diff --git a/src/backend/access/spgist/spgquadtreeproc.c b/src/backend/access/spgist/spgquadtreeproc.c
index ce6464f7e04..d9bf2ed0176 100644
--- a/src/backend/access/spgist/spgquadtreeproc.c
+++ b/src/backend/access/spgist/spgquadtreeproc.c
@@ -39,6 +39,32 @@ spg_quad_config(PG_FUNCTION_ARGS)
 #define SPTEST(f, x, y) \
 	DatumGetBool(DirectFunctionCall2(f, PointPGetDatum(x), PointPGetDatum(y)))
 
+/*
+ * Compare one coordinate of a point against that of the centroid
+ */
+static int
+getAxisSide(float8 tst, float8 centroid)
+{
+	/*
+	 * Test FPeq first so that points on a centroid axis get the same result
+	 * as getQuadrant.
+	 */
+	if (FPeq(tst, centroid))
+		return 0;
+	if (FPgt(tst, centroid))
+		return 1;
+	if (FPlt(tst, centroid))
+		return -1;
+	/* fall back to exact comparison */
+	if (tst > centroid)
+		return 1;
+	if (tst < centroid)
+		return -1;
+
+	elog(ERROR, "getQuadrant: impossible case");
+	return 0;
+}
+
 /*
  * Determine which quadrant a point falls into, relative to the centroid.
  *
@@ -54,6 +80,9 @@ spg_quad_config(PG_FUNCTION_ARGS)
 static int16
 getQuadrant(Point *centroid, Point *tst)
 {
+	int			xside;
+	int			yside;
+
 	if ((SPTEST(point_above, tst, centroid) ||
 		 SPTEST(point_horiz, tst, centroid)) &&
 		(SPTEST(point_right, tst, centroid) ||
@@ -74,8 +103,18 @@ getQuadrant(Point *centroid, Point *tst)
 		SPTEST(point_left, tst, centroid))
 		return 4;
 
-	elog(ERROR, "getQuadrant: impossible case");
-	return 0;
+	/*
+	 * The fuzzy tests above can all fail on one axis, because FPeq() rounds a
+	 * difference while FPlt() and FPgt() round a sum.  Compare exactly only
+	 * on that axis.  On the other axis, keep the result of the fuzzy tests,
+	 * because searches route by that result too.
+	 */
+	xside = getAxisSide(tst->x, centroid->x);
+	yside = getAxisSide(tst->y, centroid->y);
+
+	if (xside >= 0)
+		return (yside >= 0) ? 1 : 2;
+	return (yside <= 0) ? 3 : 4;
 }
 
 /* Returns bounding box of a given quadrant inside given bounding box */
diff --git a/src/test/regress/expected/spgist.out b/src/test/regress/expected/spgist.out
index 2e911285600..064b0a69cae 100644
--- a/src/test/regress/expected/spgist.out
+++ b/src/test/regress/expected/spgist.out
@@ -94,3 +94,19 @@ select box(point(i,j))
   from generate_series(1,100,5) i,
        generate_series(1,10,5) j;
 -- leave this table around, to help in testing dump/restore
+-- Bug #19597: coverage of getQuadrant for a point just over EPSILON from the centroid
+create table spgist_quad_fp_tbl(p point);
+insert into spgist_quad_fp_tbl select point(i, 5) from generate_series(1, 300) i;
+create index spgist_quad_fp_idx on spgist_quad_fp_tbl using spgist(p);
+insert into spgist_quad_fp_tbl values (point(3, 5 + 1e-6));
+-- Bug #19597: needs exact comparison on one axis, within EPSILON on the other
+insert into spgist_quad_fp_tbl values (point(114 - 5e-7, 5 + 1e-6));
+set enable_seqscan = off;
+select count(*) from spgist_quad_fp_tbl
+  where p <@ box(point(113.999999, 5.0000004), point(113.9999999, 5.0000016));
+ count 
+-------
+     1
+(1 row)
+
+reset enable_seqscan;
diff --git a/src/test/regress/sql/spgist.sql b/src/test/regress/sql/spgist.sql
index 4828ede68c3..c7bfae347db 100644
--- a/src/test/regress/sql/spgist.sql
+++ b/src/test/regress/sql/spgist.sql
@@ -89,3 +89,16 @@ select box(point(i,j))
   from generate_series(1,100,5) i,
        generate_series(1,10,5) j;
 -- leave this table around, to help in testing dump/restore
+
+-- Bug #19597: coverage of getQuadrant for a point just over EPSILON from the centroid
+create table spgist_quad_fp_tbl(p point);
+insert into spgist_quad_fp_tbl select point(i, 5) from generate_series(1, 300) i;
+create index spgist_quad_fp_idx on spgist_quad_fp_tbl using spgist(p);
+insert into spgist_quad_fp_tbl values (point(3, 5 + 1e-6));
+
+-- Bug #19597: needs exact comparison on one axis, within EPSILON on the other
+insert into spgist_quad_fp_tbl values (point(114 - 5e-7, 5 + 1e-6));
+set enable_seqscan = off;
+select count(*) from spgist_quad_fp_tbl
+  where p <@ box(point(113.999999, 5.0000004), point(113.9999999, 5.0000016));
+reset enable_seqscan;