diff --git a/java/com/google/re2j/Compiler.java b/java/com/google/re2j/Compiler.java
index 2fc4b82e..ce056dfa 100644
--- a/java/com/google/re2j/Compiler.java
+++ b/java/com/google/re2j/Compiler.java
@@ -24,6 +24,7 @@ class Compiler {
   private static class Frag {
     final int i; // an instruction address (pc).
     int out; // a patch list; see explanation in Prog.java
+    boolean nullable; // whether the fragment can match the empty string
 
     Frag() {
       this(0, 0);
@@ -34,8 +35,13 @@ private static class Frag {
     }
 
     Frag(int i, int out) {
+      this(i, out, false);
+    }
+
+    Frag(int i, int out, boolean nullable) {
       this.i = i;
       this.out = out;
+      this.nullable = nullable;
     }
   }
 
@@ -56,7 +62,7 @@ static Prog compileRegexp(Regexp re) {
   private Frag newInst(int op) {
     // TODO(rsc): impose length limit.
     prog.addInst(op);
-    return new Frag(prog.numInst() - 1);
+    return new Frag(prog.numInst() - 1, 0, true);
   }
 
   // Returns a no-op fragment.  Sometimes unavoidable.
@@ -90,7 +96,7 @@ private Frag cat(Frag f1, Frag f2) {
     }
     // TODO(rsc): elide nop
     prog.patch(f1.out, f2.i);
-    return new Frag(f1.i, f2.out);
+    return new Frag(f1.i, f2.out, f1.nullable && f2.nullable);
   }
 
   // Given fragments for a and b, returns fragment for a|b.
@@ -107,11 +113,16 @@ private Frag alt(Frag f1, Frag f2) {
     i.out = f1.i;
     i.arg = f2.i;
     f.out = prog.append(f1.out, f2.out);
+    f.nullable = f1.nullable || f2.nullable;
     return f;
   }
 
-  // Given a fragment for a, returns a fragment for a? or a?? (if nongreedy)
-  private Frag quest(Frag f1, boolean nongreedy) {
+  // loop returns the fragment for the main loop of a plus or star.
+  // For plus, it can be used directly. with f1.i as the entry.
+  // For star, it can be used directly when f1 can't match an empty string.
+  // (When f1 can match an empty string, f1* must be implemented as (f1+)?
+  // to get the priority match order correct.)
+  private Frag loop(Frag f1, boolean nongreedy) {
     Frag f = newInst(Inst.ALT);
     Inst i = prog.getInst(f.i);
     if (nongreedy) {
@@ -121,12 +132,12 @@ private Frag quest(Frag f1, boolean nongreedy) {
       i.out = f1.i;
       f.out = f.i << 1 | 1;
     }
-    f.out = prog.append(f.out, f1.out);
+    prog.patch(f1.out, f.i);
     return f;
   }
 
-  // Given a fragment a, returns a fragment for a* or a*? (if nongreedy)
-  private Frag star(Frag f1, boolean nongreedy) {
+  // Given a fragment for a, returns a fragment for a? or a?? (if nongreedy)
+  private Frag quest(Frag f1, boolean nongreedy) {
     Frag f = newInst(Inst.ALT);
     Inst i = prog.getInst(f.i);
     if (nongreedy) {
@@ -136,13 +147,21 @@ private Frag star(Frag f1, boolean nongreedy) {
       i.out = f1.i;
       f.out = f.i << 1 | 1;
     }
-    prog.patch(f1.out, f.i);
+    f.out = prog.append(f.out, f1.out);
     return f;
   }
 
+  // Given a fragment a, returns a fragment for a* or a*? (if nongreedy)
+  private Frag star(Frag f1, boolean nongreedy) {
+    if (f1.nullable) {
+      return quest(plus(f1, nongreedy), nongreedy);
+    }
+    return loop(f1, nongreedy);
+  }
+
   // Given a fragment for a, returns a fragment for a+ or a+? (if nongreedy)
   private Frag plus(Frag f1, boolean nongreedy) {
-    return new Frag(f1.i, star(f1, nongreedy).out);
+    return new Frag(f1.i, loop(f1, nongreedy).out, f1.nullable);
   }
 
   // op is a bitmask of EMPTY_* flags.
@@ -160,6 +179,7 @@ private Frag rune(int rune, int flags) {
   // flags : parser flags
   private Frag rune(int[] runes, int flags) {
     Frag f = newInst(Inst.RUNE);
+    f.nullable = false;
     Inst i = prog.getInst(f.i);
     i.runes = runes;
     flags &= RE2.FOLD_CASE; // only relevant flag is FoldCase
diff --git a/javatests/com/google/re2j/FindTest.java b/javatests/com/google/re2j/FindTest.java
index 58b83d3e..12a983a9 100644
--- a/javatests/com/google/re2j/FindTest.java
+++ b/javatests/com/google/re2j/FindTest.java
@@ -249,6 +249,7 @@ public String toString() {
         35,
         35,
         36),
+    new Test("(|a)*", "aa", 3, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2),
   };
 
   @Parameters
@@ -538,7 +539,14 @@ private void testSubmatch(String testName, Test test, int n, String[] result) {
       System.err.println(testName + "  " + test + " " + n + " " + k + " ");
       String expect = test.submatchString(n, k / 2);
       if (!expect.equals(result[k / 2])) {
-        fail(String.format("%s %d: expected %s got %s: %s", testName, n, expect, result, test));
+        fail(
+            String.format(
+                "%s %d: expected %s got %s: %s",
+                testName,
+                n,
+                expect,
+                Arrays.toString(result),
+                test));
       }
     }
   }
diff --git a/javatests/com/google/re2j/ProgTest.java b/javatests/com/google/re2j/ProgTest.java
index da70615f..4abc3ecf 100644
--- a/javatests/com/google/re2j/ProgTest.java
+++ b/javatests/com/google/re2j/ProgTest.java
@@ -94,6 +94,25 @@ public class ProgTest {
       "(?:(?:^).)",
       "0       fail\n" + "1*      empty 4 -> 2\n" + "2       anynotnl -> 3\n" + "3       match\n"
     },
+    {
+      "(?:|a)+",
+      "0       fail\n"
+          + "1       nop -> 4\n"
+          + "2       rune1 \"a\" -> 4\n"
+          + "3*      alt -> 1, 2\n"
+          + "4       alt -> 3, 5\n"
+          + "5       match\n"
+    },
+    {
+      "(?:|a)*",
+      "0       fail\n"
+          + "1       nop -> 4\n"
+          + "2       rune1 \"a\" -> 4\n"
+          + "3       alt -> 1, 2\n"
+          + "4       alt -> 3, 6\n"
+          + "5*      alt -> 3, 6\n"
+          + "6       match\n"
+    },
   };
 
   private final String input;
diff --git a/javatests/com/google/re2j/SimplifyTest.java b/javatests/com/google/re2j/SimplifyTest.java
index bd5b5e64..7f29c377 100644
--- a/javatests/com/google/re2j/SimplifyTest.java
+++ b/javatests/com/google/re2j/SimplifyTest.java
@@ -11,6 +11,7 @@
 
 import static org.junit.Assert.assertEquals;
 
+import com.google.common.truth.Truth;
 import org.junit.Test;
 import org.junit.runner.RunWith;
 import org.junit.runners.Parameterized;
diff --git a/testdata/basic.dat b/testdata/basic.dat
index 7859290b..1776b1ff 100644
--- a/testdata/basic.dat
+++ b/testdata/basic.dat
@@ -124,24 +124,20 @@ E	((a))			abc		(0,1)(0,1)(0,1)
 E	(a)b(c)			abc		(0,3)(0,1)(2,3)
 E	a+b+c			aabbabc		(4,7)
 E	a*			aaa		(0,3)
-#E	(a*)*			-		(0,0)(0,0)
-E	(a*)*			-		(0,0)(?,?)	RE2/Go
+E	(a*)*			-		(0,0)(0,0)
 E	(a*)+			-		(0,0)(0,0)
-#E	(a*|b)*			-		(0,0)(0,0)
-E	(a*|b)*			-		(0,0)(?,?)	RE2/Go
+E	(a*|b)*			-		(0,0)(0,0)
 E	(a+|b)*			ab		(0,2)(1,2)
 E	(a+|b)+			ab		(0,2)(1,2)
 E	(a+|b)?			ab		(0,1)(0,1)
 BE	[^ab]*			cde		(0,3)
-#E	(^)*			-		(0,0)(0,0)
-E	(^)*			-		(0,0)(?,?)	RE2/Go
+E	(^)*			-		(0,0)(0,0)
 BE	a*			NULL		(0,0)
 E	([abc])*d		abbbcd		(0,6)(4,5)
 E	([abc])*bcd		abcd		(0,4)(0,1)
 E	a|b|c|d|e		e		(0,1)
 E	(a|b|c|d|e)f		ef		(0,2)(0,1)
-#E	((a*|b))*		-		(0,0)(0,0)(0,0)
-E	((a*|b))*		-		(0,0)(?,?)(?,?)	RE2/Go
+E	((a*|b))*		-		(0,0)(0,0)(0,0)
 BE	abcd*efg		abcdefg		(0,7)
 BE	ab*			xabyabbbz	(1,3)
 BE	ab*			xayabbbz	(1,2)
diff --git a/testdata/nullsubexpr.dat b/testdata/nullsubexpr.dat
index 2e18fbb9..68d9c999 100644
--- a/testdata/nullsubexpr.dat
+++ b/testdata/nullsubexpr.dat
@@ -1,8 +1,7 @@
 NOTE	null subexpression matches : 2002-06-06
 
 E	(a*)*		a		(0,1)(0,1)
-#E	SAME		x		(0,0)(0,0)
-E	SAME		x		(0,0)(?,?)	RE2/Go
+E	SAME		x		(0,0)(0,0)
 E	SAME		aaaaaa		(0,6)(0,6)
 E	SAME		aaaaaax		(0,6)(0,6)
 E	(a*)+		a		(0,1)(0,1)
@@ -19,8 +18,7 @@ E	SAME		aaaaaa		(0,6)(0,6)
 E	SAME		aaaaaax		(0,6)(0,6)
 
 E	([a]*)*		a		(0,1)(0,1)
-#E	SAME		x		(0,0)(0,0)
-E	SAME		x		(0,0)(?,?)	RE2/Go
+E	SAME		x		(0,0)(0,0)
 E	SAME		aaaaaa		(0,6)(0,6)
 E	SAME		aaaaaax		(0,6)(0,6)
 E	([a]*)+		a		(0,1)(0,1)
@@ -28,8 +26,7 @@ E	SAME		x		(0,0)(0,0)
 E	SAME		aaaaaa		(0,6)(0,6)
 E	SAME		aaaaaax		(0,6)(0,6)
 E	([^b]*)*	a		(0,1)(0,1)
-#E	SAME		b		(0,0)(0,0)
-E	SAME		b		(0,0)(?,?)	RE2/Go
+E	SAME		b		(0,0)(0,0)
 E	SAME		aaaaaa		(0,6)(0,6)
 E	SAME		aaaaaab		(0,6)(0,6)
 E	([ab]*)*	a		(0,1)(0,1)
@@ -41,11 +38,9 @@ E	SAME		bbbbbb		(0,6)(0,6)
 E	SAME		aaaabcde	(0,5)(0,5)
 E	([^a]*)*	b		(0,1)(0,1)
 E	SAME		bbbbbb		(0,6)(0,6)
-#E	SAME		aaaaaa		(0,0)(0,0)
-E	SAME		aaaaaa		(0,0)(?,?)	RE2/Go
+E	SAME		aaaaaa		(0,0)(0,0)
 E	([^ab]*)*	ccccxx		(0,6)(0,6)
-#E	SAME		ababab		(0,0)(0,0)
-E	SAME		ababab		(0,0)(?,?)	RE2/Go
+E	SAME		ababab		(0,0)(0,0)
 
 E	((z)+|a)*	zabcde		(0,2)(1,2)
 
@@ -65,8 +60,7 @@ B	\(a*\)*\(x\)\(\1\)	axa	(0,3)(0,1)(1,2)(2,3)
 B	\(a*\)*\(x\)\(\1\)\(x\)	axax	(0,4)(0,1)(1,2)(2,3)(3,4)
 B	\(a*\)*\(x\)\(\1\)\(x\)	axxa	(0,3)(1,1)(1,2)(2,2)(2,3)
 
-#E	(a*)*(x)		x	(0,1)(0,0)(0,1)
-E	(a*)*(x)		x	(0,1)(?,?)(0,1)	RE2/Go
+E	(a*)*(x)		x	(0,1)(0,0)(0,1)
 E	(a*)*(x)		ax	(0,2)(0,1)(1,2)
 E	(a*)*(x)		axa	(0,2)(0,1)(1,2)
 
diff --git a/testdata/re2-exhaustive.txt.gz b/testdata/re2-exhaustive.txt.gz
index c6eebf00..4482caf9 100644
Binary files a/testdata/re2-exhaustive.txt.gz and b/testdata/re2-exhaustive.txt.gz differ
diff --git a/testdata/re2-search.txt b/testdata/re2-search.txt
index f648e552..8c4098a4 100644
--- a/testdata/re2-search.txt
+++ b/testdata/re2-search.txt
@@ -1,5 +1,5 @@
 # RE2 basic search tests built by make log
-# Thu Sep 8 13:43:43 EDT 2011
+# Wed May 12 12:13:22 EDT 2021
 Regexp.SearchTests
 strings
 ""
@@ -227,22 +227,6 @@ regexps
 0-0;0-0;0-0;0-0
 strings
 ""
-""
-regexps
-"a*"
-0-0;0-0;0-0;0-0
-0-0;0-0;0-0;0-0
-"^(?:a*)$"
-0-0;0-0;0-0;0-0
-0-0;0-0;0-0;0-0
-"^(?:a*)"
-0-0;0-0;0-0;0-0
-0-0;0-0;0-0;0-0
-"(?:a*)$"
-0-0;0-0;0-0;0-0
-0-0;0-0;0-0;0-0
-strings
-""
 "xabcdx"
 regexps
 "ab|cd"
@@ -3651,6 +3635,86 @@ regexps
 0-1;0-1;0-1;0-1
 strings
 ""
+"a"
+regexps
+"a\\C+"
+-;-;-;-
+-;-;-;-
+"^(?:a\\C+)$"
+-;-;-;-
+-;-;-;-
+"^(?:a\\C+)"
+-;-;-;-
+-;-;-;-
+"(?:a\\C+)$"
+-;-;-;-
+-;-;-;-
+strings
+""
+"a"
+regexps
+"a\\C?"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"^(?:a\\C?)$"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"^(?:a\\C?)"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"(?:a\\C?)$"
+-;-;-;-
+0-1;0-1;0-1;0-1
+strings
+""
+"a"
+regexps
+"a\\C*?"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"^(?:a\\C*?)$"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"^(?:a\\C*?)"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"(?:a\\C*?)$"
+-;-;-;-
+0-1;0-1;0-1;0-1
+strings
+""
+"a"
+regexps
+"a\\C+?"
+-;-;-;-
+-;-;-;-
+"^(?:a\\C+?)$"
+-;-;-;-
+-;-;-;-
+"^(?:a\\C+?)"
+-;-;-;-
+-;-;-;-
+"(?:a\\C+?)$"
+-;-;-;-
+-;-;-;-
+strings
+""
+"a"
+regexps
+"a\\C??"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"^(?:a\\C??)$"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"^(?:a\\C??)"
+-;-;-;-
+0-1;0-1;0-1;0-1
+"(?:a\\C??)$"
+-;-;-;-
+0-1;0-1;0-1;0-1
+strings
+""
 "baba"
 regexps
 "a\\C*|ba\\C"
@@ -3665,3 +3729,51 @@ regexps
 "(?:a\\C*|ba\\C)$"
 -;-;-;-
 -;1-4;-;1-4
+strings
+""
+"Inc."
+regexps
+"\\w*I\\w*"
+-;-;-;-
+-;0-3;-;0-3
+"^(?:\\w*I\\w*)$"
+-;-;-;-
+-;-;-;-
+"^(?:\\w*I\\w*)"
+-;-;-;-
+-;0-3;-;0-3
+"(?:\\w*I\\w*)$"
+-;-;-;-
+-;-;-;-
+strings
+""
+"aaa"
+regexps
+"(?:|a)*"
+0-0;0-0;0-0;0-0
+0-3;0-0;0-3;0-3
+"^(?:(?:|a)*)$"
+0-0;0-0;0-0;0-0
+0-3;0-3;0-3;0-3
+"^(?:(?:|a)*)"
+0-0;0-0;0-0;0-0
+0-3;0-0;0-3;0-3
+"(?:(?:|a)*)$"
+0-0;0-0;0-0;0-0
+0-3;0-3;0-3;0-3
+strings
+""
+"aaa"
+regexps
+"(?:|a)+"
+0-0;0-0;0-0;0-0
+0-3;0-0;0-3;0-3
+"^(?:(?:|a)+)$"
+0-0;0-0;0-0;0-0
+0-3;0-3;0-3;0-3
+"^(?:(?:|a)+)"
+0-0;0-0;0-0;0-0
+0-3;0-0;0-3;0-3
+"(?:(?:|a)+)$"
+0-0;0-0;0-0;0-0
+0-3;0-3;0-3;0-3