diff --git a/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequest.java b/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequest.java
index 95f9d79354268..1ad5c9e65a737 100644
--- a/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequest.java
+++ b/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequest.java
@@ -24,4 +24,6 @@ protected EsqlQueryRequest(StreamInput in) throws IOException {
     public abstract String query();
 
     public abstract QueryBuilder filter();
+
+    public abstract boolean approximate();
 }
diff --git a/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequestBuilder.java b/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequestBuilder.java
index 0975f827b828a..43560bcbb7781 100644
--- a/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequestBuilder.java
+++ b/x-pack/plugin/core/src/main/java/org/elasticsearch/xpack/core/esql/action/EsqlQueryRequestBuilder.java
@@ -39,6 +39,8 @@ public final ActionType<Response> action() {
 
     public abstract EsqlQueryRequestBuilder<Request, Response> filter(QueryBuilder filter);
 
+    public abstract EsqlQueryRequestBuilder<Request, Response> approximate(boolean approximate);
+
     public abstract EsqlQueryRequestBuilder<Request, Response> allowPartialResults(boolean allowPartialResults);
 
 }
diff --git a/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvSpecReader.java b/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvSpecReader.java
index ba0d11059a69b..9ae1877d1574b 100644
--- a/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvSpecReader.java
+++ b/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvSpecReader.java
@@ -31,6 +31,7 @@ public static class CsvSpecParser implements SpecReader.Parser {
         private final StringBuilder query = new StringBuilder();
         private final StringBuilder data = new StringBuilder();
         private final List<String> requiredCapabilities = new ArrayList<>();
+        private boolean approximate = false;
         private CsvTestCase testCase;
 
         private CsvSpecParser() {}
@@ -44,6 +45,8 @@ public Object parse(String line) {
                     earlySchema.append(line.substring(SCHEMA_PREFIX.length()).trim());
                 } else if (line.toLowerCase(Locale.ROOT).startsWith("required_capability:")) {
                     requiredCapabilities.add(line.substring("required_capability:".length()).trim());
+                } else if (line.toLowerCase(Locale.ROOT).startsWith("approximate:")) {
+                    approximate = Boolean.parseBoolean(line.substring("approximate:".length()).trim());
                 } else {
                     if (line.endsWith(";")) {
                         // pick up the query
@@ -52,9 +55,11 @@ public Object parse(String line) {
                         testCase.query = query.toString();
                         testCase.earlySchema = earlySchema.toString();
                         testCase.requiredCapabilities = List.copyOf(requiredCapabilities);
+                        testCase.approximate = approximate;
                         requiredCapabilities.clear();
                         earlySchema.setLength(0);
                         query.setLength(0);
+                        approximate = false;
                     }
                     // keep reading the query
                     else {
@@ -116,6 +121,7 @@ public static class CsvTestCase {
         private final List<Pattern> expectedWarningsRegex = new ArrayList<>();
         public boolean ignoreOrder;
         public List<String> requiredCapabilities = List.of();
+        public boolean approximate = false;
 
         /**
          * Returns the warning headers expected to be added by the test. To declare such a header, use the `warning:definition` format
diff --git a/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvTestsDataLoader.java b/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvTestsDataLoader.java
index 96f8ffdcccbbe..54ba30586a677 100644
--- a/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvTestsDataLoader.java
+++ b/x-pack/plugin/esql/qa/testFixtures/src/main/java/org/elasticsearch/xpack/esql/CsvTestsDataLoader.java
@@ -168,6 +168,7 @@ public class CsvTestsDataLoader {
     private static final TestDataset MV_TEXT = new TestDataset("mv_text");
     private static final TestDataset DENSE_VECTOR = new TestDataset("dense_vector");
     private static final TestDataset COLORS = new TestDataset("colors");
+    private static final TestDataset MANY_NUMBERS = new TestDataset("many_numbers");
 
     public static final Map<String, TestDataset> CSV_DATASET_MAP = Map.ofEntries(
         Map.entry(EMPLOYEES.indexName, EMPLOYEES),
@@ -234,7 +235,8 @@ public class CsvTestsDataLoader {
         Map.entry(DENSE_VECTOR.indexName, DENSE_VECTOR),
         Map.entry(COLORS.indexName, COLORS),
         Map.entry(MULTI_COLUMN_JOINABLE.indexName, MULTI_COLUMN_JOINABLE),
-        Map.entry(MULTI_COLUMN_JOINABLE_LOOKUP.indexName, MULTI_COLUMN_JOINABLE_LOOKUP)
+        Map.entry(MULTI_COLUMN_JOINABLE_LOOKUP.indexName, MULTI_COLUMN_JOINABLE_LOOKUP),
+        Map.entry(MANY_NUMBERS.indexName, MANY_NUMBERS)
     );
 
     private static final EnrichConfig LANGUAGES_ENRICH = new EnrichConfig("languages_policy", "enrich-policy-languages.json");
diff --git a/x-pack/plugin/esql/qa/testFixtures/src/main/resources/approximate.csv-spec b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/approximate.csv-spec
new file mode 100644
index 0000000000000..bfd652d5efaf0
--- /dev/null
+++ b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/approximate.csv-spec
@@ -0,0 +1,194 @@
+// Tests focused on query approximation.
+//
+// The data set "many_numbers" contains 10,000 rows with a multi-valued field
+// "value" containing 550 integers each. Aggregated all over rows, there are
+// 1x1, 2x2, ..., 999x999 and 1000x1000. It's not one number per row, because
+// that's too slow to ingest.
+//
+// Note: this tests only basic behavior, because of limitations of the CSV tests.
+// Most tests assert that the count, average and sum of some values are within a
+// range. All ranges are very loose, so that the tests should practically never fail.
+// The range checks are done in ES|QL, resulting in one boolean value (is_expected),
+// because the CSV tests don't support such assertions.
+
+approximate stats on large data
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | STATS count=COUNT(), avg=AVG(value), sum=SUM(value)
+    | EVAL is_expected = count >= 450000 AND count <= 550000 AND
+                         avg >= 600 AND avg <= 750 AND
+                         sum >= 300000000 AND sum <= 380000000
+    | KEEP is_expected
+;
+
+is_expected:boolean
+true
+;
+
+
+exact stats on small data
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | WHERE value <= 25
+    | STATS count=COUNT(), avg=AVG(value), sum=SUM(value)
+;
+
+count:long | avg:double | sum:long
+325        | 17.0       | 5525
+;
+
+
+with where
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | WHERE value >= 500
+    | STATS count=COUNT(), avg=AVG(value), sum=SUM(value)
+    | EVAL is_expected = count >= 300000 AND count <= 450000 AND
+                         avg >= 700 AND avg <= 850 AND
+                         sum >= 270000000 AND sum <= 310000000
+    | KEEP is_expected
+;
+
+is_expected:boolean
+true
+;
+
+
+with stats where
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | STATS count=COUNT() WHERE value >= 500,
+            avg=AVG(value) WHERE value >= 500,
+            sum=SUM(value) WHERE value >= 500
+    | EVAL is_expected = count >= 300000 AND count <= 450000 AND
+                         avg >= 700 AND avg <= 850 AND
+                         sum >= 270000000 AND sum <= 310000000
+    | KEEP is_expected
+;
+
+is_expected:boolean
+true
+;
+
+
+with sample
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | SAMPLE 0.5
+    | STATS count=COUNT(), avg=AVG(value), sum=SUM(value)
+    | EVAL is_expected = count >= 200000 AND count <= 300000 AND
+                         avg >= 600 AND avg <= 750 AND
+                         sum >= 140000000 AND sum <= 200000000
+    | KEEP is_expected
+;
+
+is_expected:boolean
+true
+;
+
+
+with commands inbetween
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | EVAL value2 = 2 * value
+    | DROP value
+    | SORT value2 DESC
+    | RENAME value2 AS value3
+    | MV_EXPAND value3
+    | EVAL value4 = TO_STRING(value3)
+    | SORT value4 ASC
+    | GROK value4 "%{NUMBER:value5}"
+    | EVAL value5 = TO_INTEGER(value5)
+    | KEEP value3, value5
+    | STATS count=COUNT(), avg=AVG(value5), sum=SUM(value5)
+    | EVAL is_expected = count >= 450000 AND count <= 550000 AND
+                         avg >= 1200 AND avg <= 1500 AND
+                         sum >= 600000000 AND sum <= 760000000
+    | KEEP is_expected
+;
+
+is_expected:boolean
+true
+;
+
+
+with commands after
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | STATS count=COUNT(), avg=AVG(value), sum=SUM(value)
+    | EVAL avg2 = 2 * avg
+    | LIMIT 10
+    | MV_EXPAND avg
+    | SORT count ASC
+    | EVAL is_expected = count >= 450000 AND count <= 550000 AND
+                         avg2 >= 1200 AND avg2 <= 1500 AND
+                         sum >= 300000000 AND sum <= 380000000
+    | KEEP is_expected
+;
+
+is_expected:boolean
+true
+;
+
+
+approximate stats by on large data
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | STATS count=COUNT() BY value
+    | SORT value DESC
+    | LIMIT 5
+    | EVAL is_expected = count >= 100 AND count <= 2000
+    | KEEP value, is_expected
+;
+
+value:integer | is_expected:boolean
+1000        | true
+999         | true
+998         | true
+997         | true
+996         | true
+;
+
+
+exact stats by on small data
+required_capability: approximate
+approximate: true
+
+FROM many_numbers
+    | MV_EXPAND value
+    | WHERE value <= 5
+    | STATS count=COUNT() BY value
+    | SORT value
+;
+
+count:long | value:integer
+1          | 1
+2          | 2
+3          | 3
+4          | 4
+5          | 5
+;
diff --git a/x-pack/plugin/esql/qa/testFixtures/src/main/resources/data/many_numbers.csv b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/data/many_numbers.csv
new file mode 100644
index 0000000000000..54a50cef55da3
--- /dev/null
+++ b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/data/many_numbers.csv
@@ -0,0 +1,10002 @@
+value:integer
+[969, 737, 416, 869, 786, 411, 738, 410, 934, 686, 973, 701, 732, 930, 134, 443, 369, 858, 799, 418, 890, 154, 920, 884, 536, 749, 821, 278, 733, 905, 507, 776, 752, 534, 952, 620, 993, 920, 529, 692, 981, 500, 425, 637, 667, 974, 993, 862, 680, 319]
+[926, 617, 770, 693, 813, 769, 674, 745, 391, 389, 417, 683, 598, 477, 158, 730, 834, 538, 582, 683, 546, 744, 880, 930, 966, 586, 388, 336, 750, 339, 800, 304, 588, 615, 720, 443, 631, 668, 495, 677, 460, 960, 348, 226, 563, 442, 870, 899, 480, 828]
+[912, 936, 460, 754, 891, 781, 674, 406, 851, 814, 816, 332, 934, 842, 955, 306, 754, 390, 771, 558, 911, 844, 367, 977, 451, 822, 825, 574, 292, 702, 906, 913, 676, 476, 764, 297, 267, 309, 876, 38, 127, 528, 744, 465, 510, 580, 491, 873, 999, 598]
+[834, 923, 781, 968, 844, 348, 370, 914, 971, 885, 690, 567, 464, 592, 523, 367, 660, 642, 361, 360, 861, 953, 861, 899, 920, 884, 478, 586, 957, 940, 846, 959, 436, 968, 836, 273, 680, 700, 988, 795, 396, 380, 642, 830, 848, 779, 941, 881, 367, 840]
+[825, 431, 972, 311, 805, 786, 855, 693, 615, 622, 675, 408, 912, 835, 724, 998, 911, 744, 543, 844, 641, 748, 562, 679, 364, 389, 726, 539, 635, 519, 307, 680, 932, 799, 182, 601, 985, 910, 539, 970, 534, 799, 292, 675, 471, 788, 719, 268, 528, 979]
+[954, 289, 852, 966, 752, 989, 975, 238, 998, 766, 747, 525, 792, 865, 761, 693, 511, 992, 490, 619, 771, 986, 388, 653, 426, 577, 865, 118, 980, 546, 1000, 417, 552, 622, 520, 734, 773, 842, 739, 71, 177, 195, 924, 229, 906, 761, 919, 515, 809, 521]
+[410, 434, 720, 977, 542, 548, 428, 653, 558, 878, 818, 204, 971, 315, 938, 977, 897, 531, 227, 375, 765, 895, 913, 531, 640, 345, 651, 893, 889, 975, 679, 632, 763, 536, 269, 630, 488, 995, 889, 310, 770, 815, 746, 898, 897, 903, 847, 793, 112, 604]
+[964, 648, 613, 380, 622, 439, 769, 390, 833, 341, 844, 727, 308, 333, 530, 495, 680, 883, 755, 818, 441, 879, 815, 823, 627, 678, 291, 945, 407, 923, 938, 248, 824, 874, 837, 972, 446, 910, 604, 890, 469, 772, 838, 794, 507, 884, 348, 384, 692, 853]
+[371, 530, 246, 507, 840, 169, 428, 379, 877, 928, 595, 920, 704, 589, 838, 347, 691, 770, 891, 322, 997, 179, 769, 772, 352, 950, 497, 401, 921, 717, 730, 374, 220, 438, 996, 815, 991, 926, 581, 693, 121, 847, 623, 696, 273, 207, 948, 623, 954, 694]
+[590, 528, 732, 835, 310, 835, 722, 621, 921, 683, 995, 547, 578, 967, 894, 999, 637, 626, 875, 672, 925, 371, 131, 223, 645, 871, 895, 326, 877, 734, 716, 515, 434, 216, 490, 825, 400, 471, 809, 938, 788, 333, 904, 743, 525, 921, 915, 629, 626, 558]
+[559, 663, 826, 643, 756, 927, 557, 872, 607, 957, 962, 296, 278, 990, 463, 288, 366, 679, 713, 951, 851, 943, 975, 797, 404, 197, 161, 905, 533, 924, 540, 286, 694, 976, 274, 953, 846, 283, 781, 626, 882, 272, 428, 885, 678, 968, 523, 489, 787, 329]
+[651, 795, 318, 874, 804, 784, 556, 531, 344, 602, 353, 991, 517, 789, 438, 469, 435, 320, 379, 899, 647, 878, 416, 922, 546, 701, 864, 847, 225, 850, 739, 374, 651, 776, 681, 688, 701, 128, 904, 711, 840, 582, 813, 688, 353, 614, 635, 639, 916, 704]
+[269, 755, 937, 869, 628, 835, 834, 796, 946, 830, 660, 831, 599, 916, 817, 954, 690, 349, 738, 982, 351, 913, 589, 658, 885, 961, 716, 904, 571, 736, 498, 397, 754, 625, 818, 606, 955, 509, 302, 515, 799, 651, 821, 834, 948, 588, 221, 436, 850, 987]
+[762, 432, 978, 668, 289, 985, 243, 853, 708, 522, 795, 715, 491, 462, 930, 991, 503, 825, 603, 745, 907, 117, 959, 801, 877, 889, 948, 689, 569, 887, 423, 733, 452, 564, 797, 221, 848, 640, 586, 872, 382, 348, 923, 180, 932, 901, 587, 994, 935, 519]
+[803, 913, 648, 923, 651, 26, 328, 851, 901, 878, 909, 173, 966, 758, 631, 952, 354, 311, 541, 482, 626, 957, 921, 396, 483, 538, 133, 711, 190, 653, 519, 836, 767, 990, 540, 807, 991, 932, 938, 794, 583, 740, 661, 429, 498, 272, 611, 774, 732, 747]
+[476, 904, 432, 160, 390, 558, 871, 678, 654, 443, 682, 998, 643, 422, 561, 50, 470, 522, 391, 300, 984, 847, 857, 883, 796, 963, 941, 333, 574, 162, 947, 873, 217, 848, 667, 263, 671, 361, 378, 406, 453, 190, 387, 709, 81, 547, 577, 459, 372, 794]
+[332, 345, 613, 971, 949, 639, 563, 661, 626, 360, 211, 957, 505, 639, 264, 703, 603, 133, 565, 608, 945, 325, 333, 1000, 538, 471, 918, 635, 561, 840, 702, 177, 999, 624, 518, 601, 573, 638, 534, 601, 961, 900, 321, 658, 713, 534, 112, 652, 895, 873]
+[351, 964, 847, 632, 267, 489, 468, 354, 970, 617, 611, 582, 785, 607, 764, 665, 670, 865, 198, 249, 806, 884, 954, 743, 830, 682, 346, 621, 394, 458, 817, 925, 803, 418, 1000, 680, 736, 996, 658, 768, 821, 377, 622, 513, 756, 338, 505, 720, 401, 446]
+[214, 795, 513, 844, 684, 496, 522, 66, 879, 534, 658, 545, 613, 259, 684, 788, 513, 673, 981, 328, 672, 987, 805, 817, 963, 612, 671, 597, 723, 852, 620, 912, 459, 589, 298, 790, 724, 483, 645, 887, 144, 414, 846, 412, 279, 356, 781, 71, 882, 714]
+[565, 838, 246, 973, 810, 476, 839, 34, 816, 817, 692, 761, 898, 566, 959, 770, 476, 234, 640, 922, 609, 713, 938, 724, 364, 783, 742, 897, 675, 891, 991, 634, 193, 295, 692, 989, 814, 397, 675, 931, 805, 984, 378, 927, 643, 617, 907, 275, 569, 742]
+[918, 528, 706, 389, 906, 773, 410, 745, 999, 453, 535, 692, 422, 839, 934, 258, 571, 907, 393, 749, 469, 973, 560, 955, 946, 612, 520, 442, 561, 838, 730, 833, 998, 885, 781, 995, 46, 452, 924, 141, 340, 706, 472, 441, 668, 41, 536, 219, 440, 717]
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+[975, 887, 794, 326, 307, 766, 611, 484, 933, 876, 980, 610, 901, 600, 362, 822, 771, 948, 650, 313, 969, 936, 290, 452, 925, 874, 274, 772, 403, 257, 714, 599, 986, 598, 886, 872, 589, 948, 289, 827, 752, 899, 231, 841, 971, 642, 535, 449, 781, 548]
+[308, 286, 847, 826, 601, 393, 666, 937, 587, 590, 585, 407, 462, 878, 944, 344, 845, 696, 809, 881, 555, 514, 368, 494, 908, 207, 530, 432, 120, 778, 482, 793, 535, 939, 819, 894, 865, 801, 751, 887, 710, 395, 851, 707, 824, 831, 981, 196, 686, 303]
+[680, 705, 689, 753, 998, 899, 922, 886, 792, 459, 885, 335, 894, 589, 148, 735, 678, 763, 601, 191, 888, 772, 597, 643, 782, 987, 824, 894, 930, 534, 694, 933, 693, 982, 92, 643, 846, 827, 225, 655, 612, 360, 971, 314, 728, 370, 964, 120, 840, 378]
+[837, 488, 603, 996, 539, 650, 976, 880, 835, 22, 690, 945, 515, 414, 721, 757, 145, 348, 986, 915, 991, 798, 951, 950, 761, 366, 696, 886, 491, 542, 143, 444, 427, 506, 557, 643, 692, 295, 762, 910, 487, 617, 339, 617, 974, 137, 846, 845, 812, 648]
+[944, 966, 527, 865, 398, 896, 687, 400, 431, 716, 702, 292, 399, 635, 844, 884, 619, 769, 406, 475, 954, 787, 958, 637, 700, 681, 948, 706, 509, 564, 730, 959, 282, 473, 899, 131, 666, 492, 633, 843, 934, 980, 645, 616, 415, 824, 946, 692, 504, 247]
+[575, 915, 790, 948, 952, 480, 287, 538, 432, 1000, 257, 676, 919, 743, 486, 631, 303, 589, 967, 982, 673, 299, 413, 735, 653, 809, 612, 924, 949, 343, 989, 244, 567, 997, 141, 282, 815, 424, 929, 675, 602, 909, 711, 977, 417, 789, 865, 302, 626, 820]
+[701, 790, 439, 941, 763, 868, 833, 723, 717, 442, 620, 754, 270, 758, 845, 923, 879, 451, 878, 820, 994, 324, 637, 638, 968, 839, 978, 818, 905, 254, 541, 274, 964, 136, 603, 565, 676, 920, 938, 775, 310, 878, 905, 991, 321, 850, 229, 796, 644, 761]
+[849, 773, 935, 609, 248, 754, 924, 806, 641, 205, 594, 886, 850, 327, 132, 460, 769, 676, 876, 497, 720, 680, 294, 949, 779, 749, 913, 967, 816, 431, 837, 941, 701, 999, 681, 657, 910, 886, 736, 764, 901, 570, 920, 721, 615, 577, 467, 756, 317, 161]
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diff --git a/x-pack/plugin/esql/qa/testFixtures/src/main/resources/mapping-many_numbers.json b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/mapping-many_numbers.json
new file mode 100644
index 0000000000000..bbd4275176fa3
--- /dev/null
+++ b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/mapping-many_numbers.json
@@ -0,0 +1,7 @@
+{
+    "properties": {
+        "value": {
+            "type": "integer"
+        }
+    }
+}
diff --git a/x-pack/plugin/esql/qa/testFixtures/src/main/resources/sample.csv-spec b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/sample.csv-spec
index 405e618b91707..5e2c9a7d9557c 100644
--- a/x-pack/plugin/esql/qa/testFixtures/src/main/resources/sample.csv-spec
+++ b/x-pack/plugin/esql/qa/testFixtures/src/main/resources/sample.csv-spec
@@ -1,10 +1,7 @@
 // Tests focused on the SAMPLE command
 // Note: this tests only basic behavior, because of limitations of the CSV tests.
 // Most tests assert that the count, average and sum of some values are within a
-// range. These stats should be correctly adjusted for the sampling. Furthermore,
-// they also assert the value of MV_COUNT(VALUES(...)), which is not adjusted for
-// the sampling and therefore gives the size of the sample.
-// All ranges are very loose, so that the tests should practically never fail.
+// range. All ranges are very loose, so that the tests should practically never fail.
 // The range checks are done in ES|QL, resulting in one boolean value (is_expected),
 // because the CSV tests don't support such assertions.
 
diff --git a/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalDoubleEvaluator.java b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalDoubleEvaluator.java
new file mode 100644
index 0000000000000..3a9b6f618429e
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalDoubleEvaluator.java
@@ -0,0 +1,173 @@
+// Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+// or more contributor license agreements. Licensed under the Elastic License
+// 2.0; you may not use this file except in compliance with the Elastic License
+// 2.0.
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import java.lang.Override;
+import java.lang.String;
+import org.apache.lucene.util.RamUsageEstimator;
+import org.elasticsearch.compute.data.Block;
+import org.elasticsearch.compute.data.DoubleBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.operator.DriverContext;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.compute.operator.Warnings;
+import org.elasticsearch.core.Releasables;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+
+/**
+ * {@link EvalOperator.ExpressionEvaluator} implementation for {@link ConfidenceInterval}.
+ * This class is generated. Edit {@code EvaluatorImplementer} instead.
+ */
+public final class ConfidenceIntervalDoubleEvaluator implements EvalOperator.ExpressionEvaluator {
+  private static final long BASE_RAM_BYTES_USED = RamUsageEstimator.shallowSizeOfInstance(ConfidenceIntervalDoubleEvaluator.class);
+
+  private final Source source;
+
+  private final EvalOperator.ExpressionEvaluator bestEstimateBlock;
+
+  private final EvalOperator.ExpressionEvaluator estimatesBlock;
+
+  private final EvalOperator.ExpressionEvaluator trialCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator bucketCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator confidenceLevelBlock;
+
+  private final DriverContext driverContext;
+
+  private Warnings warnings;
+
+  public ConfidenceIntervalDoubleEvaluator(Source source,
+      EvalOperator.ExpressionEvaluator bestEstimateBlock,
+      EvalOperator.ExpressionEvaluator estimatesBlock,
+      EvalOperator.ExpressionEvaluator trialCountBlock,
+      EvalOperator.ExpressionEvaluator bucketCountBlock,
+      EvalOperator.ExpressionEvaluator confidenceLevelBlock, DriverContext driverContext) {
+    this.source = source;
+    this.bestEstimateBlock = bestEstimateBlock;
+    this.estimatesBlock = estimatesBlock;
+    this.trialCountBlock = trialCountBlock;
+    this.bucketCountBlock = bucketCountBlock;
+    this.confidenceLevelBlock = confidenceLevelBlock;
+    this.driverContext = driverContext;
+  }
+
+  @Override
+  public Block eval(Page page) {
+    try (DoubleBlock bestEstimateBlockBlock = (DoubleBlock) bestEstimateBlock.eval(page)) {
+      try (DoubleBlock estimatesBlockBlock = (DoubleBlock) estimatesBlock.eval(page)) {
+        try (IntBlock trialCountBlockBlock = (IntBlock) trialCountBlock.eval(page)) {
+          try (IntBlock bucketCountBlockBlock = (IntBlock) bucketCountBlock.eval(page)) {
+            try (DoubleBlock confidenceLevelBlockBlock = (DoubleBlock) confidenceLevelBlock.eval(page)) {
+              return eval(page.getPositionCount(), bestEstimateBlockBlock, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock, confidenceLevelBlockBlock);
+            }
+          }
+        }
+      }
+    }
+  }
+
+  @Override
+  public long baseRamBytesUsed() {
+    long baseRamBytesUsed = BASE_RAM_BYTES_USED;
+    baseRamBytesUsed += bestEstimateBlock.baseRamBytesUsed();
+    baseRamBytesUsed += estimatesBlock.baseRamBytesUsed();
+    baseRamBytesUsed += trialCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += bucketCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += confidenceLevelBlock.baseRamBytesUsed();
+    return baseRamBytesUsed;
+  }
+
+  public DoubleBlock eval(int positionCount, DoubleBlock bestEstimateBlockBlock,
+      DoubleBlock estimatesBlockBlock, IntBlock trialCountBlockBlock,
+      IntBlock bucketCountBlockBlock, DoubleBlock confidenceLevelBlockBlock) {
+    try(DoubleBlock.Builder result = driverContext.blockFactory().newDoubleBlockBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        boolean allBlocksAreNulls = true;
+        if (!bestEstimateBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!estimatesBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!trialCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!bucketCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!confidenceLevelBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (allBlocksAreNulls) {
+          result.appendNull();
+          continue position;
+        }
+        ConfidenceInterval.process(result, p, bestEstimateBlockBlock, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock, confidenceLevelBlockBlock);
+      }
+      return result.build();
+    }
+  }
+
+  @Override
+  public String toString() {
+    return "ConfidenceIntervalDoubleEvaluator[" + "bestEstimateBlock=" + bestEstimateBlock + ", estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + ", confidenceLevelBlock=" + confidenceLevelBlock + "]";
+  }
+
+  @Override
+  public void close() {
+    Releasables.closeExpectNoException(bestEstimateBlock, estimatesBlock, trialCountBlock, bucketCountBlock, confidenceLevelBlock);
+  }
+
+  private Warnings warnings() {
+    if (warnings == null) {
+      this.warnings = Warnings.createWarnings(
+              driverContext.warningsMode(),
+              source.source().getLineNumber(),
+              source.source().getColumnNumber(),
+              source.text()
+          );
+    }
+    return warnings;
+  }
+
+  static class Factory implements EvalOperator.ExpressionEvaluator.Factory {
+    private final Source source;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bestEstimateBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory estimatesBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory trialCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bucketCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory confidenceLevelBlock;
+
+    public Factory(Source source, EvalOperator.ExpressionEvaluator.Factory bestEstimateBlock,
+        EvalOperator.ExpressionEvaluator.Factory estimatesBlock,
+        EvalOperator.ExpressionEvaluator.Factory trialCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory bucketCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory confidenceLevelBlock) {
+      this.source = source;
+      this.bestEstimateBlock = bestEstimateBlock;
+      this.estimatesBlock = estimatesBlock;
+      this.trialCountBlock = trialCountBlock;
+      this.bucketCountBlock = bucketCountBlock;
+      this.confidenceLevelBlock = confidenceLevelBlock;
+    }
+
+    @Override
+    public ConfidenceIntervalDoubleEvaluator get(DriverContext context) {
+      return new ConfidenceIntervalDoubleEvaluator(source, bestEstimateBlock.get(context), estimatesBlock.get(context), trialCountBlock.get(context), bucketCountBlock.get(context), confidenceLevelBlock.get(context), context);
+    }
+
+    @Override
+    public String toString() {
+      return "ConfidenceIntervalDoubleEvaluator[" + "bestEstimateBlock=" + bestEstimateBlock + ", estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + ", confidenceLevelBlock=" + confidenceLevelBlock + "]";
+    }
+  }
+}
diff --git a/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalIntEvaluator.java b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalIntEvaluator.java
new file mode 100644
index 0000000000000..18525b1d80fe1
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalIntEvaluator.java
@@ -0,0 +1,173 @@
+// Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+// or more contributor license agreements. Licensed under the Elastic License
+// 2.0; you may not use this file except in compliance with the Elastic License
+// 2.0.
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import java.lang.Override;
+import java.lang.String;
+import org.apache.lucene.util.RamUsageEstimator;
+import org.elasticsearch.compute.data.Block;
+import org.elasticsearch.compute.data.DoubleBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.operator.DriverContext;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.compute.operator.Warnings;
+import org.elasticsearch.core.Releasables;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+
+/**
+ * {@link EvalOperator.ExpressionEvaluator} implementation for {@link ConfidenceInterval}.
+ * This class is generated. Edit {@code EvaluatorImplementer} instead.
+ */
+public final class ConfidenceIntervalIntEvaluator implements EvalOperator.ExpressionEvaluator {
+  private static final long BASE_RAM_BYTES_USED = RamUsageEstimator.shallowSizeOfInstance(ConfidenceIntervalIntEvaluator.class);
+
+  private final Source source;
+
+  private final EvalOperator.ExpressionEvaluator bestEstimateBlock;
+
+  private final EvalOperator.ExpressionEvaluator estimatesBlock;
+
+  private final EvalOperator.ExpressionEvaluator trialCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator bucketCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator confidenceLevelBlock;
+
+  private final DriverContext driverContext;
+
+  private Warnings warnings;
+
+  public ConfidenceIntervalIntEvaluator(Source source,
+      EvalOperator.ExpressionEvaluator bestEstimateBlock,
+      EvalOperator.ExpressionEvaluator estimatesBlock,
+      EvalOperator.ExpressionEvaluator trialCountBlock,
+      EvalOperator.ExpressionEvaluator bucketCountBlock,
+      EvalOperator.ExpressionEvaluator confidenceLevelBlock, DriverContext driverContext) {
+    this.source = source;
+    this.bestEstimateBlock = bestEstimateBlock;
+    this.estimatesBlock = estimatesBlock;
+    this.trialCountBlock = trialCountBlock;
+    this.bucketCountBlock = bucketCountBlock;
+    this.confidenceLevelBlock = confidenceLevelBlock;
+    this.driverContext = driverContext;
+  }
+
+  @Override
+  public Block eval(Page page) {
+    try (IntBlock bestEstimateBlockBlock = (IntBlock) bestEstimateBlock.eval(page)) {
+      try (IntBlock estimatesBlockBlock = (IntBlock) estimatesBlock.eval(page)) {
+        try (IntBlock trialCountBlockBlock = (IntBlock) trialCountBlock.eval(page)) {
+          try (IntBlock bucketCountBlockBlock = (IntBlock) bucketCountBlock.eval(page)) {
+            try (DoubleBlock confidenceLevelBlockBlock = (DoubleBlock) confidenceLevelBlock.eval(page)) {
+              return eval(page.getPositionCount(), bestEstimateBlockBlock, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock, confidenceLevelBlockBlock);
+            }
+          }
+        }
+      }
+    }
+  }
+
+  @Override
+  public long baseRamBytesUsed() {
+    long baseRamBytesUsed = BASE_RAM_BYTES_USED;
+    baseRamBytesUsed += bestEstimateBlock.baseRamBytesUsed();
+    baseRamBytesUsed += estimatesBlock.baseRamBytesUsed();
+    baseRamBytesUsed += trialCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += bucketCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += confidenceLevelBlock.baseRamBytesUsed();
+    return baseRamBytesUsed;
+  }
+
+  public IntBlock eval(int positionCount, IntBlock bestEstimateBlockBlock,
+      IntBlock estimatesBlockBlock, IntBlock trialCountBlockBlock, IntBlock bucketCountBlockBlock,
+      DoubleBlock confidenceLevelBlockBlock) {
+    try(IntBlock.Builder result = driverContext.blockFactory().newIntBlockBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        boolean allBlocksAreNulls = true;
+        if (!bestEstimateBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!estimatesBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!trialCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!bucketCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!confidenceLevelBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (allBlocksAreNulls) {
+          result.appendNull();
+          continue position;
+        }
+        ConfidenceInterval.process(result, p, bestEstimateBlockBlock, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock, confidenceLevelBlockBlock);
+      }
+      return result.build();
+    }
+  }
+
+  @Override
+  public String toString() {
+    return "ConfidenceIntervalIntEvaluator[" + "bestEstimateBlock=" + bestEstimateBlock + ", estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + ", confidenceLevelBlock=" + confidenceLevelBlock + "]";
+  }
+
+  @Override
+  public void close() {
+    Releasables.closeExpectNoException(bestEstimateBlock, estimatesBlock, trialCountBlock, bucketCountBlock, confidenceLevelBlock);
+  }
+
+  private Warnings warnings() {
+    if (warnings == null) {
+      this.warnings = Warnings.createWarnings(
+              driverContext.warningsMode(),
+              source.source().getLineNumber(),
+              source.source().getColumnNumber(),
+              source.text()
+          );
+    }
+    return warnings;
+  }
+
+  static class Factory implements EvalOperator.ExpressionEvaluator.Factory {
+    private final Source source;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bestEstimateBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory estimatesBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory trialCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bucketCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory confidenceLevelBlock;
+
+    public Factory(Source source, EvalOperator.ExpressionEvaluator.Factory bestEstimateBlock,
+        EvalOperator.ExpressionEvaluator.Factory estimatesBlock,
+        EvalOperator.ExpressionEvaluator.Factory trialCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory bucketCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory confidenceLevelBlock) {
+      this.source = source;
+      this.bestEstimateBlock = bestEstimateBlock;
+      this.estimatesBlock = estimatesBlock;
+      this.trialCountBlock = trialCountBlock;
+      this.bucketCountBlock = bucketCountBlock;
+      this.confidenceLevelBlock = confidenceLevelBlock;
+    }
+
+    @Override
+    public ConfidenceIntervalIntEvaluator get(DriverContext context) {
+      return new ConfidenceIntervalIntEvaluator(source, bestEstimateBlock.get(context), estimatesBlock.get(context), trialCountBlock.get(context), bucketCountBlock.get(context), confidenceLevelBlock.get(context), context);
+    }
+
+    @Override
+    public String toString() {
+      return "ConfidenceIntervalIntEvaluator[" + "bestEstimateBlock=" + bestEstimateBlock + ", estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + ", confidenceLevelBlock=" + confidenceLevelBlock + "]";
+    }
+  }
+}
diff --git a/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalLongEvaluator.java b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalLongEvaluator.java
new file mode 100644
index 0000000000000..1a78540dee5e9
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceIntervalLongEvaluator.java
@@ -0,0 +1,174 @@
+// Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+// or more contributor license agreements. Licensed under the Elastic License
+// 2.0; you may not use this file except in compliance with the Elastic License
+// 2.0.
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import java.lang.Override;
+import java.lang.String;
+import org.apache.lucene.util.RamUsageEstimator;
+import org.elasticsearch.compute.data.Block;
+import org.elasticsearch.compute.data.DoubleBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.LongBlock;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.operator.DriverContext;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.compute.operator.Warnings;
+import org.elasticsearch.core.Releasables;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+
+/**
+ * {@link EvalOperator.ExpressionEvaluator} implementation for {@link ConfidenceInterval}.
+ * This class is generated. Edit {@code EvaluatorImplementer} instead.
+ */
+public final class ConfidenceIntervalLongEvaluator implements EvalOperator.ExpressionEvaluator {
+  private static final long BASE_RAM_BYTES_USED = RamUsageEstimator.shallowSizeOfInstance(ConfidenceIntervalLongEvaluator.class);
+
+  private final Source source;
+
+  private final EvalOperator.ExpressionEvaluator bestEstimateBlock;
+
+  private final EvalOperator.ExpressionEvaluator estimatesBlock;
+
+  private final EvalOperator.ExpressionEvaluator trialCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator bucketCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator confidenceLevelBlock;
+
+  private final DriverContext driverContext;
+
+  private Warnings warnings;
+
+  public ConfidenceIntervalLongEvaluator(Source source,
+      EvalOperator.ExpressionEvaluator bestEstimateBlock,
+      EvalOperator.ExpressionEvaluator estimatesBlock,
+      EvalOperator.ExpressionEvaluator trialCountBlock,
+      EvalOperator.ExpressionEvaluator bucketCountBlock,
+      EvalOperator.ExpressionEvaluator confidenceLevelBlock, DriverContext driverContext) {
+    this.source = source;
+    this.bestEstimateBlock = bestEstimateBlock;
+    this.estimatesBlock = estimatesBlock;
+    this.trialCountBlock = trialCountBlock;
+    this.bucketCountBlock = bucketCountBlock;
+    this.confidenceLevelBlock = confidenceLevelBlock;
+    this.driverContext = driverContext;
+  }
+
+  @Override
+  public Block eval(Page page) {
+    try (LongBlock bestEstimateBlockBlock = (LongBlock) bestEstimateBlock.eval(page)) {
+      try (LongBlock estimatesBlockBlock = (LongBlock) estimatesBlock.eval(page)) {
+        try (IntBlock trialCountBlockBlock = (IntBlock) trialCountBlock.eval(page)) {
+          try (IntBlock bucketCountBlockBlock = (IntBlock) bucketCountBlock.eval(page)) {
+            try (DoubleBlock confidenceLevelBlockBlock = (DoubleBlock) confidenceLevelBlock.eval(page)) {
+              return eval(page.getPositionCount(), bestEstimateBlockBlock, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock, confidenceLevelBlockBlock);
+            }
+          }
+        }
+      }
+    }
+  }
+
+  @Override
+  public long baseRamBytesUsed() {
+    long baseRamBytesUsed = BASE_RAM_BYTES_USED;
+    baseRamBytesUsed += bestEstimateBlock.baseRamBytesUsed();
+    baseRamBytesUsed += estimatesBlock.baseRamBytesUsed();
+    baseRamBytesUsed += trialCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += bucketCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += confidenceLevelBlock.baseRamBytesUsed();
+    return baseRamBytesUsed;
+  }
+
+  public LongBlock eval(int positionCount, LongBlock bestEstimateBlockBlock,
+      LongBlock estimatesBlockBlock, IntBlock trialCountBlockBlock, IntBlock bucketCountBlockBlock,
+      DoubleBlock confidenceLevelBlockBlock) {
+    try(LongBlock.Builder result = driverContext.blockFactory().newLongBlockBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        boolean allBlocksAreNulls = true;
+        if (!bestEstimateBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!estimatesBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!trialCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!bucketCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!confidenceLevelBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (allBlocksAreNulls) {
+          result.appendNull();
+          continue position;
+        }
+        ConfidenceInterval.process(result, p, bestEstimateBlockBlock, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock, confidenceLevelBlockBlock);
+      }
+      return result.build();
+    }
+  }
+
+  @Override
+  public String toString() {
+    return "ConfidenceIntervalLongEvaluator[" + "bestEstimateBlock=" + bestEstimateBlock + ", estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + ", confidenceLevelBlock=" + confidenceLevelBlock + "]";
+  }
+
+  @Override
+  public void close() {
+    Releasables.closeExpectNoException(bestEstimateBlock, estimatesBlock, trialCountBlock, bucketCountBlock, confidenceLevelBlock);
+  }
+
+  private Warnings warnings() {
+    if (warnings == null) {
+      this.warnings = Warnings.createWarnings(
+              driverContext.warningsMode(),
+              source.source().getLineNumber(),
+              source.source().getColumnNumber(),
+              source.text()
+          );
+    }
+    return warnings;
+  }
+
+  static class Factory implements EvalOperator.ExpressionEvaluator.Factory {
+    private final Source source;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bestEstimateBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory estimatesBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory trialCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bucketCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory confidenceLevelBlock;
+
+    public Factory(Source source, EvalOperator.ExpressionEvaluator.Factory bestEstimateBlock,
+        EvalOperator.ExpressionEvaluator.Factory estimatesBlock,
+        EvalOperator.ExpressionEvaluator.Factory trialCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory bucketCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory confidenceLevelBlock) {
+      this.source = source;
+      this.bestEstimateBlock = bestEstimateBlock;
+      this.estimatesBlock = estimatesBlock;
+      this.trialCountBlock = trialCountBlock;
+      this.bucketCountBlock = bucketCountBlock;
+      this.confidenceLevelBlock = confidenceLevelBlock;
+    }
+
+    @Override
+    public ConfidenceIntervalLongEvaluator get(DriverContext context) {
+      return new ConfidenceIntervalLongEvaluator(source, bestEstimateBlock.get(context), estimatesBlock.get(context), trialCountBlock.get(context), bucketCountBlock.get(context), confidenceLevelBlock.get(context), context);
+    }
+
+    @Override
+    public String toString() {
+      return "ConfidenceIntervalLongEvaluator[" + "bestEstimateBlock=" + bestEstimateBlock + ", estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + ", confidenceLevelBlock=" + confidenceLevelBlock + "]";
+    }
+  }
+}
diff --git a/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableDoubleEvaluator.java b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableDoubleEvaluator.java
new file mode 100644
index 0000000000000..2447b6d225a2e
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableDoubleEvaluator.java
@@ -0,0 +1,144 @@
+// Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+// or more contributor license agreements. Licensed under the Elastic License
+// 2.0; you may not use this file except in compliance with the Elastic License
+// 2.0.
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import java.lang.Override;
+import java.lang.String;
+import org.apache.lucene.util.RamUsageEstimator;
+import org.elasticsearch.compute.data.Block;
+import org.elasticsearch.compute.data.BooleanBlock;
+import org.elasticsearch.compute.data.DoubleBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.operator.DriverContext;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.compute.operator.Warnings;
+import org.elasticsearch.core.Releasables;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+
+/**
+ * {@link EvalOperator.ExpressionEvaluator} implementation for {@link Reliable}.
+ * This class is generated. Edit {@code EvaluatorImplementer} instead.
+ */
+public final class ReliableDoubleEvaluator implements EvalOperator.ExpressionEvaluator {
+  private static final long BASE_RAM_BYTES_USED = RamUsageEstimator.shallowSizeOfInstance(ReliableDoubleEvaluator.class);
+
+  private final Source source;
+
+  private final EvalOperator.ExpressionEvaluator estimatesBlock;
+
+  private final EvalOperator.ExpressionEvaluator trialCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator bucketCountBlock;
+
+  private final DriverContext driverContext;
+
+  private Warnings warnings;
+
+  public ReliableDoubleEvaluator(Source source, EvalOperator.ExpressionEvaluator estimatesBlock,
+      EvalOperator.ExpressionEvaluator trialCountBlock,
+      EvalOperator.ExpressionEvaluator bucketCountBlock, DriverContext driverContext) {
+    this.source = source;
+    this.estimatesBlock = estimatesBlock;
+    this.trialCountBlock = trialCountBlock;
+    this.bucketCountBlock = bucketCountBlock;
+    this.driverContext = driverContext;
+  }
+
+  @Override
+  public Block eval(Page page) {
+    try (DoubleBlock estimatesBlockBlock = (DoubleBlock) estimatesBlock.eval(page)) {
+      try (IntBlock trialCountBlockBlock = (IntBlock) trialCountBlock.eval(page)) {
+        try (IntBlock bucketCountBlockBlock = (IntBlock) bucketCountBlock.eval(page)) {
+          return eval(page.getPositionCount(), estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock);
+        }
+      }
+    }
+  }
+
+  @Override
+  public long baseRamBytesUsed() {
+    long baseRamBytesUsed = BASE_RAM_BYTES_USED;
+    baseRamBytesUsed += estimatesBlock.baseRamBytesUsed();
+    baseRamBytesUsed += trialCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += bucketCountBlock.baseRamBytesUsed();
+    return baseRamBytesUsed;
+  }
+
+  public BooleanBlock eval(int positionCount, DoubleBlock estimatesBlockBlock,
+      IntBlock trialCountBlockBlock, IntBlock bucketCountBlockBlock) {
+    try(BooleanBlock.Builder result = driverContext.blockFactory().newBooleanBlockBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        boolean allBlocksAreNulls = true;
+        if (!estimatesBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!trialCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!bucketCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (allBlocksAreNulls) {
+          result.appendNull();
+          continue position;
+        }
+        Reliable.process(result, p, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock);
+      }
+      return result.build();
+    }
+  }
+
+  @Override
+  public String toString() {
+    return "ReliableDoubleEvaluator[" + "estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + "]";
+  }
+
+  @Override
+  public void close() {
+    Releasables.closeExpectNoException(estimatesBlock, trialCountBlock, bucketCountBlock);
+  }
+
+  private Warnings warnings() {
+    if (warnings == null) {
+      this.warnings = Warnings.createWarnings(
+              driverContext.warningsMode(),
+              source.source().getLineNumber(),
+              source.source().getColumnNumber(),
+              source.text()
+          );
+    }
+    return warnings;
+  }
+
+  static class Factory implements EvalOperator.ExpressionEvaluator.Factory {
+    private final Source source;
+
+    private final EvalOperator.ExpressionEvaluator.Factory estimatesBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory trialCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bucketCountBlock;
+
+    public Factory(Source source, EvalOperator.ExpressionEvaluator.Factory estimatesBlock,
+        EvalOperator.ExpressionEvaluator.Factory trialCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory bucketCountBlock) {
+      this.source = source;
+      this.estimatesBlock = estimatesBlock;
+      this.trialCountBlock = trialCountBlock;
+      this.bucketCountBlock = bucketCountBlock;
+    }
+
+    @Override
+    public ReliableDoubleEvaluator get(DriverContext context) {
+      return new ReliableDoubleEvaluator(source, estimatesBlock.get(context), trialCountBlock.get(context), bucketCountBlock.get(context), context);
+    }
+
+    @Override
+    public String toString() {
+      return "ReliableDoubleEvaluator[" + "estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + "]";
+    }
+  }
+}
diff --git a/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableIntEvaluator.java b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableIntEvaluator.java
new file mode 100644
index 0000000000000..aa98934050c52
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableIntEvaluator.java
@@ -0,0 +1,143 @@
+// Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+// or more contributor license agreements. Licensed under the Elastic License
+// 2.0; you may not use this file except in compliance with the Elastic License
+// 2.0.
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import java.lang.Override;
+import java.lang.String;
+import org.apache.lucene.util.RamUsageEstimator;
+import org.elasticsearch.compute.data.Block;
+import org.elasticsearch.compute.data.BooleanBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.operator.DriverContext;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.compute.operator.Warnings;
+import org.elasticsearch.core.Releasables;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+
+/**
+ * {@link EvalOperator.ExpressionEvaluator} implementation for {@link Reliable}.
+ * This class is generated. Edit {@code EvaluatorImplementer} instead.
+ */
+public final class ReliableIntEvaluator implements EvalOperator.ExpressionEvaluator {
+  private static final long BASE_RAM_BYTES_USED = RamUsageEstimator.shallowSizeOfInstance(ReliableIntEvaluator.class);
+
+  private final Source source;
+
+  private final EvalOperator.ExpressionEvaluator estimatesBlock;
+
+  private final EvalOperator.ExpressionEvaluator trialCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator bucketCountBlock;
+
+  private final DriverContext driverContext;
+
+  private Warnings warnings;
+
+  public ReliableIntEvaluator(Source source, EvalOperator.ExpressionEvaluator estimatesBlock,
+      EvalOperator.ExpressionEvaluator trialCountBlock,
+      EvalOperator.ExpressionEvaluator bucketCountBlock, DriverContext driverContext) {
+    this.source = source;
+    this.estimatesBlock = estimatesBlock;
+    this.trialCountBlock = trialCountBlock;
+    this.bucketCountBlock = bucketCountBlock;
+    this.driverContext = driverContext;
+  }
+
+  @Override
+  public Block eval(Page page) {
+    try (IntBlock estimatesBlockBlock = (IntBlock) estimatesBlock.eval(page)) {
+      try (IntBlock trialCountBlockBlock = (IntBlock) trialCountBlock.eval(page)) {
+        try (IntBlock bucketCountBlockBlock = (IntBlock) bucketCountBlock.eval(page)) {
+          return eval(page.getPositionCount(), estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock);
+        }
+      }
+    }
+  }
+
+  @Override
+  public long baseRamBytesUsed() {
+    long baseRamBytesUsed = BASE_RAM_BYTES_USED;
+    baseRamBytesUsed += estimatesBlock.baseRamBytesUsed();
+    baseRamBytesUsed += trialCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += bucketCountBlock.baseRamBytesUsed();
+    return baseRamBytesUsed;
+  }
+
+  public BooleanBlock eval(int positionCount, IntBlock estimatesBlockBlock,
+      IntBlock trialCountBlockBlock, IntBlock bucketCountBlockBlock) {
+    try(BooleanBlock.Builder result = driverContext.blockFactory().newBooleanBlockBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        boolean allBlocksAreNulls = true;
+        if (!estimatesBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!trialCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!bucketCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (allBlocksAreNulls) {
+          result.appendNull();
+          continue position;
+        }
+        Reliable.process(result, p, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock);
+      }
+      return result.build();
+    }
+  }
+
+  @Override
+  public String toString() {
+    return "ReliableIntEvaluator[" + "estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + "]";
+  }
+
+  @Override
+  public void close() {
+    Releasables.closeExpectNoException(estimatesBlock, trialCountBlock, bucketCountBlock);
+  }
+
+  private Warnings warnings() {
+    if (warnings == null) {
+      this.warnings = Warnings.createWarnings(
+              driverContext.warningsMode(),
+              source.source().getLineNumber(),
+              source.source().getColumnNumber(),
+              source.text()
+          );
+    }
+    return warnings;
+  }
+
+  static class Factory implements EvalOperator.ExpressionEvaluator.Factory {
+    private final Source source;
+
+    private final EvalOperator.ExpressionEvaluator.Factory estimatesBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory trialCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bucketCountBlock;
+
+    public Factory(Source source, EvalOperator.ExpressionEvaluator.Factory estimatesBlock,
+        EvalOperator.ExpressionEvaluator.Factory trialCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory bucketCountBlock) {
+      this.source = source;
+      this.estimatesBlock = estimatesBlock;
+      this.trialCountBlock = trialCountBlock;
+      this.bucketCountBlock = bucketCountBlock;
+    }
+
+    @Override
+    public ReliableIntEvaluator get(DriverContext context) {
+      return new ReliableIntEvaluator(source, estimatesBlock.get(context), trialCountBlock.get(context), bucketCountBlock.get(context), context);
+    }
+
+    @Override
+    public String toString() {
+      return "ReliableIntEvaluator[" + "estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + "]";
+    }
+  }
+}
diff --git a/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableLongEvaluator.java b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableLongEvaluator.java
new file mode 100644
index 0000000000000..d07f67bbcc863
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ReliableLongEvaluator.java
@@ -0,0 +1,144 @@
+// Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+// or more contributor license agreements. Licensed under the Elastic License
+// 2.0; you may not use this file except in compliance with the Elastic License
+// 2.0.
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import java.lang.Override;
+import java.lang.String;
+import org.apache.lucene.util.RamUsageEstimator;
+import org.elasticsearch.compute.data.Block;
+import org.elasticsearch.compute.data.BooleanBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.LongBlock;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.operator.DriverContext;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.compute.operator.Warnings;
+import org.elasticsearch.core.Releasables;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+
+/**
+ * {@link EvalOperator.ExpressionEvaluator} implementation for {@link Reliable}.
+ * This class is generated. Edit {@code EvaluatorImplementer} instead.
+ */
+public final class ReliableLongEvaluator implements EvalOperator.ExpressionEvaluator {
+  private static final long BASE_RAM_BYTES_USED = RamUsageEstimator.shallowSizeOfInstance(ReliableLongEvaluator.class);
+
+  private final Source source;
+
+  private final EvalOperator.ExpressionEvaluator estimatesBlock;
+
+  private final EvalOperator.ExpressionEvaluator trialCountBlock;
+
+  private final EvalOperator.ExpressionEvaluator bucketCountBlock;
+
+  private final DriverContext driverContext;
+
+  private Warnings warnings;
+
+  public ReliableLongEvaluator(Source source, EvalOperator.ExpressionEvaluator estimatesBlock,
+      EvalOperator.ExpressionEvaluator trialCountBlock,
+      EvalOperator.ExpressionEvaluator bucketCountBlock, DriverContext driverContext) {
+    this.source = source;
+    this.estimatesBlock = estimatesBlock;
+    this.trialCountBlock = trialCountBlock;
+    this.bucketCountBlock = bucketCountBlock;
+    this.driverContext = driverContext;
+  }
+
+  @Override
+  public Block eval(Page page) {
+    try (LongBlock estimatesBlockBlock = (LongBlock) estimatesBlock.eval(page)) {
+      try (IntBlock trialCountBlockBlock = (IntBlock) trialCountBlock.eval(page)) {
+        try (IntBlock bucketCountBlockBlock = (IntBlock) bucketCountBlock.eval(page)) {
+          return eval(page.getPositionCount(), estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock);
+        }
+      }
+    }
+  }
+
+  @Override
+  public long baseRamBytesUsed() {
+    long baseRamBytesUsed = BASE_RAM_BYTES_USED;
+    baseRamBytesUsed += estimatesBlock.baseRamBytesUsed();
+    baseRamBytesUsed += trialCountBlock.baseRamBytesUsed();
+    baseRamBytesUsed += bucketCountBlock.baseRamBytesUsed();
+    return baseRamBytesUsed;
+  }
+
+  public BooleanBlock eval(int positionCount, LongBlock estimatesBlockBlock,
+      IntBlock trialCountBlockBlock, IntBlock bucketCountBlockBlock) {
+    try(BooleanBlock.Builder result = driverContext.blockFactory().newBooleanBlockBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        boolean allBlocksAreNulls = true;
+        if (!estimatesBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!trialCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (!bucketCountBlockBlock.isNull(p)) {
+          allBlocksAreNulls = false;
+        }
+        if (allBlocksAreNulls) {
+          result.appendNull();
+          continue position;
+        }
+        Reliable.process(result, p, estimatesBlockBlock, trialCountBlockBlock, bucketCountBlockBlock);
+      }
+      return result.build();
+    }
+  }
+
+  @Override
+  public String toString() {
+    return "ReliableLongEvaluator[" + "estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + "]";
+  }
+
+  @Override
+  public void close() {
+    Releasables.closeExpectNoException(estimatesBlock, trialCountBlock, bucketCountBlock);
+  }
+
+  private Warnings warnings() {
+    if (warnings == null) {
+      this.warnings = Warnings.createWarnings(
+              driverContext.warningsMode(),
+              source.source().getLineNumber(),
+              source.source().getColumnNumber(),
+              source.text()
+          );
+    }
+    return warnings;
+  }
+
+  static class Factory implements EvalOperator.ExpressionEvaluator.Factory {
+    private final Source source;
+
+    private final EvalOperator.ExpressionEvaluator.Factory estimatesBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory trialCountBlock;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bucketCountBlock;
+
+    public Factory(Source source, EvalOperator.ExpressionEvaluator.Factory estimatesBlock,
+        EvalOperator.ExpressionEvaluator.Factory trialCountBlock,
+        EvalOperator.ExpressionEvaluator.Factory bucketCountBlock) {
+      this.source = source;
+      this.estimatesBlock = estimatesBlock;
+      this.trialCountBlock = trialCountBlock;
+      this.bucketCountBlock = bucketCountBlock;
+    }
+
+    @Override
+    public ReliableLongEvaluator get(DriverContext context) {
+      return new ReliableLongEvaluator(source, estimatesBlock.get(context), trialCountBlock.get(context), bucketCountBlock.get(context), context);
+    }
+
+    @Override
+    public String toString() {
+      return "ReliableLongEvaluator[" + "estimatesBlock=" + estimatesBlock + ", trialCountBlock=" + trialCountBlock + ", bucketCountBlock=" + bucketCountBlock + "]";
+    }
+  }
+}
diff --git a/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/random/RandomEvaluator.java b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/random/RandomEvaluator.java
new file mode 100644
index 0000000000000..2a76975e89dd5
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/generated/org/elasticsearch/xpack/esql/expression/function/scalar/random/RandomEvaluator.java
@@ -0,0 +1,134 @@
+// Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+// or more contributor license agreements. Licensed under the Elastic License
+// 2.0; you may not use this file except in compliance with the Elastic License
+// 2.0.
+package org.elasticsearch.xpack.esql.expression.function.scalar.random;
+
+import java.lang.IllegalArgumentException;
+import java.lang.Override;
+import java.lang.String;
+import org.apache.lucene.util.RamUsageEstimator;
+import org.elasticsearch.compute.data.Block;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.IntVector;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.operator.DriverContext;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.compute.operator.Warnings;
+import org.elasticsearch.core.Releasables;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+
+/**
+ * {@link EvalOperator.ExpressionEvaluator} implementation for {@link Random}.
+ * This class is generated. Edit {@code EvaluatorImplementer} instead.
+ */
+public final class RandomEvaluator implements EvalOperator.ExpressionEvaluator {
+  private static final long BASE_RAM_BYTES_USED = RamUsageEstimator.shallowSizeOfInstance(RandomEvaluator.class);
+
+  private final Source source;
+
+  private final EvalOperator.ExpressionEvaluator bound;
+
+  private final DriverContext driverContext;
+
+  private Warnings warnings;
+
+  public RandomEvaluator(Source source, EvalOperator.ExpressionEvaluator bound,
+      DriverContext driverContext) {
+    this.source = source;
+    this.bound = bound;
+    this.driverContext = driverContext;
+  }
+
+  @Override
+  public Block eval(Page page) {
+    try (IntBlock boundBlock = (IntBlock) bound.eval(page)) {
+      IntVector boundVector = boundBlock.asVector();
+      if (boundVector == null) {
+        return eval(page.getPositionCount(), boundBlock);
+      }
+      return eval(page.getPositionCount(), boundVector).asBlock();
+    }
+  }
+
+  @Override
+  public long baseRamBytesUsed() {
+    long baseRamBytesUsed = BASE_RAM_BYTES_USED;
+    baseRamBytesUsed += bound.baseRamBytesUsed();
+    return baseRamBytesUsed;
+  }
+
+  public IntBlock eval(int positionCount, IntBlock boundBlock) {
+    try(IntBlock.Builder result = driverContext.blockFactory().newIntBlockBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        switch (boundBlock.getValueCount(p)) {
+          case 0:
+              result.appendNull();
+              continue position;
+          case 1:
+              break;
+          default:
+              warnings().registerException(new IllegalArgumentException("single-value function encountered multi-value"));
+              result.appendNull();
+              continue position;
+        }
+        int bound = boundBlock.getInt(boundBlock.getFirstValueIndex(p));
+        result.appendInt(Random.process(bound));
+      }
+      return result.build();
+    }
+  }
+
+  public IntVector eval(int positionCount, IntVector boundVector) {
+    try(IntVector.FixedBuilder result = driverContext.blockFactory().newIntVectorFixedBuilder(positionCount)) {
+      position: for (int p = 0; p < positionCount; p++) {
+        int bound = boundVector.getInt(p);
+        result.appendInt(p, Random.process(bound));
+      }
+      return result.build();
+    }
+  }
+
+  @Override
+  public String toString() {
+    return "RandomEvaluator[" + "bound=" + bound + "]";
+  }
+
+  @Override
+  public void close() {
+    Releasables.closeExpectNoException(bound);
+  }
+
+  private Warnings warnings() {
+    if (warnings == null) {
+      this.warnings = Warnings.createWarnings(
+              driverContext.warningsMode(),
+              source.source().getLineNumber(),
+              source.source().getColumnNumber(),
+              source.text()
+          );
+    }
+    return warnings;
+  }
+
+  static class Factory implements EvalOperator.ExpressionEvaluator.Factory {
+    private final Source source;
+
+    private final EvalOperator.ExpressionEvaluator.Factory bound;
+
+    public Factory(Source source, EvalOperator.ExpressionEvaluator.Factory bound) {
+      this.source = source;
+      this.bound = bound;
+    }
+
+    @Override
+    public RandomEvaluator get(DriverContext context) {
+      return new RandomEvaluator(source, bound.get(context), context);
+    }
+
+    @Override
+    public String toString() {
+      return "RandomEvaluator[" + "bound=" + bound + "]";
+    }
+  }
+}
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlCapabilities.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlCapabilities.java
index 88738d9084f8a..dc7bdf2d54291 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlCapabilities.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlCapabilities.java
@@ -1493,7 +1493,12 @@ public enum Cap {
         /**
          * Support for dots in FUSE attributes
          */
-        DOTS_IN_FUSE;
+        DOTS_IN_FUSE,
+
+        /**
+         * Support query approximation.
+         */
+        APPROXIMATE;
 
         private final boolean enabled;
 
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequest.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequest.java
index 1f90362cda38d..e6d9013187f7e 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequest.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequest.java
@@ -49,6 +49,8 @@ public class EsqlQueryRequest extends org.elasticsearch.xpack.core.esql.action.E
     private Boolean includeExecutionMetadata;
     private Locale locale;
     private QueryBuilder filter;
+    // TODO: discuss how to wire the approximation functionality in the API
+    private boolean approximate;
     private QueryPragmas pragmas = new QueryPragmas(Settings.EMPTY);
     private QueryParams params = new QueryParams();
     private TimeValue waitForCompletionTimeout = DEFAULT_WAIT_FOR_COMPLETION;
@@ -176,6 +178,16 @@ public QueryBuilder filter() {
         return filter;
     }
 
+    public EsqlQueryRequest approximate(boolean approximate) {
+        this.approximate = approximate;
+        return this;
+    }
+
+    @Override
+    public boolean approximate() {
+        return approximate;
+    }
+
     public EsqlQueryRequest pragmas(QueryPragmas pragmas) {
         this.pragmas = pragmas;
         return this;
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequestBuilder.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequestBuilder.java
index 09f01b3c60a20..08e4993f8b37e 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequestBuilder.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/EsqlQueryRequestBuilder.java
@@ -46,6 +46,12 @@ public EsqlQueryRequestBuilder filter(QueryBuilder filter) {
         return this;
     }
 
+    @Override
+    public EsqlQueryRequestBuilder approximate(boolean approximate) {
+        request.approximate(approximate);
+        return this;
+    }
+
     public EsqlQueryRequestBuilder pragmas(QueryPragmas pragmas) {
         request.pragmas(pragmas);
         return this;
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/RequestXContent.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/RequestXContent.java
index fea94c27963b4..8cba084770b3f 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/RequestXContent.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/action/RequestXContent.java
@@ -74,6 +74,7 @@ String fields() {
     static final ParseField QUERY_FIELD = new ParseField("query");
     private static final ParseField COLUMNAR_FIELD = new ParseField("columnar");
     private static final ParseField FILTER_FIELD = new ParseField("filter");
+    private static final ParseField APPROMIXATE_FIELD = new ParseField("approximate");
     static final ParseField PRAGMA_FIELD = new ParseField("pragma");
     private static final ParseField PARAMS_FIELD = new ParseField("params");
     private static final ParseField LOCALE_FIELD = new ParseField("locale");
@@ -104,6 +105,7 @@ private static void objectParserCommon(ObjectParser<EsqlQueryRequest, ?> parser)
         parser.declareString(EsqlQueryRequest::query, QUERY_FIELD);
         parser.declareBoolean(EsqlQueryRequest::columnar, COLUMNAR_FIELD);
         parser.declareObject(EsqlQueryRequest::filter, (p, c) -> AbstractQueryBuilder.parseTopLevelQuery(p), FILTER_FIELD);
+        parser.declareBoolean(EsqlQueryRequest::approximate, APPROMIXATE_FIELD);
         parser.declareBoolean(EsqlQueryRequest::acceptedPragmaRisks, ACCEPT_PRAGMA_RISKS);
         parser.declareBoolean(EsqlQueryRequest::includeCCSMetadata, INCLUDE_CCS_METADATA_FIELD);
         parser.declareBoolean(EsqlQueryRequest::includeExecutionMetadata, INCLUDE_EXECUTION_METADATA_FIELD);
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/approximate/Approximate.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/approximate/Approximate.java
new file mode 100644
index 0000000000000..4db6649e56e14
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/approximate/Approximate.java
@@ -0,0 +1,679 @@
+/*
+ * Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+ * or more contributor license agreements. Licensed under the Elastic License
+ * 2.0; you may not use this file except in compliance with the Elastic License
+ * 2.0.
+ */
+
+package org.elasticsearch.xpack.esql.approximate;
+
+import org.apache.logging.log4j.LogManager;
+import org.apache.logging.log4j.Logger;
+import org.elasticsearch.action.ActionListener;
+import org.elasticsearch.compute.data.LongBlock;
+import org.elasticsearch.xpack.esql.VerificationException;
+import org.elasticsearch.xpack.esql.common.Failure;
+import org.elasticsearch.xpack.esql.core.expression.Alias;
+import org.elasticsearch.xpack.esql.core.expression.Attribute;
+import org.elasticsearch.xpack.esql.core.expression.Expression;
+import org.elasticsearch.xpack.esql.core.expression.Literal;
+import org.elasticsearch.xpack.esql.core.expression.NameId;
+import org.elasticsearch.xpack.esql.core.expression.NamedExpression;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+import org.elasticsearch.xpack.esql.core.util.Holder;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.AggregateFunction;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.Avg;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.Count;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.Median;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.MedianAbsoluteDeviation;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.Percentile;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.StdDev;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.Sum;
+import org.elasticsearch.xpack.esql.expression.function.aggregate.WeightedAvg;
+import org.elasticsearch.xpack.esql.expression.function.scalar.EsqlScalarFunction;
+import org.elasticsearch.xpack.esql.expression.function.scalar.approximate.ConfidenceInterval;
+import org.elasticsearch.xpack.esql.expression.function.scalar.approximate.Reliable;
+import org.elasticsearch.xpack.esql.expression.function.scalar.convert.ToLong;
+import org.elasticsearch.xpack.esql.expression.function.scalar.multivalue.MvAppend;
+import org.elasticsearch.xpack.esql.expression.function.scalar.multivalue.MvSlice;
+import org.elasticsearch.xpack.esql.expression.function.scalar.random.Random;
+import org.elasticsearch.xpack.esql.expression.predicate.logical.And;
+import org.elasticsearch.xpack.esql.expression.predicate.nulls.IsNotNull;
+import org.elasticsearch.xpack.esql.expression.predicate.operator.arithmetic.Div;
+import org.elasticsearch.xpack.esql.expression.predicate.operator.comparison.Equals;
+import org.elasticsearch.xpack.esql.expression.predicate.operator.comparison.NotEquals;
+import org.elasticsearch.xpack.esql.plan.logical.Aggregate;
+import org.elasticsearch.xpack.esql.plan.logical.ChangePoint;
+import org.elasticsearch.xpack.esql.plan.logical.Dissect;
+import org.elasticsearch.xpack.esql.plan.logical.Drop;
+import org.elasticsearch.xpack.esql.plan.logical.Enrich;
+import org.elasticsearch.xpack.esql.plan.logical.EsRelation;
+import org.elasticsearch.xpack.esql.plan.logical.Eval;
+import org.elasticsearch.xpack.esql.plan.logical.Filter;
+import org.elasticsearch.xpack.esql.plan.logical.Grok;
+import org.elasticsearch.xpack.esql.plan.logical.Insist;
+import org.elasticsearch.xpack.esql.plan.logical.Keep;
+import org.elasticsearch.xpack.esql.plan.logical.LeafPlan;
+import org.elasticsearch.xpack.esql.plan.logical.Limit;
+import org.elasticsearch.xpack.esql.plan.logical.LogicalPlan;
+import org.elasticsearch.xpack.esql.plan.logical.MvExpand;
+import org.elasticsearch.xpack.esql.plan.logical.OrderBy;
+import org.elasticsearch.xpack.esql.plan.logical.Project;
+import org.elasticsearch.xpack.esql.plan.logical.RegexExtract;
+import org.elasticsearch.xpack.esql.plan.logical.Rename;
+import org.elasticsearch.xpack.esql.plan.logical.Row;
+import org.elasticsearch.xpack.esql.plan.logical.Sample;
+import org.elasticsearch.xpack.esql.plan.logical.TopN;
+import org.elasticsearch.xpack.esql.plan.logical.inference.Completion;
+import org.elasticsearch.xpack.esql.plan.logical.inference.Rerank;
+import org.elasticsearch.xpack.esql.plan.logical.local.EsqlProject;
+import org.elasticsearch.xpack.esql.session.Result;
+
+import java.util.ArrayList;
+import java.util.HashMap;
+import java.util.List;
+import java.util.Locale;
+import java.util.Map;
+import java.util.Set;
+import java.util.stream.Collectors;
+
+/**
+ * This class computes approximate results for certain classes of ES|QL queries.
+ * Approximate results are usually much faster to compute than exact results.
+ * <p>
+ * A query is currently suitable for approximation if:
+ * <ul>
+ *   <li> it contains exactly one {@code STATS} command
+ *   <li> the other processing commands are from the supported set
+ *        ({@link Approximate#SUPPORTED_COMMANDS}); this set contains almost all
+ *        unary commands, but most notably not {@code FORK} or {@code JOIN}.
+ *   <li> the aggregate functions are from the supported set
+ *        ({@link Approximate#SUPPORTED_SINGLE_VALUED_AGGS} and
+ *         {@link Approximate#SUPPORTED_MULTIVALUED_AGGS})
+ * </ul>
+ * Some of these restrictions may be lifted in the future.
+ * <p>
+ *
+ * When these conditions are met, the query is replaced by an approximation query
+ * that samples documents before the STATS, and extrapolates the aggregate
+ * functions if needed. This new logical plan is generated by
+ * {@link Approximate#approximatePlan}.
+ * <p>
+ * In addition to approximate results, confidence intervals are also computed.
+ * This is done by dividing the sampled rows {@link Approximate#TRIAL_COUNT}
+ * times into {@link Approximate#BUCKET_COUNT} buckets, computing the aggregate
+ * functions for each bucket, and using these sampled values to compute
+ * confidence intervals.
+ * <p>
+ * To obtain an appropriate sample probability, first a target number of rows
+ * is set. For now this is a fixed number ({@link Approximate#SAMPLE_ROW_COUNT}).
+ * Next, the total number of rows in the source index is counted via the plan
+ * {@link Approximate#sourceCountPlan}. This plan should execute fast. When
+ * there are no commands that can change the number of rows, the sample
+ * probability can be directly computed as a ratio of the target number of rows
+ * and this total number.
+ * <p>
+ * In the presence of commands that can change to number of rows, another step
+ * is needed. The initial sample probability is set to the ratio above and the
+ * number of rows is sampled with the plan {@link Approximate#countPlan}. As
+ * long as the sampled number of rows is smaller than intended, the probability
+ * is scaled up until a good probability is reached. This final probability is
+ * used for approximating the original plan.
+ */
+public class Approximate {
+
+    public interface LogicalPlanRunner {
+        void run(LogicalPlan plan, ActionListener<Result> listener);
+    }
+
+    /**
+     * These processing commands are supported.
+     */
+    private static final Set<Class<? extends LogicalPlan>> SUPPORTED_COMMANDS = Set.of(
+        Aggregate.class,
+        ChangePoint.class,
+        Completion.class,
+        Dissect.class,
+        Drop.class,
+        Enrich.class,
+        EsqlProject.class,
+        EsRelation.class,
+        Eval.class,
+        Filter.class,
+        Grok.class,
+        Insist.class,
+        Keep.class,
+        Limit.class,
+        MvExpand.class,
+        OrderBy.class,
+        Project.class,
+        RegexExtract.class,
+        Rename.class,
+        Rerank.class,
+        Row.class,
+        Sample.class,
+        TopN.class
+    );
+
+    /**
+     * These commands preserve all rows, making it easy to predict the number of output rows.
+     */
+    private static final Set<Class<? extends LogicalPlan>> ROW_PRESERVING_COMMANDS = Set.of(
+        ChangePoint.class,
+        Dissect.class,
+        Drop.class,
+        Enrich.class,
+        EsqlProject.class,
+        Eval.class,
+        Grok.class,
+        Insist.class,
+        Keep.class,
+        OrderBy.class,
+        Project.class,
+        Rename.class
+    );
+
+    /**
+     * These aggregate functions behave well with random sampling, in the sense
+     * that they converge to the true value as the sample size increases.
+     */
+    private static final Set<Class<? extends AggregateFunction>> SUPPORTED_SINGLE_VALUED_AGGS = Set.of(
+        Avg.class,
+        Count.class,
+        Median.class,
+        MedianAbsoluteDeviation.class,
+        Percentile.class,
+        StdDev.class,
+        Sum.class,
+        WeightedAvg.class
+    );
+
+    /**
+     * These aggregate functions need to be corrected for random sampling, by
+     * scaling up the sampled value by the inverse of the sampling probability.
+     * Other aggregate functions do not need any correction.
+     */
+    private static final Set<Class<? extends AggregateFunction>> SAMPLE_CORRECTED_AGGS = Set.of(Count.class, Sum.class);
+
+    /**
+     * These multivalued aggregate functions work well with random sampling.
+     */
+    private static final Set<Class<? extends AggregateFunction>> SUPPORTED_MULTIVALUED_AGGS = Set.of(
+        org.elasticsearch.xpack.esql.expression.function.aggregate.Sample.class
+    );
+
+    /**
+     * These scalar functions produce multivalued output. When they are applied
+     * to an approximated field, confidence intervals make no sense anymore and
+     * are dropped.
+     */
+    private static final Set<Class<? extends EsqlScalarFunction>> MULTIVALUED_OUTPUT_FUNCTIONS = Set.of(MvAppend.class);
+
+    // TODO: find a good default value, or alternative ways of setting it
+    private static final int SAMPLE_ROW_COUNT = 100000;
+
+    /**
+     * The number of times (trials) the sampled rows are divided into buckets.
+     */
+    private static final int TRIAL_COUNT = 3;
+
+    /**
+     * The number of buckets to use for computing confidence intervals.
+     */
+    private static final int BUCKET_COUNT = 16;
+
+    private static final Logger logger = LogManager.getLogger(Approximate.class);
+
+    private final LogicalPlan logicalPlan;
+    private final boolean preservesRows;
+
+    public Approximate(LogicalPlan logicalPlan) {
+        this.logicalPlan = logicalPlan;
+        this.preservesRows = verifyPlan();
+    }
+
+    /**
+     * Computes approximate results for the logical plan.
+     */
+    public void approximate(LogicalPlanRunner runner, ActionListener<Result> listener) {
+        runner.run(sourceCountPlan(), sourceCountListener(runner, listener));
+    }
+
+    /**
+     * Verifies that a plan is suitable for approximation.
+     *
+     * @return whether the part of the query until the STATS command preserves all rows
+     * @throws VerificationException if the plan is not suitable for approximation
+     */
+    private boolean verifyPlan() throws VerificationException {
+        if (logicalPlan.preOptimized() == false) {
+            throw new IllegalStateException("Expected pre-optimized plan");
+        }
+        // The plan must contain a STATS command.
+        if (logicalPlan.anyMatch(plan -> plan instanceof Aggregate) == false) {
+            throw new VerificationException(
+                List.of(Failure.fail(logicalPlan.collectLeaves().getFirst(), "query without [STATS] cannot be approximated"))
+            );
+        }
+        // Verify that all commands are supported.
+        logicalPlan.forEachUp(plan -> {
+            if (SUPPORTED_COMMANDS.contains(plan.getClass()) == false) {
+                // TODO: better way of obtaining the command name
+                throw new VerificationException(
+                    List.of(Failure.fail(plan, "query with [" + plan.nodeName().toUpperCase(Locale.ROOT) + "] cannot be approximated"))
+                );
+            }
+        });
+
+        Holder<Boolean> encounteredStats = new Holder<>(false);
+        Holder<Boolean> preservesRows = new Holder<>(true);
+        logicalPlan.transformUp(plan -> {
+            if (encounteredStats.get() == false) {
+                if (plan instanceof Aggregate) {
+                    // Verify that the aggregate functions are supported.
+                    encounteredStats.set(true);
+                    plan.transformExpressionsOnly(AggregateFunction.class, aggFn -> {
+                        if (SUPPORTED_SINGLE_VALUED_AGGS.contains(aggFn.getClass()) == false
+                            && SUPPORTED_MULTIVALUED_AGGS.contains(aggFn.getClass()) == false) {
+                            // TODO: better way of obtaining the agg name
+                            throw new VerificationException(
+                                List.of(
+                                    Failure.fail(
+                                        aggFn,
+                                        "aggregation function [" + aggFn.nodeName().toUpperCase(Locale.ROOT) + "] cannot be approximated"
+                                    )
+                                )
+                            );
+                        }
+                        return aggFn;
+                    });
+                } else if (plan instanceof LeafPlan == false && ROW_PRESERVING_COMMANDS.contains(plan.getClass()) == false) {
+                    // Keep track of whether the plan until the STATS preserves all rows.
+                    preservesRows.set(false);
+                }
+            } else {
+                // Multiple STATS commands are not supported.
+                if (plan instanceof Aggregate) {
+                    throw new VerificationException(List.of(Failure.fail(plan, "query with multiple [STATS] cannot be approximated")));
+                }
+            }
+            return plan;
+        });
+
+        return preservesRows.get();
+    }
+
+    /**
+     * Plan that counts the number of rows in the source index.
+     * This is the ES|QL query:
+     * <pre>
+     * {@code FROM index | STATS COUNT(*)}
+     * </pre>
+     */
+    private LogicalPlan sourceCountPlan() {
+        LogicalPlan sourceCountPlan = logicalPlan.transformUp(plan -> {
+            if (plan instanceof LeafPlan) {
+                // Append the leaf plan by a STATS COUNT(*).
+                plan = new Aggregate(
+                    Source.EMPTY,
+                    plan,
+                    List.of(),
+                    List.of(new Alias(Source.EMPTY, "$count", new Count(Source.EMPTY, Literal.keyword(Source.EMPTY, "*"))))
+                );
+            } else {
+                // Strip everything after the leaf.
+                plan = plan.children().getFirst();
+            }
+            return plan;
+        });
+
+        sourceCountPlan.setPreOptimized();
+        return sourceCountPlan;
+    }
+
+    /**
+     * Receives the total number of rows, and runs either the
+     * {@link Approximate#approximatePlan} or {@link Approximate#countPlan}
+     * depending on whether the original query preserves all rows or not.
+     */
+    private ActionListener<Result> sourceCountListener(LogicalPlanRunner runner, ActionListener<Result> listener) {
+        return listener.delegateFailureAndWrap((countListener, countResult) -> {
+            long rowCount = rowCount(countResult);
+            logger.debug("sourceCountPlan result: {} rows", rowCount);
+            double sampleProbability = rowCount <= SAMPLE_ROW_COUNT ? 1.0 : (double) SAMPLE_ROW_COUNT / rowCount;
+            if (preservesRows || sampleProbability == 1.0) {
+                runner.run(approximatePlan(sampleProbability), listener);
+            } else {
+                runner.run(countPlan(sampleProbability), countListener(runner, sampleProbability, listener));
+            }
+        });
+    }
+
+    /**
+     * Plan that counts a sampled number of rows reaching the STATS function.
+     * This is the ES|QL query
+     * <pre>
+     * {@code FROM index | SAMPLE prob | (...) | STATS COUNT(*)}
+     * </pre>
+     * where {@code prob} is the given sample probability.
+     */
+    private LogicalPlan countPlan(double sampleProbability) {
+        Holder<Boolean> encounteredStats = new Holder<>(false);
+        LogicalPlan countPlan = logicalPlan.transformUp(plan -> {
+            if (plan instanceof LeafPlan) {
+                // The leaf plan should be appended by a SAMPLE.
+                plan = new Sample(Source.EMPTY, Literal.fromDouble(Source.EMPTY, sampleProbability), plan);
+            } else if (encounteredStats.get() == false) {
+                if (plan instanceof Aggregate aggregate) {
+                    // The STATS function should be replaced by a STATS COUNT(*).
+                    encounteredStats.set(true);
+                    plan = new Aggregate(
+                        Source.EMPTY,
+                        aggregate.child(),
+                        List.of(),
+                        List.of(new Alias(Source.EMPTY, "$count", new Count(Source.EMPTY, Literal.keyword(Source.EMPTY, "*"))))
+                    );
+                }
+            } else {
+                // Strip everything after the STATS command.
+                plan = plan.children().getFirst();
+            }
+            return plan;
+        });
+
+        countPlan.setPreOptimized();
+        return countPlan;
+    }
+
+    /**
+     * Receives the sampled number of rows reaching the STATS function.
+     * Runs either the {@link Approximate#approximatePlan} or a next iteration
+     * {@link Approximate#countPlan} depending on whether the current count is
+     * sufficient.
+     */
+    private ActionListener<Result> countListener(LogicalPlanRunner runner, double sampleProbability, ActionListener<Result> listener) {
+        return listener.delegateFailureAndWrap((countListener, countResult) -> {
+            long rowCount = rowCount(countResult);
+            logger.debug("countPlan result (p={}): {} rows", sampleProbability, rowCount);
+            double newSampleProbability = sampleProbability * SAMPLE_ROW_COUNT / Math.max(1, rowCount);
+            if (rowCount <= SAMPLE_ROW_COUNT / 2 && newSampleProbability < 1.0) {
+                runner.run(countPlan(newSampleProbability), countListener(runner, newSampleProbability, listener));
+            } else {
+                runner.run(approximatePlan(newSampleProbability), listener);
+            }
+        });
+    }
+
+    /**
+     * Returns the row count in the result and closes the result.
+     */
+    private long rowCount(Result countResult) {
+        long rowCount = ((LongBlock) (countResult.pages().getFirst().getBlock(0))).getLong(0);
+        countResult.pages().getFirst().close();
+        return rowCount;
+    }
+
+    /**
+     * Returns a plan that approximates the original plan and computes confidence intervals.
+     * This approximation query consists of the following:
+     * <ul>
+     *     <li> Source command
+     *     <li> {@code SAMPLE} with the provided sample probability
+     *     <li> All commands before the {@code STATS} command
+     *     <li> {@code EVAL} adding a new column with a random bucket ID for each trial
+     *     <li> {@code STATS} command with:
+     *          <ul>
+     *              <li> Each aggregate function replaced by a sample-corrected version (if needed)
+     *              <li> {@link Approximate#TRIAL_COUNT} * {@link Approximate#BUCKET_COUNT} additional columns
+     *                   with a sampled values for each aggregate function, sample-corrected (if needed)
+     *          </ul>
+     *     <li> A filter to remove rows with empty buckets
+     *     <li> All commands after the {@code STATS} command, modified to also process
+     *          the additional bucket columns where possible
+     *     <li> {@code EVAL} to compute confidence intervals for all fields with buckets
+     *     <li> {@code PROJECT} to drop all bucket columns
+     * </ul>
+     *
+     * As an example, the simple query:
+     * <pre>
+     *     {@code
+     *         FROM index
+     *             | EVAL x = 2*x
+     *             | STATS s = SUM(x) BY group
+     *             | EVAL s2 = s*s
+     *     }
+     * </pre>
+     * is rewritten to (prob=sampleProbability, T=trialCount, B=bucketCount):
+     * <pre>
+     *     {@code
+     *         FROM index
+     *             | SAMPLE prob
+     *             | EVAL x = 2*x
+     *             | EVAL bucketId = MV_APPEND(RANDOM(B), ... , RANDOM(B))  // T times
+     *             | STATS s = SUM(x) / prob,
+     *                     `s$0` = SUM(x) / (prob/B)) WHERE MV_SLICE(bucketId, 0, 0) == 0
+     *                     ...,
+     *                     `s$T*B-1` = SUM(x) / (prob/B) WHERE MV_SLICE(bucketId, T-1, T-1) == B-1
+     *               BY group
+     *             | WHERE `s$0` IS NOT NULL AND ... AND `s$T*B-1` IS NOT NULL
+     *             | EVAL t = s*s, `t$0` = `s$0`*`s$0`, ..., `t$T*B-1` = `s$T*B-1`*`s$T*B-1`
+     *             | EVAL `CONFIDENCE_INTERVAL(s)` = CONFIDENCE_INTERVAL(s, MV_APPEND(`s$0`, ... `s$T*B-1`), T, B, 0.95),
+     *                    `CONFIDENCE_INTERVAL(t)` = CONFIDENCE_INTERVAL(t, MV_APPEND(`t$0`, ... `t$T*B-1`), T, B, 0.95)
+     *             | KEEP s, t, `CONFIDENCE_INTERVAL(s)`, `CONFIDENCE_INTERVAL(t)`
+     *     }
+     * </pre>
+     */
+    private LogicalPlan approximatePlan(double sampleProbability) {
+        if (sampleProbability >= 1.0) {
+            logger.debug("using original plan (too few rows)");
+            return logicalPlan;
+        }
+
+        logger.debug("generating approximate plan (p={})", sampleProbability);
+        Holder<Boolean> encounteredStats = new Holder<>(false);
+        Map<NameId, List<Alias>> fieldBuckets = new HashMap<>();
+
+        LogicalPlan approximatePlan = logicalPlan.transformUp(plan -> {
+            if (plan instanceof LeafPlan) {
+                // The leaf plan should be appended by a SAMPLE.
+                plan = new Sample(Source.EMPTY, Literal.fromDouble(Source.EMPTY, sampleProbability), plan);
+
+            } else if (encounteredStats.get() == false && plan instanceof Aggregate aggregate) {
+                // The STATS function should be replaced by a sample-corrected STATS and buckets
+                // to compute confidence intervals.
+                encounteredStats.set(true);
+
+                Expression bucketIds = new Random(Source.EMPTY, Literal.integer(Source.EMPTY, BUCKET_COUNT));
+                for (int trialId = 1; trialId < TRIAL_COUNT; trialId++) {
+                    bucketIds = new MvAppend(
+                        Source.EMPTY,
+                        bucketIds,
+                        new Random(Source.EMPTY, Literal.integer(Source.EMPTY, BUCKET_COUNT))
+                    );
+                }
+                Alias bucketIdField = new Alias(Source.EMPTY, "$bucket_id", bucketIds);
+
+                List<NamedExpression> aggregates = new ArrayList<>();
+                // Ensure that all buckets are non-empty, because empty buckets
+                // would mess up the confidence interval computation.
+                Expression allBucketsNonEmpty = Literal.TRUE;
+                for (NamedExpression aggOrKey : aggregate.aggregates()) {
+                    if ((aggOrKey instanceof Alias alias && alias.child() instanceof AggregateFunction) == false) {
+                        // This is a grouping key, not an aggregate function.
+                        aggregates.add(aggOrKey);
+                        continue;
+                    }
+
+                    // Replace the original aggregation by a sample-corrected one.
+                    Alias aggAlias = (Alias) aggOrKey;
+                    AggregateFunction aggFn = (AggregateFunction) aggAlias.child();
+                    aggregates.add(aggAlias.replaceChild(correctForSampling(aggFn, sampleProbability)));
+
+                    if (SUPPORTED_MULTIVALUED_AGGS.contains(aggFn.getClass()) == false) {
+                        // For the supported single-valued aggregations, add buckets with sampled
+                        // values, that will be used to compute a confidence interval.
+                        // For multivalued aggregations, confidence intervals do not make sense.
+                        List<Alias> buckets = new ArrayList<>();
+                        for (int trialId = 0; trialId < TRIAL_COUNT; trialId++) {
+                            for (int bucketId = 0; bucketId < BUCKET_COUNT; bucketId++) {
+                                Alias bucket = new Alias(
+                                    Source.EMPTY,
+                                    aggOrKey.name() + "$" + (trialId * BUCKET_COUNT + bucketId),
+                                    correctForSampling(
+                                        aggFn.withFilter(
+                                            new Equals(
+                                                Source.EMPTY,
+                                                new MvSlice(
+                                                    Source.EMPTY,
+                                                    bucketIdField.toAttribute(),
+                                                    Literal.integer(Source.EMPTY, trialId),
+                                                    Literal.integer(Source.EMPTY, trialId)
+                                                ),
+                                                Literal.integer(Source.EMPTY, bucketId)
+                                            )
+                                        ),
+                                        sampleProbability / BUCKET_COUNT
+                                    )
+                                );
+                                buckets.add(bucket);
+                                aggregates.add(bucket);
+                                allBucketsNonEmpty = new And(
+                                    Source.EMPTY,
+                                    allBucketsNonEmpty,
+                                    aggFn instanceof Count
+                                        ? new NotEquals(Source.EMPTY, bucket.toAttribute(), Literal.integer(Source.EMPTY, 0))
+                                        : new IsNotNull(Source.EMPTY, bucket.toAttribute())
+                                );
+                            }
+                        }
+                        fieldBuckets.put(aggOrKey.id(), buckets);
+                    }
+                }
+
+                // Add the bucket ID, do the aggregations (sampled corrected, including the buckets),
+                // and filter out rows with empty buckets.
+                plan = new Eval(Source.EMPTY, aggregate.child(), List.of(bucketIdField));
+                plan = aggregate.with(plan, aggregate.groupings(), aggregates);
+                plan = new Filter(Source.EMPTY, plan, allBucketsNonEmpty);
+
+            } else if (encounteredStats.get()) {
+                // After the STATS function, any processing of fields that have buckets, should
+                // also process the buckets, so that confidence intervals for the dependent fields
+                // can be computed.
+                switch (plan) {
+                    case Eval eval:
+                        // For EVAL, if any of the evaluated expressions depends on a field with buckets,
+                        // create corresponding expressions for the buckets as well.
+                        // If the results are non-numeric or multivalued, though, confidence intervals don't
+                        // apply anymore, and the buckets not computed.
+                        List<Alias> fields = new ArrayList<>(eval.fields());
+                        for (Alias field : eval.fields()) {
+                            // Don't create buckets for non-numeric or multivalued fields.
+                            if (field.dataType().isNumeric() == false
+                                || field.child().anyMatch(expr -> MULTIVALUED_OUTPUT_FUNCTIONS.contains(expr.getClass()))) {
+                                continue;
+                            }
+                            // If any of the field's dependencies has buckets, create buckets for this field as well.
+                            if (field.child().anyMatch(e -> e instanceof NamedExpression ne && fieldBuckets.containsKey(ne.id()))) {
+                                List<Alias> buckets = new ArrayList<>();
+                                for (int bucketId = 0; bucketId < TRIAL_COUNT * BUCKET_COUNT; bucketId++) {
+                                    final int finalBucketId = bucketId;
+                                    Expression bucket = field.child()
+                                        .transformDown(
+                                            e -> e instanceof NamedExpression ne && fieldBuckets.containsKey(ne.id())
+                                                ? fieldBuckets.get(ne.id()).get(finalBucketId).toAttribute()
+                                                : e
+                                        );
+                                    buckets.add(new Alias(Source.EMPTY, field.name() + "$" + bucketId, bucket));
+                                }
+                                fields.addAll(buckets);
+                                fieldBuckets.put(field.id(), buckets);
+                            }
+                        }
+                        plan = new Eval(Source.EMPTY, eval.child(), fields);
+                        break;
+
+                    case Project project:
+                        // For PROJECT, if it renames a fields with buckets, add the renamed field
+                        // to the map of fields with buckets.
+                        for (NamedExpression projection : project.projections()) {
+                            if (projection instanceof Alias alias
+                                && alias.child() instanceof NamedExpression named
+                                && fieldBuckets.containsKey(named.id())) {
+                                fieldBuckets.put(alias.id(), fieldBuckets.get(named.id()));
+                            }
+                        }
+                        break;
+                    default:
+                }
+            }
+            return plan;
+        });
+
+        Expression trialCount = Literal.integer(Source.EMPTY, TRIAL_COUNT);
+        Expression bucketCount = Literal.integer(Source.EMPTY, BUCKET_COUNT);
+        Expression confidenceLevel = Literal.fromDouble(Source.EMPTY, 0.95);
+
+        // Compute the confidence interval for all output fields that have buckets.
+        List<Alias> confidenceIntervalsAndReliable = new ArrayList<>();
+        for (Attribute output : logicalPlan.output()) {
+            if (fieldBuckets.containsKey(output.id())) {
+                List<Alias> buckets = fieldBuckets.get(output.id());
+                // Collect a multivalued expression with all bucket values, and pass that to the
+                // confidence interval computation.
+                Expression bucketsMv = buckets.getFirst().toAttribute();
+                for (int i = 1; i < TRIAL_COUNT * BUCKET_COUNT; i++) {
+                    bucketsMv = new MvAppend(Source.EMPTY, bucketsMv, buckets.get(i).toAttribute());
+                }
+                confidenceIntervalsAndReliable.add(
+                    new Alias(
+                        Source.EMPTY,
+                        "CONFIDENCE_INTERVAL(" + output.name() + ")",
+                        new ConfidenceInterval(Source.EMPTY, output, bucketsMv, trialCount, bucketCount, confidenceLevel)
+                    )
+                );
+                confidenceIntervalsAndReliable.add(
+                    new Alias(
+                        Source.EMPTY,
+                        "RELIABLE(" + output.name() + ")",
+                        new Reliable(Source.EMPTY, bucketsMv, trialCount, bucketCount)
+                    )
+                );
+            }
+        }
+        approximatePlan = new Eval(Source.EMPTY, approximatePlan, confidenceIntervalsAndReliable);
+
+        // Finally, drop all bucket fields from the output.
+        Set<Attribute> dropAttributes = fieldBuckets.values()
+            .stream()
+            .flatMap(List::stream)
+            .map(Alias::toAttribute)
+            .collect(Collectors.toSet());
+        List<Attribute> keepAttributes = new ArrayList<>(approximatePlan.output());
+        keepAttributes.removeAll(dropAttributes);
+        approximatePlan = new Project(Source.EMPTY, approximatePlan, keepAttributes);
+
+        approximatePlan.setPreOptimized();
+
+        logger.debug("approximate plan:\n{}", approximatePlan);
+
+        return approximatePlan;
+    }
+
+    /**
+     * Corrects an aggregation function for random sampling, if needed.
+     * Some functions (like COUNT and SUM) need to be scaled up by the inverse of
+     * the sampling probability, while others (like AVG and MEDIAN) do not.
+     */
+    private static Expression correctForSampling(AggregateFunction agg, double sampleProbability) {
+        if (SAMPLE_CORRECTED_AGGS.contains(agg.getClass()) == false) {
+            return agg;
+        }
+        Expression correctedAgg = new Div(agg.source(), agg, Literal.fromDouble(Source.EMPTY, sampleProbability));
+        return switch (agg.dataType()) {
+            case DOUBLE -> correctedAgg;
+            case LONG -> new ToLong(agg.source(), correctedAgg);
+            default -> throw new IllegalStateException("unexpected data type [" + agg.dataType() + "]");
+        };
+    }
+}
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/ExpressionWritables.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/ExpressionWritables.java
index ed7e8ebb57003..51a8a256cd12f 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/ExpressionWritables.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/ExpressionWritables.java
@@ -62,6 +62,7 @@
 import org.elasticsearch.xpack.esql.expression.function.scalar.math.Tan;
 import org.elasticsearch.xpack.esql.expression.function.scalar.math.Tanh;
 import org.elasticsearch.xpack.esql.expression.function.scalar.multivalue.MvFunctionWritables;
+import org.elasticsearch.xpack.esql.expression.function.scalar.random.Random;
 import org.elasticsearch.xpack.esql.expression.function.scalar.spatial.SpatialContains;
 import org.elasticsearch.xpack.esql.expression.function.scalar.spatial.SpatialDisjoint;
 import org.elasticsearch.xpack.esql.expression.function.scalar.spatial.SpatialIntersects;
@@ -190,6 +191,7 @@ public static List<NamedWriteableRegistry.Entry> unaryScalars() {
         entries.add(LTrim.ENTRY);
         entries.add(Neg.ENTRY);
         entries.add(Not.ENTRY);
+        entries.add(Random.ENTRY);
         entries.add(RLike.ENTRY);
         entries.add(RLikeList.ENTRY);
         entries.add(RTrim.ENTRY);
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/EsqlFunctionRegistry.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/EsqlFunctionRegistry.java
index c6520c8563d6d..e4db47f854b05 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/EsqlFunctionRegistry.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/EsqlFunctionRegistry.java
@@ -158,6 +158,7 @@
 import org.elasticsearch.xpack.esql.expression.function.scalar.multivalue.MvSum;
 import org.elasticsearch.xpack.esql.expression.function.scalar.multivalue.MvZip;
 import org.elasticsearch.xpack.esql.expression.function.scalar.nulls.Coalesce;
+import org.elasticsearch.xpack.esql.expression.function.scalar.random.Random;
 import org.elasticsearch.xpack.esql.expression.function.scalar.score.Decay;
 import org.elasticsearch.xpack.esql.expression.function.scalar.spatial.SpatialContains;
 import org.elasticsearch.xpack.esql.expression.function.scalar.spatial.SpatialDisjoint;
@@ -534,8 +535,9 @@ private static FunctionDefinition[][] functions() {
                 def(LastOverTime.class, uni(LastOverTime::new), "last_over_time"),
                 def(FirstOverTime.class, uni(FirstOverTime::new), "first_over_time"),
                 // dense vector function
-                def(TextEmbedding.class, bi(TextEmbedding::new), "text_embedding") } };
-
+                def(TextEmbedding.class, bi(TextEmbedding::new), "text_embedding") },
+            // random functions
+            new FunctionDefinition[] { def(Random.class, uni(Random::new), "random") } };
     }
 
     private static FunctionDefinition[][] snapshotFunctions() {
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/ScalarFunctionWritables.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/ScalarFunctionWritables.java
index 961d577692aa0..558f653d64f0e 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/ScalarFunctionWritables.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/ScalarFunctionWritables.java
@@ -9,6 +9,8 @@
 
 import org.elasticsearch.common.io.stream.NamedWriteableRegistry;
 import org.elasticsearch.xpack.esql.expression.function.grouping.GroupingWritables;
+import org.elasticsearch.xpack.esql.expression.function.scalar.approximate.ConfidenceInterval;
+import org.elasticsearch.xpack.esql.expression.function.scalar.approximate.Reliable;
 import org.elasticsearch.xpack.esql.expression.function.scalar.conditional.Case;
 import org.elasticsearch.xpack.esql.expression.function.scalar.conditional.ClampMax;
 import org.elasticsearch.xpack.esql.expression.function.scalar.conditional.ClampMin;
@@ -73,6 +75,7 @@ public static List<NamedWriteableRegistry.Entry> getNamedWriteables() {
         entries.add(CIDRMatch.ENTRY);
         entries.add(Coalesce.ENTRY);
         entries.add(Concat.ENTRY);
+        entries.add(ConfidenceInterval.ENTRY);
         entries.add(Contains.ENTRY);
         entries.add(E.ENTRY);
         entries.add(EndsWith.ENTRY);
@@ -102,6 +105,7 @@ public static List<NamedWriteableRegistry.Entry> getNamedWriteables() {
         entries.add(Or.ENTRY);
         entries.add(Pi.ENTRY);
         entries.add(Pow.ENTRY);
+        entries.add(Reliable.ENTRY);
         entries.add(Right.ENTRY);
         entries.add(Repeat.ENTRY);
         entries.add(Replace.ENTRY);
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceInterval.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceInterval.java
new file mode 100644
index 0000000000000..6d97c0b1ccc1e
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/ConfidenceInterval.java
@@ -0,0 +1,347 @@
+/*
+ * Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+ * or more contributor license agreements. Licensed under the Elastic License
+ * 2.0; you may not use this file except in compliance with the Elastic License
+ * 2.0.
+ */
+
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import org.apache.commons.math3.distribution.NormalDistribution;
+import org.apache.commons.math3.stat.descriptive.moment.Mean;
+import org.apache.commons.math3.stat.descriptive.moment.Skewness;
+import org.apache.commons.math3.stat.descriptive.moment.StandardDeviation;
+import org.elasticsearch.common.io.stream.NamedWriteableRegistry;
+import org.elasticsearch.common.io.stream.StreamInput;
+import org.elasticsearch.common.io.stream.StreamOutput;
+import org.elasticsearch.compute.ann.Evaluator;
+import org.elasticsearch.compute.ann.Position;
+import org.elasticsearch.compute.data.DoubleBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.LongBlock;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.xpack.esql.EsqlIllegalArgumentException;
+import org.elasticsearch.xpack.esql.approximate.Approximate;
+import org.elasticsearch.xpack.esql.core.expression.Expression;
+import org.elasticsearch.xpack.esql.core.expression.Nullability;
+import org.elasticsearch.xpack.esql.core.tree.NodeInfo;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+import org.elasticsearch.xpack.esql.core.type.DataType;
+import org.elasticsearch.xpack.esql.expression.function.EsqlFunctionRegistry;
+import org.elasticsearch.xpack.esql.expression.function.FunctionInfo;
+import org.elasticsearch.xpack.esql.expression.function.Param;
+import org.elasticsearch.xpack.esql.expression.function.scalar.EsqlScalarFunction;
+import org.elasticsearch.xpack.esql.io.stream.PlanStreamInput;
+import org.elasticsearch.xpack.esql.planner.PlannerUtils;
+
+import java.io.IOException;
+import java.util.List;
+import java.util.Objects;
+
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.FIFTH;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.FIRST;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.FOURTH;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.SECOND;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.THIRD;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.isType;
+import static org.elasticsearch.xpack.esql.core.type.DataType.isRepresentable;
+
+/**
+ * This function is used internally by {@link Approximate}, and is not exposed
+ * to users via the {@link EsqlFunctionRegistry}.
+ */
+public class ConfidenceInterval extends EsqlScalarFunction {
+    public static final NamedWriteableRegistry.Entry ENTRY = new NamedWriteableRegistry.Entry(
+        Expression.class,
+        "ConfidenceInterval",
+        ConfidenceInterval::new
+    );
+
+    private static final NormalDistribution normal = new NormalDistribution();
+
+    private final Expression bestEstimate;
+    private final Expression estimates;
+    private final Expression trialCount;
+    private final Expression bucketCount;
+    private final Expression confidenceLevel;
+
+    @FunctionInfo(returnType = { "double", }, description = "...")
+    public ConfidenceInterval(
+        Source source,
+        @Param(name = "bestEstimate", type = { "double", "int", "long" }) Expression bestEstimate,
+        @Param(name = "estimates", type = { "double", "int", "long" }) Expression estimates,
+        @Param(name = "trialCount", type = { "int" }) Expression trialCount,
+        @Param(name = "bucketCount", type = { "int" }) Expression bucketCount,
+        @Param(name = "confidenceLevel", type = { "double" }) Expression confidenceLevel
+    ) {
+        super(source, List.of(bestEstimate, estimates));
+        this.bestEstimate = bestEstimate;
+        this.estimates = estimates;
+        this.trialCount = trialCount;
+        this.bucketCount = bucketCount;
+        this.confidenceLevel = confidenceLevel;
+    }
+
+    private ConfidenceInterval(StreamInput in) throws IOException {
+        this(
+            Source.readFrom((PlanStreamInput) in),
+            in.readNamedWriteable(Expression.class),
+            in.readNamedWriteable(Expression.class),
+            in.readNamedWriteable(Expression.class),
+            in.readNamedWriteable(Expression.class),
+            in.readNamedWriteable(Expression.class)
+        );
+    }
+
+    @Override
+    public void writeTo(StreamOutput out) throws IOException {
+        Source.EMPTY.writeTo(out);
+        out.writeNamedWriteable(bestEstimate);
+        out.writeNamedWriteable(estimates);
+        out.writeNamedWriteable(trialCount);
+        out.writeNamedWriteable(bucketCount);
+        out.writeNamedWriteable(confidenceLevel);
+    }
+
+    @Override
+    public String getWriteableName() {
+        return ENTRY.name;
+    }
+
+    @Override
+    protected TypeResolution resolveType() {
+        return isType(bestEstimate, t -> t.isNumeric() && isRepresentable(t), sourceText(), FIRST, "numeric").and(
+            isType(estimates, t -> t.isNumeric() && isRepresentable(t), sourceText(), SECOND, "numeric")
+        )
+            .and(isType(trialCount, t -> t == DataType.INTEGER, sourceText(), THIRD, "integer"))
+            .and(isType(bucketCount, t -> t == DataType.INTEGER, sourceText(), FOURTH, "integer"))
+            .and(isType(confidenceLevel, t -> t == DataType.DOUBLE, sourceText(), FIFTH, "double"));
+    }
+
+    @Override
+    public boolean foldable() {
+        return bestEstimate.foldable()
+            && estimates.foldable()
+            && trialCount.foldable()
+            && bucketCount.foldable()
+            && confidenceLevel.foldable();
+    }
+
+    @Override
+    public EvalOperator.ExpressionEvaluator.Factory toEvaluator(ToEvaluator toEvaluator) {
+        return switch (PlannerUtils.toElementType(bestEstimate.dataType())) {
+            case DOUBLE -> new ConfidenceIntervalDoubleEvaluator.Factory(
+                source(),
+                toEvaluator.apply(bestEstimate),
+                toEvaluator.apply(estimates),
+                toEvaluator.apply(trialCount),
+                toEvaluator.apply(bucketCount),
+                toEvaluator.apply(confidenceLevel)
+            );
+            case INT -> new ConfidenceIntervalIntEvaluator.Factory(
+                source(),
+                toEvaluator.apply(bestEstimate),
+                toEvaluator.apply(estimates),
+                toEvaluator.apply(trialCount),
+                toEvaluator.apply(bucketCount),
+                toEvaluator.apply(confidenceLevel)
+            );
+            case LONG -> new ConfidenceIntervalLongEvaluator.Factory(
+                source(),
+                toEvaluator.apply(bestEstimate),
+                toEvaluator.apply(estimates),
+                toEvaluator.apply(trialCount),
+                toEvaluator.apply(bucketCount),
+                toEvaluator.apply(confidenceLevel)
+            );
+            default -> throw EsqlIllegalArgumentException.illegalDataType(bestEstimate.dataType());
+        };
+    }
+
+    @Override
+    public Expression replaceChildren(List<Expression> newChildren) {
+        return new ConfidenceInterval(
+            source(),
+            newChildren.get(0),
+            newChildren.get(1),
+            newChildren.get(2),
+            newChildren.get(3),
+            newChildren.get(4)
+        );
+    }
+
+    @Override
+    protected NodeInfo<? extends Expression> info() {
+        return NodeInfo.create(this, ConfidenceInterval::new, bestEstimate, estimates, trialCount, bucketCount, confidenceLevel);
+    }
+
+    @Override
+    public DataType dataType() {
+        return bestEstimate.dataType();
+    }
+
+    @Override
+    public int hashCode() {
+        return Objects.hash(bestEstimate, estimates, trialCount, bucketCount, confidenceLevel);
+    }
+
+    @Override
+    public boolean equals(Object obj) {
+        if (obj == null || obj.getClass() != getClass()) {
+            return false;
+        }
+        ConfidenceInterval other = (ConfidenceInterval) obj;
+        return Objects.equals(other.bestEstimate, bestEstimate)
+            && Objects.equals(other.estimates, estimates)
+            && Objects.equals(other.trialCount, trialCount)
+            && Objects.equals(other.bucketCount, bucketCount)
+            && Objects.equals(other.confidenceLevel, confidenceLevel);
+    }
+
+    @Evaluator(extraName = "Double")
+    static void process(
+        DoubleBlock.Builder builder,
+        @Position int position,
+        DoubleBlock bestEstimateBlock,
+        DoubleBlock estimatesBlock,
+        IntBlock trialCountBlock,
+        IntBlock bucketCountBlock,
+        DoubleBlock confidenceLevelBlock
+    ) {
+        if (bestEstimateBlock.getValueCount(position) != 1
+            || trialCountBlock.getValueCount(position) != 1
+            || bucketCountBlock.getValueCount(position) != 1
+            || confidenceLevelBlock.getValueCount(position) != 1) {
+            builder.appendNull();
+            return;
+        }
+        double bestEstimate = bestEstimateBlock.getDouble(bestEstimateBlock.getFirstValueIndex(position));
+        double[] estimates = new double[estimatesBlock.getValueCount(position)];
+        for (int i = 0; i < estimatesBlock.getValueCount(position); i++) {
+            estimates[i] = estimatesBlock.getDouble(estimatesBlock.getFirstValueIndex(position) + i);
+        }
+        int trialCount = trialCountBlock.getInt(trialCountBlock.getFirstValueIndex(position));
+        int bucketCount = bucketCountBlock.getInt(bucketCountBlock.getFirstValueIndex(position));
+        double confidenceLevel = confidenceLevelBlock.getDouble(confidenceLevelBlock.getFirstValueIndex(position));
+        double[] confidenceInterval = computeConfidenceInterval(bestEstimate, estimates, trialCount, bucketCount, confidenceLevel);
+        builder.beginPositionEntry();
+        for (double v : confidenceInterval) {
+            builder.appendDouble(v);
+        }
+        builder.endPositionEntry();
+    }
+
+    @Evaluator(extraName = "Int")
+    static void process(
+        IntBlock.Builder builder,
+        @Position int position,
+        IntBlock bestEstimateBlock,
+        IntBlock estimatesBlock,
+        IntBlock trialCountBlock,
+        IntBlock bucketCountBlock,
+        DoubleBlock confidenceLevelBlock
+    ) {
+        if (bestEstimateBlock.getValueCount(position) != 1
+            || trialCountBlock.getValueCount(position) != 1
+            || bucketCountBlock.getValueCount(position) != 1
+            || confidenceLevelBlock.getValueCount(position) != 1) {
+            builder.appendNull();
+            return;
+        }
+        double bestEstimate = bestEstimateBlock.getInt(bestEstimateBlock.getFirstValueIndex(position));
+        double[] estimates = new double[estimatesBlock.getValueCount(position)];
+        for (int i = 0; i < estimatesBlock.getValueCount(position); i++) {
+            estimates[i] = estimatesBlock.getInt(estimatesBlock.getFirstValueIndex(position) + i);
+        }
+        int trialCount = trialCountBlock.getInt(trialCountBlock.getFirstValueIndex(position));
+        int bucketCount = bucketCountBlock.getInt(bucketCountBlock.getFirstValueIndex(position));
+        double confidenceLevel = confidenceLevelBlock.getDouble(confidenceLevelBlock.getFirstValueIndex(position));
+        double[] confidenceInterval = computeConfidenceInterval(bestEstimate, estimates, trialCount, bucketCount, confidenceLevel);
+        builder.beginPositionEntry();
+        for (double v : confidenceInterval) {
+            builder.appendInt((int) Math.round(v));
+        }
+        builder.endPositionEntry();
+    }
+
+    @Evaluator(extraName = "Long")
+    static void process(
+        LongBlock.Builder builder,
+        @Position int position,
+        LongBlock bestEstimateBlock,
+        LongBlock estimatesBlock,
+        IntBlock trialCountBlock,
+        IntBlock bucketCountBlock,
+        DoubleBlock confidenceLevelBlock
+    ) {
+        if (bestEstimateBlock.getValueCount(position) != 1
+            || trialCountBlock.getValueCount(position) != 1
+            || bucketCountBlock.getValueCount(position) != 1
+            || confidenceLevelBlock.getValueCount(position) != 1) {
+            builder.appendNull();
+            return;
+        }
+        double bestEstimate = bestEstimateBlock.getLong(bestEstimateBlock.getFirstValueIndex(position));
+        double[] estimates = new double[estimatesBlock.getValueCount(position)];
+        for (int i = 0; i < estimatesBlock.getValueCount(position); i++) {
+            estimates[i] = estimatesBlock.getLong(estimatesBlock.getFirstValueIndex(position) + i);
+        }
+        int trialCount = trialCountBlock.getInt(trialCountBlock.getFirstValueIndex(position));
+        int bucketCount = bucketCountBlock.getInt(bucketCountBlock.getFirstValueIndex(position));
+        double confidenceLevel = confidenceLevelBlock.getDouble(confidenceLevelBlock.getFirstValueIndex(position));
+        double[] confidenceInterval = computeConfidenceInterval(bestEstimate, estimates, trialCount, bucketCount, confidenceLevel);
+        builder.beginPositionEntry();
+        for (double v : confidenceInterval) {
+            builder.appendLong(Math.round(v));
+        }
+        builder.endPositionEntry();
+    }
+
+    public static double[] computeConfidenceInterval(
+        double bestEstimate,
+        double[] estimates,
+        int trialCount,
+        int bucketCount,
+        double confidenceLevel
+    ) {
+        System.out.println("bestEstimate = " + bestEstimate + ", estimates = " + java.util.Arrays.toString(estimates));
+        Mean means = new Mean();
+        Mean stddevs = new Mean();
+        Mean skews = new Mean();
+        for (int trial = 0; trial < trialCount; trial++) {
+            Mean mean = new Mean();
+            StandardDeviation stdDev = new StandardDeviation(false);
+            Skewness skew = new Skewness();
+            for (int bucket = 0; bucket < bucketCount; bucket++) {
+                double estimate = estimates[trial * bucketCount + bucket];
+                mean.increment(estimate);
+                stdDev.increment(estimate);
+                skew.increment(estimate);
+            }
+            means.increment(mean.getResult());
+            stddevs.increment(stdDev.getResult());
+            skews.increment(skew.getResult());
+            System.out.println(
+                "trial " + trial + ": mean = " + mean.getResult() + ", stddev = " + stdDev.getResult() + ", skew = " + skew.getResult()
+            );
+        }
+        System.out.println("total: mean = " + means.getResult() + ", stddev = " + stddevs.getResult() + ", skew = " + skews.getResult());
+        double sm = stddevs.getResult();
+        if (sm == 0.0) {
+            return new double[] { bestEstimate, bestEstimate };
+        }
+        double mm = means.getResult();
+        double a = skews.getResult() / (6.0 * Math.sqrt(bucketCount));
+        double z0 = (bestEstimate - mm) / sm;
+        double dz = normal.inverseCumulativeProbability((1.0 + confidenceLevel) / 2.0);
+        double zl = z0 + (z0 - dz) / (1.0 - Math.min(a * (z0 - dz), 0.9));
+        double zu = z0 + (z0 + dz) / (1.0 - Math.min(a * (z0 + dz), 0.9));
+        double scale = Math.max(1.0 / Math.sqrt(bucketCount), z0 < 0.0 ? z0 / zl : z0 / zu);
+        return new double[] { mm + scale * sm * zl, mm + sm * scale * zu };
+    }
+
+    @Override
+    public Nullability nullable() {
+        return Nullability.TRUE;
+    }
+}
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/Reliable.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/Reliable.java
new file mode 100644
index 0000000000000..4ffd91572f0d9
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/approximate/Reliable.java
@@ -0,0 +1,251 @@
+/*
+ * Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+ * or more contributor license agreements. Licensed under the Elastic License
+ * 2.0; you may not use this file except in compliance with the Elastic License
+ * 2.0.
+ */
+
+package org.elasticsearch.xpack.esql.expression.function.scalar.approximate;
+
+import org.apache.commons.math3.stat.descriptive.moment.Kurtosis;
+import org.apache.commons.math3.stat.descriptive.moment.Skewness;
+import org.elasticsearch.common.io.stream.NamedWriteableRegistry;
+import org.elasticsearch.common.io.stream.StreamInput;
+import org.elasticsearch.common.io.stream.StreamOutput;
+import org.elasticsearch.compute.ann.Evaluator;
+import org.elasticsearch.compute.ann.Position;
+import org.elasticsearch.compute.data.BooleanBlock;
+import org.elasticsearch.compute.data.DoubleBlock;
+import org.elasticsearch.compute.data.IntBlock;
+import org.elasticsearch.compute.data.LongBlock;
+import org.elasticsearch.compute.operator.EvalOperator;
+import org.elasticsearch.xpack.esql.EsqlIllegalArgumentException;
+import org.elasticsearch.xpack.esql.approximate.Approximate;
+import org.elasticsearch.xpack.esql.core.expression.Expression;
+import org.elasticsearch.xpack.esql.core.expression.Nullability;
+import org.elasticsearch.xpack.esql.core.tree.NodeInfo;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+import org.elasticsearch.xpack.esql.core.type.DataType;
+import org.elasticsearch.xpack.esql.expression.function.EsqlFunctionRegistry;
+import org.elasticsearch.xpack.esql.expression.function.FunctionInfo;
+import org.elasticsearch.xpack.esql.expression.function.Param;
+import org.elasticsearch.xpack.esql.expression.function.scalar.EsqlScalarFunction;
+import org.elasticsearch.xpack.esql.io.stream.PlanStreamInput;
+import org.elasticsearch.xpack.esql.planner.PlannerUtils;
+
+import java.io.IOException;
+import java.util.Arrays;
+import java.util.List;
+import java.util.Objects;
+
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.FOURTH;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.SECOND;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.ParamOrdinal.THIRD;
+import static org.elasticsearch.xpack.esql.core.expression.TypeResolutions.isType;
+import static org.elasticsearch.xpack.esql.core.type.DataType.isRepresentable;
+
+/**
+ * This function is used internally by {@link Approximate}, and is not exposed
+ * to users via the {@link EsqlFunctionRegistry}.
+ */
+public class Reliable extends EsqlScalarFunction {
+    public static final NamedWriteableRegistry.Entry ENTRY = new NamedWriteableRegistry.Entry(Expression.class, "Reliable", Reliable::new);
+
+    private final Expression estimates;
+    private final Expression trialCount;
+    private final Expression bucketCount;
+
+    @FunctionInfo(returnType = { "boolean", }, description = "...")
+    public Reliable(
+        Source source,
+        @Param(name = "estimates", type = { "double", "int", "long" }) Expression estimates,
+        @Param(name = "trialCount", type = { "int" }) Expression trialCount,
+        @Param(name = "bucketCount", type = { "int" }) Expression bucketCount
+    ) {
+        super(source, List.of(estimates));
+        this.estimates = estimates;
+        this.trialCount = trialCount;
+        this.bucketCount = bucketCount;
+    }
+
+    private Reliable(StreamInput in) throws IOException {
+        this(
+            Source.readFrom((PlanStreamInput) in),
+            in.readNamedWriteable(Expression.class),
+            in.readNamedWriteable(Expression.class),
+            in.readNamedWriteable(Expression.class)
+        );
+    }
+
+    @Override
+    public void writeTo(StreamOutput out) throws IOException {
+        Source.EMPTY.writeTo(out);
+        out.writeNamedWriteable(estimates);
+        out.writeNamedWriteable(trialCount);
+        out.writeNamedWriteable(bucketCount);
+    }
+
+    @Override
+    public String getWriteableName() {
+        return ENTRY.name;
+    }
+
+    @Override
+    protected TypeResolution resolveType() {
+        return isType(estimates, t -> t.isNumeric() && isRepresentable(t), sourceText(), SECOND, "numeric").and(
+            isType(trialCount, t -> t == DataType.INTEGER, sourceText(), THIRD, "integer")
+        ).and(isType(bucketCount, t -> t == DataType.INTEGER, sourceText(), FOURTH, "integer"));
+    }
+
+    @Override
+    public boolean foldable() {
+        return estimates.foldable() && trialCount.foldable() && bucketCount.foldable();
+    }
+
+    @Override
+    public EvalOperator.ExpressionEvaluator.Factory toEvaluator(ToEvaluator toEvaluator) {
+        return switch (PlannerUtils.toElementType(estimates.dataType())) {
+            case DOUBLE -> new ReliableDoubleEvaluator.Factory(
+                source(),
+                toEvaluator.apply(estimates),
+                toEvaluator.apply(trialCount),
+                toEvaluator.apply(bucketCount)
+            );
+            case INT -> new ReliableIntEvaluator.Factory(
+                source(),
+                toEvaluator.apply(estimates),
+                toEvaluator.apply(trialCount),
+                toEvaluator.apply(bucketCount)
+            );
+            case LONG -> new ReliableLongEvaluator.Factory(
+                source(),
+                toEvaluator.apply(estimates),
+                toEvaluator.apply(trialCount),
+                toEvaluator.apply(bucketCount)
+            );
+            default -> throw EsqlIllegalArgumentException.illegalDataType(estimates.dataType());
+        };
+    }
+
+    @Override
+    public Expression replaceChildren(List<Expression> newChildren) {
+        return new Reliable(source(), newChildren.get(0), newChildren.get(1), newChildren.get(2));
+    }
+
+    @Override
+    protected NodeInfo<? extends Expression> info() {
+        return NodeInfo.create(this, Reliable::new, estimates, trialCount, bucketCount);
+    }
+
+    @Override
+    public DataType dataType() {
+        return DataType.BOOLEAN;
+    }
+
+    @Override
+    public int hashCode() {
+        return Objects.hash(estimates, trialCount, bucketCount);
+    }
+
+    @Override
+    public boolean equals(Object obj) {
+        if (obj == null || obj.getClass() != getClass()) {
+            return false;
+        }
+        Reliable other = (Reliable) obj;
+        return Objects.equals(other.estimates, estimates)
+            && Objects.equals(other.trialCount, trialCount)
+            && Objects.equals(other.bucketCount, bucketCount);
+    }
+
+    @Evaluator(extraName = "Double")
+    static void process(
+        BooleanBlock.Builder builder,
+        @Position int position,
+        DoubleBlock estimatesBlock,
+        IntBlock trialCountBlock,
+        IntBlock bucketCountBlock
+    ) {
+        if (trialCountBlock.getValueCount(position) != 1 || bucketCountBlock.getValueCount(position) != 1) {
+            builder.appendNull();
+            return;
+        }
+        double[] estimates = new double[estimatesBlock.getValueCount(position)];
+        for (int i = 0; i < estimatesBlock.getValueCount(position); i++) {
+            estimates[i] = estimatesBlock.getDouble(estimatesBlock.getFirstValueIndex(position) + i);
+        }
+        int trialCount = trialCountBlock.getInt(trialCountBlock.getFirstValueIndex(position));
+        int bucketCount = bucketCountBlock.getInt(bucketCountBlock.getFirstValueIndex(position));
+        builder.appendBoolean(computeReliable(estimates, trialCount, bucketCount));
+    }
+
+    @Evaluator(extraName = "Int")
+    static void process(
+        BooleanBlock.Builder builder,
+        @Position int position,
+        IntBlock estimatesBlock,
+        IntBlock trialCountBlock,
+        IntBlock bucketCountBlock
+    ) {
+        if (trialCountBlock.getValueCount(position) != 1 || bucketCountBlock.getValueCount(position) != 1) {
+            builder.appendNull();
+            return;
+        }
+        double[] estimates = new double[estimatesBlock.getValueCount(position)];
+        for (int i = 0; i < estimatesBlock.getValueCount(position); i++) {
+            estimates[i] = estimatesBlock.getInt(estimatesBlock.getFirstValueIndex(position) + i);
+        }
+        int trialCount = trialCountBlock.getInt(trialCountBlock.getFirstValueIndex(position));
+        int bucketCount = bucketCountBlock.getInt(bucketCountBlock.getFirstValueIndex(position));
+        builder.appendBoolean(computeReliable(estimates, trialCount, bucketCount));
+    }
+
+    @Evaluator(extraName = "Long")
+    static void process(
+        BooleanBlock.Builder builder,
+        @Position int position,
+        LongBlock estimatesBlock,
+        IntBlock trialCountBlock,
+        IntBlock bucketCountBlock
+    ) {
+        if (trialCountBlock.getValueCount(position) != 1 || bucketCountBlock.getValueCount(position) != 1) {
+            builder.appendNull();
+            return;
+        }
+        double[] estimates = new double[estimatesBlock.getValueCount(position)];
+        for (int i = 0; i < estimatesBlock.getValueCount(position); i++) {
+            estimates[i] = estimatesBlock.getLong(estimatesBlock.getFirstValueIndex(position) + i);
+        }
+        int trialCount = trialCountBlock.getInt(trialCountBlock.getFirstValueIndex(position));
+        int bucketCount = bucketCountBlock.getInt(bucketCountBlock.getFirstValueIndex(position));
+        builder.appendBoolean(computeReliable(estimates, trialCount, bucketCount));
+    }
+
+    public static boolean computeReliable(double[] estimates, int trialCount, int B) {
+        if (B < 5) {
+            return false;
+        }
+        double maxSkew = Math.sqrt(6.0 * B * (B - 1) / ((B - 2) * (B + 1) * (B + 3))) * 1.96;
+        double maxKurtosis = Math.sqrt(24.0 * B * (B - 1) * (B - 1) / ((B - 3) * (B - 2) * (B + 3) * (B + 5))) * 1.96;
+        int reliableCount = 0;
+        for (int trial = 0; trial < trialCount; trial++) {
+            Skewness skew = new Skewness();
+            Kurtosis kurtosis = new Kurtosis();
+            for (int bucket = 0; bucket < B; bucket++) {
+                double estimate = estimates[trial * B + bucket];
+                skew.increment(estimate);
+                kurtosis.increment(estimate);
+            }
+            if (Math.abs(skew.getResult()) < maxSkew && Math.abs(kurtosis.getResult()) < maxKurtosis) {
+                reliableCount++;
+            }
+        }
+        System.out.println("estimates = " + Arrays.toString(estimates) + " -> reliableCount = " + reliableCount + " / " + trialCount);
+        return 2 * reliableCount > trialCount;
+    }
+
+    @Override
+    public Nullability nullable() {
+        return Nullability.TRUE;
+    }
+}
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/random/Random.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/random/Random.java
new file mode 100644
index 0000000000000..4ba78a82179b7
--- /dev/null
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/expression/function/scalar/random/Random.java
@@ -0,0 +1,74 @@
+/*
+ * Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+ * or more contributor license agreements. Licensed under the Elastic License
+ * 2.0; you may not use this file except in compliance with the Elastic License
+ * 2.0.
+ */
+
+package org.elasticsearch.xpack.esql.expression.function.scalar.random;
+
+import org.elasticsearch.common.Randomness;
+import org.elasticsearch.common.io.stream.NamedWriteableRegistry;
+import org.elasticsearch.common.io.stream.StreamInput;
+import org.elasticsearch.compute.ann.Evaluator;
+import org.elasticsearch.compute.operator.EvalOperator.ExpressionEvaluator;
+import org.elasticsearch.xpack.esql.core.expression.Expression;
+import org.elasticsearch.xpack.esql.core.tree.NodeInfo;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+import org.elasticsearch.xpack.esql.expression.function.FunctionInfo;
+import org.elasticsearch.xpack.esql.expression.function.FunctionType;
+import org.elasticsearch.xpack.esql.expression.function.Param;
+import org.elasticsearch.xpack.esql.expression.function.scalar.UnaryScalarFunction;
+
+import java.io.IOException;
+import java.util.List;
+
+public class Random extends UnaryScalarFunction {
+    public static final NamedWriteableRegistry.Entry ENTRY = new NamedWriteableRegistry.Entry(Expression.class, "Random", Random::new);
+
+    @FunctionInfo(
+        returnType = { "integer" },
+        description = "Returns a pseudorandom number, uniformly distributed between 0 (inclusive) and bound (exclusive).",
+        type = FunctionType.SCALAR
+    )
+    public Random(
+        Source source,
+        @Param(name = "bound", type = { "integer" }, description = "The upper bound (exclusive).") Expression bound
+    ) {
+        super(source, bound);
+    }
+
+    private Random(StreamInput in) throws IOException {
+        super(in);
+    }
+
+    @Override
+    public String getWriteableName() {
+        return ENTRY.name;
+    }
+
+    @Evaluator
+    static int process(int bound) {
+        return Randomness.get().nextInt(bound);
+    }
+
+    @Override
+    public ExpressionEvaluator.Factory toEvaluator(ToEvaluator toEvaluator) {
+        return new RandomEvaluator.Factory(source(), toEvaluator.apply(field));
+    }
+
+    @Override
+    public final Expression replaceChildren(List<Expression> newChildren) {
+        return new Random(source(), newChildren.get(0));
+    }
+
+    @Override
+    protected NodeInfo<? extends Expression> info() {
+        return NodeInfo.create(this, Random::new, field());
+    }
+
+    @Override
+    public boolean foldable() {
+        return false;
+    }
+}
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/plan/QueryPlan.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/plan/QueryPlan.java
index 81a89950b0a02..3d77a2bd86097 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/plan/QueryPlan.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/plan/QueryPlan.java
@@ -102,6 +102,10 @@ public PlanType transformExpressionsOnly(Function<Expression, ? extends Expressi
         return transformPropertiesOnly(Object.class, e -> doTransformExpression(e, exp -> exp.transformDown(rule)));
     }
 
+    public PlanType transformExpressionsOnlyUp(Function<Expression, ? extends Expression> rule) {
+        return transformPropertiesOnly(Object.class, e -> doTransformExpression(e, exp -> exp.transformUp(rule)));
+    }
+
     public <E extends Expression> PlanType transformExpressionsOnly(Class<E> typeToken, Function<E, ? extends Expression> rule) {
         return transformPropertiesOnly(Object.class, e -> doTransformExpression(e, exp -> exp.transformDown(typeToken, rule)));
     }
diff --git a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/session/EsqlSession.java b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/session/EsqlSession.java
index ded34107f5aec..3839fa0547728 100644
--- a/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/session/EsqlSession.java
+++ b/x-pack/plugin/esql/src/main/java/org/elasticsearch/xpack/esql/session/EsqlSession.java
@@ -44,6 +44,7 @@
 import org.elasticsearch.xpack.esql.analysis.EnrichResolution;
 import org.elasticsearch.xpack.esql.analysis.PreAnalyzer;
 import org.elasticsearch.xpack.esql.analysis.Verifier;
+import org.elasticsearch.xpack.esql.approximate.Approximate;
 import org.elasticsearch.xpack.esql.core.expression.Attribute;
 import org.elasticsearch.xpack.esql.core.expression.ReferenceAttribute;
 import org.elasticsearch.xpack.esql.core.tree.NodeUtils;
@@ -197,7 +198,12 @@ public void onResponse(LogicalPlan analyzedPlan) {
                     ThreadPool.Names.SEARCH_COORDINATION,
                     ThreadPool.Names.SYSTEM_READ
                 );
-                SubscribableListener.<LogicalPlan>newForked(l -> preOptimizedPlan(analyzedPlan, l))
+                SubscribableListener.<LogicalPlan>newForked(l -> preOptimizedPlan(analyzedPlan, l)).<LogicalPlan>andThen((l, p) -> {
+                    if (request.approximate()) {
+                        new Approximate(p);  // to verify whether the pre-optimized plan is suitable for approximation
+                    }
+                    l.onResponse(p);
+                })
                     .<LogicalPlan>andThen((l, p) -> preMapper.preMapper(optimizedPlan(p), l))
                     .<Result>andThen((l, p) -> executeOptimizedPlan(request, executionInfo, planRunner, p, l))
                     .addListener(listener);
@@ -268,6 +274,11 @@ private void executeSubPlans(
                 // Ensure we don't have subplan flag stuck in there on failure
                 ActionListener.runAfter(listener, executionInfo::finishSubPlans)
             );
+        } else if (request.approximate()) {
+            new Approximate(optimizedPlan).approximate(
+                (p, l) -> runner.run(logicalPlanToPhysicalPlan(optimizedPlan(p), request), l),
+                listener
+            );
         } else {
             PhysicalPlan physicalPlan = logicalPlanToPhysicalPlan(optimizedPlan, request);
             // execute main plan
diff --git a/x-pack/plugin/esql/src/test/java/org/elasticsearch/xpack/esql/CsvTests.java b/x-pack/plugin/esql/src/test/java/org/elasticsearch/xpack/esql/CsvTests.java
index 1c822a11a811f..a8a53e0110c25 100644
--- a/x-pack/plugin/esql/src/test/java/org/elasticsearch/xpack/esql/CsvTests.java
+++ b/x-pack/plugin/esql/src/test/java/org/elasticsearch/xpack/esql/CsvTests.java
@@ -604,7 +604,7 @@ private ActualResults executePlan(BigArrays bigArrays) throws Exception {
 
         session.preOptimizedPlan(analyzed, listener.delegateFailureAndWrap((l, preOptimized) -> {
             session.executeOptimizedPlan(
-                new EsqlQueryRequest(),
+                new EsqlQueryRequest().approximate(testCase.approximate),
                 new EsqlExecutionInfo(randomBoolean()),
                 planRunner(bigArrays, foldCtx, physicalOperationProviders),
                 session.optimizedPlan(preOptimized),
diff --git a/x-pack/plugin/esql/src/test/java/org/elasticsearch/xpack/esql/approximate/ApproximateTests.java b/x-pack/plugin/esql/src/test/java/org/elasticsearch/xpack/esql/approximate/ApproximateTests.java
new file mode 100644
index 0000000000000..d6bea328be4b3
--- /dev/null
+++ b/x-pack/plugin/esql/src/test/java/org/elasticsearch/xpack/esql/approximate/ApproximateTests.java
@@ -0,0 +1,321 @@
+/*
+ * Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
+ * or more contributor license agreements. Licensed under the Elastic License
+ * 2.0; you may not use this file except in compliance with the Elastic License
+ * 2.0.
+ */
+
+package org.elasticsearch.xpack.esql.approximate;
+
+import org.apache.lucene.util.SetOnce;
+import org.elasticsearch.action.ActionListener;
+import org.elasticsearch.common.breaker.CircuitBreaker;
+import org.elasticsearch.common.unit.ByteSizeValue;
+import org.elasticsearch.common.util.BigArrays;
+import org.elasticsearch.common.util.MockBigArrays;
+import org.elasticsearch.common.util.PageCacheRecycler;
+import org.elasticsearch.compute.data.LongBlock;
+import org.elasticsearch.compute.data.Page;
+import org.elasticsearch.compute.test.MockBlockFactory;
+import org.elasticsearch.test.ESTestCase;
+import org.elasticsearch.xpack.esql.VerificationException;
+import org.elasticsearch.xpack.esql.analysis.Analyzer;
+import org.elasticsearch.xpack.esql.analysis.AnalyzerTestUtils;
+import org.elasticsearch.xpack.esql.core.expression.FoldContext;
+import org.elasticsearch.xpack.esql.core.expression.Literal;
+import org.elasticsearch.xpack.esql.core.expression.NamedExpression;
+import org.elasticsearch.xpack.esql.core.tree.Source;
+import org.elasticsearch.xpack.esql.expression.Foldables;
+import org.elasticsearch.xpack.esql.inference.InferenceService;
+import org.elasticsearch.xpack.esql.optimizer.LogicalPlanPreOptimizer;
+import org.elasticsearch.xpack.esql.optimizer.LogicalPreOptimizerContext;
+import org.elasticsearch.xpack.esql.parser.EsqlParser;
+import org.elasticsearch.xpack.esql.plan.logical.Eval;
+import org.elasticsearch.xpack.esql.plan.logical.Filter;
+import org.elasticsearch.xpack.esql.plan.logical.LogicalPlan;
+import org.elasticsearch.xpack.esql.plan.logical.Sample;
+import org.elasticsearch.xpack.esql.session.Result;
+import org.hamcrest.Description;
+import org.hamcrest.Matcher;
+import org.hamcrest.TypeSafeMatcher;
+
+import java.util.ArrayList;
+import java.util.List;
+import java.util.function.Predicate;
+
+import static org.elasticsearch.xpack.esql.EsqlTestUtils.TEST_CFG;
+import static org.elasticsearch.xpack.esql.EsqlTestUtils.withDefaultLimitWarning;
+import static org.hamcrest.CoreMatchers.allOf;
+import static org.hamcrest.CoreMatchers.not;
+import static org.hamcrest.Matchers.equalTo;
+import static org.hamcrest.Matchers.hasSize;
+import static org.mockito.Mockito.mock;
+
+public class ApproximateTests extends ESTestCase {
+
+    private static final EsqlParser parser = new EsqlParser();
+    private static final Analyzer analyzer = AnalyzerTestUtils.defaultAnalyzer();
+    private static final LogicalPlanPreOptimizer preOptimizer = new LogicalPlanPreOptimizer(
+        new LogicalPreOptimizerContext(FoldContext.small(), mock(InferenceService.class))
+    );
+    private static final CircuitBreaker breaker = newLimitedBreaker(ByteSizeValue.ofGb(1));
+    private static final BigArrays bigArrays = new MockBigArrays(PageCacheRecycler.NON_RECYCLING_INSTANCE, ByteSizeValue.ofGb(1));
+    private static final MockBlockFactory blockFactory = new MockBlockFactory(breaker, bigArrays);
+
+    /**
+     * Runner that simulates the execution of an ESQL query.
+     * <p>
+     * The runner always returns a result with one field: the number of rows.
+     * <p>
+     * The runner is initialized with a total number of rows (returned when
+     * there are no filters in the query), and a number of filtered rows
+     * (returned when there are filters in the query). If there's random
+     * sampling in the query, the returned number of rows is multiplied by
+     * the sampling probability.
+     * <p>
+     * The runner collects all its invocations.
+     */
+    private static class TestRunner implements Approximate.LogicalPlanRunner {
+
+        private final long totalRows;
+        private final long filteredRows;
+        private final List<LogicalPlan> invocations;
+
+        static ActionListener<Result> resultCloser = ActionListener.wrap(result -> result.pages().getFirst().close(), e -> {});
+
+        TestRunner(long totalRows, long filteredRows) {
+            this.totalRows = totalRows;
+            this.filteredRows = filteredRows;
+            this.invocations = new ArrayList<>();
+        }
+
+        @Override
+        public void run(LogicalPlan logicalPlan, ActionListener<Result> listener) {
+            invocations.add(logicalPlan);
+            List<LogicalPlan> filters = logicalPlan.collect(plan -> plan instanceof Filter);
+            long numResults = filters.isEmpty() ? totalRows : filteredRows;
+            List<LogicalPlan> samples = logicalPlan.collect(plan -> plan instanceof Sample);
+            for (LogicalPlan sample : samples) {
+                numResults = (long) (numResults * (double) ((Literal) ((Sample) sample).probability()).value());
+            }
+            LongBlock block = blockFactory.newConstantLongBlockWith(numResults, 1);
+            listener.onResponse(new Result(null, List.of(new Page(block)), null, null));
+        }
+    }
+
+    @Override
+    protected List<String> filteredWarnings() {
+        return withDefaultLimitWarning(super.filteredWarnings());
+    }
+
+    public void testVerify_validQuery() throws Exception {
+        createApproximate("FROM test | WHERE emp_no<99 | SORT last_name | MV_EXPAND salary | STATS COUNT() BY gender");
+        createApproximate("FROM test | CHANGE_POINT salary ON emp_no | EVAL x=1 | DROP emp_no | STATS SUM(salary) BY x");
+        createApproximate("FROM test | LIMIT 1000 | KEEP gender, emp_no | RENAME gender AS whatever | STATS MEDIAN(emp_no)");
+        createApproximate("FROM test | EVAL blah=1 | GROK last_name \"%{IP:x}\" | SAMPLE 0.1 | STATS a=COUNT() | LIMIT 100 | SORT a");
+        createApproximate("ROW i=[1,2,3] | EVAL x=TO_STRING(i) | DISSECT x \"%{x}\" | STATS i=10*POW(PERCENTILE(i, 0.5), 2) | LIMIT 10");
+    }
+
+    public void testVerify_exactlyOneStats() {
+        assertError(
+            "FROM test | EVAL x = 1 | SORT emp_no | LIMIT 100 | MV_EXPAND x",
+            equalTo("line 1:1: query without [STATS] cannot be approximated")
+        );
+        assertError(
+            "FROM test | STATS COUNT() BY emp_no | STATS COUNT()",
+            equalTo("line 1:39: query with multiple [STATS] cannot be approximated")
+        );
+    }
+
+    public void testVerify_incompatibleSourceCommand() {
+        assertError("SHOW INFO | STATS COUNT()", equalTo("line 1:1: query with [SHOWINFO] cannot be approximated"));
+        assertError("TS test | STATS COUNT(emp_no)", equalTo("line 1:11: query with [TIMESERIESAGGREGATE] cannot be approximated"));
+    }
+
+    public void testVerify_incompatibleProcessingCommand() {
+        assertError(
+            "FROM test | FORK (EVAL x=1) (EVAL y=1) | STATS COUNT()",
+            equalTo("line 1:13: query with [FORK] cannot be approximated")
+        );
+        assertError(
+            "FROM test | STATS COUNT() | FORK (EVAL x=1) (EVAL y=1)",
+            equalTo("line 1:29: query with [FORK] cannot be approximated")
+        );
+        assertError(
+            "FROM test | INLINE STATS COUNT() | STATS COUNT()",
+            equalTo("line 1:13: query with [INLINESTATS] cannot be approximated")
+        );
+        assertError(
+            "FROM test | STATS COUNT() | INLINE STATS COUNT()",
+            equalTo("line 1:29: query with [INLINESTATS] cannot be approximated")
+        );
+        assertError(
+            "FROM test | LOOKUP JOIN test_lookup ON emp_no | FORK (EVAL x=1) (EVAL y=1) | STATS COUNT()",
+            equalTo("line 1:13: query with [LOOKUPJOIN] cannot be approximated")
+        );
+        assertError(
+            "FROM test | STATS emp_no=COUNT() | LOOKUP JOIN test_lookup ON emp_no | FORK (EVAL x=1) (EVAL y=1)",
+            equalTo("line 1:36: query with [LOOKUPJOIN] cannot be approximated")
+        );
+    }
+
+    public void testVerify_incompatibleAggregation() {
+        assertError(
+            "FROM test | SORT emp_no STATS MIN(emp_no) | LIMIT 100",
+            equalTo("line 1:19: aggregation function [MIN] cannot be approximated")
+        );
+        assertError(
+            "FROM test | STATS SUM(emp_no), VALUES(emp_no), TOP(emp_no, 2, \"ASC\"), COUNT()",
+            equalTo("line 1:32: aggregation function [VALUES] cannot be approximated")
+        );
+        assertError(
+            "FROM test | STATS 5+10*POW(MAX(emp_no), 2) BY gender",
+            equalTo("line 1:28: aggregation function [MAX] cannot be approximated")
+        );
+    }
+
+    public void testCountPlan_largeDataNoFilters() throws Exception {
+        Approximate approximate = createApproximate("FROM test | STATS SUM(emp_no)");
+        TestRunner runner = new TestRunner(1_000_000_000, 1_000_000_000);
+        approximate.approximate(runner, TestRunner.resultCloser);
+        // One pass is needed to get the number of rows, and approximation is executed immediately
+        // after that with the correct sample probability.
+        assertThat(runner.invocations, hasSize(2));
+        assertThat(runner.invocations.get(0), allOf(not(hasFilter("emp_no")), not(hasSample())));
+        assertThat(runner.invocations.get(1), allOf(not(hasFilter("emp_no")), hasSample(1e-4)));
+    }
+
+    public void testCountPlan_smallDataNoFilters() throws Exception {
+        Approximate approximate = createApproximate("FROM test | STATS SUM(emp_no)");
+        TestRunner runner = new TestRunner(1_000, 1_000);
+        approximate.approximate(runner, TestRunner.resultCloser);
+        // One pass is needed to get the number of rows, and the original query is executed
+        // immediately after that without sampling.
+        assertThat(runner.invocations, hasSize(2));
+        assertThat(runner.invocations.get(0), allOf(not(hasFilter("emp_no")), not(hasSample())));
+        assertThat(runner.invocations.get(1), allOf(not(hasFilter("emp_no")), not(hasSample())));
+    }
+
+    public void testCountPlan_largeDataAfterFiltering() throws Exception {
+        Approximate approximate = createApproximate("FROM test | WHERE emp_no < 1 | STATS SUM(emp_no)");
+        TestRunner runner = new TestRunner(1_000_000_000_000L, 1_000_000_000);
+        approximate.approximate(runner, TestRunner.resultCloser);
+        // One pass is needed to get the number of rows, then a few passes to get a good sample
+        // probability, and finally approximation is executed.
+        assertThat(runner.invocations, hasSize(4));
+        assertThat(runner.invocations.get(0), allOf(not(hasFilter("emp_no")), not(hasSample())));
+        assertThat(runner.invocations.get(1), allOf(hasFilter("emp_no"), hasSample(1e-7)));
+        assertThat(runner.invocations.get(2), allOf(hasFilter("emp_no"), hasSample(1e-4)));
+        assertThat(runner.invocations.get(3), allOf(hasFilter("emp_no"), hasSample(1e-4)));
+    }
+
+    public void testCountPlan_smallDataAfterFiltering() throws Exception {
+        Approximate approximate = createApproximate("FROM test | WHERE emp_no < 1 | STATS SUM(emp_no)");
+        TestRunner runner = new TestRunner(1_000_000_000_000_000_000L, 100);
+        approximate.approximate(runner, TestRunner.resultCloser);
+        // One pass is needed to get the number of rows, then a few passes to get a good sample
+        // probability, and finally the original query is executed without sampling.
+        assertThat(runner.invocations, hasSize(5));
+        assertThat(runner.invocations.get(0), allOf(not(hasFilter("emp_no")), not(hasSample())));
+        assertThat(runner.invocations.get(1), allOf(hasFilter("emp_no"), hasSample(1e-13)));
+        assertThat(runner.invocations.get(2), allOf(hasFilter("emp_no"), hasSample(1e-8)));
+        assertThat(runner.invocations.get(3), allOf(hasFilter("emp_no"), hasSample(1e-3)));
+        assertThat(runner.invocations.get(4), allOf(hasFilter("emp_no"), not(hasSample())));
+    }
+
+    public void testCountPlan_smallDataBeforeFiltering() throws Exception {
+        Approximate approximate = createApproximate("FROM test | WHERE emp_no < 1 | STATS SUM(emp_no)");
+        TestRunner runner = new TestRunner(1_000, 10);
+        approximate.approximate(runner, TestRunner.resultCloser);
+        // One pass is needed to get the number of rows, and the original query is executed
+        // immediately after that without sampling.
+        assertThat(runner.invocations, hasSize(2));
+        assertThat(runner.invocations.get(0), allOf(not(hasFilter("emp_no")), not(hasSample())));
+        assertThat(runner.invocations.get(1), allOf(hasFilter("emp_no"), not(hasSample())));
+    }
+
+    public void testApproximatePlan_createsConfidenceInterval() throws Exception {
+        Approximate approximate = createApproximate("FROM test | STATS SUM(emp_no)");
+        TestRunner runner = new TestRunner(1_000_000_000, 1_000_000_000);
+        approximate.approximate(runner, TestRunner.resultCloser);
+        // One pass is needed to get the number of rows, and approximation is executed immediately
+        // after that with the correct sample probability.
+        assertThat(runner.invocations, hasSize(2));
+
+        LogicalPlan approximatePlan = runner.invocations.get(1);
+        assertThat(approximatePlan, hasSample(1e-4));
+        assertThat(approximatePlan, hasEval("CONFIDENCE_INTERVAL(SUM(emp_no))"));
+    }
+
+    public void testApproximatePlan_dependentConfidenceIntervals() throws Exception {
+        Approximate approximate = createApproximate(
+            "FROM test | STATS x=COUNT() | EVAL a=x*x, b=7, c=TO_STRING(x), d=MV_APPEND(x, 1::LONG), e=a+POW(b, 2)"
+        );
+        TestRunner runner = new TestRunner(1_000_000_000, 1_000_000_000);
+        approximate.approximate(runner, TestRunner.resultCloser);
+        // One pass is needed to get the number of rows, and approximation is executed immediately
+        // after that with the correct sample probability.
+        assertThat(runner.invocations, hasSize(2));
+
+        LogicalPlan approximatePlan = runner.invocations.get(1);
+        assertThat(approximatePlan, hasPlan(Sample.class, s -> Foldables.literalValueOf(s.probability()).equals(1e-4)));
+        assertThat(approximatePlan, hasEval("CONFIDENCE_INTERVAL(x)"));
+        assertThat(approximatePlan, hasEval("CONFIDENCE_INTERVAL(a)"));
+        assertThat(approximatePlan, not(hasEval("CONFIDENCE_INTERVAL(b)")));
+        assertThat(approximatePlan, not(hasEval("CONFIDENCE_INTERVAL(c)")));
+        assertThat(approximatePlan, not(hasEval("CONFIDENCE_INTERVAL(d)")));
+        assertThat(approximatePlan, hasEval("CONFIDENCE_INTERVAL(e)"));
+    }
+
+    private Matcher<? super LogicalPlan> hasFilter(String field) {
+        return hasPlan(
+            Filter.class,
+            filter -> filter.condition().anyMatch(expr -> expr instanceof NamedExpression ne && ne.name().equals(field))
+        );
+    }
+
+    private Matcher<? super LogicalPlan> hasEval(String field) {
+        return hasPlan(Eval.class, eval -> eval.fields().stream().anyMatch(alias -> alias.name().equals(field)));
+    }
+
+    private Matcher<? super LogicalPlan> hasSample() {
+        return hasPlan(Sample.class, sample -> true);
+    }
+
+    private Matcher<? super LogicalPlan> hasSample(Double probability) {
+        return hasPlan(Sample.class, sample -> sample.probability().equals(Literal.fromDouble(Source.EMPTY, probability)));
+    }
+
+    private <E extends LogicalPlan> Matcher<? super LogicalPlan> hasPlan(Class<E> typeToken, Predicate<? super E> predicate) {
+        return new TypeSafeMatcher<>() {
+            @Override
+            @SuppressWarnings("unchecked")
+            protected boolean matchesSafely(LogicalPlan logicalPlan) {
+                return logicalPlan.anyMatch(plan -> plan.getClass() == typeToken && predicate.test((E) plan));
+            }
+
+            @Override
+            public void describeTo(Description description) {
+                description.appendText("a plan containing [" + typeToken.getSimpleName() + "] matching the predicate");
+            }
+        };
+    }
+
+    private void assertError(String esql, Matcher<String> matcher) {
+        Exception e = assertThrows(VerificationException.class, () -> createApproximate(esql));
+        assertThat(e.getMessage().substring("Found 1 problem\n".length()), matcher);
+    }
+
+    private Approximate createApproximate(String query) throws Exception {
+        SetOnce<LogicalPlan> resultHolder = new SetOnce<>();
+        SetOnce<Exception> exceptionHolder = new SetOnce<>();
+        LogicalPlan plan = parser.createStatement(query, TEST_CFG);
+        plan = analyzer.analyze(plan);
+        plan.setAnalyzed();
+        preOptimizer.preOptimize(plan, ActionListener.wrap(resultHolder::set, exceptionHolder::set));
+        if (exceptionHolder.get() != null) {
+            throw exceptionHolder.get();
+        }
+        return new Approximate(resultHolder.get());
+    }
+}