decimal128 floating-point format

In computing, decimal128 is a decimal floating-point number format that occupies 128 bits in memory. Formally introduced in IEEE 754-2008,^[1].

Purpose and use

Like the binary128 formats, decimal128 takes place where extreme precision or ranges are to be handeled.

In contrast to the binaryxxx data formats the decimalxxx formats provide exact representation of decimal fractions, exact calculations with them and enable human common 'ties away from zero' rounding^[2] (in some range, to some precision, to some degree). In a trade-off for reduced performance, which is especially harming decimal128 computations on common 64- or 32-bit hardware. They are intended for applications where exact schoolhouse math is requested, such as financial and tax computations. (In short they avoid plenty of problems like 0.2 + 0.1 -> 0.30000000000000000000000000000000004 which happen with binary128 datatypes.)

Format

decimal128 supports 'normal' values that can have 34 digit precision from ±1.000000000000000000000000000000000×10^⁻⁶¹⁴³ to ±9.999999999999999999999999999999999×10^⁺⁶¹⁴⁴, plus 'denormal' values with ramp-down relative precision down to ±1 × 10⁻⁶¹⁷⁶, signed zeros, signed infinities and NaN (Not a Number).

The binary format of the same size supports a range from denormal-min ±6×10^⁻⁴⁹⁶⁶, over normal-min with full 113-bit precision ±3.3621031431120935062626778173217526×10^⁻⁴⁹³² to max ±1.189731495357231765085759326628007×10^⁺⁴⁹³².

Because the significand for the IEEE 754 decimal formats is not normalized, most values with less than 34 significant digits have multiple possible representations; 1000000 × 10^-2=100000 × 10^-1=10000 × 10⁰=1000 × 10¹ all have the value 10000. These sets of representations for a same value are called cohorts, the different members can be used to denote how many digits of the value are known precisely. Each signed zero has 12288 possible representations (24576 for all zeros, in two different cohorts).

Representation / encoding of decimal128 values

decimal128 values are represented in a 'not normalized' near to 'scientific format', with combining some bits of the exponent with the leading bits of the significand in a 'combination field'.

Sign	Combination	Trailing significand bits
1 bit	17 bits	110 bits
s	mmmmmmmmmmmmmmmmm	tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt

The IEEE 754 standard allows two alternative encodings for decimal128 values:

The binary encoding, based on binary integer decimal (BID): The significand is encoded as an unsigned integer written in binary.
The decimal encoding, based on densely packed decimal (DPD): The significand is encoded as an unsigned integer written in decimal, with groups of 3 digits packed together in a declet and a special rule for the most significant digit.

This standard does not specify how to signify which encoding is used, for instance in a situation where decimal128 values are communicated between systems.

Both alternatives provide exactly the same set of representable numbers: 34 digits of significand and $3 \times 2 12 = 12288$ 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 5 bits in the combination field. The remaining combinations encode infinities and NaNs.

In the case of Infinity and NaN, all other bits of the encoding are ignored. Thus, it is possible to initialize an array to Infinities or NaNs by filling it with a single byte value.

Binary integer significand field

This format uses a binary significand from 0 to $1034 - 1$ = 9999999999999999999999999999999999 = 1ED09BEAD87C0378D8E63FFFFFFFF₁₆ = 011110110100001001101111101010110110000111110000000011011110001101100011100110001111111111111111111111111111111111₂. The encoding can represent binary significands up to $10 \times 2 110 - 1$ = 12980742146337069071326240823050239 but values larger than $1034 - 1$ are illegal (and the standard requires implementations to treat them as 0, if encountered on input).

If the 2 bits after the sign bit are "00", "01", or "10", then the exponent field consists of the 14 bits following the sign bit, and the significand is the remaining 113 bits, with an implicit leading 0 bit:

This includes subnormal numbers where the leading significand digit is 0.

If the 2 bits after the sign bit are "11", then the 14-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 111 bits. In this case there is an implicit (that is, not stored) leading 3-bit sequence "100" in the true 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.

For the decimal128 format, all of these significands are out of the valid range (they begin with $2113 > 1.038 \times 10 34$ ), and are thus decoded as zero, but the pattern is same as decimal32 and decimal64.

Be aware that the bit numbering used in the tables for e.g. m₁₆ … m₀ is in opposite direction than that used in the paper for the IEEE 754 standard G₀ … G₁₇.

BID Encoding
Combination Field																	Exponent	Significand / Description
m₁₆	m₁₅	m₁₄	m₁₃	m₁₂	m₁₁	m₁₀	m₉	m₈	m₇	m₆	m₅	m₄	m₃	m₂	m₁	m₀	Exponent	Significand / Description
combination field not! starting with '11', bits ab = 00, 01 or 10
a	b	c	d	m	m	m	m	m	m	m	m	m	m	e	f	g	abcdmmmmmmmmmm	(0)efgtttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt Finite number, all 'legal' significands 0 .. 9999999999999999999999999999999999 fit here.
combination field starting with '11', but not 1111, bits ab = 11, bits cd = 00, 01 or 10
1	1	c	d	m	m	m	m	m	m	m	m	m	m	e	f	g	cdmmmmmmmmmmef	100gtttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt Theoretical case, all these signifiands are > 1.0384593717069655257060992658440191 × 10^34, thus > 10^34 - 1, 'illegal' and to be treated as zero.
combination field starting with '1111', bits abcd = 1111
1	1	1	1	0														±Infinity
1	1	1	1	1	0													quiet NaN
1	1	1	1	1	1													signaling NaN (with payload in significand)

In the above cases, the value represented is

(−1)^sign × 10^{exponent−6176} × 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 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).

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.

This twelve bits after that are the exponent continuation field, providing the less-significant bits of the exponent.

The last 110 bits are the significand continuation field, consisting of eleven 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):

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):

DPD Encoding
Combination Field																	Exponent	Significand / Description
m₁₆	m₁₅	m₁₄	m₁₃	m₁₂	m₁₁	m₁₀	m₉	m₈	m₇	m₆	m₅	m₄	m₃	m₂	m₁	m₀	Exponent	Significand / Description
combination field not! starting with '11', bits ab = 00, 01 or 10
a	b	c	d	e	m	m	m	m	m	m	m	m	m	m	m	m	abmmmmmmmmmmmm	(0)cde tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt Finite number with small first digit of significand (0 … 7).
combination field starting with '11', but not 1111, bits ab = 11, bits cd = 00, 01 or 10
1	1	c	d	e	m	m	m	m	m	m	m	m	m	m	m	m	cdmmmmmmmmmmmm	100e tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt Finite number with big first digit of significand (8 or 9).
combination field starting with '1111', bits abcd = 1111
1	1	1	1	0														±Infinity
1	1	1	1	1	0													quiet NaN
1	1	1	1	1	1													signaling NaN (with payload in significand)

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]
Code space (1024 states)	b9	b8	b7	b6	b5	b4	b3	b2	b1	b0	d2	d1	d0	Values encoded	Description	Occurrences (1000 states)
DPD encoded value											Decimal digits
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 $8 \times 3 = 24$ 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)^{\text{signbit}}\times 10^{{\text{exponentbits}}_{2}-6176_{10}}\times {\text{truesignificand}}_{10}

References

^ 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.
^ Cowlishaw, Mike (2007). "Decimal Arithmetic FAQ – Part 1 – General Questions". speleotrove.com. IBM Corporation. Retrieved 2022-07-29.
^ ^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.

[IEEE-754_2008-1] 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.

[2] Cowlishaw, Mike (2007). "Decimal Arithmetic FAQ – Part 1 – General Questions". speleotrove.com. IBM Corporation. Retrieved 2022-07-29.

[Muller_2010-3] 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.

[Cowlishaw_2000-4] 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.

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