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base-10 real values are represented as character strings in ISO 6093 format; binary real values are represented in a binary format that includes the mantissa, the base (2, 8, or 16), and the exponent; the special values NaN, -INF, +INF, and negative zero are also supported
A more efficient encoding can be designed using the fact that the exponent range is of the form 3×2 k, so the exponent never starts with 11. Using the Decimal32 encoding (with a significand of 3*2+1 decimal digits) as an example (e stands for exponent, m for mantissa, i.e. significand):
MBF numbers consist of an 8-bit base-2 exponent, a sign bit (positive mantissa: s = 0; negative mantissa: s = 1) and a 23-, [43] [8] 31-[8] or 55-bit [43] mantissa of the significand. There is always a 1-bit implied to the left of the explicit mantissa, and the radix point is located before this assumed bit.
Binary search Visualization of the binary search algorithm where 7 is the target value Class Search algorithm Data structure Array Worst-case performance O (log n) Best-case performance O (1) Average performance O (log n) Worst-case space complexity O (1) Optimal Yes In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search ...
Similar binary floating-point formats can be defined for computers. There is a number of such schemes, the most popular has been defined by Institute of Electrical and Electronics Engineers (IEEE). The IEEE 754-2008 standard specification defines a 64 bit floating-point format with: an 11-bit binary exponent, using "excess-1023" format.
Notice that for a binary radix, the leading binary digit is always 1. In a subnormal number, since the exponent is the least that it can be, zero is the leading significant digit (0.m 1 m 2 m 3...m p−2 m p−1), allowing the representation of numbers closer to zero than the smallest normal number. A floating-point number may be recognized as ...
Thus, only 10 bits of the significand appear in the memory format but the total precision is 11 bits. In IEEE 754 parlance, there are 10 bits of significand, but there are 11 bits of significand precision (log 10 (2 11) ≈ 3.311 decimal digits, or 4 digits ± slightly less than 5 units in the last place).
Because superscript exponents like 10 7 can be inconvenient to display or type, the letter "E" or "e" (for "exponent") is often used to represent "times ten raised to the power of", so that the notation m E n for a decimal significand m and integer exponent n means the same as m × 10 n. For example 6.022 × 10 23 is written as 6.022E23 or 6 ...