ArcTan[x] - ArcCos[y] == -ArcTan[(Sqrt[1 + x^2] Sqrt[(1 + x^2 - (x y - Sqrt[1 - y^2])^2)/(1 + x^2)])/ ((-1)^Floor[1/2 - Arg[(y + x Sqrt[1 - y^2])/Sqrt[1 + x^2]]/Pi] (x y - Sqrt[1 - y^2]))] + (1/2) Pi (-1 + (-1)^(Floor[(Pi - Arg[1 + x^2] + 2 Arg[x y - Sqrt[1 - y^2]])/ (2 Pi)] + Floor[(Pi + Arg[1 + x^2] - 2 Arg[y + x Sqrt[1 - y^2]])/ (2 Pi)] - 2 Floor[(2 Pi + Arg[1 + x^2] - 2 Arg[y + x Sqrt[1 - y^2]])/ (4 Pi)]) + (-1)^Floor[1/2 - Arg[(y + x Sqrt[1 - y^2])/Sqrt[1 + x^2]]/ Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[(y + x Sqrt[1 - y^2])/Sqrt[1 + x^2]]/Pi]) Floor[(Arg[(I - x)/Sqrt[1 + x^2]] + Arg[I y + Sqrt[1 - y^2]])/(2 Pi)] - 2 (-1 + (-1)^Floor[1/2 - Arg[(y + x Sqrt[1 - y^2])/Sqrt[1 + x^2]]/Pi]) Floor[1/2 - (Arg[(I - x)/Sqrt[1 + x^2]] + Arg[I y + Sqrt[1 - y^2]])/ (2 Pi)])

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