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6. Prove this slight improvement of Inequality (8) x-x 1 +224 for the representation of numbers in the Marc-32. 7. How many normalized machine numbers are there in the Marc-32? (Do not count 0.) 8. Does each machine number in the Marc-32 have a unique normalized representation? 9. Let (I.11.. 1110002 x 216, in which the fractional part has 26 1's followed by 0's. For the Marc-32, determine x-, x+, fl(x), x-x-,,-x, x,-., and Ir-fl(x )1/r. 10. Let x 2+2-19 +2-22. Find the machine numbers on the Marc-32 that are just to the right and just to the left of x. Determine fl(x), the absolute error f(x)I, and the relative error x fx)l/Lxl. Verify that the relative error in this case does not exceed 2-24 11. Find the machine number just to the right of 1/9 in a binary computer with a 43-bit normalized mantissa. what is the exact value of x.-x, ifx-261 2-n and x. is the nearest machine number on the Marc-32? 12. Let S,, where each xi is a machine number. Let S, be what the machine computes. Then S: = fl(S-1 + x.). Prove that on the Marc-32, 14. Which of these is not necessarily true on the Marc-32? (Here x, y, and z are machine numbers and 2-24.) b. fl(x y) (x +y)(1 + 8) 15. Use the Marc-32 for this problem. Determine a bound on the relative error in computing (a .b)/(c.d) for machine numbers a, b, c, d. 16. Are these machine numbers in the Marc-32? a. 1040 b. 2-1 +2-26 6. Prove this slight improvement of Inequality (8) x-x 1 +224 for the representation of numbers in the Marc-32. 7. How many normalized machine numbers are there in the Marc-32? (Do not count 0.) 8. Does each machine number in the Marc-32 have a unique normalized representation? 9. Let (I.11.. 1110002 x 216, in which the fractional part has 26 1's followed by 0's. For the Marc-32, determine x-, x+, fl(x), x-x-,,-x, x,-., and Ir-fl(x )1/r. 10. Let x 2+2-19 +2-22. Find the machine numbers on the Marc-32 that are just to the right and just to the left of x. Determine fl(x), the absolute error f(x)I, and the relative error x fx)l/Lxl. Verify that the relative error in this case does not exceed 2-24 11. Find the machine number just to the right of 1/9 in a binary computer with a 43-bit normalized mantissa. what is the exact value of x.-x, ifx-261 2-n and x. is the nearest machine number on the Marc-32? 12. Let S,, where each xi is a machine number. Let S, be what the machine computes. Then S: = fl(S-1 + x.). Prove that on the Marc-32, 14. Which of these is not necessarily true on the Marc-32? (Here x, y, and z are machine numbers and 2-24.) b. fl(x y) (x +y)(1 + 8) 15. Use the Marc-32 for this problem. Determine a bound on the relative error in computing (a .b)/(c.d) for machine numbers a, b, c, d. 16. Are these machine numbers in the Marc-32? a. 1040 b. 2-1 +2-26