World To Screenspace for Mode7 effect (fragment shader) - opengl

Question

I have a fragment shader that transforms the view into something resembling mode7.
I want to know the Screen-Space x,y coordinates given a world position.
As the transformation happens in the fragment shader, I can't simply inverse a matrix. This is the fragment shader code:
uniform float Fov; //1.4
uniform float Horizon; //0.6
uniform float Scaling; //0.8
void main() {
vec2 pos = uv.xy - vec2(0.5, Horizon);
vec3 p = vec3(pos.x, pos.y, pos.y + Fov);
vec2 s = vec2(p.x/p.z, p.y/p.z) * Scaling;
s.x += 0.5;
s.y += screenRatio;
gl_FragColor = texture2D(ColorTexture, s);
}
It transforms pixels in a pseudo 3d way:
-
What I want to do is get a screen-space coordinate for a given world position (in normal code, not shaders).
How do I reverse the order of operations above?
This is what I have right now:
(GAME_WIDTH and GAME_HEIGHT are constants and hold pixel values, e.g. 320x240)
vec2 WorldToScreenspace(float x, float y) {
// normalize coordinates 0..1, as x,y are in pixels
x = x/GAME_WIDTH - 0.5;
y = y/GAME_HEIGHT - Horizon;
// as z depends on a y value I have yet to calculate, how can I calc it?
float z = ??;
// invert: vec2 s = vec2(p.x/p.z, p.y/p.z) * Scaling;
float sx = x*z / Scaling;
float sy = y*z / Scaling;
// invert: pos = uv.xy - vec2(0.5, Horizon);
sx += 0.5;
sy += screenRatio;
// convert back to screen space
return new vec2(sx * GAME_WIDTH, sy * GAME_HEIGHT);
}