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floating-point - CS 312 Spring 2009 Some Notes on Floating...

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CS 312, Spring 2009 : Some Notes on Floating Point Arithmetic. [email protected] 1 Introduction Blah, blah, say some words to introduce the topic, maybe provide an example, whatever. On second thought, let’s just dive head-first into things and see where the current takes us. 2 Normalized Real Numbers in Decimal Before we hit the meat of the topic too terribly hard, let’s briefly review scientific notation, signifi- cant places, and the process for normalizing real numbers in the base that we’re familiar with: base 10. Additionally, we say real number to mean any number x in R , the set of real numbers. Scientific notation is a way of representing a real number that would otherwise be cumbersome to be written conventionally. We’ve all seen this several thousand times already, so I’ll save a lot of the detail except for the form of the representation and what it means for a real number to be normalized. That said, a normalized real number x in scientific notation has the form x = a · 10 b , (1) where 1 ≤ | a | < 10. We call the coefficient a one of either the significand , mantissa , or fraction and b the characteristic or exponent . In 312, we used the terms fraction and exponent . The process of normalization itself is braindead simple to the point that we can probably do it in our sleep. Informally, we just shift the decimal place as appropriate to ensure 1 ≤ | a | < 10 and keep count of how many places we shift it. The count of places is, hence, b . 3 Normalized Real Numbers in Some Base β Now that we’ve hit the material with which we’re all familiar, we can finally abstract the select few parts so that we can glean more insight as to how this mess works as a whole. Really, the changes to the definition we discussed earlier are minimal. A normalized real number x in a base β in floating-point notation has the form x β = a · β b , (2) where 1 ≤ | a | < β . The same vocabulary applies, and so on. The process of normalization is even
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