The average frequency shift versus ω₀ for the four formulas. Parameters σ = 1, τ = 0.01, φ = 0.1. The red curve represented by direct integral shown in Eq. (22), the green dashed curve represented by nonlinear formula with the modified weak value shown in Eq. (23), the black dot-dashed curve represented by the nonlinear formula with the normal weak value shown in Eq. (25) and the blue curve represented by linear formula shown in Eq. (24).