Question

Show that when approximating a positive real number E IR+ in normalized floating-point form with machine number x*=rd(E, Rt,s) also in normalized floating-point form, the relative error of the approximation can be bounded as.

x-x*/x = x*-x/x < 2⁻ᵗ

Hint:
1. Estimate the numerator from above using the formula by separating it into two cases: Case 1: b₋ₜ₋₁ < 0 and Case 2: b₋ₜ₋₁ = 1.

Answer 1

The relative error of approximating a positive real number E in floating-point form can be bounded by 2^-t, considering the worst-case rounding scenario where the error is the difference between the approximation and the next representable floating-point number.

Step-by-step explanation:

To show that the relative error of approximating a positive real number E in normalized floating-point form with a machine number x* can be bounded by 2-t, we start by separating the error expression into two cases based on the last bit b-t-1.

Case 1: If b-t-1 is less than 0, we round down and the error is negative. Case 2: If b-t-1 equals 1, we round up and the error is positive, leading to an overestimation. In both scenarios, the absolute value of the relative error is less than the unit in the last place (ulp), which is directly related to 2-t.

To bound the relative error (x - x*)/x from above, we consider the worst-case scenario where we lose the most precision due to rounding. This occurs when the rounding causes us to increase to the next representable floating-point number, which has a difference of 2-t from the true value. Therefore, the relative error is strictly less than 2-t.

Thus, for any positive real number E in IR+, when approximated by its nearest floating-point form x*, the relative error is bounded by < 2-t.