decimal32 floating-point format

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 ±9.999999×10⁹⁶. (Equivalently, ±0000001×10⁻¹⁰¹ to ±9999999×10⁹⁰.) 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 × 10²=0.1 × 10³=0.01 × 10⁴, 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]

Representation of decimal32 values

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 × 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.

Encoding of the Combination Field

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

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 = 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. 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 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.

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 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.

Densely packed decimal encoding rules[4]

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)	Three 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)	Two small digits, one large	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)	One small digit, two large	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)	Three large digits, bits b9 and b8 are 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 10³ = 1000 and 2¹⁰ = 1024.)

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

${\displaystyle (-1)^{\text{signbit}}\times 10^{{\text{exponentbits}}_{2}-101_{10}}\times {\text{truesignificand}}_{10}}$

References

1. 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.
2. "ISO/IEC/IEEE 60559:2011". 2011. Retrieved 2016-02-08. Cite journal requires |journal= (help)
3. 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.
4. 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.