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A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom. For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times.
For example, the difference of the two smallest positive normal floating-point numbers = (+) + and = + is = + which is necessarily subnormal. In floating-point number systems without subnormal numbers , such as CPUs in nonstandard flush-to-zero mode instead of the standard gradual underflow, the Sterbenz lemma does not apply.
The number 0.15625 represented as a single-precision IEEE 754-1985 floating-point number. See text for explanation. The three fields in a 64bit IEEE 754 float. Floating-point numbers in IEEE 754 format consist of three fields: a sign bit, a biased exponent, and a fraction. The following example illustrates the meaning of each.
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). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and ...
The significand (or mantissa) of an IEEE floating-point number is the part of a floating-point number that represents the significant digits. For a positive normalised number, it can be represented as m 0.m 1 m 2 m 3...m p−2 m p−1 (where m represents a significant digit, and p is the precision) with non-zero m 0.
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 IEEE 854, the Radix-Independent floating-point standard was withdrawn in December 2008.
For example, a PowerPC 750 running at 800 MHz would be rated at 0.00024 WT due to being able to execute one floating point instruction per cycle and not having a vector unit. Note that only 64 bit (or wider) floating point instructions count. Notes: Processors without 64 bit (or better) floating point support have an FPO of zero.
A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2 31 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2 −23) × 2 127 ≈ 3.4028235 ...