float square(float a)
{
    return a*a;
}

float normal_distribution_ggx(vec3 normal, vec3 halfway, float roughness) 
{
    float a2 = square(square(roughness));
    float dotnh = max(dot(normal, halfway), 0.0);
    float dotnh2 = square(dotnh);

    return a2 / (PI * square(dotnh2 * (a2 - 1.0) + 1.0));
}

float k_direct(float roughness)
{
    return square(roughness + 1) / 8;
}

float k_ibl(float roughness)
{
    return square(roughness) / 2;
}

float geometry_schlick_ggx(float dotnv, float k)
{
    return dotnv / (dotnv * (1.0 - k) + k);
}

float geometry_smith(vec3 normal, vec3 vdir, vec3 point_to_light, float k)
{
    float dotnv = max(dot(normal, vdir), 0.0);
    float dotnl = max(dot(normal, point_to_light), 0.0);

    return geometry_schlick_ggx(dotnv, k) *
           geometry_schlick_ggx(dotnl, k);
}

float fresnel_schlick(float cos_theta, float F0)
{
    return F0 + (1.0 - F0) * pow(1.0 - cos_theta, 5.0);
}