Backward Error Analysis Floating-point
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the Z3, included floating-point arithmetic (replica on display at Deutsches Museum in Munich). In computing, floating point is the formulaic representation that approximates a real number so as to support a trade-off
Floating Point Arithmetic Examples
between range and precision. A number is, in general, represented approximately to a catastrophic cancellation fixed number of significant digits (the significand) and scaled using an exponent in some fixed base; the base for the
Machine Epsilon
scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form: significand × base exponent , {\displaystyle {\text{significand}}\times {\text{base}}^{\text{exponent}},} where significand ∈ Z, base is wolfram alpha an integer ≥ 2, and exponent ∈ Z. For example: 1.2345 = 12345 ⏟ significand × 10 ⏟ base − 4 ⏞ exponent {\displaystyle 1.2345=\underbrace {12345} _{\text{significand}}\times \underbrace {10} _{\text{base}}\!\!\!\!\!\!^{\overbrace {-4} ^{\text{exponent}}}} The term floating point refers to the fact that a number's radix point (decimal point, or, more commonly in computers, binary point) can "float"; that is, it can be placed anywhere relative to the significant digits of the number. This position is indicated as the exponent component, and thus the floating-point representation can be thought of as a kind of scientific notation. A floating-point system can be used to represent, with a fixed number of digits, numbers of different orders of magnitude: e.g. the distance between galaxies or the diameter of an atomic nucleus can be expressed with the same unit of length. The result of this dynamic range is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers grows with the chosen scale.[1] Over the years, a variety of floating-point representations have been used in computers. However, since the 1990s, the most commonly encountered representation is that defined by the IEEE 754 Standard. The speed of floating-point operations, commonly measured in terms of FLOPS, is an important characteristic of a computer system, especially for applications that involve intensive mathematical calculations. Contents 1 Overview 1.1 Floating-point numbers 1.2 Alternatives to floating-point numbers 1.3 History 2 Range of floating-point numbers 3 IEEE 754: floating point in modern computers 3.1 Internal representation 3.1.1 Piecewise linear approximation to exponential and logarithm 3.2 Special values 3.2.1 Sign
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