Octuple-precision floating-point format

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In computing, octuple precision is a binary floating-point-based computer number format that occupies 32 bytes (256 bits or 64 nibbles) in computer memory. This 256 bit octuple precision is for applications requiring results in higher than quadruple precision. Disclaimer: This format is rarely (if ever) used and very few things support it.

IEEE 754 octuple-precision binary floating-point format: binary128

The IEEE 754 standard specifies a binary128 as having:

• Sign bit: 1 bit
• Exponent width: 18 bits
• Significand precision: 238 bits (237 explicitly stored)

This gives from 33 - 36 significant decimal digits precision (if a decimal string with at most 33 significant decimal is converted to IEEE 754 octuple precision and then converted back to the same number of significant decimal, then the final string should match the original; and if an IEEE 754 octuple precision is converted to a decimal string with at least 36 significant decimal and then converted back to octuple, then the final number must match the original ^[1]).

The format is written with an implicit lead bit with value 1 unless the exponent is stored with all zeros. Thus only 112 bits of the significand appear in the memory format, but the total precision is 113 bits (approximately 34 decimal digits, ${\displaystyle \log _{10}(2^{113})\approx 34.016}$ ). The bits are laid out as follows: [[|thumbnail|center]]  

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 = −262143
• E_max = 262143
• Exponent bias = 3FFF₁₆ = 16383

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 0000₁₆ and 7FFF₁₆ are interpreted specially.

Exponent	Significand zero	Significand non-zero	Equation
000016	0, −0	subnormal numbers	${\displaystyle (-1)^{\text{signbit}}\times 2^{-16382}\times 0.{\text{significandbits}}_{2}}$
000116, ..., 7FFE16	normalized value		${\displaystyle (-1)^{\text{signbit}}\times 2^{{{\text{exponentbits}}_{2}}-16383}\times 1.{\text{significandbits}}_{2}}$
7FFF16	±∞	NaN (quiet, signalling)

The minimum strictly positive (subnormal) value is 2⁻¹⁶⁴⁹⁴ ≈ 10⁻⁴⁹⁶⁵ and has a precision of only one bit. The minimum positive normal value is 2⁻¹⁶³⁸² ≈ 3.3621 × 10⁻⁴⁹³² and has a precision of 112 bits, i.e. ±2 ⁻¹⁶⁴⁹⁴ as well. The maximum representable value is 2¹⁶³⁸⁴ - 2¹⁶²⁷² ≈ 1.1897 × 10⁴⁹³².

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 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000   = infinity
ffff 0000 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 if ever implemented in to software since usage of it is extremely rare. One can use general arbitrary-precision arithmetic libraries to obtain octuple (or higher) precision, but specialized octruple-precision implementations may achieve higher performance.

Computer-language support

In C++, It is possible to make a library to handle Octuple-precision floating-point arithmetic. Controversially, in theory it is possible to do Octuple-precision floating-point arithmic in binary (but it would be incredibly hard, painful torture).

Hardware support

There is little to no hardware support for octuple precision arithmetic.

See also

• IEEE Standard for Floating-Point Arithmetic (IEEE 754)
• Extended precision (80-bit)
• ISO/IEC 10967, Language Independent Arithmetic
• Primitive data type
• long double
• Single-precision floating point format
• Octuple-precision_floating-point_format
• Half-precision_floating-point_format
• Double-precision_floating-point_format
• Quadruple-precision_floating-point_format

References

1. William Kahan (1 October 1987). "Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic" (PDF).

Source For The Layout Of This Entire Page

• The entire layout of this page is based off of Quadruple-precision floating-point format/