# Method of continuity

Template:Lowercase Template:Floating-point In computing, decimal128 is a decimal floating-point computer numbering format that occupies 16 bytes (128 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations.

Decimal128 supports 34 decimal digits of significand and an exponent range of −6143 to +6144, i.e. Template:Gaps to Template:Gaps. (Equivalently, Template:Gaps to Template:Gaps.) Therefore, decimal128 has the greatest range of values compared with other IEEE basic floating point formats. Because the significand is not normalized, most values with less than 34 significant digits have multiple possible representations; Template:Gaps, etc. Zero has Template:Gaps possible representations (Template:Gaps if you include both signed zeros).

Decimal128 floating point is a relatively new decimal floating-point format, formally introduced in the 2008 version of IEEE 754.

## Representation of decimal128 values

IEEE 754 allows two alternative representation methods for decimal128 values. The standard does not specify how to signify which representation is used, for instance in a situation where decimal128 values are communicated between systems.

In one representation method, based on binary integer decimal, the significand is represented as binary coded positive integer.

The other, alternative, representation method is based on densely packed decimal for most of the significand (except the most significant digit).

Both alternatives provide exactly the same range of representable numbers: 34 digits of significand and 3×212 = Template:Gaps 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. The remaining combinations encode infinities and NaNs.

If the leading 4 bits of the significand is between 0 and 7, the number begins as follows

s 00mmm xxx   Exponent begins with 00, significand with 0mmm
s 01mmm xxx   Exponent begins with 01, significand with 0mmm
s 10mmm xxx   Exponent begins with 10, significand with 0mmm


If the leading 4 bits of the significand are binary 1000 or 1001 (decimal 8 or 9), the number begins as follows:

s 1100m xxx   Exponent begins with 00, significand with 100m
s 1101m xxx   Exponent begins with 01, significand with 100m
s 1110m xxx   Exponent begins with 10, significand with 100m


The following bits (xxx in the above) encode the additional exponent bits and the remainder of the most significant digit, but the details vary depending on the encoding alternative used. There is no particular reason for this difference, other than historical reasons in the eight-year long development of IEEE 754-2008.

The final combinations are used for infinities and NaNs, and are the same for both alternative encodings:

s 11110 x   ±Infinity (see Extended real number line)
s 11111 0   quiet NaN (sign bit ignored)
s 11111 1   signaling NaN (sign bit ignored)


In the latter cases, all other bits of the encoding are ignored. Thus, it is possible to initialize an array to NaNs by filling it with a single byte value.

### Binary integer significand field

This format uses a binary significand from 0 to 1034−1 = Template:Gaps = 1ED09BEAD87C0378D8E63FFFFFFFF16 = Template:Gaps. The encoding can represent binary significands up to 10×2110−1 = Template:Gaps but values larger than 1034−1 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 14 bits following the sign bit, and the significand is the remaining 113 bits, with an implicit leading 0 bit:

s 00eeeeeeeeeeee (0)TTTtttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt
s 01eeeeeeeeeeee (0)TTTtttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt
s 10eeeeeeeeeeee (0)TTTtttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt


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

If the 4 bits after the sign bit are "1100", "1101", or "1110", 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.

s 11 00eeeeeeeeeeee (100)Ttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt
s 11 01eeeeeeeeeeee (100)Ttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt
s 11 10eeeeeeeeeeee (100)Ttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt tttttttttttttttttttttttttttttttt


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. Note also that 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 2^113 > 1.038×1034), and are thus decoded as zero, but the pattern is same as decimal32 and decimal64.

In the above cases, the value represented is

(−1)sign × 10exponent−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 encoding.

Unlike the binary integer significand version, where the exponent changed position and came before the significand, this encoding combines the leading 2 bits of the exponent and the leading digit (3 or 4 bits) of the significand 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 11 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 after that are interpreted as the leading decimal digit (0 to 7):

s 00 TTT (00)eeeeeeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]
s 01 TTT (01)eeeeeeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]
s 10 TTT (10)eeeeeeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]


If the 4 bits after the sign bit are "1100", "1101", or "1110", 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)eeeeeeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]
s 1101 T (01)eeeeeeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt]
s 1110 T (10)eeeeeeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt][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.

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

$(-1)^{\text{signbit}}\times 10^{{\text{exponentbits}}_{2}-6176_{10}}\times {\text{truesignificand}}_{10}$ 