AIUI, you don't need to "compute" sqrt5 to high precision in order to use that closed form! Just "adjoin" it to Q and compute symbolically over Q(sqrt5).
Personally I like the version better where you explicitly work in terms of integers like a + bφ or rational numbers like (a + bφ)/c. https://observablehq.com/@jrus/zome-arithmetic very handy when working with the symmetry system of the icosahedron. https://observablehq.com/d/cae3fea57d7cf2d4