Octuple-precision floating-point format

In computing, octuple precision is a binary floating-point-based computer number format that occupies 32 bytes (256 bits) in computer memory. This 256-bit octuple precision is for applications requiring results in higher than quadruple precision. This format is rarely (if ever) used and very few things support it.

IEEE 754 octuple-precision binary floating-point format: binary256

In its 2008 revision, the IEEE 754 standard specifies a binary256 format among the interchange formats (it is not a basic format), as having:

• Sign bit: 1 bit
• Exponent width: 19 bits
• Significand precision: 237 bits (236 explicitly stored)

The format is written with an implicit lead bit with value 1 unless the exponent is all zeros. Thus only 236 bits of the significand appear in the memory format, but the total precision is 237 bits (approximately 71 decimal digits: log₁₀(2²³⁷) ≈ 71.344). The bits are laid out as follows:

Exponent encoding

The octuple-precision binary floating-point exponent is encoded using an offset binary representation, with the zero offset being 262143; also known as exponent bias in the IEEE 754 standard.

• E_min = −262142
• E_max = 262143
• Exponent bias = 3FFFF₁₆ = 262143

Thus, as defined by the offset binary representation, in order to get the true exponent the offset of 16383 has to be subtracted from the stored exponent.

The stored exponents 00000₁₆ and 7FFFF₁₆ are interpreted specially.

Exponent	Significand zero	Significand non-zero	Equation
0000016	0, −0	subnormal numbers	$(-1)^{\text{signbit}}\times 2^{-262142}\times 0.{\text{significandbits}}_{2}$
0000116, ..., 7FFFE16	normalized value		$(-1)^{\text{signbit}}\times 2^{{{\text{exponentbits}}_{2}}-262143}\times 1.{\text{significandbits}}_{2}$
7FFFF16	±∞	NaN (quiet, signalling)

The minimum strictly positive (subnormal) value is 2^−262378 ≈ 10⁻⁷⁸⁹⁸⁴ and has a precision of only one bit. The minimum positive normal value is 2^−262142 ≈ 2.4824 × 10⁻⁷⁸⁹¹³. The maximum representable value is 2²⁶²¹⁴⁴ − 2²⁶¹⁹⁰⁷ ≈ 1.6113 × 10⁷⁸⁹¹³.

Octuple-precision examples

These examples are given in bit representation, in hexadecimal, of the floating-point value. This includes the sign, (biased) exponent, and significand.

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000  = +0
8000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000  = −0

7fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000   = +infinity
ffff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000   = −infinity

By default, 1/3 rounds down like double precision, because of the odd number of bits in the significand. So the bits beyond the rounding point are 0101... which is less than 1/2 of a unit in the last place.

Implementations

Octuple precision is rarely implemented since usage of it is extremely rare. Apple Inc. had an implementation of addition, subtraction and multiplication of octuple-precision numbers with a 224-bit two's complement significand and a 32-bit exponent. One can use general arbitrary-precision arithmetic libraries to obtain octuple (or higher) precision, but specialized octuple-precision implementations may achieve higher performance.

Hardware support

There is little to no hardware support for it. Octuple-precision arithmetic is too impractical for most commercial uses of it, making implementation of it very rare (if any).

Processing Statistics

Since an octuple precision numeral takes up 32 bytes of storage, the requirements in-order to transport this piece of data are as follows.

• 8-bit architecture – 32 separate packets¹ of information (at least) in order to transport this across the main data bus
• x16 architecture – 16 separate packets of information (at least) in order to transport this across the main data bus
• x86 architecture – 8 separate packets of information (at least) in order to transport this across the main data bus
• x64 architecture – 4 separate packets of information (at least) in order to transport this across the main data bus

¹statistical extrapolation since it would take 1/8 of the entire memory just to store 1 binary256 numeral, making it completely impractical.

See also

• IEEE Standard for Floating-Point Arithmetic (IEEE 754)
• ISO/IEC 10967, Language-independent arithmetic
• Primitive data type