def compute_dependency_graph(exprs):
"""
Given an ordered list of :class:`Expression`, build a mapper from lvalues
to reads occurring in the rvalues.
"""
deps_graph = OrderedDict()
writes = [e.output for e in exprs if e.is_tensor]
for i in writes:
for e in exprs:
i_reads = [j for j in e.reads if as_symbol(i) == as_symbol(j)]
deps_graph.setdefault(i, []).extend(i_reads)
return deps_graph
def __init__(self, exprs, **kwargs):
# Check input legality
mapper = OrderedDict([(i.lhs, i) for i in exprs])
if len(set(mapper)) != len(mapper):
raise DSEException(
"Found redundant node, cannot build TemporariesGraph.")
# Construct Temporaries, tracking reads and readby
tensor_map = DefaultOrderedDict(list)
for i in mapper:
tensor_map[as_symbol(i)].append(i)
reads = DefaultOrderedDict(set)
readby = DefaultOrderedDict(set)
for k, v in mapper.items():
handle = retrieve_terminals(v.rhs)
for i in list(handle):
if i.is_Indexed:
for idx in i.indices:
handle |= retrieve_terminals(idx)
reads[k].update(
set(flatten([tensor_map.get(as_symbol(i), [])
for i in handle])))
for i in reads[k]:
readby[i].add(k)
# Make sure read-after-writes are honored for scalar temporaries
processed = [i for i in mapper if i.is_Indexed]
queue = [i for i in mapper if i not in processed]
while queue:
k = queue.pop(0)
if not readby[k]:
processed.insert(0, k)
elif all(i in processed for i in readby[k]):
index = min(processed.index(i) for i in readby[k])
processed.insert(index, k)
else:
queue.append(k)
# Build up the TemporariesGraph
temporaries = [(i,
Temporary(*mapper[i].args,
inc=q_inc(mapper[i]),
reads=reads[i],
readby=readby[i])) for i in processed]
super(TemporariesGraph, self).__init__(temporaries, **kwargs)
# Determine indices along the space and time dimensions
terms = [
v for k, v in self.items() if v.is_tensor and not q_indirect(k)
]
indices = filter_ordered(flatten([i.function.indices for i in terms]))
self.space_indices = tuple(i for i in indices if i.is_Space)
self.time_indices = tuple(i for i in indices if i.is_Time)
def __init__(self, exprs, **kwargs):
# Always convert to SSA
exprs = convert_to_SSA(exprs)
mapper = OrderedDict([(i.lhs, i) for i in exprs])
assert len(set(mapper)) == len(exprs), "not SSA Cluster?"
# Construct the Nodes, tracking reads and readby
tensor_map = DefaultOrderedDict(list)
for i in mapper:
tensor_map[as_symbol(i)].append(i)
reads = DefaultOrderedDict(set)
readby = DefaultOrderedDict(set)
for k, v in mapper.items():
handle = retrieve_terminals(v.rhs)
for i in list(handle):
if i.is_Indexed:
for idx in i.indices:
handle |= retrieve_terminals(idx)
reads[k].update(
set(flatten([tensor_map.get(as_symbol(i), [])
for i in handle])))
for i in reads[k]:
readby[i].add(k)
# Make sure read-after-writes are honored for scalar temporaries
processed = [i for i in mapper if i.is_Indexed]
queue = [i for i in mapper if i not in processed]
while queue:
k = queue.pop(0)
if not readby[k]:
processed.insert(0, k)
elif all(i in processed for i in readby[k]):
index = min(processed.index(i) for i in readby[k])
processed.insert(index, k)
else:
queue.append(k)
# Build up the FlowGraph
temporaries = [(i,
Node(*mapper[i].args,
inc=q_inc(mapper[i]),
reads=reads[i],
readby=readby[i])) for i in processed]
super(FlowGraph, self).__init__(temporaries, **kwargs)
# Determine indices along the space and time dimensions
terms = [
v for k, v in self.items() if v.is_tensor and not q_indirect(k)
]
indices = filter_ordered(flatten([i.function.indices for i in terms]))
self.space_indices = tuple(i for i in indices if i.is_Space)
self.time_indices = tuple(i for i in indices if i.is_Time)
def symbolic_bounds(self):
"""A 2-tuple representing the symbolic bounds [min, max] of the Iteration."""
_min = self.limits[0]
_max = self.limits[1]
try:
_min = as_symbol(_min)
except TypeError:
# A symbolic expression
pass
try:
_max = as_symbol(_max)
except TypeError:
# A symbolic expression
pass
return (_min, _max)
def symbolic_bounds(self):
"""Return a 2-tuple representing the symbolic bounds of the object."""
start = self.limits[0]
end = self.limits[1]
try:
start = as_symbol(start)
except TypeError:
# Already a symbolic expression
pass
try:
end = as_symbol(end)
except TypeError:
# Already a symbolic expression
pass
return (start + as_symbol(self.offsets[0]), end + as_symbol(self.offsets[1]))
def __init__(self, exprs, **kwargs):
# Always convert to SSA
exprs = makeit_ssa(exprs)
mapper = OrderedDict([(i.lhs, i) for i in exprs])
assert len(set(mapper)) == len(exprs), "not SSA Cluster?"
# Construct the Nodes, tracking reads and readby
tensor_map = DefaultOrderedDict(list)
for i in mapper:
tensor_map[as_symbol(i)].append(i)
reads = DefaultOrderedDict(set)
readby = DefaultOrderedDict(set)
for k, v in mapper.items():
handle = retrieve_terminals(v.rhs)
for i in list(handle):
if i.is_Indexed:
for idx in i.indices:
handle |= retrieve_terminals(idx)
reads[k].update(set(flatten([tensor_map.get(as_symbol(i), [])
for i in handle])))
for i in reads[k]:
readby[i].add(k)
# Make sure read-after-writes are honored for scalar nodes
processed = [i for i in mapper if i.is_Indexed]
queue = [i for i in mapper if i not in processed]
while queue:
k = queue.pop(0)
if not readby[k] or k in readby[k]:
processed.insert(0, k)
elif all(i in processed for i in readby[k]):
index = min(processed.index(i) for i in readby[k])
processed.insert(index, k)
else:
queue.append(k)
# Build up the FlowGraph
nodes = [(i, Node(mapper[i], reads=reads[i], readby=readby[i]))
for i in processed]
super(FlowGraph, self).__init__(nodes, **kwargs)
# Determine indices along the space and time dimensions
terms = [v for k, v in self.items() if v.is_Tensor and not q_indirect(k)]
indices = filter_ordered(flatten([i.function.indices for i in terms]))
self.space_indices = tuple(i for i in indices if i.is_Space)
self.time_indices = tuple(i for i in indices if i.is_Time)
def __init__(self, index, start=0, step=None, dim=None, expr=None):
self.name = index
self.index = index
self.dim = dim
self.expr = expr
try:
self.start = as_symbol(start)
except TypeError:
self.start = start
try:
if step is None:
self.step = index + 1
else:
self.step = as_symbol(step)
except TypeError:
self.step = step
def __init__(self, exprs, **kwargs):
# Always convert to SSA
exprs = makeit_ssa(exprs)
mapper = OrderedDict([(i.lhs, i) for i in exprs])
assert len(set(mapper)) == len(exprs), "not SSA Cluster?"
# Construct the Nodes, tracking reads and readby
tensor_map = DefaultOrderedDict(list)
for i in mapper:
tensor_map[as_symbol(i)].append(i)
reads = DefaultOrderedDict(set)
readby = DefaultOrderedDict(set)
for k, v in mapper.items():
handle = retrieve_terminals(v.rhs, deep=True)
reads[k].update(
set(flatten([tensor_map.get(as_symbol(i), [])
for i in handle])))
for i in reads[k]:
readby[i].add(k)
# Make sure read-after-writes are honored for scalar nodes
processed = [i for i in mapper if i.is_Indexed]
queue = [i for i in mapper if i not in processed]
while queue:
k = queue.pop(0)
if not readby[k] or k in readby[k]:
processed.insert(0, k)
elif all(i in processed for i in readby[k]):
index = min(processed.index(i) for i in readby[k])
processed.insert(index, k)
else:
queue.append(k)
# Build up the FlowGraph
nodes = [(i, Node(mapper[i], reads=reads[i], readby=readby[i]))
for i in processed]
super(FlowGraph, self).__init__(nodes, **kwargs)
def optimize_unfolded_tree(unfolded, root):
"""
Transform folded trees to reduce the memory footprint.
Examples
--------
Given:
.. code-block::
for i = 1 to N - 1 # Folded tree
for j = 1 to N - 1
tmp[i,j] = ...
for i = 2 to N - 2 # Root
for j = 2 to N - 2
... = ... tmp[i,j] ...
The temporary ``tmp`` has shape ``(N-1, N-1)``. However, as soon as the
iteration space is blocked, with blocks of shape ``(i_bs, j_bs)``, the
``tmp`` shape can be shrunk to ``(i_bs-1, j_bs-1)``. The resulting
iteration tree becomes:
.. code-block::
for i = 1 to i_bs + 1 # Folded tree
for j = 1 to j_bs + 1
i' = i + i_block - 2
j' = j + j_block - 2
tmp[i,j] = ... # use i' and j'
for i = i_block to i_block + i_bs # Root
for j = j_block to j_block + j_bs
i' = i - x_block
j' = j - j_block
... = ... tmp[i',j'] ...
"""
processed = []
for i, tree in enumerate(unfolded):
assert len(tree) == len(root)
# We can optimize the folded trees only iff:
# test0 := they compute temporary arrays, but not if they compute input data
# test1 := the outer Iterations have actually been blocked
exprs = FindNodes(Expression).visit(tree)
writes = [j.write for j in exprs if j.is_tensor]
test0 = not all(j.is_Array for j in writes)
test1 = any(not isinstance(j.limits[0], BlockDimension) for j in root)
if test0 or test1:
processed.append(compose_nodes(tree))
root = compose_nodes(root)
continue
# Shrink the iteration space
modified_tree = []
modified_root = []
modified_dims = {}
mapper = {}
for t, r in zip(tree, root):
udim0 = IncrDimension(t.dim, t.symbolic_min, 1, "%ss%d" % (t.index, i))
modified_tree.append(t._rebuild(limits=(0, t.limits[1] - t.limits[0], t.step),
uindices=t.uindices + (udim0,)))
mapper[t.dim] = udim0
udim1 = IncrDimension(t.dim, 0, 1, "%ss%d" % (t.index, i))
modified_root.append(r._rebuild(uindices=r.uindices + (udim1,)))
d = r.limits[0]
assert isinstance(d, BlockDimension)
modified_dims[d.root] = d
# Temporary arrays can now be moved onto the stack
for w in writes:
dims = tuple(modified_dims.get(d, d) for d in w.dimensions)
shape = tuple(d.symbolic_size for d in dims)
w.update(shape=shape, dimensions=dims, scope='stack')
# Substitute iteration variables within the folded trees
modified_tree = compose_nodes(modified_tree)
replaced = xreplace_indices([j.expr for j in exprs], mapper,
lambda i: i.function not in writes, True)
subs = [j._rebuild(expr=k) for j, k in zip(exprs, replaced)]
processed.append(Transformer(dict(zip(exprs, subs))).visit(modified_tree))
# Introduce the new iteration variables within /root/
modified_root = compose_nodes(modified_root)
exprs = FindNodes(Expression).visit(modified_root)
candidates = [as_symbol(j.output) for j in subs]
replaced = xreplace_indices([j.expr for j in exprs], mapper, candidates)
subs = [j._rebuild(expr=k) for j, k in zip(exprs, replaced)]
root = Transformer(dict(zip(exprs, subs))).visit(modified_root)
return processed + [root]
def _create_elemental_functions(self, nodes, state):
"""
Extract :class:`Iteration` sub-trees and move them into :class:`Callable`s.
Currently, only tagged, elementizable Iteration objects are targeted.
"""
noinline = self._compiler_decoration('noinline',
c.Comment('noinline?'))
functions = OrderedDict()
mapper = {}
for tree in retrieve_iteration_tree(nodes, mode='superset'):
# Search an elementizable sub-tree (if any)
tagged = filter_iterations(tree, lambda i: i.tag is not None,
'asap')
if not tagged:
continue
root = tagged[0]
if not root.is_Elementizable:
continue
target = tree[tree.index(root):]
# Elemental function arguments
args = [] # Found so far (scalars, tensors)
maybe_required = set() # Scalars that *may* have to be passed in
not_required = set() # Elemental function locally declared scalars
# Build a new Iteration/Expression tree with free bounds
free = []
for i in target:
name, bounds = i.dim.name, i.bounds_symbolic
# Iteration bounds
start = Scalar(name='%s_start' % name, dtype=np.int32)
finish = Scalar(name='%s_finish' % name, dtype=np.int32)
args.extend(zip([ccode(j) for j in bounds], (start, finish)))
# Iteration unbounded indices
ufunc = [
Scalar(name='%s_ub%d' % (name, j), dtype=np.int32)
for j in range(len(i.uindices))
]
args.extend(zip([ccode(j.start) for j in i.uindices], ufunc))
limits = [Symbol(start.name), Symbol(finish.name), 1]
uindices = [
UnboundedIndex(j.index, i.dim + as_symbol(k))
for j, k in zip(i.uindices, ufunc)
]
free.append(
i._rebuild(limits=limits, offsets=None, uindices=uindices))
not_required.update({i.dim}, set(j.index for j in i.uindices))
# Construct elemental function body, and inspect it
free = NestedTransformer(dict((zip(target, free)))).visit(root)
expressions = FindNodes(Expression).visit(free)
fsymbols = FindSymbols('symbolics').visit(free)
# Add all definitely-required arguments
not_required.update({i.output for i in expressions if i.is_scalar})
for i in fsymbols:
if i in not_required:
continue
elif i.is_Array:
args.append(
("(%s*)%s" % (c.dtype_to_ctype(i.dtype), i.name), i))
elif i.is_TensorFunction:
args.append(("%s_vec" % i.name, i))
elif i.is_Scalar:
args.append((i.name, i))
# Add all maybe-required arguments that turn out to be required
maybe_required.update(
set(FindSymbols(mode='free-symbols').visit(free)))
for i in fsymbols:
not_required.update({as_symbol(i), i.indexify()})
for j in i.symbolic_shape:
maybe_required.update(j.free_symbols)
required = filter_sorted(maybe_required - not_required,
key=attrgetter('name'))
args.extend([(i.name, Scalar(name=i.name, dtype=i.dtype))
for i in required])
call, params = zip(*args)
handle = flatten([p.rtargs for p in params])
name = "f_%d" % root.tag
# Produce the new Call
mapper[root] = List(header=noinline, body=Call(name, call))
# Produce the new Callable
functions.setdefault(
name, Callable(name, free, 'void', handle, ('static', )))
# Transform the main tree
processed = Transformer(mapper).visit(nodes)
return processed, {'elemental_functions': functions.values()}
def optimize_unfolded_tree(unfolded, root):
"""
Transform folded trees to reduce the memory footprint.
Examples
========
Given:
.. code-block::
for i = 1 to N - 1 # Folded tree
for j = 1 to N - 1
tmp[i,j] = ...
for i = 2 to N - 2 # Root
for j = 2 to N - 2
... = ... tmp[i,j] ...
The temporary ``tmp`` has shape ``(N-1, N-1)``. However, as soon as the
iteration space is blocked, with blocks of shape ``(i_bs, j_bs)``, the
``tmp`` shape can be shrunk to ``(i_bs-1, j_bs-1)``. The resulting
iteration tree becomes:
.. code-block::
for i = 1 to i_bs + 1 # Folded tree
for j = 1 to j_bs + 1
i' = i + i_block - 2
j' = j + j_block - 2
tmp[i,j] = ... # use i' and j'
for i = i_block to i_block + i_bs # Root
for j = j_block to j_block + j_bs
i' = i - x_block
j' = j - j_block
... = ... tmp[i',j'] ...
"""
processed = []
for i, tree in enumerate(unfolded):
assert len(tree) == len(root)
modified_tree = []
modified_root = []
mapper = {}
# "Shrink" the iteration space
for t1, t2 in zip(tree, root):
index = Symbol('%ss%d' % (t1.index, i))
mapper[t1.dim] = index
t1_uindex = (UnboundedIndex(index, t1.limits[0]), )
t2_uindex = (UnboundedIndex(index, -t1.limits[0]), )
limits = (0, t1.limits[1] - t1.limits[0], t1.incr_symbolic)
modified_tree.append(
t1._rebuild(limits=limits, uindices=t1.uindices + t1_uindex))
modified_root.append(t2._rebuild(uindices=t2.uindices + t2_uindex))
# Temporary arrays can now be moved onto the stack
exprs = FindNodes(Expression).visit(modified_tree[-1])
if all(not j.is_Remainder for j in modified_tree):
dimensions = tuple(j.limits[0] for j in modified_root)
for j in exprs:
if j.write.is_Array:
j_dimensions = dimensions + j.write.dimensions[
len(modified_root):]
j_shape = tuple(k.symbolic_size for k in j_dimensions)
j.write.update(shape=j_shape,
dimensions=j_dimensions,
onstack=True)
# Substitute iteration variables within the folded trees
modified_tree = compose_nodes(modified_tree)
replaced = xreplace_indices([j.expr for j in exprs],
mapper,
only_rhs=True)
subs = [j._rebuild(expr=k) for j, k in zip(exprs, replaced)]
processed.append(
Transformer(dict(zip(exprs, subs))).visit(modified_tree))
# Introduce the new iteration variables within /root/
modified_root = compose_nodes(modified_root)
exprs = FindNodes(Expression).visit(modified_root)
candidates = [as_symbol(j.output) for j in subs]
replaced = xreplace_indices([j.expr for j in exprs], mapper,
candidates)
subs = [j._rebuild(expr=k) for j, k in zip(exprs, replaced)]
root = Transformer(dict(zip(exprs, subs))).visit(modified_root)
return processed + [root]
def _create_elemental_functions(self, nodes, state):
"""
Extract :class:`Iteration` sub-trees and move them into :class:`Callable`s.
Currently, only tagged, elementizable Iteration objects are targeted.
"""
noinline = self._compiler_decoration('noinline',
c.Comment('noinline?'))
functions = OrderedDict()
mapper = {}
for tree in retrieve_iteration_tree(nodes, mode='superset'):
# Search an elementizable sub-tree (if any)
tagged = filter_iterations(tree, lambda i: i.tag is not None,
'asap')
if not tagged:
continue
root = tagged[0]
if not root.is_Elementizable:
continue
target = tree[tree.index(root):]
# Elemental function arguments
args = [] # Found so far (scalars, tensors)
defined_args = {} # Map of argument values defined by loop bounds
# Build a new Iteration/Expression tree with free bounds
free = []
for i in target:
name, bounds = i.dim.name, i.bounds_symbolic
# Iteration bounds
start = Scalar(name='%s_start' % name, dtype=np.int32)
finish = Scalar(name='%s_finish' % name, dtype=np.int32)
defined_args[start.name] = bounds[0]
defined_args[finish.name] = bounds[1]
# Iteration unbounded indices
ufunc = [
Scalar(name='%s_ub%d' % (name, j), dtype=np.int32)
for j in range(len(i.uindices))
]
defined_args.update(
{uf.name: j.start
for uf, j in zip(ufunc, i.uindices)})
limits = [
Scalar(name=start.name, dtype=np.int32),
Scalar(name=finish.name, dtype=np.int32), 1
]
uindices = [
UnboundedIndex(j.index, i.dim + as_symbol(k))
for j, k in zip(i.uindices, ufunc)
]
free.append(
i._rebuild(limits=limits, offsets=None, uindices=uindices))
# Construct elemental function body, and inspect it
free = NestedTransformer(dict((zip(target, free)))).visit(root)
# Insert array casts for all non-defined
f_symbols = FindSymbols('symbolics').visit(free)
defines = [s.name for s in FindSymbols('defines').visit(free)]
casts = [
ArrayCast(f) for f in f_symbols
if f.is_Tensor and f.name not in defines
]
free = (List(body=casts), free)
for i in derive_parameters(free):
if i.name in defined_args:
args.append((defined_args[i.name], i))
elif i.is_Dimension:
d = Scalar(name=i.name, dtype=i.dtype)
args.append((d, d))
else:
args.append((i, i))
call, params = zip(*args)
name = "f_%d" % root.tag
# Produce the new Call
mapper[root] = List(header=noinline, body=Call(name, call))
# Produce the new Callable
functions.setdefault(
name,
Callable(name, free, 'void', flatten(params), ('static', )))
# Transform the main tree
processed = Transformer(mapper).visit(nodes)
return processed, {'elemental_functions': functions.values()}
def _create_efuncs(self, nodes, state):
"""
Extract Iteration sub-trees and turn them into Calls+Callables.
Currently, only tagged, elementizable Iteration objects are targeted.
"""
noinline = self._compiler_decoration('noinline',
c.Comment('noinline?'))
efuncs = OrderedDict()
mapper = {}
for tree in retrieve_iteration_tree(nodes, mode='superset'):
# Search an elementizable sub-tree (if any)
tagged = filter_iterations(tree, lambda i: i.tag is not None,
'asap')
if not tagged:
continue
root = tagged[0]
if not root.is_Elementizable:
continue
target = tree[tree.index(root):]
# Build a new Iteration/Expression tree with free bounds
free = []
defined_args = {} # Map of argument values defined by loop bounds
for i in target:
name, bounds = i.dim.name, i.symbolic_bounds
# Iteration bounds
_min = Scalar(name='%sf_m' % name,
dtype=np.int32,
is_const=True)
_max = Scalar(name='%sf_M' % name,
dtype=np.int32,
is_const=True)
defined_args[_min.name] = bounds[0]
defined_args[_max.name] = bounds[1]
# Iteration unbounded indices
ufunc = [
Scalar(name='%s_ub%d' % (name, j), dtype=np.int32)
for j in range(len(i.uindices))
]
defined_args.update({
uf.name: j.symbolic_min
for uf, j in zip(ufunc, i.uindices)
})
uindices = [
IncrDimension(j.parent, i.dim + as_symbol(k), 1, j.name)
for j, k in zip(i.uindices, ufunc)
]
free.append(
i._rebuild(limits=(_min, _max, 1),
offsets=None,
uindices=uindices))
# Construct elemental function body
free = Transformer(dict((zip(target, free))),
nested=True).visit(root)
items = FindSymbols().visit(free)
# Insert array casts
casts = [ArrayCast(i) for i in items if i.is_Tensor]
free = List(body=casts + [free])
# Insert declarations
external = [i for i in items if i.is_Array]
free = iet_insert_C_decls(free, external)
# Create the Callable
name = "f_%d" % root.tag
params = derive_parameters(free)
efuncs.setdefault(name,
Callable(name, free, 'void', params, 'static'))
# Create the Call
args = [defined_args.get(i.name, i) for i in params]
mapper[root] = List(header=noinline, body=Call(name, args))
# Transform the main tree
processed = Transformer(mapper).visit(nodes)
return processed, {'efuncs': efuncs.values()}
def optimize_unfolded_tree(unfolded, root):
"""
Transform folded trees to reduce the memory footprint.
Examples
--------
Given:
.. code-block::
for i = 1 to N - 1 # Folded tree
for j = 1 to N - 1
tmp[i,j] = ...
for i = 2 to N - 2 # Root
for j = 2 to N - 2
... = ... tmp[i,j] ...
The temporary ``tmp`` has shape ``(N-1, N-1)``. However, as soon as the
iteration space is blocked, with blocks of shape ``(i_bs, j_bs)``, the
``tmp`` shape can be shrunk to ``(i_bs-1, j_bs-1)``. The resulting
iteration tree becomes:
.. code-block::
for i = 1 to i_bs + 1 # Folded tree
for j = 1 to j_bs + 1
i' = i + i_block - 2
j' = j + j_block - 2
tmp[i,j] = ... # use i' and j'
for i = i_block to i_block + i_bs # Root
for j = j_block to j_block + j_bs
i' = i - x_block
j' = j - j_block
... = ... tmp[i',j'] ...
"""
processed = []
for i, tree in enumerate(unfolded):
assert len(tree) == len(root)
# We can optimize the folded trees only if they compute temporary
# arrays, but not if they compute input data
exprs = FindNodes(Expression).visit(tree[-1])
writes = [j.write for j in exprs if j.is_tensor]
if not all(j.is_Array for j in writes):
processed.append(compose_nodes(tree))
root = compose_nodes(root)
continue
modified_tree = []
modified_root = []
mapper = {}
# "Shrink" the iteration space
for t1, t2 in zip(tree, root):
t1_udim = IncrDimension(t1.dim, t1.symbolic_min, 1, "%ss%d" % (t1.index, i))
limits = (0, t1.limits[1] - t1.limits[0], t1.step)
modified_tree.append(t1._rebuild(limits=limits,
uindices=t1.uindices + (t1_udim,)))
t2_udim = IncrDimension(t1.dim, 0, 1, "%ss%d" % (t1.index, i))
modified_root.append(t2._rebuild(uindices=t2.uindices + (t2_udim,)))
mapper[t1.dim] = t1_udim
# Temporary arrays can now be moved onto the stack
dimensions = tuple(j.limits[0] for j in modified_root)
for j in writes:
if j.is_Array:
j_dimensions = dimensions + j.dimensions[len(modified_root):]
j_shape = tuple(k.symbolic_size for k in j_dimensions)
j.update(shape=j_shape, dimensions=j_dimensions, scope='stack')
# Substitute iteration variables within the folded trees
modified_tree = compose_nodes(modified_tree)
replaced = xreplace_indices([j.expr for j in exprs], mapper, only_rhs=True)
subs = [j._rebuild(expr=k) for j, k in zip(exprs, replaced)]
processed.append(Transformer(dict(zip(exprs, subs))).visit(modified_tree))
# Introduce the new iteration variables within /root/
modified_root = compose_nodes(modified_root)
exprs = FindNodes(Expression).visit(modified_root)
candidates = [as_symbol(j.output) for j in subs]
replaced = xreplace_indices([j.expr for j in exprs], mapper, candidates)
subs = [j._rebuild(expr=k) for j, k in zip(exprs, replaced)]
root = Transformer(dict(zip(exprs, subs))).visit(modified_root)
return processed + [root]