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 environments support it.
In its 2008 revision, the IEEE 754 standard specifies a binary256 format among the interchange formats (it is not a basic format), as having:
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: log_{10}(2^{237}) ≈ 71.344). The bits are laid out as follows:
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.
Thus, as defined by the offset binary representation, in order to get the true exponent the offset of 262143 has to be subtracted from the stored exponent.
The stored exponents 00000_{16} and 7FFFF_{16} are interpreted specially.
Exponent | Significand zero | Significand non-zero | Equation |
---|---|---|---|
00000_{16} | 0, −0 | subnormal numbers | (-1)^{signbit} x 2^{−262142} x 0.significandbits_{2} |
00001_{16}, ..., 7FFFE_{16} | normalized value | (-1)^{signbit} x 2^{exponent bits2} x 1.significandbits_{2} | |
7FFFF_{16} | ±∞ | 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 2^{262144} − 2^{261907} ≈ 1.6113 × 10^{78913}.
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.
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.^{[1]} One can use general arbitrary-precision arithmetic libraries to obtain octuple (or higher) precision, but specialized octuple-precision implementations may achieve higher performance.
There is no hardware support for octuple precision. Apart from astrophysical simulations, octuple-precision arithmetic is too impractical for most commercial uses, making its implementation very rare.
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