Mesh created with Geometry Nodes is slightly shifted compared to mesh created with XYZ Math Surface

Question

I was trying to complete my answer for this question. I'm creating an XYZ Math Surface via Add > Mesh > Math Function > XYZ Math Surface using the following parametric equations:

X: (g*f)/16*(3/2*((u**2 + v**2)*(1 + (u + v - 1)**2) - 2*(u**2*v**2 - 2*(u*v - u - v) - 1))/(u**2 + u*v + v**2 - u - v + 1)**2) + (sqrt(3)/6*(3*u + 3*v - 2)/sqrt(u**2 + v**2 + u*v - u - v + 1))*f/2
Y: (g*f)/16*(3/2*sqrt(3)*((u**2 - v**2)*(1 + (u + v - 1)**2))/(u**2 + u*v + v**2 - u - v + 1)**2) - (1/2*(u - v)/sqrt(u**2 + v**2 + u*v - u - v + 1))*f/2
Z: (g*f)/16*(3/2*sqrt(2)*((u**2 + v**2)*(2 - (u + v - 1)**2) - (u**2*v**2 - 2*(u*v - u - v) - 1))/(u**2 + u*v + v**2 - u - v + 1)**2 - sqrt(2)) - (sqrt(6)/3*(-1)/sqrt(u**2 + u*v + v**2 - u - v + 1) - 1)*f/2

parameters:

U min:  0.00
U max:  1.00
U step: 72
V min:  0.00
V max:  1.00
V step: 72
F:  2.00
G: 0

Which look like this if I'm not mistaken: $$ x(u,v) = \frac{g \cdot f}{16} \cdot \frac{3}{2} \cdot \frac{(u^2 + v^2) \cdot (1 + (u + v - 1)^2) - 2 \cdot (u^2 \cdot v^2 - 2 \cdot (u \cdot v - u - v) - 1)}{(u^2 + u \cdot v + v^2 - u - v + 1)^2} + \frac{\sqrt{3}}{6} \cdot \frac{(3 \cdot u + 3 \cdot v - 2)}{\sqrt{u^2 + v^2 + u \cdot v - u - v + 1}} \cdot \frac{f}{2} $$

$$ y(u,v) = \frac{g \cdot f}{16} \left( \frac{3}{2} \cdot \sqrt{3} \cdot \frac{(u^2 - v^2) \cdot (1 + (u + v - 1)^2)}{(u^2 + u \cdot v + v^2 - u - v + 1)^2} \right) - \frac{1}{2} \cdot \frac{(u - v)}{\sqrt{u^2 + v^2 + u \cdot v - u - v + 1}} \cdot \frac{f}{2} $$

$$ z(u,v) = \frac{g \cdot f}{16} \left( \frac{3}{2} \sqrt{2} \cdot \frac{(u^2 + v^2) \cdot \left(2 - (u + v - 1)^2\right) - \left(u^2 v^2 - 2 \cdot (u v - u - v) - 1\right)}{(u^2 + u v + v^2 - u - v + 1)^2} - \sqrt{2} \right) - \left( \frac{\sqrt{6}}{3} \cdot \left( -\frac{1}{\sqrt{u^2 + u v + v^2 - u - v + 1}} - 1 \right) \right) \cdot \frac{f}{2} $$

When I try to map the nodes in a custom node group named XYZ-Surface, I get a surface that is lower on the z-axis. They should be coinciding with each other:

All the other surfaces I made worked perfectly except this one. Since it is the Z that is shifted, the problem must probably lie in the Z equation. However, no matter how many times I recheck the node tree, I can't understand why it's not positioned correctly. I have checked this several times now to no avail forcing me to seek for extra eyes to look at this.

Here's what's inside the main Geometry Nodes modifier:

Answer 1

There is an error in transposing the equation from XYZ Math Surface code to LaTeX. The last part of $z(u,v)$, computed in the frame "Z Equation Term 2", is written as: $$ z(u,v) = (...) - \left( \frac{\sqrt{6}}{3} \cdot \left( -\frac{1}{\sqrt{u^2 + u v + v^2 - u - v + 1}} - 1 \right) \right) \cdot \frac{f}{2} $$ whereas it should be (from the pseudo-code): $$ z(u,v) = (...) - \left( \frac{\sqrt{6}}{3} \cdot \frac{-1}{\sqrt{u^2 + u v + v^2 - u - v + 1}} - 1 \right) \cdot \frac{f}{2} $$ To debug, taking advantage that $g=0$, I specified $u=v=0$ for every grid point, yielding constant functions easier to track, after I noticed that the shift is almost the same at the four corners.