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In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by rounding any result that is not a floating-point number itself to a nearby floating-point number.
The otherwise binary Wang VS machine supported a 64-bit decimal floating-point format in 1977. [2] The Motorola 68881 supported a format with 17 digits of mantissa and 3 of exponent in 1984, with the floating-point support library for the Motorola 68040 processor providing a compatible 96-bit decimal floating-point storage format in 1990. [2]
00000000000 2 =000 16 is used to represent a signed zero (if F = 0) and subnormal numbers (if F ≠ 0); and; 11111111111 2 =7ff 16 is used to represent ∞ (if F = 0) and NaNs (if F ≠ 0), where F is the fractional part of the significand. All bit patterns are valid encoding. Except for the above exceptions, the entire double-precision number ...
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 standard defines five basic formats that are named for their numeric base and the number of bits used in their interchange encoding. There are three binary floating-point basic formats (encoded with 32, 64 or 128 bits) and two decimal floating-point basic formats (encoded with 64 or 128 bits).
The smallest possible float size that follows all IEEE principles, including normalized numbers, subnormal numbers, signed zero, signed infinity, and multiple NaN values, is a 4-bit float with 1-bit sign, 2-bit exponent, and 1-bit mantissa. [11]
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The 80-bit floating-point format has a range (including subnormals) from approximately 3.65 × 10 −4951 to 1.18 × 10 +4932. Although log 10 ( 2 64) ≈ 19.266, this format is usually described as giving approximately eighteen significant digits of precision (the floor of log 10 ( 2 63), the minimum guaranteed precision).