Colour-coded relative difference log₁₀(|ℰ_R|) between the exact and approximate ratio calculation in a bin, Eq. (8), as a function of the numerator and denominator spectral difference, α_den − α_num (x-axis), and as a function of the rigidity bin width, log₁₀(R_max/R_min) (y-axis). For instance, a value of 2 (red) corresponds to a relative difference of 100%, and a value of −1 (blue) to a relative difference of 0.1%. The left and right panels correspond to calculations using and respectively, see Eq. (5). The rows correspond to three different values of the denominator spectral index α_den. In each panel, the dash-dotted green contour delimits regions in which the relative difference is above or below 3%. See text for discussion.