diff --git a/libs/exponential-histogram/src/main/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtils.java b/libs/exponential-histogram/src/main/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtils.java
index a9a93cf023369..a27fbeea89093 100644
--- a/libs/exponential-histogram/src/main/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtils.java
+++ b/libs/exponential-histogram/src/main/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtils.java
@@ -234,9 +234,19 @@ static double exponentiallyScaledToDoubleValue(long index, int scale) {
      */
     public static double getPointOfLeastRelativeError(long bucketIndex, int scale) {
         checkIndexAndScaleBounds(bucketIndex, scale);
-        double upperBound = getUpperBucketBoundary(bucketIndex, scale);
         double histogramBase = Math.pow(2, Math.scalb(1, -scale));
-        return 2 / (histogramBase + 1) * upperBound;
+        if (Double.isFinite(histogramBase)) {
+            double upperBound = getUpperBucketBoundary(bucketIndex, scale);
+            return 2 / (histogramBase + 1) * upperBound;
+        } else {
+            if (bucketIndex >= 0) {
+                // the bucket is (1, +inf), approximate point of least error as inf
+                return Double.POSITIVE_INFINITY;
+            } else {
+                // the bucket is (1/(Inf), 1), approximate point of least error as 0
+                return 0;
+            }
+        }
     }
 
     /**
diff --git a/libs/exponential-histogram/src/test/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtilsTests.java b/libs/exponential-histogram/src/test/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtilsTests.java
index a75aedbf35231..68371395a4df2 100644
--- a/libs/exponential-histogram/src/test/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtilsTests.java
+++ b/libs/exponential-histogram/src/test/java/org/elasticsearch/exponentialhistogram/ExponentialScaleUtilsTests.java
@@ -79,6 +79,12 @@ public void testExtremeValueIndexing() {
         }
     }
 
+    public void testPointOfLeastErrorAtInfinity() {
+        assertThat(getPointOfLeastRelativeError(0, MIN_SCALE), equalTo(Double.POSITIVE_INFINITY));
+        assertThat(getPointOfLeastRelativeError(-1, MIN_SCALE), equalTo(0.0));
+        assertThat(getPointOfLeastRelativeError(10, MIN_SCALE + 2), equalTo(Double.POSITIVE_INFINITY));
+    }
+
     public void testRandomValueIndexing() {
         for (int i = 0; i < 100_000; i++) {
             // generate values in the range 10^-100 to 10^100