By Michael L. Overton
Are you acquainted with the IEEE floating element mathematics average? do you want to appreciate it greater? This e-book supplies a huge review of numerical computing, in a old context, with a distinct specialise in the IEEE regular for binary floating aspect mathematics. Key rules are built step-by-step, taking the reader from floating element illustration, appropriately rounded mathematics, and the IEEE philosophy on exceptions, to an knowing of the the most important techniques of conditioning and balance, defined in an easy but rigorous context. It supplies technical info that aren't on hand somewhere else and comprises hard workouts that transcend the themes lined within the textual content.
Numerical Computing with IEEE Floating aspect mathematics offers an simply obtainable but specified dialogue of IEEE Std 754-1985, arguably an important commonplace within the desktop undefined. the results of an unheard of cooperation among educational machine scientists and the innovative of undefined, it truly is supported by means of almost each glossy machine. different themes comprise the floating aspect structure of the Intel microprocessors and a dialogue of programming language help for a standard.
The publication might be available to scholars at any point, in addition to to any reader with an curiosity in pcs and arithmetic. It offers sufficient number of content material that every one however the so much specialist readers will locate anything of curiosity.
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Extra resources for Numerical Computing with IEEE Floating Point Arithmetic
Example text
5 Suppose that x > Nmax. What is abserr(a;)7 for each of the four rounding modes? Look carefully at the definition of round (x}. 3, using the rounding mode round down ? 6) hold when \x\ < N min ? Explain. 4), is in the normalized range and is not a floating point number, we have Therefore, for all rounding modes, the relative rounding error satisfies the bound CHAPTER 5. 5). 9). The same inequalities hold when x is a floating point number in the normalized range, since then relerr(x) = abserr(x) = 0.
Make up some more examples where a guard bit is required. The following is a particularly interesting example. 1111... 1)2 x 2~l, where the fraction field for y contains 23 ones after the binary point. ) Aligning the significands, we obtain This is an example of cancellation, since almost all the bits in the two numbers cancel each other. 0)2 x 2~ 24 , which is a floating point number. As in the previous example, we need a guard bit to get the correct answer; indeed, without it, we would get a completely wrong answer.
2 What is the gap between 2 and the first IEEE single format number larger than 2 ? What is the gap between 1024 and the first IEEE single format number larger than 1024? 3 Give an algorithm that, given two IEEE single format floating point numbers x and y, determines whether x is less than, equal to, or greater than y, by comparing their representations bitwise from left to right, stopping as soon as the first differing bit is encountered. Assume that neither x nory is ±0, ±00 , or NaN. The fact that such a comparison can be done easily motivates biased exponent representation.