diff --git a/examples/dagrt_fusion.py b/examples/dagrt_fusion.py
new file mode 100644
index 0000000000000000000000000000000000000000..1099be1163f99f4a1a11b7a654cce6060984d9b7
--- /dev/null
+++ b/examples/dagrt_fusion.py
@@ -0,0 +1,170 @@
+import pyopencl as cl
+from grudge.models.wave import StrongWaveOperator
+from leap.rk import LSRK4Method
+import numpy as np  # noqa
+
+import dagrt.language as lang
+import pymbolic.primitives as p
+import grudge.symbolic.mappers as gmap
+from pymbolic.mapper.evaluator import EvaluationMapper \
+        as PymbolicEvaluationMapper
+
+from grudge import sym, bind, DGDiscretizationWithBoundaries
+
+
+def topological_sort(stmts, root_deps):
+    id_to_stmt = {stmt.id: stmt for stmt in stmts}
+
+    ordered_stmts = []
+    satisfied = set()
+
+    def satisfy_dep(name):
+        if name in satisfied:
+            return
+
+        stmt = id_to_stmt[name]
+        for dep in stmt.depends_on:
+            satisfy_dep(dep)
+        ordered_stmts.append(stmt)
+        satisfied.add(name)
+
+    for d in root_deps:
+        satisfy_dep(d)
+
+    return ordered_stmts
+
+
+# Use evaluation, not identity mappers to propagate symbolic vectors to
+# outermost level.
+
+class DagrtToGrudgeRewriter(PymbolicEvaluationMapper):
+    def __init__(self, context):
+        self.context = context
+
+    def map_variable(self, expr):
+        return self.context[expr.name]
+
+    def map_call(self, expr):
+        raise ValueError("function call not expected")
+
+
+class GrudgeArgSubstitutor(gmap.SymbolicEvaluator):
+    def __init__(self, args):
+        self.args = args
+
+    def map_grudge_variable(self, expr):
+        if expr.name in self.args:
+            return self.args[expr.name]
+        else:
+            return super().map_variable(expr)
+
+
+def transcribe_phase(dag, phase_name, rhs_name, sym_operator):
+    sym_operator = gmap.OperatorBinder()(sym_operator)
+
+    phase = dag.phases[phase_name]
+
+    ctx = {
+            "<t>": sym.var("input_t", sym.DD_SCALAR),
+            "<dt>": sym.var("input_dt", sym.DD_SCALAR),
+            "<state>w": sym.make_sym_array("input_w", 3),
+            "<p>residual_w": sym.var("input_residual")
+          }
+    output_vars = [v for v in ctx.keys()]
+    yielded_states = []
+
+    from dagrt.codegen.transform import isolate_function_calls_in_phase
+    ordered_stmts = topological_sort(
+        isolate_function_calls_in_phase(
+            phase,
+            dag.get_stmt_id_generator(),
+            dag.get_var_name_generator()).statements,
+        phase.depends_on)
+
+    for stmt in ordered_stmts:
+        if stmt.condition is not True:
+            raise NotImplementedError(
+                "non-True condition (in statement '%s') not supported"
+                % stmt.id)
+
+        if isinstance(stmt, lang.Nop):
+            pass
+        elif isinstance(stmt, lang.AssignExpression):
+            if not isinstance(stmt.lhs, p.Variable):
+                raise NotImplementedError("lhs of statement %s is not a variable: %s"
+                        % (stmt.id, stmt.lhs))
+            ctx[stmt.lhs.name] = sym.cse(
+                DagrtToGrudgeRewriter(ctx)(stmt.rhs),
+                (
+                    stmt.lhs.name
+                    .replace("<", "")
+                    .replace(">", "")))
+        elif isinstance(stmt, lang.AssignFunctionCall):
+            if stmt.function_id != rhs_name:
+                raise NotImplementedError(
+                        "statement '%s' calls unsupported function '%s'"
+                        % (stmt.id, stmt.function_id))
+            if stmt.parameters:
+                raise NotImplementedError(
+                    "statement '%s' calls function '%s' with positional arguments"
+                    % (stmt.id, stmt.function_id))
+            kwargs = {name: sym.cse(DagrtToGrudgeRewriter(ctx)(arg))
+                      for name, arg in stmt.kw_parameters.items()}
+
+            if len(stmt.assignees) != 1:
+                raise NotImplementedError(
+                    "statement '%s' calls function '%s' "
+                    "with more than one LHS"
+                    % (stmt.id, stmt.function_id))
+
+            assignee, = stmt.assignees
+            ctx[assignee] = GrudgeArgSubstitutor(
+                kwargs)(sym_operator)
+        elif isinstance(stmt, lang.YieldState):
+            d2g = DagrtToGrudgeRewriter(ctx)
+            yielded_states.append(
+                (stmt.time_id, d2g(stmt.time), stmt.component_id,
+                    d2g(stmt.expression)))
+        else:
+            raise NotImplementedError("statement %s is of unsupported type ''%s'"
+                        % (stmt.id, type(stmt).__name__))
+
+    return output_vars, [ctx[ov] for ov in output_vars], yielded_states
+
+
+def main():
+    dt_method = LSRK4Method(component_id="w")
+    dt_code = dt_method.generate()
+
+    from meshmode.mesh import BTAG_ALL, BTAG_NONE
+    op = StrongWaveOperator(-0.1, 2,
+            dirichlet_tag=BTAG_NONE,
+            neumann_tag=BTAG_NONE,
+            radiation_tag=BTAG_ALL,
+            flux_type="upwind")
+    sym_op = op.sym_operator()
+
+    ov, results, ys = transcribe_phase(dt_code, "primary", "<func>w", sym_op)
+    for ov_i, res_i in zip(ov, results):
+        print(75*"#")
+        print(sym.pretty(res_i))
+        print(f"{ov_i} IS THE ABOVE ^")
+
+    from meshmode.mesh.generation import generate_regular_rect_mesh
+    mesh = generate_regular_rect_mesh(
+            a=(-0.5,)*2,
+            b=(0.5,)*2,
+            n=(16,)*2)
+
+    cl_ctx = cl.create_some_context()
+    discr = DGDiscretizationWithBoundaries(cl_ctx, mesh, order=3)
+
+    from pytools.obj_array import join_fields
+    results = join_fields(results[0], results[1], *results[2], *results[3])
+    bound_op = bind(discr, results)
+
+    print(bound_op.eval_code)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/grudge/execution.py b/grudge/execution.py
index c20aa4bc6271f1b9a8f5620c8b3021c22563880c..89db0a1aaae757d8fb9f54089328757fa4a9897f 100644
--- a/grudge/execution.py
+++ b/grudge/execution.py
@@ -31,6 +31,7 @@ from pytools import memoize_in
 
 import grudge.symbolic.mappers as mappers
 from grudge import sym
+from grudge.function_registry import base_function_registry
 
 import logging
 logger = logging.getLogger(__name__)
@@ -50,6 +51,7 @@ class ExecutionMapper(mappers.Evaluator,
         super(ExecutionMapper, self).__init__(context)
         self.discrwb = bound_op.discrwb
         self.bound_op = bound_op
+        self.function_registry = bound_op.function_registry
         self.queue = queue
 
     # {{{ expression mappings -------------------------------------------------
@@ -95,45 +97,8 @@ class ExecutionMapper(mappers.Evaluator,
         return value
 
     def map_call(self, expr):
-        from pymbolic.primitives import Variable
-        assert isinstance(expr.function, Variable)
-
-        # FIXME: Make a way to register functions
-
         args = [self.rec(p) for p in expr.parameters]
-        from numbers import Number
-        representative_arg = args[0]
-        if (
-                isinstance(representative_arg,  Number)
-                or (isinstance(representative_arg, np.ndarray)
-                    and representative_arg.shape == ())):
-            func = getattr(np, expr.function.name)
-            return func(*args)
-
-        cached_name = "map_call_knl_"
-
-        i = Variable("i")
-        func = Variable(expr.function.name)
-        if expr.function.name == "fabs":  # FIXME
-            func = Variable("abs")
-            cached_name += "abs"
-        else:
-            cached_name += expr.function.name
-
-        @memoize_in(self.bound_op, cached_name)
-        def knl():
-            knl = lp.make_kernel(
-                "{[i]: 0<=i<n}",
-                [
-                    lp.Assignment(Variable("out")[i],
-                        func(Variable("a")[i]))
-                ], default_offset=lp.auto)
-            return lp.split_iname(knl, "i", 128, outer_tag="g.0", inner_tag="l.0")
-
-        assert len(args) == 1
-        evt, (out,) = knl()(self.queue, a=args[0])
-
-        return out
+        return self.function_registry[expr.function.name](self.queue, *args)
 
     def map_nodal_sum(self, op, field_expr):
         # FIXME: Could allow array scalars
@@ -473,12 +438,15 @@ class MPISendFuture(object):
 # {{{ bound operator
 
 class BoundOperator(object):
-    def __init__(self, discrwb, discr_code, eval_code, debug_flags, allocator=None):
+    
+    def __init__(self, discrwb, discr_code, eval_code, debug_flags,
+            function_registry, allocator=None):
         self.discrwb = discrwb
         self.discr_code = discr_code
         self.eval_code = eval_code
         self.operator_data_cache = {}
         self.debug_flags = debug_flags
+        self.function_registry = function_registry
         self.allocator = allocator
 
     def __str__(self):
@@ -625,7 +593,8 @@ def process_sym_operator(discrwb, sym_operator, post_bind_mapper=None,
 
 
 def bind(discr, sym_operator, post_bind_mapper=lambda x: x,
-        debug_flags=set(), allocator=None):
+        debug_flags=set(), allocator=None,
+        function_registry=base_function_registry):
     # from grudge.symbolic.mappers import QuadratureUpsamplerRemover
     # sym_operator = QuadratureUpsamplerRemover(self.quad_min_degrees)(
     #         sym_operator)
@@ -648,9 +617,10 @@ def bind(discr, sym_operator, post_bind_mapper=lambda x: x,
             dumper=dump_sym_operator)
 
     from grudge.symbolic.compiler import OperatorCompiler
-    discr_code, eval_code = OperatorCompiler(discr)(sym_operator)
+    discr_code, eval_code = OperatorCompiler(discr, function_registry)(sym_operator)
 
     bound_op = BoundOperator(discr, discr_code, eval_code,
+            function_registry=function_registry,
             debug_flags=debug_flags, allocator=allocator)
 
     if "dump_op_code" in debug_flags:
diff --git a/grudge/function_registry.py b/grudge/function_registry.py
new file mode 100644
index 0000000000000000000000000000000000000000..bac92cc34e10f6b4d517849ff4237131e76db656
--- /dev/null
+++ b/grudge/function_registry.py
@@ -0,0 +1,196 @@
+from __future__ import division, with_statement
+
+__copyright__ = """
+Copyright (C) 2013 Andreas Kloeckner
+Copyright (C) 2019 Matt Wala
+"""
+
+__license__ = """
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in
+all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
+THE SOFTWARE.
+"""
+
+
+import loopy as lp
+import numpy as np
+
+from pytools import RecordWithoutPickling, memoize_in
+
+
+# {{{ function
+
+class FunctionNotFound(KeyError):
+    pass
+
+
+class Function(RecordWithoutPickling):
+    """
+    .. attribute:: identifier
+    .. attribute:: supports_codegen
+    .. automethod:: __call__
+    .. automethod:: get_result_dofdesc
+    """
+
+    def __init__(self, identifier, **kwargs):
+        super(Function, self).__init__(identifier=identifier, **kwargs)
+
+    def __call__(self, queue, *args, **kwargs):
+        """Call the function implementation, if available."""
+        raise TypeError("function '%s' is not callable" % self.identifier)
+
+    def get_result_dofdesc(self, arg_dds):
+        """Return the :class:`grudge.symbolic.primitives.DOFDesc` for the return value
+        of the function.
+
+        :arg arg_dds: A list of :class:`grudge.symbolic.primitives.DOFDesc` instances
+            for each argument
+        """
+        raise NotImplementedError
+
+
+class CElementwiseUnaryFunction(Function):
+
+    supports_codegen = True
+
+    def get_result_dofdesc(self, arg_dds):
+        assert len(arg_dds) == 1
+        return arg_dds[0]
+
+    def __call__(self, queue, *args):
+        assert len(args) == 1
+
+        func_name = self.identifier
+
+        arg, = args
+        from numbers import Number
+        if (
+                isinstance(arg, Number)
+                or (isinstance(arg, np.ndarray)
+                    and arg.shape == ())):
+            func = getattr(np, func_name)
+            return func(arg)
+
+        cached_name = "map_call_knl_"
+
+        from pymbolic.primitives import Variable
+        i = Variable("i")
+
+        if self.identifier == "fabs":  # FIXME
+            func_name = "abs"
+
+        cached_name += func_name
+
+        @memoize_in(self, cached_name)
+        def knl():
+            knl = lp.make_kernel(
+                "{[i]: 0<=i<n}",
+                [
+                    lp.Assignment(Variable("out")[i],
+                        Variable(func_name)(Variable("a")[i]))
+                ], default_offset=lp.auto)
+            return lp.split_iname(knl, "i", 128, outer_tag="g.0", inner_tag="l.0")
+
+        evt, (out,) = knl()(queue, a=arg)
+        return out
+
+
+class CBesselFunction(Function):
+
+    supports_codegen = True
+
+    def get_result_dofdesc(self, arg_dds):
+        assert len(arg_dds) == 2
+        return arg_dds[1]
+
+
+class FixedDOFDescExternalFunction(Function):
+
+    def __init__(self, identifier, implementation, dd):
+        super(FixedDOFDescExternalFunction, self).__init__(
+                identifier,
+                implementation=implementation,
+                dd=dd)
+
+    def __call__(self, queue, *args, **kwargs):
+        return self.implementation(queue, *args, **kwargs)
+
+    def get_result_dofdesc(self, arg_dds):
+        return self.dd
+
+# }}}
+
+
+# {{{ function registry
+
+class FunctionRegistry(RecordWithoutPickling):
+    def __init__(self, id_to_function=None):
+        if id_to_function is None:
+            id_to_function = {}
+
+        super(FunctionRegistry, self).__init__(
+                id_to_function=id_to_function)
+
+    def register(self, function):
+        """Return a copy of *self* with *function* registered."""
+
+        if function.identifier in self.id_to_function:
+            raise ValueError("function '%s' is already registered"
+                    % function.identifier)
+
+        new_id_to_function = self.id_to_function.copy()
+        new_id_to_function[function.identifier] = function
+        return self.copy(id_to_function=new_id_to_function)
+
+    def __getitem__(self, function_id):
+        try:
+            return self.id_to_function[function_id]
+        except KeyError:
+            raise FunctionNotFound(
+                    "unknown function: '%s'"
+                    % function_id)
+
+    def __contains__(self, function_id):
+        return function_id in self.id_to_function
+
+# }}}
+
+
+def _make_bfr():
+    bfr = FunctionRegistry()
+
+    bfr = bfr.register(CElementwiseUnaryFunction("sqrt"))
+    bfr = bfr.register(CElementwiseUnaryFunction("exp"))
+    bfr = bfr.register(CElementwiseUnaryFunction("fabs"))
+    bfr = bfr.register(CElementwiseUnaryFunction("sin"))
+    bfr = bfr.register(CElementwiseUnaryFunction("cos"))
+    bfr = bfr.register(CBesselFunction("bessel_j"))
+    bfr = bfr.register(CBesselFunction("bessel_y"))
+
+    return bfr
+
+
+base_function_registry = _make_bfr()
+
+
+def register_external_function(
+        function_registry, identifier, implementation, dd):
+    return function_registry.register(
+            FixedDOFDescExternalFunction(
+                identifier, implementation, dd))
+
+# vim: foldmethod=marker
diff --git a/grudge/symbolic/compiler.py b/grudge/symbolic/compiler.py
index b27d55855237b5e0200b094c2aa164e192022f0b..7672444574915d2bdb26fe5518a4f555e9c1712d 100644
--- a/grudge/symbolic/compiler.py
+++ b/grudge/symbolic/compiler.py
@@ -31,7 +31,7 @@ from six.moves import zip, reduce
 from pytools import Record, memoize_method, memoize
 from grudge import sym
 import grudge.symbolic.mappers as mappers
-from pymbolic.primitives import Variable, Subscript
+from pymbolic.primitives import Variable, Subscript, Call
 from six.moves import intern
 
 from loopy.version import LOOPY_USE_LANGUAGE_VERSION_2018_1  # noqa: F401
@@ -447,7 +447,8 @@ class Code(object):
         available_insns = [
                 (insn, insn.priority) for insn in self.instructions
                 if insn not in done_insns
-                and all(dep.name in available_names
+                and all((dep.aggregate.name if isinstance(dep, Subscript)
+                    else dep.name) in available_names
                     for dep in insn.get_dependencies())]
 
         if not available_insns:
@@ -455,7 +456,8 @@ class Code(object):
 
         from pytools import flatten
         discardable_vars = set(available_names) - set(flatten(
-            [dep.name for dep in insn.get_dependencies()]
+            [dep.aggregate.name if isinstance(dep, Subscript) else dep.name
+            for dep in insn.get_dependencies()]
             for insn in self.instructions
             if insn not in done_insns))
 
@@ -607,6 +609,8 @@ def aggregate_assignments(inf_mapper, instructions, result,
         max_vectors_in_batch_expr):
     from pymbolic.primitives import Variable
 
+    function_registry = inf_mapper.function_registry
+
     # {{{ aggregation helpers
 
     def get_complete_origins_set(insn, skip_levels=0):
@@ -674,6 +678,7 @@ def aggregate_assignments(inf_mapper, instructions, result,
                 isinstance(insn, Assign)
                 and not isinstance(insn, ToDiscretizationScopedAssign)
                 and not isinstance(insn, FromDiscretizationScopedAssign)
+                and not is_external_call(insn.exprs[0], function_registry)
                 and not any(
                     inf_mapper.infer_for_name(n).domain_tag == DTAG_SCALAR
                     for n in insn.names)),
@@ -824,9 +829,21 @@ def aggregate_assignments(inf_mapper, instructions, result,
 
 # {{{ to-loopy mapper
 
+def is_external_call(expr, function_registry):
+    if not isinstance(expr, Call):
+        return False
+    return not is_function_loopyable(expr.function, function_registry)
+
+
+def is_function_loopyable(function, function_registry):
+    assert isinstance(function, Variable)
+    return function_registry[function.name].supports_codegen
+
+
 class ToLoopyExpressionMapper(mappers.IdentityMapper):
     def __init__(self, dd_inference_mapper, temp_names, iname):
         self.dd_inference_mapper = dd_inference_mapper
+        self.function_registry = dd_inference_mapper.function_registry
         self.temp_names = temp_names
         self.iname = iname
         from pymbolic import var
@@ -887,8 +904,9 @@ class ToLoopyExpressionMapper(mappers.IdentityMapper):
                 "%s_%d" % (expr.aggregate.name, subscript))
 
     def map_call(self, expr):
-        if isinstance(expr.function, sym.CFunction):
+        if is_function_loopyable(expr.function, self.function_registry):
             from pymbolic import var
+
             func_name = expr.function.name
             if func_name == "fabs":
                 func_name = "abs"
@@ -968,10 +986,17 @@ def bessel_function_mangler(kernel, name, arg_dtypes):
 class ToLoopyInstructionMapper(object):
     def __init__(self, dd_inference_mapper):
         self.dd_inference_mapper = dd_inference_mapper
+        self.function_registry = dd_inference_mapper.function_registry
 
     def map_insn_assign(self, insn):
         from grudge.symbolic.primitives import OperatorBinding
-        if len(insn.exprs) == 1 and isinstance(insn.exprs[0], OperatorBinding):
+
+        if (
+                len(insn.exprs) == 1
+                and (
+                    isinstance(insn.exprs[0], OperatorBinding)
+                    or is_external_call(
+                        insn.exprs[0], self.function_registry))):
             return insn
 
         iname = "grdg_i"
@@ -1093,7 +1118,7 @@ class CodeGenerationState(Record):
 
 
 class OperatorCompiler(mappers.IdentityMapper):
-    def __init__(self, discr, prefix="_expr", max_vectors_in_batch_expr=None):
+    def __init__(self, discr, function_registry, prefix="_expr", max_vectors_in_batch_expr=None):
         super(OperatorCompiler, self).__init__()
         self.prefix = prefix
 
@@ -1109,6 +1134,7 @@ class OperatorCompiler(mappers.IdentityMapper):
         self.assigned_names = set()
 
         self.discr = discr
+        self.function_registry = function_registry
 
         from pytools import UniqueNameGenerator
         self.name_gen = UniqueNameGenerator()
@@ -1142,7 +1168,8 @@ class OperatorCompiler(mappers.IdentityMapper):
         del self.discr_code
 
         from grudge.symbolic.dofdesc_inference import DOFDescInferenceMapper
-        inf_mapper = DOFDescInferenceMapper(discr_code + eval_code)
+        inf_mapper = DOFDescInferenceMapper(
+                discr_code + eval_code, self.function_registry)
 
         eval_code = aggregate_assignments(
                 inf_mapper, eval_code, result, self.max_vectors_in_batch_expr)
@@ -1262,21 +1289,19 @@ class OperatorCompiler(mappers.IdentityMapper):
             return result_var
 
     def map_call(self, expr, codegen_state):
-        from grudge.symbolic.primitives import CFunction
-        if isinstance(expr.function, CFunction):
+        if is_function_loopyable(expr.function, self.function_registry):
             return super(OperatorCompiler, self).map_call(expr, codegen_state)
         else:
             # If it's not a C-level function, it shouldn't get muddled up into
             # a vector math expression.
-
             return self.assign_to_new_var(
-                    codegen_state,
-                    type(expr)(
-                        expr.function,
-                        [self.assign_to_new_var(
-                            codegen_state,
-                            self.rec(par, codegen_state))
-                            for par in expr.parameters]))
+                        codegen_state,
+                        type(expr)(
+                            expr.function,
+                            [self.assign_to_new_var(
+                                codegen_state,
+                                self.rec(par, codegen_state))
+                                for par in expr.parameters]))
 
     def map_ref_diff_op_binding(self, expr, codegen_state):
         try:
diff --git a/grudge/symbolic/dofdesc_inference.py b/grudge/symbolic/dofdesc_inference.py
index 92be126f7f3f081b668247e1fe25a73b122a4887..c44a7940b9ec142171a0b0a67885c041ec9d43cb 100644
--- a/grudge/symbolic/dofdesc_inference.py
+++ b/grudge/symbolic/dofdesc_inference.py
@@ -76,7 +76,8 @@ class InferrableMultiAssignment(object):
 
 
 class DOFDescInferenceMapper(RecursiveMapper, CSECachingMapperMixin):
-    def __init__(self, assignments, name_to_dofdesc=None, check=True):
+    def __init__(self, assignments, function_registry,
+                name_to_dofdesc=None, check=True):
         """
         :arg assignments: a list of objects adhering to
             :class:`InferrableMultiAssignment`.
@@ -98,6 +99,8 @@ class DOFDescInferenceMapper(RecursiveMapper, CSECachingMapperMixin):
 
         self.name_to_dofdesc = name_to_dofdesc
 
+        self.function_registry = function_registry
+
     def infer_for_name(self, name):
         try:
             return self.name_to_dofdesc[name]
@@ -186,10 +189,9 @@ class DOFDescInferenceMapper(RecursiveMapper, CSECachingMapperMixin):
                 self.rec(par)
                 for par in expr.parameters]
 
-        assert arg_dds
-
-        # FIXME
-        return arg_dds[0]
+        return (
+                self.function_registry[expr.function.name]
+                .get_result_dofdesc(arg_dds))
 
     # }}}
 
diff --git a/grudge/symbolic/mappers/__init__.py b/grudge/symbolic/mappers/__init__.py
index d52f7ac5bf99adbb16a618a8c3cb44fb8c65c515..82d5cc289a91714fb561a18e4ec4f325e82b7102 100644
--- a/grudge/symbolic/mappers/__init__.py
+++ b/grudge/symbolic/mappers/__init__.py
@@ -210,8 +210,6 @@ class IdentityMapperMixin(LocalOpReducerMixin, FluxOpReducerMixin):
         # it's a leaf--no changing children
         return expr
 
-    map_c_function = map_grudge_variable
-
     map_ones = map_grudge_variable
     map_node_coordinate_component = map_grudge_variable
 
@@ -278,7 +276,7 @@ class FlopCounter(
     def map_grudge_variable(self, expr):
         return 0
 
-    def map_c_function(self, expr):
+    def map_call(self, expr):
         return 1
 
     def map_ones(self, expr):
@@ -840,9 +838,6 @@ class StringifyMapper(pymbolic.mapper.stringifier.StringifyMapper):
                 self.rec(expr.op, PREC_NONE),
                 self.rec(expr.field, PREC_NONE))
 
-    def map_c_function(self, expr, enclosing_prec):
-        return expr.name
-
     def map_grudge_variable(self, expr, enclosing_prec):
         return "%s:%s" % (expr.name, self._format_dd(expr.dd))
 
@@ -1224,8 +1219,8 @@ class CollectorMixin(OperatorReducerMixin, LocalOpReducerMixin, FluxOpReducerMix
     def map_constant(self, expr, *args, **kwargs):
         return set()
 
+    map_variable = map_constant
     map_grudge_variable = map_constant
-    map_c_function = map_grudge_variable
 
     map_ones = map_grudge_variable
     map_node_coordinate_component = map_grudge_variable
diff --git a/grudge/symbolic/operators.py b/grudge/symbolic/operators.py
index 53fb1422619db271786abb9c2c17df5b3f9a97c0..f01266895e95021f4dd6fd6bbf22cb03763aa7e5 100644
--- a/grudge/symbolic/operators.py
+++ b/grudge/symbolic/operators.py
@@ -602,16 +602,16 @@ def norm(p, arg, dd=None):
 
     if p == 2:
         norm_squared = sym.NodalSum(dd_in=dd)(
-                sym.CFunction("fabs")(
+                sym.Variable("fabs")(
                     arg * sym.MassOperator()(arg)))
 
         if isinstance(norm_squared, np.ndarray):
             norm_squared = norm_squared.sum()
 
-        return sym.CFunction("sqrt")(norm_squared)
+        return sym.sqrt(norm_squared)
 
     elif p == np.Inf:
-        result = sym.NodalMax(dd_in=dd)(sym.CFunction("fabs")(arg))
+        result = sym.NodalMax(dd_in=dd)(sym.Variable("fabs")(arg))
         from pymbolic.primitives import Max
 
         if isinstance(result, np.ndarray):
diff --git a/grudge/symbolic/primitives.py b/grudge/symbolic/primitives.py
index 231a70dfa374d8c0e60e2830ff13427f5d33c0dd..98b13fa2f7e7533dfdcac98a16bfd89ec0b4a3cd 100644
--- a/grudge/symbolic/primitives.py
+++ b/grudge/symbolic/primitives.py
@@ -34,12 +34,13 @@ from meshmode.discretization.connection import (  # noqa
 
 from pymbolic.primitives import (  # noqa
         cse_scope as cse_scope_base,
-        make_common_subexpression as cse, If, Comparison)
+        make_common_subexpression as cse, If, Comparison, Expression)
 from pymbolic.geometric_algebra import MultiVector
 from pytools.obj_array import join_fields, make_obj_array  # noqa
 
 
-class ExpressionBase(pymbolic.primitives.Expression):
+class GrudgeStringifiable(object):
+
     def stringifier(self):
         from grudge.symbolic.mappers import StringifyMapper
         return StringifyMapper
@@ -74,9 +75,9 @@ Symbols
 
 .. autoclass:: Variable
 .. autoclass:: ScalarVariable
+.. autoclass:: ExternalCall
 .. autoclass:: make_sym_array
 .. autoclass:: make_sym_mv
-.. autoclass:: CFunction
 
 .. function :: sqrt(arg)
 .. function :: exp(arg)
@@ -289,7 +290,16 @@ class HasDOFDesc(object):
         discretization on which this property is given.
     """
 
-    def __init__(self, dd):
+    def __init__(self, *args, **kwargs):
+        # The remaining arguments are passed to the chained superclass.
+
+        if "dd" in kwargs:
+            dd = kwargs.pop("dd")
+        else:
+            dd = args[-1]
+            args = args[:-1]
+
+        super(HasDOFDesc, self).__init__(*args, **kwargs)
         self.dd = dd
 
     def __getinitargs__(self):
@@ -298,9 +308,7 @@ class HasDOFDesc(object):
     def with_dd(self, dd):
         """Return a copy of *self*, modified to the given DOF descriptor.
         """
-        return type(self)(
-                *self.__getinitargs__()[:-1],
-                dd=dd or self.dd)
+        return type(self)(*self.__getinitargs__())
 
 # }}}
 
@@ -311,7 +319,7 @@ class cse_scope(cse_scope_base):  # noqa
     DISCRETIZATION = "grudge_discretization"
 
 
-class Variable(HasDOFDesc, ExpressionBase, pymbolic.primitives.Variable):
+class Variable(HasDOFDesc, GrudgeStringifiable, pymbolic.primitives.Variable):
     """A user-supplied input variable with a known :class:`DOFDesc`.
     """
     init_arg_names = ("name", "dd")
@@ -320,8 +328,7 @@ class Variable(HasDOFDesc, ExpressionBase, pymbolic.primitives.Variable):
         if dd is None:
             dd = DD_VOLUME
 
-        HasDOFDesc.__init__(self, dd)
-        pymbolic.primitives.Variable.__init__(self, name)
+        super(Variable, self).__init__(name, dd)
 
     def __getinitargs__(self):
         return (self.name, self.dd,)
@@ -349,37 +356,20 @@ def make_sym_mv(name, dim, var_factory=None):
             make_sym_array(name, dim, var_factory))
 
 
-class CFunction(pymbolic.primitives.Variable):
-    """A symbol representing a C-level function, to be used as the function
-    argument of :class:`pymbolic.primitives.Call`.
-    """
-    def stringifier(self):
-        from grudge.symbolic.mappers import StringifyMapper
-        return StringifyMapper
-
-    def __call__(self, *exprs):
-        from pytools.obj_array import with_object_array_or_scalar_n_args
-        from functools import partial
-        return with_object_array_or_scalar_n_args(
-                partial(pymbolic.primitives.Expression.__call__, self),
-                *exprs)
-
-    mapper_method = "map_c_function"
-
-
-sqrt = CFunction("sqrt")
-exp = CFunction("exp")
-sin = CFunction("sin")
-cos = CFunction("cos")
-bessel_j = CFunction("bessel_j")
-bessel_y = CFunction("bessel_y")
+# function symbols
+sqrt = Variable("sqrt")
+exp = Variable("exp")
+sin = Variable("sin")
+cos = Variable("cos")
+bessel_j = Variable("bessel_j")
+bessel_y = Variable("bessel_y")
 
 # }}}
 
 
 # {{{ technical helpers
 
-class OperatorBinding(ExpressionBase):
+class OperatorBinding(GrudgeStringifiable, pymbolic.primitives.Expression):
     init_arg_names = ("op", "field")
 
     def __init__(self, op, field):
@@ -424,16 +414,16 @@ class PrioritizedSubexpression(pymbolic.primitives.CommonSubexpression):
 # }}}
 
 
-class Ones(ExpressionBase, HasDOFDesc):
+class Ones(GrudgeStringifiable, HasDOFDesc, Expression):
     def __getinitargs__(self):
-        return ()
+        return (self.dd,)
 
     mapper_method = intern("map_ones")
 
 
 # {{{ geometry data
 
-class DiscretizationProperty(ExpressionBase, HasDOFDesc):
+class DiscretizationProperty(GrudgeStringifiable, HasDOFDesc, Expression):
     pass
 
 
diff --git a/test/test_grudge.py b/test/test_grudge.py
index cf820f4c89f259a03e74b1ef37436f4c45833afc..47ed3efc8aed5f9ad5fcddf099778c18dbb90264 100644
--- a/test/test_grudge.py
+++ b/test/test_grudge.py
@@ -514,6 +514,41 @@ def test_bessel(ctx_factory):
     assert z < 1e-15
 
 
+def test_external_call(ctx_factory):
+    cl_ctx = ctx_factory()
+    queue = cl.CommandQueue(cl_ctx)
+
+    def double(x):
+        return 2 * x
+
+    from meshmode.mesh.generation import generate_regular_rect_mesh
+
+    dims = 2
+
+    mesh = generate_regular_rect_mesh(a=(0,) * dims, b=(1,) * dims, n=(4,) * dims)
+    discr = DGDiscretizationWithBoundaries(cl_ctx, mesh, order=1)
+
+    ones = sym.Ones(sym.DD_VOLUME)
+    from pymbolic.primitives import Variable
+    op = (
+            ones * 3
+            + Variable("double")(ones))
+
+    from grudge.function_registry import (
+            base_function_registry, register_external_function)
+
+    freg = register_external_function(
+            base_function_registry,
+            "double",
+            implementation=double,
+            dd=sym.DD_VOLUME)
+
+    bound_op = bind(discr, op, function_registry=freg)
+
+    result = bound_op(queue, double=double)
+    assert (result == 5).get().all()
+
+
 # You can test individual routines by typing
 # $ python test_grudge.py 'test_routine()'