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This article is about the method of representing a number. For this reason, floating-point computation is often found in systems which include very small and very large real numbers, which require fast processing times. Over the years, a variety of floating-point representations have been used in computers. There are several mechanisms by which strings of digits can represent numbers. So a fixed-point scheme might be to use a string of 8 decimal digits with the decimal point in the middle, whereby “00012345” would represent 0001. 1 and 10, with the radix point appearing immediately after the first digit.
The scaling factor, as a power of ten, is then indicated separately at the end of the number. Floating-point representation is similar in concept to scientific notation. The significand is assumed to have a binary point to the right of the leftmost bit. The floating-point representation is by far the most common way of representing in computers an approximation to real numbers. 6 bits or digits from the right. The hardware to manipulate these representations is less costly than floating point, and it can be used to perform normal integer operations, too.
Binary fixed point is usually used in special-purpose applications on embedded processors that can only do integer arithmetic, but decimal fixed point is common in commercial applications. It is possible to implement a floating-point system with BCD encoding. Conversely to floating-point arithmetic, in a logarithmic number system multiplication, division and exponentiation are simple to implement, but addition and subtraction are complex. Clenshaw, Olver, and Turner is a scheme based on a generalized logarithm representation. NaN representations, anticipating features of the IEEE Standard by four decades. The arithmetic is actually implemented in software, but with a one megahertz clock rate, the speed of floating-point and fixed-point operations in this machine were initially faster than those of many competing computers.
36 bits, organized as a 1-bit sign, an 8-bit exponent, and a 27-bit significand. 72 bits, organized as a 1-bit sign, an 11-bit exponent, and a 60-bit significand. 1962, supports single-precision and double-precision representations, but with no relation to the UNIVAC’s representations. 2005, IBM also added IEEE-compatible decimal floating-point arithmetic.
Initially, computers used many different representations for floating-point numbers. A precisely specified floating-point representation at the bit-string level, so that all compliant computers interpret bit patterns the same way. A precisely specified behavior for the arithmetic operations: A result is required to be produced as if infinitely precise arithmetic were used to yield a value that is then rounded according to specific rules. This means that a compliant computer program would always produce the same result when given a particular input, thus mitigating the almost mystical reputation that floating-point computation had developed for its hitherto seemingly non-deterministic behavior. Whereas components linearly depend on their range, the floating-point range linearly depends on the significant range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number.
11 bits, and one sign bit. 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent. 1 as the value for each digit of the significand and the largest possible value for the exponent. This first standard is followed by almost all modern machines. The standard provides for many closely related formats, differing in only a few details. On other processors, “long double” may be a synonym for “double” if any form of extended precision is not available, or may stand for a larger format, such as quadruple precision.