Accuracy loss in `normalize`/`cross`

Hi!

I've noticed a small loss in accuracy when computing mat * vec:

Without StaticArrays:

julia> lookMat = Float32[ -0.707107   0.707107  0.0       -0.0
        -0.485071  -0.485071  0.727607   0.485071
         0.514496   0.514496  0.685994  -3.42997
         0.0        0.0       0.0        1.0]
4×4 Matrix{Float32}:
 -0.707107   0.707107  0.0       -0.0
 -0.485071  -0.485071  0.727607   0.485071
  0.514496   0.514496  0.685994  -3.42997
  0.0        0.0       0.0        1.0

julia> lookMat * Float32[0.5, 0.5, 0, 1]
4-element Vector{Float32}:
  0.0
  0.0
 -2.9154742
  1.0

With StaticArrays loaded (no matter whether it's Float32 or Float64, the accuracy loss is the same - note the nonzero y component):

julia> Array(lookMat)
4×4 Matrix{Float64}:
 -0.707107   0.707107  0.0       -0.0
 -0.485071  -0.485071  0.727607   0.485071
  0.514496   0.514496  0.685994  -3.42997
  0.0        0.0       0.0        1.0

julia> Array(lookMat) * [0.5, 0.5, 0, 1]
4-element Vector{Float64}:
  0.0
  2.9802322387695312e-8
 -2.915476143360138
  1.0

The matrix is generated by this function:

function lookAt(eye, target, up)
    # up is local up of the camera
    zaxis = -normalize(target - eye)
    xaxis = normalize(cross(up, zaxis))
    yaxis = cross(zaxis, xaxis)

    res = @SMatrix Float32[
        xaxis[1]  xaxis[2]  xaxis[3]  -dot(eye, xaxis);
        yaxis[1]  yaxis[2]  yaxis[3]  -dot(eye, yaxis);
        zaxis[1]  zaxis[2]  zaxis[3]  -dot(eye, zaxis);
               0         0         0                 1
    ]

    return res
end

What this does, in essence, is orient a coordinate system with origin at (2,2,2) to point at target in the -z direction. So lookMat * target should be exactly a vector in -z direction.

Accuracy loss of the StaticArray version aside, is it expected that this also happens for the regular Array multiplication?

Start with the reported lookAt function and its normalize, cross, and matrix-vector multiplication calls. Reproduce the Julia examples with and without StaticArrays, then compare the Float32 and Float64 results. Done means determining whether the nonzero y component and precision difference are expected, and documenting or correcting the behavior as appropriate.