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  3. 1 let a 02337123 104 b ...

Question: 1 let a 02337123 104 b ...

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1) Let a = 0.2337123 × 10−4 , b = 0.3368678 × 102 , and c = −0.3395375 × 102 . Assuming a 6-digit decimal computer. Calculate:

a) S = fl(fl(a + b) + c);

b) S = fl(a + fl(b + c));

where fl(x) uses symmetric rounding. Compare the results a) and b) and comment on your finding.

2. Consider a miniature binary computer whose floating-point words consist of 4 binary digits for the mantissa and 3 binary digits for the exponent (plus sign bits). Let x = (.1011)2 × 2 0 , y = (.1100)2 × 2 0 .

(1) Mark in the provided table whether the machine operation indicated (with the result Z assumed normalized) is exact, rounded (i.e., subject to nonzero rounding error), overflows, or underflows. Operation exact rounded overflow underflow Z = fl(x − y) Z = fl((x−y) 10) Z = fl(x + y) Z = fl(y+(x/4)) Z = fl(x+(y/4))

3. Prove: Based on best possible relative error bounds, the floating-point addition fl(fl(x + y) + z) is more accurate than fl(x + fl(y + z)), if and only if |x + y| < |y + z|. As applications, formulate addition rules in the cases: a) 0 < x < y < z; b) x > 0, y < 0, z > 0; c) x < 0, y > 0, z < 0.

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