Precision errors with raytracing normal

Question

I'm writing a raytracer in Java that draws a scene containing the elliptic paraboloid defined by the equation $F(x,y,z)=x^2+z^2-y=0$, as well as the hyperbolic paraboloid defined by $G(x,y,z)=x^2-z^2-y=0$.

To compute the surface normal to this, I'm taking the normalized gradients of the equations: $\vec{\nabla} F(x,y,z) = \left<2x, -1, 2z\right>\\ \vec{\nabla} G(x,y,z) = \left<2x, -1, -2z\right>$.

To ensure that the normal is on the correct surface, I take the dot product with the incident ray and flip the normal if the dot product is negative.

However, when I render the image, I observe this strange outline around the object:

I can tell that this is a normal-related issue because when I color the objects based on their normals, I can see discontinuities:

I have no idea why this is, but I believe it has something to do with rounding errors during this sign-change step. I'm not sure why this would be though; I'm using double-precision floating-point numbers (Java's double type) for all calculations. When I remove this check, the normals are broken, but the outline disappears. How might I be able to fix this?

Here's where the normal is computed:

Vec3 dP = pt.sub(pos);
Vec3 normal = new Vec3(2 * dP.x, -1, 2 * dP.z)
       .getNormalized()
       .faceForward(ray.rd);

where pos is the object's local position, and ray.rd is the ray's direction.

Here's the code responsible for the check (the faceForward function in the above code):

public Vec3 faceForward(Vec3 incident) {
    boolean outside = (this.dot(incident) < 0);
    return (outside ? this : this.negate());
}