One　of　the　difficult　problems　encountered　when　implementing　artificial　neural　networks　based　on　FPGA　is　the　approximation　of　the　activation　function.　The　sigmoid　function　is　the　most　widely　used　and　is　difficult　to　approximate.　This　paper　is　devoted　to　show　a　saving　hardware　resources　and　accurate　way　to　compute　the　sigmoid　function　based　on　FPGA　by　non-linear　approximation.　This　is　done　by　subsection　analysis　involved　a　new　low-leakage　FPGA　Look-up　Tables　(LUTs),　introducing　a　non-linear　approximation　algorithm　in　detail,　analyzing　the　approximating　accuracy　and　the　FPGA　hardware　resources,　which　can　achieve　some　kind　of　balance　between　the　approximating　precision　and　the　limited　hardware　resources　of　FPGA,　shows　improvements　over　the　previous　known　algorithms.　The　implementation　of　sigmoid　function　and　the　simulation　are　completed　by　the　development　software　of　QUARTUSII.