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 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.

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'.

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

IEEE 754 allows two alternative encodings for decimal64 values. The standard does not specify how to signify which representation is used, for instance in a situation where decimal64 values are communicated between systems:

• In the binary encoding, the 16-digit significand is represented as a binary coded positive integer, based on binary integer decimal (BID).
• In the decimal encoding, the 16-digit significand is represented as a decimal coded positive integer, based on densely packed decimal (DPD) with 5 groups of 3 digits (except the most significant digit encoded specially) are each represented in declets (10-bit sequences). This is pretty efficient, because 2¹⁰ = 1024, is only little more than needed to still contain all numbers from 0 to 999.

Both alternatives provide exactly the same set of representable numbers: 16 digits of significand and 3 × 2⁸ = 768 possible decimal exponent values. (All the possible decimal exponent values storable in a binary64 number are representable in decimal64, and most bits of the significand of a binary64 are stored keeping roughly the same number of decimal digits in the significand.)

In both cases, the most significant 4 bits of the significand (which actually only have 10 possible values) are combined with two bits of the exponent (3 possible values) to use 30 of the 32 possible values of a 5-bit field. The remaining combinations encode infinities and NaNs. BID and DPD use different bits of the combination field for that.

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.

Because the significand for the IEEE 754 decimal formats is not normalized, most values with less than 16 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.

BID encoding

This format uses a binary significand from 0 to 10¹⁶ − 1 = 9999999999999999 = 2386F26FC0FFFF₁₆ = 100011100001101111001001101111110000001111111111111111₂.The encoding, completely stored on 64 bits, can represent binary significands up to 10 × 2⁵⁰ − 1 = 11258999068426239 = 27FFFFFFFFFFFF₁₆, 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
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 × 10^{exponent−398} × 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. Each declet encodes three decimal digits 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
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

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 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 of the IEEE 754 standard, which was taken over into the ISO/IEC/IEEE 60559:2011 standard.

Side effects, more info

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 states, and the 4 MSBs of the significand stay within 0000 … 1001 (10 states). In total that is 3 × 10 = 30 possible values when combined in one encoding, which is representable in 5 bits (⁠$2^{5}=32$⁠).

See also

• ISO/IEC 10967, Language Independent Arithmetic
• Primitive data type
• D (E) notation (scientific notation)