In computing, **decimal32** is a decimal floating-point 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 floating-point format, formally introduced in the 2008 version^{[1]} of IEEE 754 as well as with ISO/IEC/IEEE 60559:2011.^{[2]}

Sign | Combination | Exponent continuation | Coefficient continuation |
---|---|---|---|

1 bit | 5 bits | 6 bits | 20 bits |

s | mmmmm | xxxxxx | cccccccccccccccccccc |

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 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.

Combination Field | MSBs of | Code Value |
Description | ||||||
---|---|---|---|---|---|---|---|---|---|

m4 | m3 | m2 | m1 | m0 | Exp. | Significand | |||

0 |
0 |
a |
b |
c |
00 |
0abc |
(0-7) | Digit up to 7 | |

0 |
1 |
a |
b |
c |
01 |
0abc
| |||

1 |
0 |
a |
b |
c |
10 |
0abc
| |||

1 | 1 | 0 |
0 |
c |
00 |
100c |
(8-9) | Digit greater than 7 | |

1 | 1 | 0 |
1 |
c |
01 |
100c
| |||

1 | 1 | 1 |
0 |
c |
10 |
100c
| |||

1 | 1 | 1 | 1 | 0 | ±Infinity | ||||

1 | 1 | 1 | 1 | 1 | NaN |

**Note:** The sign bit of NaNs is ignored. The first bit of the remaining exponent determines whether the NaN is quiet or signaling.

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 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 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

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*.^{[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 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.

DPD encoded value | Decimal digits | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

b9 | b8 | b7 | b6 | b5 | b4 | b3 | b2 | b1 | b0 | d2 | d1 | d0 | Values encoded | Description | |

a | b | c | d | e | f | 0 |
g | h | i | 0abc |
0def |
0ghi |
(0–7) (0–7) (0–7) | Three small digits | |

a | b | c | d | e | f | 1 |
0 |
0 |
i | 0abc |
0def |
100i |
(0–7) (0–7) (8–9) | Two small digits, one large | |

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) | ||

g | h | c | 0 |
0 |
f | 1 |
1 |
1 |
i | 100c |
100f |
0ghi |
(8–9) (8–9) (0–7) | One small digit, two large | |

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) | ||

x | x | c | 1 |
1 |
f | 1 |
1 |
1 |
i | 100c |
100f |
100i |
(8–9) (8–9) (8–9) | Three large 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 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

- Decimal64
- Decimal128
- IEEE Standard for Floating-Point Arithmetic (IEEE 754)
- ISO/IEC 10967, Language Independent Arithmetic
- Primitive data type

**^**IEEE Computer Society (2008-08-29).*IEEE Standard for Floating-Point Arithmetic*. IEEE. doi:10.1109/IEEESTD.2008.4610935. ISBN 978-0-7381-5753-5. IEEE Std 754-2008. Retrieved 2016-02-08.**^**"ISO/IEC/IEEE 60559:2011". 2011. Retrieved 2016-02-08.- ^
^{a}^{b}Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010).*Handbook of Floating-Point Arithmetic*(1 ed.). Birkhäuser. doi:10.1007/978-0-8176-4705-6. ISBN 978-0-8176-4704-9. LCCN 2009939668. **^**Cowlishaw, Michael Frederic (2007-02-13) [2000-10-03]. "A Summary of Densely Packed Decimal encoding". IBM. Archived from the original on 2015-09-24. Retrieved 2016-02-07.

Densely packed decimal (DPD) is an efficient method for binary encoding decimal digits.

The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in significant wastage of binary data bandwidth (since four bits can store 16 states and are being used to store only 10). Densely packed decimal is a more efficient code that packs three digits into ten bits using a scheme that allows compression from, or expansion to, BCD with only two or three gate delays in hardware.The densely packed decimal encoding is a refinement of Chen–Ho encoding; it gives the same compression and speed advantages, but the particular arrangement of bits used confers additional advantages:

Compression of one or two digits (into the optimal four or seven bits respectively) is achieved as a subset of the three-digit encoding. This means that arbitrary numbers of decimal digits (not just multiples of three digits) can be encoded efficiently. For example, 38=12×3+2 decimal digits can be encoded in 12×10+7=127 bits – that is, 12 sets of three decimal digits can be encoded using 12 sets of ten binary bits and the remaining two decimal digits can be encoded using a further seven binary bits.

The subset encoding mentioned above is simply the rightmost bits of the standard three-digit encoding; the encoded value can be widened simply by adding leading 0 bits.

All seven-bit BCD numbers (0 through 79) are encoded identically by DPD. This makes conversions of common small numbers trivial. (This must break down at 80, because that requires eight bits for BCD, but the above property requires that the DPD encoding must fit into seven bits.)

The low-order bit of each digit is copied unmodified. Thus, the non-trivial portion of the encoding can be considered a conversion from three base-5 digits to seven binary bits. Further, digit-wise logical values (in which each digit is either 0 or 1) can be manipulated directly without any encoding or decoding being necessary.

Orders of magnitude (numbers)This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.

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