An anomalously reflective polarization-independent high-efficiency multilayer slanted grating with broad spectral and angular bandwidths under normal incidence was designed by rigorous coupled wave analysis and using a simulated annealing algorithm. To enhance the efficiency and broaden the bandwidth, a four-layer sandwiched structure with a gradually changing refractive index was proposed. Based on the proposed structure, we designed a polarization-independent anomalously reflective grating structure in the visible band. It exhibits a spectral bandwidth of 130 nm (550--680 nm) with -1st efficiencies higher than 93% and polarization-dependent loss less than 0.2 dB. Moreover, simulation results suggest that the grating structure features an angular bandwidth of 47° (-2°--45°) and large range of fabrication tolerances. Therefore, the proposed multilayer slanted grating will attract great interest for its potential applications in virtual/augmented reality displays and metasurface-based devices, especially for metalenses in the visible range.