Question

Q1 (20 points) IEEE Binary Floating Point System. Suppose in a specific IEEE format, the width of the exponent field is 5 and the width of the fraction field is 5. (a) (2 points) What are the smallest positive subnormal and normal floating point numbers? (b) (2 points) Give the values of machine precision and machine epsilon. (c) (2 points) What are the smallest gap and largest gap between two consecutive finite floating point numbers? (d) (3 points) What is the smallest positive integer that is not exactly representable by this IEEE floating point format? (e) (5 points) Let a real number $x = (1.b_1b_2b_3b_4b_5b_6...)_2 \times 2^E$ be in the range of normal floating point numbers. What is the largest floating point number smaller than $x$ and what is the smallest floating point number larger than $x$? Show that for the rounding to nearest mode $\frac{|round(x) - x|}{|x|} \le \frac{1}{2} \epsilon$. (f) (3 points) Given two finite real numbers $a$ and $b$ with $a < b$, is it true that $round(a) \le round(b)$ for any rounding mode? Justify your answer. (g) (3 points) Suppose $x$ is any finite floating point number. Is it true that $x \oplus x = 2 \otimes x$ for any rounding mode? Either give a proof or a counter example.

          Q1 (20 points)
IEEE Binary Floating Point System. Suppose in a specific IEEE format, the width of the exponent field is 5 and the width of the fraction field is
5.
(a) (2 points) What are the smallest positive subnormal and normal floating point numbers?
(b) (2 points) Give the values of machine precision and machine epsilon.
(c) (2 points) What are the smallest gap and largest gap between two consecutive finite floating point numbers?
(d) (3 points) What is the smallest positive integer that is not exactly representable by this IEEE floating point format?
(e) (5 points) Let a real number $x = (1.b_1b_2b_3b_4b_5b_6...)_2 \times 2^E$ be in the range of normal floating point numbers. What is the largest
floating point number smaller than $x$ and what is the smallest floating point number larger than $x$? Show that for the rounding to nearest mode
$\frac{|round(x) - x|}{|x|} \le \frac{1}{2} \epsilon$.
(f) (3 points) Given two finite real numbers $a$ and $b$ with $a < b$, is it true that $round(a) \le round(b)$ for any rounding mode? Justify your
answer.
(g) (3 points) Suppose $x$ is any finite floating point number. Is it true that $x \oplus x = 2 \otimes x$ for any rounding mode? Either give a proof or a
counter example.

$Q1 (20 points) IEEE Binary Floating Point System. Suppose in a specific IEEE format, the width of the exponent field is 5 and the width of the fraction field is 5. (a) (2 points) What are the smallest positive subnormal and normal floating point numbers? (b) (2 points) Give the values of machine precision and machine epsilon. (c) (2 points) What are the smallest gap and largest gap between two consecutive finite floating point numbers? (d) (3 points) What is the smallest positive integer that is not exactly representable by this IEEE floating point format? (e) (5 points) Let a real number x = (1.b1b2b3b4b5b6...)2 × 2^E be in the range of normal floating point numbers. What is the largest floating point number smaller than x and what is the smallest floating point number larger than x? Show that for the rounding to nearest mode (|round(x) - x|)/(|x|)≤(1)/(2)ϵ. (f) (3 points) Given two finite real numbers a and b with a < b, is it true that round(a) ≤ round(b) for any rounding mode? Justify your answer. (g) (3 points) Suppose x is any finite floating point number. Is it true that x ⊕ x = 2 ⊗ x for any rounding mode? Either give a proof or a counter example.$

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