spherical basis change through euler angles

kirinyaga

More exactly, I want to translate polar coordinates to spherical ones, knowing the euler angles that define the polar plane in the spherical basis.

Polar coordinates: (r_p,θ_p)

it's actually the position of a satellite on its orbit

Spherical coordinates: (r_s,θ_s,φ_s)

its position relative to the planet so φ is the elevation angle(latitude), not the polar angle

Euler angles: (α,β,γ)

the orbit parameters: longitude of ascending node, inclination, argument of periapsis

What I end up with (looks correct as far as I can tell) :
r_s=r_p
θ_s=α+atan2(u,v)
φ_s=atan2(|u,v|, sin(θ_p+γ)sin(β))
with:
u=cos(θ_p+γ)
v=sin(θ_p+γ)cos(β)

But the calculation of |u,v| (length) is killing my timings with the square root.
So my question is : is there a way to simplify those equations ? And specifically to get rid of the |u,v| ?