Floating Point Error Computing
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by David Goldberg, published in the March, 1991 issue of Computing Surveys. Copyright 1991, Association for Computing Machinery, Inc., reprinted by
Floating Point Error Example
permission. Abstract Floating-point arithmetic is considered an esoteric subject by many floating point rounding error people. This is rather surprising because floating-point is ubiquitous in computer systems. Almost every language has floating point python a floating-point datatype; computers from PCs to supercomputers have floating-point accelerators; most compilers will be called upon to compile floating-point algorithms from time to time; and virtually
Floating Point Number Example
every operating system must respond to floating-point exceptions such as overflow. This paper presents a tutorial on those aspects of floating-point that have a direct impact on designers of computer systems. It begins with background on floating-point representation and rounding error, continues with a discussion of the IEEE floating-point standard, and concludes with numerous
Floating Point Arithmetic Examples
examples of how computer builders can better support floating-point. Categories and Subject Descriptors: (Primary) C.0 [Computer Systems Organization]: General -- instruction set design; D.3.4 [Programming Languages]: Processors -- compilers, optimization; G.1.0 [Numerical Analysis]: General -- computer arithmetic, error analysis, numerical algorithms (Secondary) D.2.1 [Software Engineering]: Requirements/Specifications -- languages; D.3.4 Programming Languages]: Formal Definitions and Theory -- semantics; D.4.1 Operating Systems]: Process Management -- synchronization. General Terms: Algorithms, Design, Languages Additional Key Words and Phrases: Denormalized number, exception, floating-point, floating-point standard, gradual underflow, guard digit, NaN, overflow, relative error, rounding error, rounding mode, ulp, underflow. Introduction Builders of computer systems often need information about floating-point arithmetic. There are, however, remarkably few sources of detailed information about it. One of the few books on the subject, Floating-Point Computation by Pat Sterbenz, is long out of print. This paper is a tutorial on those aspects of floating-point arithmetic (floating-point hereafter) that have a direct connection to systems building. It consists of three loosely conn
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 between range and precision. floating point calculator A number is, in general, represented approximately to a fixed number of significant digits
Double Floating Point
(the significand) and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or floating point numbers explained 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 an integer ≥ 2, and exponent ∈ Z. https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html 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, https://en.wikipedia.org/wiki/Floating_point 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 Signed zero 3.2.2 Subnormal numbers 3.2.3 Infinities 3.2.4 NaNs 3.2.5 IEEE 754 design rationale 4 Representable numbers, conversion and rounding 4.1 Rounding modes 5 Floating-poin
base 2 (binary) fractions. For example, the decimal fraction 0.125 has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction 0.001 has value 0/2 + 0/4 + 1/8. These two fractions https://docs.python.org/2/tutorial/floatingpoint.html have identical values, the only real difference being that the first is written in base 10 fractional notation, and the second in base 2. Unfortunately, most decimal fractions cannot be represented exactly as binary fractions. A consequence is that, in general, the decimal floating-point numbers you enter are only approximated by the binary floating-point numbers actually stored in the machine. The problem is easier to understand at first floating point in base 10. Consider the fraction 1/3. You can approximate that as a base 10 fraction: 0.3 or, better, 0.33 or, better, 0.333 and so on. No matter how many digits you're willing to write down, the result will never be exactly 1/3, but will be an increasingly better approximation of 1/3. In the same way, no matter how many base 2 digits you're willing to use, floating point error the decimal value 0.1 cannot be represented exactly as a base 2 fraction. In base 2, 1/10 is the infinitely repeating fraction 0.0001100110011001100110011001100110011001100110011... Stop at any finite number of bits, and you get an approximation. On a typical machine running Python, there are 53 bits of precision available for a Python float, so the value stored internally when you enter the decimal number 0.1 is the binary fraction 0.00011001100110011001100110011001100110011001100110011010 which is close to, but not exactly equal to, 1/10. It's easy to forget that the stored value is an approximation to the original decimal fraction, because of the way that floats are displayed at the interpreter prompt. Python only prints a decimal approximation to the true decimal value of the binary approximation stored by the machine. If Python were to print the true decimal value of the binary approximation stored for 0.1, it would have to display >>> 0.1 0.1000000000000000055511151231257827021181583404541015625 That is more digits than most people find useful, so Python keeps the number of digits manageable by displaying a rounded value instead >>> 0.1 0.1 It's important to realize that this is, in a real sense, an illusion: the value in the machine is not exactly 1/10, you're sim