For this problem, we represents real numbers by a modified floating point number system with the...

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For this problem, we represents real numbers by a modified floating point number system with the parameters (radix b = 2, precision s, range of exponent [m, M]), in which a modified floating point is of the form: (-1)sign x 1.f x 2⁰ where sign is either 0 or 1, f is an s-bit binary integer (note that the normalization is achieved at "1."), and e € [n, M]. For example, the real number (in decimal) (-4.125) 10 (= -(22+2-3)) is represented by the modified floating point number (with s= 6) (-1)¹ × 1.000010 × 2². (a) Assume that the truncation supporting the modified floating point system are "round down" and "round up". For a real number q in the form of (-1)sign x 1.a₁a2a+1. x 2, (1) the round-down of q, denoted by q-, is the modified floating point number (-1)sign x 1.a1a2a, x 2 (by simply chopping off all the bits after the s-th one), and (2) the round-up of q, denoted by q+, is the modified floating point number (-1)sign x 1.a1a2... as X 2+2-s (by effectively adding the last/s-th bit of q- by 1). Assume that the modified floating point system is intelligent enough to decide which rounding, q_ or q+, is closest to q- denote it by q". Give the best/tightest "error bound" for q* (that is, for lq-q*). Also, give the corresponding "absolute-error bound" and "relative-error bound". Justify/explain your answers. (b) Consider the rounding of the decimal rational number ()10 to the modified floating point number described above with b = 2 and s= 23, that is, the modified floating point number (closest to ()10) in the form of 1.a1a2a23 x 2m. What are the corresponding q- and q+? What is the roundoff error (the generated error)? What is the relative roundoff error? For this problem, we represents real numbers by a modified floating point number system with the parameters (radix b = 2, precision s, range of exponent [m, M]), in which a modified floating point is of the form: (-1)sign x 1.f x 2⁰ where sign is either 0 or 1, f is an s-bit binary integer (note that the normalization is achieved at "1."), and e € [n, M]. For example, the real number (in decimal) (-4.125) 10 (= -(22+2-3)) is represented by the modified floating point number (with s= 6) (-1)¹ × 1.000010 × 2². (a) Assume that the truncation supporting the modified floating point system are "round down" and "round up". For a real number q in the form of (-1)sign x 1.a₁a2a+1. x 2, (1) the round-down of q, denoted by q-, is the modified floating point number (-1)sign x 1.a1a2a, x 2 (by simply chopping off all the bits after the s-th one), and (2) the round-up of q, denoted by q+, is the modified floating point number (-1)sign x 1.a1a2... as X 2+2-s (by effectively adding the last/s-th bit of q- by 1). Assume that the modified floating point system is intelligent enough to decide which rounding, q_ or q+, is closest to q- denote it by q". Give the best/tightest "error bound" for q* (that is, for lq-q*). Also, give the corresponding "absolute-error bound" and "relative-error bound". Justify/explain your answers. (b) Consider the rounding of the decimal rational number ()10 to the modified floating point number described above with b = 2 and s= 23, that is, the modified floating point number (closest to ()10) in the form of 1.a1a2a23 x 2m. What are the corresponding q- and q+? What is the roundoff error (the generated error)? What is the relative roundoff error?