Question 1. [20 marks] Consider IEEE single-precision floating-point representation with 1 bit for sign, 8 bits for biased exponent, and 23 bits for mantissa.
(1.a) [6 marks] Let x = 2^5 + 2^-16 + 2^-19 + 2^-21. Find the machine number closest to x. Denote this machine number by x**. What are the absolute and relative errors, respectively, when converting x to x**?
(1.b) [8 marks] Show that the relative error in converting a real number to a machine number in this representation is no greater than the machine epsilon ε_M = 2^-24. Here it is assumed that the real number lies between the smallest machine number and the largest machine number of this representation.
(1.c) [6 marks] Consider the function f(x, y) = x + y and assume x > 0 and y > 0. Suppose that the input values x and y have errors δx and δy such that their relative errors satisfy |δx|/|x| <= Δ and |δy|/|y| <= Δ for some constant Δ > 0. Show that the relative error of the output f(x, y) thus caused is also bounded by Δ; that is, letting δf denote the error of f(x, y), there is |δf|/|f(x, y)| <= Δ.
