diff --git a/src/cairo-compositor.c b/src/cairo-compositor.c
index a633aa2e5..b31413b99 100644
--- a/src/cairo-compositor.c
+++ b/src/cairo-compositor.c
@@ -139,6 +139,10 @@ _cairo_compositor_stroke (const cairo_compositor_t	*compositor,
     cairo_int_status_t status;
 
     TRACE ((stderr, "%s\n", __FUNCTION__));
+
+    if (_cairo_pen_vertices_needed (tolerance, style->line_width/2, ctm) <= 1)
+	return CAIRO_INT_STATUS_NOTHING_TO_DO;
+
     status = _cairo_composite_rectangles_init_for_stroke (&extents, surface,
 							  op, source,
 							  path, style, ctm,
diff --git a/src/cairo-pen.c b/src/cairo-pen.c
index e71f7b561..d70a06497 100644
--- a/src/cairo-pen.c
+++ b/src/cairo-pen.c
@@ -41,11 +41,6 @@
 #include "cairo-error-private.h"
 #include "cairo-slope-private.h"
 
-static int
-_cairo_pen_vertices_needed (double tolerance,
-			    double radius,
-			    const cairo_matrix_t *matrix);
-
 static void
 _cairo_pen_compute_slopes (cairo_pen_t *pen);
 
@@ -88,10 +83,12 @@ _cairo_pen_init (cairo_pen_t	*pen,
      * is reflecting
      */
     for (i=0; i < pen->num_vertices; i++) {
-	double theta = 2 * M_PI * i / (double) pen->num_vertices;
-	double dx = radius * cos (reflect ? -theta : theta);
-	double dy = radius * sin (reflect ? -theta : theta);
 	cairo_pen_vertex_t *v = &pen->vertices[i];
+	double theta = 2 * M_PI * i / (double) pen->num_vertices, dx, dy;
+	if (reflect)
+	    theta = -theta;
+	dx = radius * cos (theta);
+	dy = radius * sin (theta);
 	cairo_matrix_transform_distance (ctm, &dx, &dy);
 	v->point.x = _cairo_fixed_from_double (dx);
 	v->point.y = _cairo_fixed_from_double (dy);
@@ -273,7 +270,7 @@ Note that this also equation works for M == m (a circle) as it
 doesn't matter where on the circle the error is computed.
 */
 
-static int
+int
 _cairo_pen_vertices_needed (double	    tolerance,
 			    double	    radius,
 			    const cairo_matrix_t  *matrix)
@@ -283,21 +280,16 @@ _cairo_pen_vertices_needed (double	    tolerance,
      * compute major axis length for a pen with the specified radius.
      * we don't need the minor axis length.
      */
+    double major_axis = _cairo_matrix_transformed_circle_major_axis (matrix,
+								     radius);
+    int num_vertices;
 
-    double  major_axis = _cairo_matrix_transformed_circle_major_axis (matrix,
-								      radius);
-
-    /*
-     * compute number of vertices needed
-     */
-    int	    num_vertices;
-
-    /* Where tolerance / M is > 1, we use 4 points */
-    if (tolerance >= major_axis) {
+    if (tolerance >= 2*major_axis) {
+	num_vertices = 1;
+    } else if (tolerance >= major_axis) {
 	num_vertices = 4;
     } else {
-	double delta = acos (1 - tolerance / major_axis);
-	num_vertices = ceil (M_PI / delta);
+	num_vertices = ceil (2*M_PI / acos (1 - tolerance / major_axis));
 
 	/* number of vertices must be even */
 	if (num_vertices % 2)
diff --git a/src/cairoint.h b/src/cairoint.h
index 26a8de149..68b25bedf 100644
--- a/src/cairoint.h
+++ b/src/cairoint.h
@@ -1524,6 +1524,11 @@ cairo_private cairo_image_color_t
 _cairo_image_analyze_color (cairo_image_surface_t      *image);
 
 /* cairo-pen.c */
+cairo_private int
+_cairo_pen_vertices_needed (double	    tolerance,
+			    double	    radius,
+			    const cairo_matrix_t  *matrix);
+
 cairo_private cairo_status_t
 _cairo_pen_init (cairo_pen_t	*pen,
 		 double		 radius,