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The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE).
Conversion of the fractional part: Consider 0.375, the fractional part of 12.375. To convert it into a binary fraction, multiply the fraction by 2, take the integer part and repeat with the new fraction by 2 until a fraction of zero is found or until the precision limit is reached which is 23 fraction digits for IEEE 754 binary32 format.
The new IEEE 754 (formally IEEE Std 754-2008, the IEEE Standard for Floating-Point Arithmetic) was published by the IEEE Computer Society on 29 August 2008, and is available from the IEEE Xplore website [4] This standard replaces IEEE 754-1985. IEEE 854, the Radix-Independent floating-point standard was withdrawn in December 2008.
IEEE 754 specifies additional floating-point formats, including 32-bit base-2 single precision and, ... where F is the fractional part of the significand. All bit ...
IEEE 754 defines the precision p to be the number of digits in the significand, including any implicit leading bit (e.g., p = 53 for the double-precision format), thus in a way independent from the encoding, and the term to express what is encoded (that is, the significand without its leading bit) is trailing significand field.
The Bfloat16 format requires the same amount of memory (16 bits) as the IEEE 754 half-precision format, but allocates 8 bits to the exponent instead of 5, thus providing the same range as a IEEE 754 single-precision number. The tradeoff is a reduced precision, as the trailing significand field is reduced from 10 to 7 bits.
The IEEE 754-2008 standard defines 32-, 64- and 128-bit decimal floating-point representations. Like the binary floating-point formats, the number is divided into a sign, an exponent, and a significand.
IEEE 754-1985 [1] is a historic industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. [2] During its 23 years, it was the most widely used format for floating-point computation.