reduce the number of flops in c and s

Previously, integers were converted to 64-bit floating-point numbers
within the square root.

Now, the numerators and denominators are first calculated as integers.
After converting to type T, they are divided in one flop.

Thus, each c and s only involves a floating-point sqrt() and a type T
division, which reduces the probability of a rounding error by about
half (compared to the previous method).

Author: MikaelSlevinsky

src/SphericalHarmonics/slowplan.jl

@@ -48,8 +48,8 @@ end
 function Pnmp2toPlm{T}(::Type{T}, n::Int, m::Int)
     G = Vector{Givens{T}}(n)
     @inbounds for ℓ = 1:n
-        c = sqrt((2ℓ+2m+3)/(ℓ+2m+3)*(2m+2)/(ℓ+2m+2))
-        s = sqrt((ℓ+1)/(ℓ+2m+3)*ℓ/(ℓ+2m+2))
+        c = sqrt(T((2m+2)*(2ℓ+2m+3))/T((ℓ+2m+2)*(ℓ+2m+3)))
+        s = sqrt(T(ℓ*(ℓ+1))/T((ℓ+2m+2)*(ℓ+2m+3)))
         G[n+1-ℓ] = Givens(ℓ, ℓ+2, c, s)
     end
     Pnmp2toPlm(G)