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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 ...
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 ...
This position is indicated by the exponent, so floating point can be considered a form of scientific notation. 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 ...
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 = 3FFFF 16 = 262143
ARM processors support (via a floating-point control register bit) an "alternative half-precision" format, which does away with the special case for an exponent value of 31 (11111 2). [10] It is almost identical to the IEEE format, but there is no encoding for infinity or NaNs; instead, an exponent of 31 encodes normalized numbers in the range ...
KCalc, Linux based scientific calculator; Maxima: a computer algebra system which bignum integers are directly inherited from its implementation language Common Lisp. In addition, it supports arbitrary-precision floating-point numbers, bigfloats. Maple, Mathematica, and several other computer algebra software include arbitrary-precision arithmetic.
When interpreting the floating-point number, the bias is subtracted to retrieve the actual exponent. For a half-precision number, the exponent is stored in the range 1 .. 30 (0 and 31 have special meanings), and is interpreted by subtracting the bias for an 5-bit exponent (15) to get an exponent value in the range −14 .. +15.
In the case of the mechanical calculators, the exponent is often treated as side information that is accounted for separately. The IBM 650 computer supported an 8-digit decimal floating-point format in 1953. [1] The otherwise binary Wang VS machine supported a 64-bit decimal floating-point format in 1977. [2]