frederic (20070213) [20001003]. "A Summary of Densely Packed Decimal encoding". IBM. Archived from the original on 20150924. Retrieved 20160207.