diff --git a/zigrtiow/src/main.zig b/zigrtiow/src/main.zig
index abcae73..82d30fc 100644
--- a/zigrtiow/src/main.zig
+++ b/zigrtiow/src/main.zig
@@ -12,11 +12,11 @@ const Ray = ray.Ray;
 
 fn hit_sphere(center: Point3, radius: f32, r: Ray) f32 {
     const oc = r.origin().sub(center);
-    const a = Vec3.dot(r.direction(), r.direction());
-    const b = Vec3.dot(oc, r.direction()) * 2;
-    const c = Vec3.dot(oc, oc) - radius * radius;
-    const discriminant = b * b - 4 * a * c;
-    return if (discriminant < 0) -1 else (-b - @sqrt(discriminant)) / (2 * a);
+    const a = r.direction().length_squared();
+    const half_b = Vec3.dot(oc, r.direction());
+    const c = oc.length_squared() - radius * radius;
+    const discriminant = half_b * half_b - a * c;
+    return if (discriminant < 0) -1 else (-half_b - @sqrt(discriminant)) / a;
 }
 
 fn ray_color(r: Ray) Color {
diff --git a/zigrtiow/src/vec.zig b/zigrtiow/src/vec.zig
index cb98032..0d785f8 100644
--- a/zigrtiow/src/vec.zig
+++ b/zigrtiow/src/vec.zig
@@ -17,7 +17,7 @@ pub const Vec3 = struct {
     pub fn length(vec: Vec3) f32 {
         return @sqrt(vec.length_squared());
     }
-    fn length_squared(vec: Vec3) f32 {
+    pub fn length_squared(vec: Vec3) f32 {
         return vec.v[0] * vec.v[0] + vec.v[1] * vec.v[1] + vec.v[2] * vec.v[2];
     }
     pub fn add(lhs: Vec3, rhs: Vec3) Vec3 {