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In August 1908, a white mob attacked Springfield, Illinois’s Black community, destroying homes and businesses and lynching two Black men. National outrage over the violence helped spur the movement that created the NAACP the following year.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
Floating-point error mitigation is the minimization of errors caused by the fact that real numbers cannot, in general, be accurately represented in a fixed space. By definition, floating-point error cannot be eliminated, and, at best, can only be managed.
Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator":[1]
Thus under some conditions, the major portion of the significant data digits may lie beyond the capacity of the registers. Therefore, the result obtained may have little meaning if not totally erroneous.
The Z1, developed by Konrad Zuse in 1936, was the first computer with floating-point arithmetic and was thus susceptible to floating-point error. Early computers, however, with operation times measured in milliseconds, could not solve large, complex problems[2] and thus were seldom plagued with floating-point error. Today, however, with supercomputer system performance measured in petaflops, floating-point error is a major concern for computational problem solvers.
The following sections describe the strengths and weaknesses of various means of mitigating floating-point error.
Though not the primary focus of numerical analysis,[3][4]: 5 numerical error analysis exists for the analysis and minimization of floating-point rounding error.
Error analysis by Monte Carlo arithmetic is accomplished by repeatedly injecting small errors into an algorithm's data values and determining the relative effect on the results.
Extension of precision is using of larger representations of real values than the one initially considered. The IEEE 754 standard defines precision as the number of digits available to represent real numbers. A programming language can include single precision (32 bits), double precision (64 bits), and quadruple precision (128 bits). While extension of precision makes the effects of error less likely or less important, the true accuracy of the results is still unknown.
Variable length arithmetic represents numbers as a string of digits of a variable's length limited only by the memory available. Variable-length arithmetic operations are considerably slower than fixed-length format floating-point instructions. When high performance is not a requirement, but high precision is, variable length arithmetic can prove useful, though the actual accuracy of the result may not be known.
The floating-point algorithm known as TwoSum[5] or 2Sum, due to Knuth and Møller, and its simpler, but restricted version FastTwoSum or Fast2Sum (3 operations instead of 6), allow one to get the (exact) error term of a floating-point addition rounded to nearest. One can also obtain the (exact) error term of a floating-point multiplication rounded to nearest in 2 operations with a fused multiply–add (FMA), or 17 operations if the FMA is not available (with an algorithm due to Dekker). These error terms can be used in algorithms in order to improve the accuracy of the final result, e.g. with floating-point expansions or compensated summations.
Operations giving the result of a floating-point addition or multiplication rounded to nearest with its error term (but slightly differing from algorithms mentioned above) have been standardized and recommended in the IEEE 754-2019 standard.
Changing the radix, in particular from binary to decimal, can help to reduce the error and better control the rounding in some applications, such as financial applications.
Interval arithmetic is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Values are intervals, which can be represented in various ways, such as:[6]
"Instead of using a single floating-point number as approximation for the value of a real variable in the mathematical model under investigation, interval arithmetic acknowledges limited precision by associating with the variable a set of reals as possible values. For ease of storage and computation, these sets are restricted to intervals."[7]
The evaluation of interval arithmetic expression may provide a large range of values,[7] and may seriously overestimate the true error boundaries.[8]: 8
Unums ("Universal Numbers") are an extension of variable length arithmetic proposed by John Gustafson.[9] Unums have variable length fields for the exponent and significand lengths and error information is carried in a single bit, the ubit, representing possible error in the least significant bit of the significand (ULP).[9]: 4
The efficacy of unums is questioned by William Kahan.[8]
Bounded floating point is a method proposed and patented by Alan Jorgensen.[10] The data structure includes the standard IEEE 754 data structure and interpretation, as well as information about the error between the true real value represented and the value stored by the floating point representation.[11]
Bounded floating point has been criticized as being derivative of Gustafson's work on unums and interval arithmetic.[10][12]
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Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
Floating-point error mitigation is the minimization of errors caused by the fact that real numbers cannot, in general, be accurately represented in a fixed space. By definition, floating-point error cannot be eliminated, and, at best, can only be managed. Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator": Thus under some conditions, the major portion of the significant data digits may lie beyond the capacity of the registers. Therefore, the result obtained may have little meaning if not totally erroneous. The Z1, developed by Konrad Zuse in 1936, was the first computer with floating-point arithmetic and was thus susceptible to floating-point error. Early computers, however, with operation times measured in milliseconds, could not solve large, complex problems and thus were seldom plagued with floating-point error. Today, however, with supercomputer system performance measured in petaflops, floating-point error is a major concern for computational problem solvers. The following sections describe the strengths and weaknesses of various means of mitigating floating-point error.
As with other spreadsheets, Microsoft Excel works only to limited accuracy because it retains only a certain number of figures to describe numbers (it has limited precision). With some exceptions regarding erroneous values, infinities, and denormalized numbers, Excel calculates in double-precision floating-point format from the IEEE 754 specification (besides numbers, Excel uses a few other data types). Although Excel allows display of up to 30 decimal places, its precision for any specific number is no more than 15 significant figures, and calculations may have an accuracy that is even less due to five issues: round off, truncation, and binary storage, accumulation of the deviations of the operands in calculations, and worst, "catastrophic cancellation" at subtraction of values with similar magnitude. Standard floating-point error mitigation techniques apply.
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a significand (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469 / 200 = 12.345 = 12345 ⏟ significand × 10 ⏟ base − 3 ⏞ exponent {\displaystyle 2469/200=12.345=\!\underbrace {12345} _{\text{significand}}\!\times \!\underbrace {10} _{\text{base}}\!\!\!\!\!\!\!\overbrace {{}^{-3}} ^{\text{exponent}}} However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. 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. For example, in a floating-point arithmetic with five base-ten digits, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345. The term floating point refers to the fact that the number's radix point can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation. A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom. For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times. 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 varies with their exponent. Over the years, a variety of floating-point representations have been used in computers. In 1985, the IEEE 754 Standard for Floating-Point Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE. 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. Floating-point numbers can be computed using software implementations (softfloat) or hardware implementations (hardfloat). Floating-point units (FPUs, colloquially math coprocessors) are specially designed to carry out operations on floating-point numbers and are part of most computer systems. When FPUs are not available, software implementations can be used instead.
Z-fighting, also called stitching or planefighting, is a phenomenon in 3D rendering that occurs when two or more primitives have very similar distance to the camera. This would cause them to have near-similar or identical values in the z-buffer, which keeps track of depth. This then means that when a specific pixel is being rendered, it is ambiguous which one of the two primitives are drawn in that pixel because the z-buffer cannot distinguish precisely which one is farther from the other. If one pixel was unambiguously closer, the less close one could be discarded. It is particularly prevalent with coplanar polygons, where two faces occupy essentially the same space, with neither in front. As a result, affected pixels are rendered with fragments from one polygon or the other arbitrarily, in a manner determined by the precision of the z-buffer. It can also vary as the scene or camera is changed, causing one polygon to "win" the z test, then another, and so on. The overall effect is flickering, noisy rasterization of two polygons which "fight" to color the screen pixels. This problem is usually caused by limited sub-pixel precision, floating point and fixed point round-off errors. The more z-buffer precision one uses, the less likely it is that z-fighting will be encountered. But for coplanar polygons, the problem is inevitable unless corrective action is taken. As the distance between near and far clip planes increases, and in particular the near plane is selected near the eye, the greater the likelihood exists that z-fighting between primitives will occur. With large virtual environments inevitably there is an inherent conflict between the need to resolve visibility in the distance and in the foreground, so for example in a space flight simulator, if a distant galaxy is drawn to scale, the viewer will not have the precision to resolve visibility on any cockpit geometry in the foreground (although even a numerical representation would present problems prior to z-buffered rendering). To mitigate these problems, z-buffer precision is weighted towards the near clip plane, but this is not the case with all visibility schemes and it is insufficient to eliminate all z-fighting issues.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Madam C.J. Walker
