decimal64 floating-point format

In computing, decimal64 is a decimal floating-point computer number format that occupies 8 bytes (64 bits) in computer memory.

Purpose and use

decimal64 fits well to replace binary64 format in applications where 'small deviations' are unwanted and speed isn't extremely crucial.

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 (in some range, to some precision, to some degree). In a trade-off for reduced performance. They are intended for applications where it's requested to come near to schoolhouse math, such as financial and tax computations. (In short they avoid plenty of problems like 0.2 + 0.1 -> 0.30000000000000004 which happen with binary64 datatypes.)

Range and precision

Decimal64 supports 'normal' values that can have 16 digit precision from ±1.000000000000000×10^⁻³⁸³ to ±9.999999999999999×10^³⁸⁴, plus 'denormal' values with ramp-down relative precision down to ±1 × 10⁻³⁹⁸, signed zeros, signed infinities and NaN (Not a Number).

The binary format of the same bit-size supports a range from denormal-min ±5×10^⁻³²⁴, over normal-min with full 53-bit precision ±2.2250738585072014×10^⁻³⁰⁸ to max ±1.7976931348623157×10^⁺³⁰⁸.

Performance

Performance comparison is inaccurate on modern IT systems for various reasons. One can roughly say that in a current 64-bit Intel(r) / linux / gcc / libdfp / BID implementation, basic arithmetic operations with decimal64 values are between factor 2 and 15 slower than with binary64 data types, while 'higher' functions like powers ( ~600 ) and trigonometric functions like tangent ( ~10 000 ) suffer more performance penalties. Perhaps the GNU gcc project and the 'libdfp' project on github could like some help to improve.

Representation / encoding of decimal64 values

decimal64 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'.

Generic encoding
Sign	Combination	Trailing significand bits
1 bit	13 bits	50 bits
s	mmmmmmmmmmmmm	tttttttttttttttttttttttttttttttttttttttttttttttttt

Besides the special cases infinities and NaNs there are four points relevant to understand the encoding of decimal64.

- BID vs. DPD encoding, Binary Integer Decimal using a positive binary integer decimal for the significand, software centric and designed by Intel(r), vs. Densely Packed Decimal based on densely packed decimal encoding for all except the first digit of the significand, hardware centric and promoted by IBM(r), differences see below. Both alternatives provide exactly the same range of representable numbers: 16 digits of significand and $3 \times 2 8 = 768$ possible exponent values. IEEE 754 allows these two different encodings, without a concept to denote which is used, for instance in a situation where decimal64 values are communicated between systems. CAUTION!: Be aware that transferring binary data between systems using different encodings will mostly produce valid decimal64 numbers, but with different value. Prefer data exchange in íntegral or ASCII 'triplets' for sign, exponent and significand.

- The significands are not 'normalized' (the leading digit(s) are allowed to be "0"), and thus most values with less than 7 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.

- The encodings combine two bits of the exponent with the leading 3 to 4 bits of the significand in a 'combination field', different for 'big' vs. 'small' significands. That enables bigger precision and range, in trade-off that some simple functions like sort and compare, very frequently used in coding, do not work on the bit pattern but require computations to extract exponent and significand and then try to obtain an exponent aligned representation. This effort is partly balanced by saving the effort for normalization, but contributes to the slower performance of the decimal datatypes. Beware: BID and DPD use different bits of the combination field for that, see below.

- Different understanding of significand as integer or fraction, and acc. different bias to apply for the exponent (for decimal64 what is stored in bits can be decoded as base to the power of 'stored value for the exponent minus bias of 383' times significand understood as d₀ . d₋₁ d₋₂ d₋₃ d₋₄ d₋₅ d₋₆ d₋₇ d₋₈ d₋₉ d₋₁₀ d₋₁₁ d₋₁₂ d₋₁₃ d₋₁₄ d₋₁₅ (note: radix dot after first digit, significand fractional), or base to the power of 'stored value for the exponent minus bias of 398' times significand understood as d15 d14 d13 d12 d11 d10 d9 d8 d7 d₆ d₅ d₄ d₃ d₂ d₁ d₀ (note: no radix dot, significand integral), both produce the same result [2019 version of IEEE 754 in clause 3.3, page 18]. For decimal datatypes the second view is more common, while for binary datatypes the first, the biases are different for each datatype.)

In all cases for decimal364, the value represented is

(−1)^sign × 10^{exponent−398} × significand, with the significand understood as positive integer.

Alternatively it can be understood as (−1)^sign × 10^{exponent−383} × significand with the significand digits understood as d₀ . d₋₁ d₋₂ d₋₃ d₋₄ d₋₅ d_{−6 ...}, note the radix dot making it a fraction.

decimal64 is superpower to binary64 in range, and head to head in precision, dec64 all normal values 16-digit, while bin64 alternates between 'not fully 16' and about 16.5 decimal digits ( 17 digits with gaps ).

BID encoding

This format uses a binary significand from 0 to $1016 - 1 = 9999999999999999 = 2386F26FC0FFFF 16 = 1000111000011011110010011011111100000011111111111111112 .$ The encoding, completely stored on 64 bits, can represent binary significands up to $10 \times 2 50 - 1 = 11258999068426239 = 27FFFFFFFFFFFF 16,$ but values larger than $1016 - 1$ 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 after the sign bit are "00", "01", or "10", then the exponent field consists of the 10 bits following the sign bit, and the significand is the remaining 53 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 10-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 51 bits. In this case there is an implicit (that is, not stored) leading 3-bit sequence "100" for the MSB bits of the true significand (in the remaining lower bits ttt...ttt of the significand, not all possible values are used).

Be aware that the bit numbering used in the tables for e.g. m₁₂ … m₀ is in opposite direction than that used in the paper for the IEEE 754 standard G₀ … G₁₂.

BID Encoding
Combination Field													Exponent	Significand / Description
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	e	f	g	abcdmmmmmm	(0)efgtttttttttttttttttttttttttttttttttttttttttttttttttt Finite number with 'small' significand, being < 9007199254740992, fits into 53 bits.
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	e	f	g	cdmmmmmmef	100gtttttttttttttttttttttttttttttttttttttttttttttttttt Finite number with 'big' significand, being > 9007199254740991, needs 54 bits.
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 contrast to DPD format below the leading bits of the significand field do not encode the most significant decimal digit; they are, combined with the implicit prefix of 100 for big significands, simply part of a larger pure-binary number.

The resulting 'raw' exponent is a 10 bit binary integer where the leading bits are not '11', thus values 0 ... 1011111111_b = 0 ... 767_d, appr. bias is to be subtracted. The resulting significand could be a positive binary integer of 54 bits up to 1001 1111111111 1111111111 1111111111 1111111111 1111111111_b = 11258999068426239_d, but values above 10¹⁶ − 1 = 9999999999999999 = 2386F26FC0FFFF₁₆ = 100011100001101111001001101111110000001111111111111111₂ are 'illegal' and have to be treated as zeroes. To obtain the individual decimal digits the significand has to be divided by 10 repeatedly.

In the above cases, the value represented is

(-1) sign \times 10 exponent-398 \times significand

DPD encoding

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.

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

The last 50 bits are the significand continuation field, consisting of five 10-bit declets.^[1] Each declet encodes three decimal digits^[1] 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 "cde" 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 2-bits are the leading bits of the exponent, and the next bit "e" is prefixed with implicit bits "100" to form the leading decimal digit of the significand (8 or 9):

The remaining two combinations (11 110 and 11 111) of the 5-bit field after the sign bit are used to represent ±infinity and NaNs, respectively.

DPD Encoding
Combination Field													Exponent	Significand / Description
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	abmmmmmmmm	(0)cde 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	cdmmmmmmmm	100e 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 resulting 'raw' exponent is a 10 bit binary integer where the leading bits are not '11', thus values 0 ... 1011111111_b = 0 ... 767_d, appr. bias is to be subtracted. The significand's leading decimal digit forms from the (0)cde or 100e bits as binary integer. The subsequent digits are encoded in the 10 bit 'declet' fields 'tttttttttt' according the DPD rules (see below). The full decimal significand is then obtained by concatenating the leading and trailing decimal digits.

The 10-bit DPD to 3-digit BCD 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. Be aware that the bit numbering used here for e.g. b₉ … b₀ is in opposite direction than that used in the paper for the IEEE 754 standard b₀ … b₉, add. the decimal digits are numbered 0-based here while in opposite direction and 1-based in the IEEE 754 paper. The bits on white background are not counting for the value, but signal how to understand / shift the other bits. The concept is to denote which digits are small (0 … 7) and encoded in three bits, and which are not, then calculated from a prefix of '100', and one bit specifying if 8 or 9.

Densely packed decimal encoding rules^[2]
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 × 3 = 24 non-standard encodings fill the unused range from 10³ = 1000 to 2¹⁰ - 1 = 1023.)

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}-398_{10}}\times {\text{truesignificand}}_{10}

History

decimal64 was formally introduced in the 2008 revision^[3] of the IEEE 754 standard, which was taken over into the ISO/IEC/IEEE 60559:2011^[4] standard.

Side effects, more info

DPD encoding is pretty efficient, not wasting more than about 2.4 percent of space vs. BID, because the 2¹⁰ = 1024 possible values in 10 bit is only little more than what is used to encode all numbers from 0 to 999.

Zero has 768 possible representations (1536 accounting signed zeroes, in two different cohorts), (even many more if you account the 'illegal' significands which have to be treated as zeroes).

The gain in range and precision by the 'combination encoding' evolves because the taken 2 bits from the exponent only use three of four possible states, and the 4 MSBs of the significand stay within 0000 … 1001 (10 of 16 possible states). In total that is $3 \times 10 = 30$ possible values when combined in one encoding, which is representable in 5 instead of 6 bits (⁠ $2^{5}=32$ ⁠).

In the cases 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.

References

^ ^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 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 JTC 1/SC 25 (June 2011). ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic. ISO. pp. 1–58.{{cite book}}: CS1 maint: numeric names: authors list (link)

[Muller_2010-1] 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_2000-2] 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_2008-3] 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-60559_2011-4] ISO/IEC JTC 1/SC 25 (June 2011). ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic. ISO. pp. 1–58.{{cite book}}: CS1 maint: numeric names: authors list (link)

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