diff --git a/sqlite_utils/db.py b/sqlite_utils/db.py
index dcd3b5f..5972032 100644
--- a/sqlite_utils/db.py
+++ b/sqlite_utils/db.py
@@ -219,6 +219,11 @@ class AlterError(Exception):
     pass
 
 
+class FunctionAlreadyRegistered(Exception):
+    "A function with this name and arity was already registered"
+    pass
+
+
 class NoObviousTable(Exception):
     "Could not tell which table this operation refers to"
     pass
@@ -409,7 +414,7 @@ class Database:
             fn_name = name or fn.__name__
             arity = len(inspect.signature(fn).parameters)
             if not replace and (fn_name, arity) in self._registered_functions:
-                return fn
+                raise FunctionAlreadyRegistered(f'Already registered function with name "{fn_name}" and identical arity')
             kwargs = {}
             registered = False
             if deterministic:
@@ -434,7 +439,7 @@ class Database:
 
     def register_fts4_bm25(self):
         "Register the ``rank_bm25(match_info)`` function used for calculating relevance with SQLite FTS4."
-        self.register_function(rank_bm25, deterministic=True)
+        self.register_function(rank_bm25, deterministic=True, replace=True)
 
     def attach(self, alias: str, filepath: Union[str, pathlib.Path]):
         """
@@ -2687,13 +2692,16 @@ class Table(Queryable):
                     return v
                 return jsonify_if_needed(fn(v))
 
-            self.db.register_function(convert_value)
+            fn_name = fn.__name__
+            if fn_name == '<lambda>':
+                fn_name = f'lambda_{hash(fn)}'
+            self.db.register_function(convert_value, name=fn_name)
             sql = "update [{table}] set {sets}{where};".format(
                 table=self.name,
                 sets=", ".join(
                     [
-                        "[{output_column}] = convert_value([{column}])".format(
-                            output_column=output or column, column=column
+                        "[{output_column}] = {fn_name}([{column}])".format(
+                            output_column=output or column, column=column, fn_name=fn_name
                         )
                         for column in columns
                     ]
diff --git a/tests/test_convert.py b/tests/test_convert.py
index 9fd29c3..8fffb4a 100644
--- a/tests/test_convert.py
+++ b/tests/test_convert.py
@@ -147,3 +147,12 @@ def test_convert_multi_exception(fresh_db):
     table.insert({"title": "Mixed Case"})
     with pytest.raises(BadMultiValues):
         table.convert("title", lambda v: v.upper(), multi=True)
+
+
+def test_convert_repeated(fresh_db):
+    table = fresh_db["table"]
+    col = "num"
+    table.insert({col: 1})
+    table.convert(col, lambda x: x*2)
+    table.convert(col, lambda _x: 0)
+    assert table.get(1) == {col: 0}
diff --git a/tests/test_register_function.py b/tests/test_register_function.py
index 5169a67..05721ea 100644
--- a/tests/test_register_function.py
+++ b/tests/test_register_function.py
@@ -3,7 +3,7 @@ import pytest
 import sys
 from unittest.mock import MagicMock, call
 from sqlite_utils.utils import sqlite3
-
+from sqlite_utils.db import FunctionAlreadyRegistered
 
 def test_register_function(fresh_db):
     @fresh_db.register_function
@@ -85,9 +85,10 @@ def test_register_function_replace(fresh_db):
     assert "one" == fresh_db.execute("select one()").fetchone()[0]
 
     # This will fail to replace the function:
-    @fresh_db.register_function()
-    def one():  # noqa
-        return "two"
+    with pytest.raises(FunctionAlreadyRegistered):
+        @fresh_db.register_function()
+        def one():  # noqa
+            return "two"
 
     assert "one" == fresh_db.execute("select one()").fetchone()[0]
 
@@ -97,3 +98,11 @@ def test_register_function_replace(fresh_db):
         return "two"
 
     assert "two" == fresh_db.execute("select one()").fetchone()[0]
+
+
+def test_register_function_duplicate(fresh_db):
+    def to_lower(s):
+        return s.lower()
+    fresh_db.register_function(to_lower)
+    with pytest.raises(FunctionAlreadyRegistered):
+        fresh_db.register_function(to_lower)