Let x and y be oating point numbers let denote the

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Unformatted text preview: be oating point numbers, let +,;, ,= denote the four standard arithmetic operations, and let , , , denote the corresponding operations as they are actually implemented on the computer. Thus, x + y may not be a oating point number, but x y is the oating point number which the computer computes as its approximation of x + y . The IEEE rule is then precisely: x y = round(x + y ) x y = round(x ; y ) x y = round(x y ) and x y = round(x=y ): 16 From the discussion of relative rounding errors given above, we see then that the computed value x y satis es x y = (x + y )(1 + ) where for all rounding modes and jj 1 2 in the case of round to nearest. The same result also holds for the other operations , and . Show that it follows from the IEEE rule for correctly rounded arithmetic that oating point addition is commutative, i.e. Exercise 13 a b=b a for any two oating point numbers a and b. Show with a simple example that oating point addition is not associative, i.e. it may not be true that a (b c) = (a b) c for some oating point numbers a, b and c. Now we ask the quest...
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