Sun shader not working - opengl

Question

I'm trying to get a sun shader to work, but I can't get it to work.
What I currently get is a quarter of a circle/elipsis on the lower left of my screen, that is really stuck to my screen (if I move the camera, it also moves).
All I do is render two triangles to form a screen-covering quad, with screen width and height in uniforms.
Vertex Shader
#version 430 core
void main(void) {
const vec4 vertices[6] = vec4[](
vec4(-1.0, -1.0, 1.0, 1.0),
vec4(-1.0, 1.0, 1.0, 1.0),
vec4(1.0, 1.0, 1.0, 1.0),
vec4(1.0, 1.0, 1.0, 1.0),
vec4(1.0, -1.0, 1.0, 1.0),
vec4(-1.0, -1.0, 1.0, 1.0)
);
gl_Position = vertices[gl_VertexID];
}
Fragment Shader
#version 430 core
layout(location = 7) uniform int screen_width;
layout(location = 8) uniform int screen_height;
layout(location = 1) uniform mat4 view_matrix;
layout(location = 2) uniform mat4 proj_matrix;
out vec4 color;
uniform vec3 light_pos = vec3(-20.0, 7.5, -20.0);
void main(void) {
//calculate light position in screen space and get x, y components
vec2 screen_space_light_pos = (proj_matrix * view_matrix * vec4(light_pos, 1.0)).xy;
//calculate fragment position in screen space
vec2 screen_space_fragment_pos = vec2(gl_FragCoord.x / screen_width, gl_FragCoord.y / screen_height);
//check if it is in the radius of the sun
if (length(screen_space_light_pos - screen_space_fragment_pos) < 0.1) {
color = vec4(1.0, 1.0, 0.0, 1.0);
}
else {
discard;
}
}
What I think it does:
Get the position of the sun (light_pos) in screen space.
Get the fragment position in screen space.
If the distance between them is below a certain value, draw fragment with yellow color;
Else discard.

Answer 1

screen_space_light_pos is not yet in clip space. You've missed perspective division:
vec3 before_division = (proj_matrix * view_matrix * vec4(light_pos, 1.0)).xyw;
vec2 screen_space_light_pos = before_division.xy / before_division.z;
With common proj_matrix configurations, screen_space_light_pos will be in [-1,1]x[-1,1]. To match screen_space_fragment_pos range, you probably need to adjust screen_space_light_pos:
screen_space_light_pos = screen_space_light_pos * 0.5 + 0.5;