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 environments 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:

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:

Layout of octuple precision floating point format

Layout of octuple precision floating point format

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.

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 0000016 and 7FFFF16 are interpreted specially.

Exponent Significand zero Significand non-zero Equation
0000016 0, −0 subnormal numbers (-1)signbit × 2−262142 × 0.significandbits2
0000116, ..., 7FFFE16 normalized value (-1)signbit × 2exponent bits2 × 1.significandbits2
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 000016 = +0
8000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = −0
7fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = +infinity
ffff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = −infinity
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000116
= 2−262142 × 2−236 = 2−262378
≈ 2.24800708647703657297018614776265182597360918266100276294348974547709294462 × 10−78984
  (smallest positive subnormal number)
0000 0fff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff16
= 2−262142 × (1 − 2−236)
≈ 2.4824279514643497882993282229138717236776877060796468692709532979137875392 × 10−78913
  (largest subnormal number)
0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016
= 2−262142
≈ 2.48242795146434978829932822291387172367768770607964686927095329791378756168 × 10−78913
  (smallest positive normal number)
7fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff16
= 2262143 × (2 − 2−236)
≈ 1.61132571748576047361957211845200501064402387454966951747637125049607182699 × 1078913
  (largest normal number)
3fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff16
= 1 − 2−237
≈ 0.999999999999999999999999999999999999999999999999999999999999999999999995472
  (largest number less than one)
3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016
= 1 (one)
3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000116
= 1 + 2−236
≈ 1.00000000000000000000000000000000000000000000000000000000000000000000000906
  (smallest number larger than one)

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.[1] 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 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.

See also

References

  1. ^ R. Crandall; J. Papadopoulos (8 May 2002). "Octuple-precision floating point on Apple G4 (archived copy on web.archive.org)" (PDF). Archived from the original on July 28, 2006.CS1 maint: Unfit url (link)

Further reading

IEEE 754

The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard.

The standard defines:

arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers (including signed zeros and subnormal numbers), infinities, and special "not a number" values (NaNs)

interchange formats: encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form

rounding rules: properties to be satisfied when rounding numbers during arithmetic and conversions

operations: arithmetic and other operations (such as trigonometric functions) on arithmetic formats

exception handling: indications of exceptional conditions (such as division by zero, overflow, etc.)The current version, IEEE 754-2008 revision published in August 2008, includes nearly all of the original IEEE 754-1985 standard plus IEEE 854-1987 Standard for Radix-Independent Floating-Point Arithmetic.

Precision (computer science)

In computer science, the precision of a numerical quantity is a measure of the detail in which the quantity is expressed. This is usually measured in bits, but sometimes in decimal digits. It is related to precision in mathematics, which describes the number of digits that are used to express a value.

Some of the standardized precision formats are

Half-precision floating-point format

Single-precision floating-point format

Double-precision floating-point format

Quadruple-precision floating-point format

Octuple-precision floating-point formatOf these, octuple-precision format is rarely used. The single- and double-precision formats are most widely used and supported on nearly all platforms. The use of half-precision format has been increasing especially in the field of machine learning since many machine learning algorithms are inherently error-tolerant.

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