Lecture 4

# b2 b1 b0 2 the binary point is between bits 4 and 5

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Unformatted text preview: int binary numbers” ¢  Let's do that, using 8-­‐bit ﬁxed point numbers as an example §  #1: the binary point is between bits 2 and 3 b7 b6 b5 b4 b3 [.] b2 b1 b0 §  #2: the binary point is between bits 4 and 5 b7 b6 b5 [.] b4 b3 b2 b1 b0 ¢  The posi-on of the binary point aﬀects the range and precision of the representa-on § range: diﬀerence between largest and smallest numbers possible §  precision: smallest possible diﬀerence between any two numbers Frac-onal Values University of Washington Fixed Point Pros and Cons ¢  Pros §  It's simple. The same hardware that does integer arithme=c can do ﬁxed point arithme=c §  In fact, the programmer can use ints with an implicit ﬁxed point §  ints are just ﬁxed point numbers with the binary point to the right of b0 ¢  Cons §  There is no good way to pick where the ﬁxed point should be §  Some=mes you need range, some=mes you need precision – the more you have of one, the less of the other. Frac-onal Values University of Washington Integer & Floa-ng Point Numbers ¢  ¢  ¢  ¢  ¢  ¢  ¢  ¢  Representa-on of integers: unsigned and signed Unsigned and signed integers in C Arithme-c and shiBing Sign extension Background: frac-onal binary numbers IEEE ﬂoa-ng-­‐point standard Floa-ng-­‐point opera-ons and rounding Floa-ng-­‐point in C IEEE Floa-ng Point Standard University of Washington IEEE Floa-ng Point ¢  Analogous to scien-ﬁc nota-on §  Not 12000000 but 1.2 x 107; not 0.0000012 but 1.2 x 10-­‐6 §  ¢  (write in C code as: 1.2e7; 1.2e-­‐6) IEEE Standard 754 §  Established in 1985 as uniform standard for ﬂoa=ng point arithme=c Before that, many idiosyncra=c formats §  Supported by all major CPUs today §  ¢  Driven by numerical concerns §  Standards for handling rounding, overﬂow, underﬂow §  Hard to make fast in hardware but numerically well-­‐behaved ¢  1989 Turin...
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