Octuple-precision floating-point format

The range greatly exceeds what is needed to describe the size of the observable universe, precise to the Planck length. However, some problems in experimental mathematics do require octuple (or higher) precision.

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 262143 has to be subtracted from the stored exponent.

The stored exponents 00000₁₆ and 7FFFF₁₆ are interpreted specially.

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<sub id="mwcw">16</sub> = +0 8000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub id="mwdA">16</sub> = −0

7fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub id="mwdg">16</sub> = +infinity ffff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub id="mwdw">16</sub> = −infinity

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001<sub id="mweQ">16</sub> = 2<sup id="mweg">−262142</sup> × 2<sup id="mwew">−236</sup> = 2<sup id="mwfA">−262378</sup> ≈ 2.24800708647703657297018614776265182597360918266100276294348974547709294462 × 10<sup id="mwfQ">−78984</sup> (smallest positive subnormal number)

0000 0fff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff<sub id="mwfw">16</sub> = 2<sup id="mwgA">−262142</sup> × (1 − 2<sup id="mwgQ">−236</sup>) ≈ 2.4824279514643497882993282229138717236776877060796468692709532979137875392 × 10<sup id="mwgg">−78913</sup> (largest subnormal number)

0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub id="mwhA">16</sub> = 2<sup id="mwhQ">−262142</sup> ≈ 2.48242795146434978829932822291387172367768770607964686927095329791378756168 × 10<sup id="mwhg">−78913</sup> (smallest positive normal number)

7fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff<sub id="mwiA">16</sub> = 2<sup id="mwiQ">262143</sup> × (2 − 2<sup id="mwig">−236</sup>) ≈ 1.61132571748576047361957211845200501064402387454966951747637125049607182699 × 10<sup id="mwiw">78913</sup> (largest normal number)

3fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff<sub id="mwjQ">16</sub> = 1 − 2<sup id="mwjg">−237</sup> ≈ 0.999999999999999999999999999999999999999999999999999999999999999999999995472 (largest number less than one)

3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000<sub id="mwkA">16</sub> = 1 (one)

3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001<sub id="mwkg">16</sub> = 1 + 2<sup id="mwkw">−236</sup> ≈ 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.