Incomputing,decimal64 is adecimal floating-pointcomputer number format that occupies 8 bytes (64 bits) in computer memory. The format was formally introduced in the2008 revision^[1] of theIEEE 754 standard, also known asISO/IEC/IEEE 60559:2011.^[2]

Decimal64 values are categorized as normal orsubnormal (denormal) numbers and can be encoded in eitherbinary integer decimal (BID) ordensely packed decimal (DPD) formats. Normal values can possess 16-digit precision ranging from ±1.000000000000000×10⁻³⁸³ to ±9.999999999999999×10³⁸⁴. In addition to normal and subnormal numbers, the format also includes signed zeros, infinities, andNaNs.

The binary format of identical size accommodates a spectrum from denormal-min ±5×10⁻³²⁴, through normal-min with complete 53-bit precision ±2.2250738585072014×10⁻³⁰⁸, to maximum ±1.7976931348623157×10⁺³⁰⁸.

Because thesignificand for theIEEE 754, decimal formats are not normalized and most values with less than 16significant 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 the same value are calledcohorts. The different members can be used to denote how many digits of the value are known precisely. Each signed zero has 768 possible representations (1536 for all zeros, in two different cohorts).

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 thebinary encoding, the 16-digit significand is represented as a binary coded positive integer, based onbinary integer decimal (BID).
• In thedecimal encoding, the 16-digit significand is represented as a decimal coded positive integer, based ondensely packed decimal (DPD) with 5 groups of 3 digits each represented indeclets, 10-bit sequences. The most significant bit is encoded separately. This is efficient because 2¹⁰ = 1024, is only slightly more than needed to contain all integers from 0 to 999.

Both alternatives provide exactly the same set of representable numbers: 16 digits of significand and3 × 2⁸ = 768 possible decimal exponent values. All the possible decimal exponent values storable in abinary64 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 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 encodeinfinities andNaNs. BID and DPD use different bits of the combination field.

For Infinity and NaN, all other bits of the encoding are not used. Thus, an array can be filled with a single byte value to set it to Infinities or NaNs.

This format uses a binary significand from 0 to10¹⁶ − 1 =9999999999999999 = 2386F26FC0FFFF₁₆ =100011100001101111001001101111110000001111111111111111₂.The encoding, completely stored on 64 bits, can represent binary significands up to10 × 2⁵⁰ − 1 =11258999068426239 = 27FFFFFFFFFFFF₁₆, but values larger than10¹⁶ − 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 significant4 bits of the significand are in the range 0 to 7 (0000₂ to 0111₂), or higher (1000₂ or 1001₂).

If the two bits after the sign bit are "00", "01", or "10", then the exponent field consists of the10 bits following the sign bit and the significand is the remaining53 bits with an implicit leading0 bit. This includessubnormal numbers where the leading significand digit is 0.

If the2 bits after the sign bit are "11", then the 10-bit exponent field is shifted2 bits to the right (after both the sign bit and the "11" bits thereafter) and the represented significand is in the remaining51 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 bitsttt...ttt of the significand, not all possible values are used).

The leading bits of the significand field donot encode the most significant decimal digit; they are simply part of a larger pure-binary number. For example, a significand of8000000000000000 is encoded as binary011100011010111111010100100110001101000000000000000000₂ with the leading4 bits encoding 7; the first significand which requires a 54th bit is2⁵³ =9007199254740992. The highest valid significant is9999999999999999 whose binary encoding is(100)011100001101111001001101111110000001111111111111111₂ (with the 3 most significant bits (100) not stored but implicit as shown above; and the next bit is always zero in valid encodings).

In the above cases, the value represented is

(−1)^sign × 10^{exponent−398} × significand

If the four bits after the sign bit are "1111" then the value is an infinity or a NaN, as described above:

0 11110 xx...x    +infinity1 11110 xx...x    -infinityx 11111 0x...x    a quiet NaNx 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 thedensely packed decimal (DPD) encoding.

The leading2 bits of the exponent and the leading digit (3 or4 bits) of the significand are combined into the five bits that follow the sign bit. The eight bits after that are the exponent continuation field, providing the less-significant bits of the exponent. The last50 bits are the significand continuation field, consisting of five 10-bit declets.^[3] Eachdeclet encodes three decimal digits^[3] using the DPD encoding.

If the initial two bits following the sign bit are "00", "01", or "10", they represent the leading bits of the exponent, whereas the subsequent three bits "cde" are regarded as the leading decimal digit (ranging from 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 (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.

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.

The 8 decimal values whose digits are all 8s or 9s have four codings each. The bits marked x in the table above areignored on input, but will always be 0 in computed results. The8 × 3 = 24 non-standard encodings fill in the gap between10³ = 1000 and 2¹⁰ = 1024.

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

${\displaystyle (-1{)}^{\text{signbit}}\times {10}^{{\text{exponentbits}}_2-{398}_{10}}\times {\text{truesignificand}}_{10}}$

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There are multiple libraries available for calculations with decimal data types. Some examples are listed below:

• Libdfp implements the ISO/IEC Technical Report "ISO/IEC TR 24732". The library implements math functions for environments built ongcc andglibc. It uses BID-encoded IEEE 754 decimal types, with the possibility to switch to Decimal64 floating-point format via recompilation ofgcc. The library can be slow, and sometimes inaccurate, for conversions and complicated operations.
• Intel Decimal Floating-Point Math Library is aC library that implements the IEEE 754-2008 Decimal Floating-Point Arithmetic specification by addressing decimal types asUINTxx items nameddecimalxx. The library improves performance by calculating more intricate functions in binary where possible, but this introduces some binary rounding errors into decimal calculations.
• decNumber provides data types and functions to calculate values using Decimal Floating-Point with arbitrary precision. It features the option to set the IEEE 754-compatible parametersdecSingle,decDouble, anddecQuad. It efficiently performs conversions between decimal numbers and strings.
• Mpdecimal is an implementation of theGeneral Decimal Arithmetic Specification proposed byMike Cowlishaw, with the option to set IEEE type-compatible parameters. It providesarbitrary precision, using an 8-bit integer to encode runtime flags and the number’s sign. The library represents numbers using seven or more double- or quad-words, storing the signed exponent, the digits, the length in words, the allocated words, a pointer todata, anddata itself.Data is stored as an array of two or more words, which holds the coefficient in 32-bit (9-digit) or 64-bit (19-digit) items. The library can handle a wide range of inputs and produces accurate results. It is implemented inPython as thedecimal module.

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

• Computer arithmetic
• Data types
• Floating point types

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