diff --git a/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesSketch.java b/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesSketch.java
index 121f1f92a..7e536bfe2 100644
--- a/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesSketch.java
+++ b/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesSketch.java
@@ -4,6 +4,8 @@
  */
 package com.yahoo.sketches.tuple;
 
+import com.yahoo.sketches.BinomialBoundsN;
+
 /**
  * This is a base class for a specialized version of a tuple sketch, where an array of double values is associated with each key.
  * A primitive array is used here, as opposed to a generic Summary object, for performance reasons.
@@ -54,7 +56,7 @@ public double getEstimate() {
    */
   public double getUpperBound(final int numStdDev) {
     if (!isEstimationMode()) return getRetainedEntries();
-    return Util.upperBound(getEstimate(), getTheta(), numStdDev);
+    return BinomialBoundsN.getUpperBound(getRetainedEntries(), getTheta(), numStdDev, isEmpty_);
   }
 
   /**
@@ -66,7 +68,7 @@ public double getUpperBound(final int numStdDev) {
    */
   public double getLowerBound(final int numStdDev) {
     if (!isEstimationMode()) return getRetainedEntries();
-    return Util.lowerBound(getEstimate(), getTheta(), numStdDev);
+    return BinomialBoundsN.getLowerBound(getRetainedEntries(), getTheta(), numStdDev, isEmpty_);
   }
 
   /**
diff --git a/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesUpdatableSketch.java b/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesUpdatableSketch.java
index 4d87f2496..478ce577e 100644
--- a/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesUpdatableSketch.java
+++ b/src/main/java/com/yahoo/sketches/tuple/ArrayOfDoublesUpdatableSketch.java
@@ -28,7 +28,7 @@ public abstract class ArrayOfDoublesUpdatableSketch extends ArrayOfDoublesSketch
    * @param values The given values
    */
   public void update(final long key, final double[] values) {
-    update(Util.longToLongArray(key), values);
+    update(new long[] {key}, values);
   }
 
   /**
diff --git a/src/main/java/com/yahoo/sketches/tuple/Sketch.java b/src/main/java/com/yahoo/sketches/tuple/Sketch.java
index a1d8c345f..55142e030 100644
--- a/src/main/java/com/yahoo/sketches/tuple/Sketch.java
+++ b/src/main/java/com/yahoo/sketches/tuple/Sketch.java
@@ -4,6 +4,8 @@
  */
 package com.yahoo.sketches.tuple;
 
+import com.yahoo.sketches.BinomialBoundsN;
+
 /**
  * This is an equivalent to com.yahoo.sketches.theta.Sketch with
  * addition of a user-defined Summary object associated with every unique entry
@@ -37,7 +39,7 @@ public double getEstimate() {
    */
   public double getUpperBound(final int numStdDev) {
     if (!isEstimationMode()) return getRetainedEntries();
-    return Util.upperBound(getEstimate(), getTheta(), numStdDev);
+    return BinomialBoundsN.getUpperBound(getRetainedEntries(), getTheta(), numStdDev, isEmpty_);
   }
 
   /**
@@ -49,7 +51,7 @@ public double getUpperBound(final int numStdDev) {
    */
   public double getLowerBound(final int numStdDev) {
     if (!isEstimationMode()) return getRetainedEntries();
-    return Util.lowerBound(getEstimate(), getTheta(), numStdDev);
+    return BinomialBoundsN.getLowerBound(getRetainedEntries(), getTheta(), numStdDev, isEmpty_);
   }
 
   /**
diff --git a/src/main/java/com/yahoo/sketches/tuple/UpdatableSketch.java b/src/main/java/com/yahoo/sketches/tuple/UpdatableSketch.java
index 4b7c6abbd..9653120df 100644
--- a/src/main/java/com/yahoo/sketches/tuple/UpdatableSketch.java
+++ b/src/main/java/com/yahoo/sketches/tuple/UpdatableSketch.java
@@ -56,7 +56,7 @@ public class UpdatableSketch<U, S extends UpdatableSummary<U>> extends QuickSele
    * @param value The given U value
    */
   public void update(final long key, final U value) {
-    update(Util.longToLongArray(key), value);
+  update(new long[] {key}, value);
   }
 
   /**
diff --git a/src/main/java/com/yahoo/sketches/tuple/Util.java b/src/main/java/com/yahoo/sketches/tuple/Util.java
index 1557632cf..a447e2508 100644
--- a/src/main/java/com/yahoo/sketches/tuple/Util.java
+++ b/src/main/java/com/yahoo/sketches/tuple/Util.java
@@ -27,21 +27,6 @@ static final int startingSubMultiple(final int lgTarget, final int lgResizeRatio
     return lgStart;
   }
 
-  static final double upperBound(final double estimate, final double theta, final double numStdDev) {
-    double dsq = numStdDev * numStdDev * ((1.0 / theta) - 1.0);
-    return estimate + (dsq / 2.0) + ((Math.sqrt(dsq) / 2.0) * Math.sqrt((4.0 * estimate) + dsq));
-  }
-
-  static final double lowerBound(final double estimate, final double theta, final double numStdDev) {
-    double dsq = numStdDev * numStdDev * ((1.0 / theta) - 1.0);
-    return estimate + (dsq / 2.0) - ((Math.sqrt(dsq) / 2.0) * Math.sqrt((4.0 * estimate) + dsq));
-  }
-
-  static final long[] longToLongArray(final long value) {
-    final long[] array = { value };
-    return array;
-  }
-
   static final long[] doubleToLongArray(final double value) {
     final double d = (value == 0.0) ? 0.0 : value; // canonicalize -0.0, 0.0
     final long[] array = { Double.doubleToLongBits(d) }; // canonicalize all NaN forms
diff --git a/src/test/java/com/yahoo/sketches/tuple/DirectArrayOfDoublesQuickSelectSketchTest.java b/src/test/java/com/yahoo/sketches/tuple/DirectArrayOfDoublesQuickSelectSketchTest.java
index 7bc8b8e72..d97efad02 100644
--- a/src/test/java/com/yahoo/sketches/tuple/DirectArrayOfDoublesQuickSelectSketchTest.java
+++ b/src/test/java/com/yahoo/sketches/tuple/DirectArrayOfDoublesQuickSelectSketchTest.java
@@ -82,10 +82,8 @@ public void estimationMode() {
     for (int i = 1; i <= 8192; i++) sketch.update(i, new double[] {1.0});
     Assert.assertTrue(sketch.isEstimationMode());
     Assert.assertEquals(sketch.getEstimate(), 8192, 8192 * 0.01);
-    Assert.assertEquals(sketch.getEstimate(), sketch.getLowerBound(0));
-    Assert.assertEquals(sketch.getEstimate(), sketch.getUpperBound(0));
     Assert.assertTrue(sketch.getEstimate() >= sketch.getLowerBound(1));
-    Assert.assertTrue(sketch.getEstimate() <= sketch.getUpperBound(1));
+    Assert.assertTrue(sketch.getEstimate() < sketch.getUpperBound(1));
 
     double[][] values = sketch.getValues();
     Assert.assertTrue(values.length >= 4096);
diff --git a/src/test/java/com/yahoo/sketches/tuple/HeapArrayOfDoublesQuickSelectSketchTest.java b/src/test/java/com/yahoo/sketches/tuple/HeapArrayOfDoublesQuickSelectSketchTest.java
index d35e0a77d..99e3a9182 100644
--- a/src/test/java/com/yahoo/sketches/tuple/HeapArrayOfDoublesQuickSelectSketchTest.java
+++ b/src/test/java/com/yahoo/sketches/tuple/HeapArrayOfDoublesQuickSelectSketchTest.java
@@ -80,10 +80,8 @@ public void estimationMode() {
     for (int i = 1; i <= 8192; i++) sketch.update(i, new double[] {1.0});
     Assert.assertTrue(sketch.isEstimationMode());
     Assert.assertEquals(sketch.getEstimate(), 8192, 8192 * 0.01);
-    Assert.assertEquals(sketch.getEstimate(), sketch.getLowerBound(0));
-    Assert.assertEquals(sketch.getEstimate(), sketch.getUpperBound(0));
     Assert.assertTrue(sketch.getEstimate() >= sketch.getLowerBound(1));
-    Assert.assertTrue(sketch.getEstimate() <= sketch.getUpperBound(1));
+    Assert.assertTrue(sketch.getEstimate() < sketch.getUpperBound(1));
     Assert.assertTrue(sketch.getRetainedEntries() > 4096);
     sketch.trim();
     Assert.assertEquals(sketch.getRetainedEntries(), 4096);
diff --git a/src/test/java/com/yahoo/sketches/tuple/UpdatableSketchWithDoubleSummaryTest.java b/src/test/java/com/yahoo/sketches/tuple/UpdatableSketchWithDoubleSummaryTest.java
index 3ab764706..9e80b82c7 100644
--- a/src/test/java/com/yahoo/sketches/tuple/UpdatableSketchWithDoubleSummaryTest.java
+++ b/src/test/java/com/yahoo/sketches/tuple/UpdatableSketchWithDoubleSummaryTest.java
@@ -81,8 +81,6 @@ public void estimationMode() {
     for (int i = 1; i <= 8192; i++) sketch.update(i, 1.0);
     Assert.assertTrue(sketch.isEstimationMode());
     Assert.assertEquals(sketch.getEstimate(), 8192, 8192 * 0.01);
-    Assert.assertEquals(sketch.getEstimate(), sketch.getLowerBound(0));
-    Assert.assertEquals(sketch.getEstimate(), sketch.getUpperBound(0));
     Assert.assertTrue(sketch.getEstimate() >= sketch.getLowerBound(1));
     Assert.assertTrue(sketch.getEstimate() < sketch.getUpperBound(1));