Abstract: The traditional mapped weighted essentially non-oscillatory (WENO) schemes commonly suffer from the drawback of low-efficiency, since they usually require the mapping processes resulting in extra computational costs. The goal of the present work is to improve the efficiency of the mapped WENO schemes. By designing a set of approximate constant mapping function which is devised using an approximation of the standard signum function, a novel mapped WENO scheme is proposed. The new mapping function is devised to meet the overall criteria for a proper mapping function required in the design of the WENO-PM6 scheme. The WENO-PM6 scheme was presented to overcome the potential loss of accuracy of the well-validated WENO-M scheme in previously published literature. The new proposed mapped WENO scheme is denoted as WENO-ACM. It maintains almost all advantages of the WENO-PM6 scheme, such as low dissipations and high resolutions. However, it decreases the number of mathematical operations remarkably in every mapping process leading to a significant improvement of efficiency. Theoretical analysis indicates that the new scheme can attain the optimal convergence rate of accuracy regardless of critical points. The investigation of approximate dispersion relation (ADR) shows that the spectral properties of the new scheme are significantly improved. A variety of benchmark-test problems, including accuracy tests, standard shock-tube problem, Mach 3 shock-entropy wave interaction, Woodward-Colella interacting blast waves, 2D Riemann problem, double Mach reflection, forward-facing step problem, Rayleigh-Taylor instability and Kelvin-Helmholtz instability are conducted. Compared to the well-established WENO-JS, WENO-M, WENO-PM6 schemes comprehensively, the present scheme exhibits significantly improved high efficiency, very high resolution and sharp discontinuity capturing. Most importantly, the extra computational cost of WENO-ACM compared to WENO-JS is much lower than those of WENO-M and WENO-PM6. Specifically, WENO-ACM can reduce the extra computational cost compared to WENO-JS more than 80% and 90% against WENO-M and WENO-PM6, respectively.