Decimal Floating Point - IEEE 754-2008 Encoding

IEEE 754-2008 Encoding

The IEEE 754-2008 standard defines 32-, 64- and 128-bit decimal floating-point representations. Like the binary floating-point formats, the number is divided into a sign, and exponent, and a significand. Unlike binary floating-point, numbers are not necessarily normalized; values with few significant digits have multiple possible representations: 1×102=0.1×103=0.01×104, etc. When the significand is zero, the exponent can be any value at all.

IEEE 754-2008 decimal floating-point formats

decimal32	decimal64	decimal128	decimal(32k)	Format
1	1	1	1	Sign field (bits)
5	5	5	5	Combination field (bits)
6	8	12	w = 2×k + 4	Exponent continuation field (bits)
20	50	110	t = 30×k−10	Coefficient continuation field (bits)
32	64	128	32×k	Total size (bits)
7	16	34	p = 3×t/10+1 = 9×k−2	Coefficient size (decimal digits)
192	768	12288	3×2w = 48×4k	Exponent range
96	384	6144	Emax = 3×2w−1	Largest value is 9.99...×10Emax
−95	−383	−6143	Emin = 1−Emax	Smallest normalized value is 1.00...×10Emin
−101	−398	−6176	Etiny = 2−p−Emax	Smallest non-zero value is 1×10Etiny

The exponent ranges were chosen so that the range available to normalized values is approximately symmetrical. Since this cannot be done exactly with an even number of possible exponent values, the extra value was given to Emax.

Two different representations are defined:

• One with a binary integer significand field encodes the significand as a large binary integer between 0 and 10p−1. This is expected to be more convenient for software implementations using a binary ALU.
• Another with a densely packed decimal significand field encodes decimal digits more directly. This makes conversion to and from binary floating-point form faster, but requires specialized hardware to manipulate efficiently. This is expected to be more convenient for hardware implementations.

Both alternatives provide exactly the same range of representable values.

The most significant two bits of the exponent are limited to the range of 0−2, and the most significant 4 bits of the significand are limited to the range of 0−9. The 30 possible combinations are encoded in a 5-bit field, along with special forms for infinity and NaN.

If the most significant 4 bits of the significand are between 0 and 7, the encoded value begins as follows:

s 00 xxxx Exponent begins with 00, significand with 0mmm s 01 xxxx Exponent begins with 01, significand with 0mmm s 10 xxxx 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 1100 xx Exponent begins with 00, significand with 100m s 1101 xx Exponent begins with 01, significand with 100m s 1110 xx Exponent begins with 10, significand with 100m

The leading bit (s in the above) is a sign bit, and 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.

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 111110 quiet NaN (sign bit ignored) s 111111 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.