def choose(aliases, exprs, mapper, selector):
"""
Analyze the detected aliases and, after applying a cost model to rule out
the aliases with a bad flops/memory trade-off, inject them into the original
expressions.
"""
tot = 0
retained = AliasMapper()
# Pass 1: a set of aliasing expressions is retained only if its cost
# exceeds the mode's threshold
candidates = OrderedDict()
aliaseds = []
others = []
for e, v in aliases.items():
score = selector(e, len(v.aliaseds))
if score > 0:
candidates[e] = score
aliaseds.extend(v.aliaseds)
else:
others.append(e)
# Do not waste time if unneccesary
if not candidates:
return exprs, retained, tot
# Project the candidate aliases into exprs to determine what the new
# working set would be
mapper = {
k: v
for k, v in mapper.items() if v.free_symbols & set(aliaseds)
}
templated = [uxreplace(e, mapper) for e in exprs]
# Pass 2: a set of aliasing expressions is retained only if the tradeoff
# between operation count reduction and working set increase is favorable
owset = wset(others + templated)
for e, v in aliases.items():
try:
score = candidates[e]
except KeyError:
score = 0
if score > 1 or \
score == 1 and max(len(wset(e)), 1) > len(wset(e) & owset):
retained[e] = v
tot += score
# Do not waste time if unneccesary
if not retained:
return exprs, retained, tot
# Substitute the chosen aliasing sub-expressions
mapper = {
k: v
for k, v in mapper.items() if v.free_symbols & set(retained.aliaseds)
}
exprs = [uxreplace(e, mapper) for e in exprs]
return exprs, retained, tot
def _buffering(expressions, callback, options):
async_degree = options['buf-async-degree']
# Locate all Function accesses within the provided `expressions`
accessmap = AccessMapper(expressions)
# Create the buffers
buffers = []
for n, (f, accessv) in enumerate(accessmap.items()):
dims = callback(f)
if dims is None:
# Not a buffer candidate
continue
if accessv.lastwrite is None:
# Read-only Functions cannot be buffering candidates
continue
buffers.append(Buffer(f, dims, accessv, n, async_degree))
# Create Eqs to initialize buffers. Note: a buffer needs to be initialized
# only if the buffered Function is read in at least one place or in the case
# of non-uniform SubDimensions, to avoid uninitialized values to be copied-back
# into the buffered Function
processed = [Eq(b.indexify(), b.function.subs(b.contraction_mapper))
for b in buffers if b.is_read or not b.has_uniform_subdims]
# Substitution rules to replace buffered Functions with buffers
subs = {}
for b in buffers:
for a in b.accessv.accesses:
subs[a] = b.indexify(a.indices)
# Create Eqs to copy back buffers into their buffered Functions
for e in expressions:
processed.append(uxreplace(e, subs))
# We also append the copy-back if `e` is the last-write of some buffers
for b in buffers:
if e is b.accessv.lastwrite:
items = list(zip(e.lhs.indices, b.function.dimensions))
lhs = b.function[[i if i in b.index_mapper else d for i, d in items]]
rhs = b.indexed[[b.index_mapper.get(i, d) for i, d in items]]
if b.subdims_mapper:
processed.append(uxreplace(Eq(lhs, rhs), b.subdims_mapper))
else:
processed.append(Eq(lhs, rhs))
break
return processed
def _compact_temporaries(exprs):
"""
Drop temporaries consisting of isolated symbols.
"""
# First of all, convert to SSA
exprs = makeit_ssa(exprs)
# What's gonna be dropped
mapper = {e.lhs: e.rhs for e in exprs
if e.lhs.is_Symbol and (q_leaf(e.rhs) or e.rhs.is_Function)}
processed = []
for e in exprs:
if e.lhs not in mapper:
# The temporary is retained, and substitutions may be applied
expr = e
while True:
handle = uxreplace(expr, mapper)
if handle == expr:
break
else:
expr = handle
processed.append(handle)
return processed
def visit_Conditional(self, o):
condition = uxreplace(o.condition, self.mapper)
then_body = self._visit(o.then_body)
else_body = self._visit(o.else_body)
return o._rebuild(condition=condition,
then_body=then_body,
else_body=else_body)
def _lower_exprs(cls, expressions, **kwargs):
"""
Expression lowering:
* Form and gather any required implicit expressions;
* Evaluate derivatives;
* Flatten vectorial equations;
* Indexify Functions;
* Apply substitution rules;
* Specialize (e.g., index shifting)
"""
# Add in implicit expressions, e.g., induced by SubDomains
expressions = cls._add_implicit(expressions)
# Unfold lazyiness
expressions = flatten([i.evaluate for i in expressions])
# Scalarize tensor expressions
expressions = [j for i in expressions for j in i._flatten]
# Indexification
# E.g., f(x - 2*h_x, y) -> f[xi + 2, yi + 4] (assuming halo_size=4)
processed = []
for expr in expressions:
if expr.subdomain:
dimension_map = expr.subdomain.dimension_map
else:
dimension_map = {}
# Handle Functions (typical case)
mapper = {
f: f.indexify(lshift=True, subs=dimension_map)
for f in retrieve_functions(expr)
}
# Handle Indexeds (from index notation)
for i in retrieve_indexed(expr):
f = i.function
# Introduce shifting to align with the computational domain
indices = [(a + o)
for a, o in zip(i.indices, f._size_nodomain.left)]
# Apply substitutions, if necessary
if dimension_map:
indices = [j.xreplace(dimension_map) for j in indices]
mapper[i] = f.indexed[indices]
subs = kwargs.get('subs')
if subs:
# Include the user-supplied substitutions, and use
# `xreplace` for constant folding
processed.append(expr.xreplace({**mapper, **subs}))
else:
processed.append(uxreplace(expr, mapper))
processed = cls._specialize_exprs(processed)
return processed
def extract(cls, n, context, min_cost, cluster, sregistry):
make = lambda: Scalar(name=sregistry.make_name(), dtype=cluster.dtype
).indexify()
exclude = {
i.source.indexed
for i in cluster.scope.d_flow.independent()
}
rule0 = lambda e: not e.free_symbols & exclude
rule1 = make_is_time_invariant(context)
rule2 = lambda e: estimate_cost(e, True) >= min_cost
rule = lambda e: rule0(e) and rule1(e) and rule2(e)
extracted = []
mapper = OrderedDict()
for e in cluster.exprs:
for i in search(e, rule, 'all', 'dfs_first_hit'):
if i not in mapper:
symbol = make()
mapper[i] = symbol
extracted.append(e.func(symbol, i))
processed = [uxreplace(e, mapper) for e in cluster.exprs]
return extracted + processed, extracted
def makeit_ssa(exprs):
"""
Convert an iterable of Eqs into Static Single Assignment (SSA) form.
"""
# Identify recurring LHSs
seen = {}
for i, e in enumerate(exprs):
seen.setdefault(e.lhs, []).append(i)
# Optimization: don't waste time reconstructing stuff if already in SSA form
if all(len(i) == 1 for i in seen.values()):
return exprs
# SSA conversion
c = 0
mapper = {}
processed = []
for i, e in enumerate(exprs):
where = seen[e.lhs]
rhs = uxreplace(e.rhs, mapper)
if len(where) > 1:
needssa = e.is_Scalar or where[-1] != i
lhs = Symbol(name='ssa%d' % c, dtype=e.dtype) if needssa else e.lhs
if e.is_Increment:
# Turn AugmentedAssignment into Assignment
processed.append(
e.func(lhs, mapper[e.lhs] + rhs, is_Increment=False))
else:
processed.append(e.func(lhs, rhs))
mapper[e.lhs] = lhs
c += 1
else:
processed.append(e.func(e.lhs, rhs))
return processed
def _compact_temporaries(exprs):
"""
Drop temporaries consisting of isolated symbols.
"""
# First of all, convert to SSA
exprs = makeit_ssa(exprs)
# Drop candidates are all exprs in the form `t0 = s` where `s` is a symbol
# Note: only CSE-captured Temps, which are by construction local objects, may
# safely be compacted; a generic Symbol could instead be accessed in a subsequent
# Cluster, for example: `for (i = ...) { a = b; for (j = a ...) ...`
mapper = {
e.lhs: e.rhs
for e in exprs
if isinstance(e.lhs, Temp) and (q_leaf(e.rhs) or e.rhs.is_Function)
}
processed = []
for e in exprs:
if e.lhs not in mapper:
# The temporary is retained, and substitutions may be applied
expr = e
while True:
handle = uxreplace(expr, mapper)
if handle == expr:
break
else:
expr = handle
processed.append(handle)
return processed
def _aliases_from_clusters(self, clusters, exclude, meta):
exprs = flatten([c.exprs for c in clusters])
# [Clusters]_n -> [Schedule]_m
variants = []
for mapper in self._generate(exprs, exclude):
# Clusters -> AliasList
found = collect(mapper.extracted, meta.ispace, self.opt_minstorage)
pexprs, aliases = choose(found, exprs, mapper, self.opt_mingain)
if not aliases:
continue
# AliasList -> Schedule
schedule = lower_aliases(aliases, meta, self.opt_maxpar)
variants.append(Variant(schedule, pexprs))
# [Schedule]_m -> Schedule (s.t. best memory/flops trade-off)
if not variants:
return []
schedule, exprs = pick_best(variants, self.opt_schedule_strategy,
self._eval_variants_delta)
# Schedule -> Schedule (optimization)
if self.opt_rotate:
schedule = optimize_schedule_rotations(schedule, self.sregistry)
schedule = optimize_schedule_padding(schedule, meta, self.platform)
# Schedule -> [Clusters]_k
processed, subs = lower_schedule(schedule, meta, self.sregistry,
self.opt_ftemps)
# [Clusters]_k -> [Clusters]_k (optimization)
if self.opt_multisubdomain:
processed = optimize_clusters_msds(processed)
# [Clusters]_k -> [Clusters]_{k+n}
for c in clusters:
n = len(c.exprs)
cexprs, exprs = exprs[:n], exprs[n:]
cexprs = [uxreplace(e, subs) for e in cexprs]
ispace = c.ispace.augment(schedule.dmapper)
ispace = ispace.augment(schedule.rmapper)
accesses = detect_accesses(cexprs)
parts = {
k: IntervalGroup(build_intervals(v)).relaxed
for k, v in accesses.items() if k
}
dspace = DataSpace(c.dspace.intervals, parts)
processed.append(
c.rebuild(exprs=cexprs, ispace=ispace, dspace=dspace))
assert len(exprs) == 0
return processed
def choose(aliases, exprs, mapper, mingain):
"""
Analyze the detected aliases and, after applying a cost model to rule out
the aliases with a bad memory/flops trade-off, inject them into the original
expressions.
"""
aliases = AliasList(aliases)
# `score < m` => discarded
# `score > M` => optimized
# `m <= score <= M` => maybe discarded, maybe optimized -- depends on heuristics
m = mingain
M = mingain * 3
if not aliases:
return exprs, aliases
# Filter off the aliases with low score
key = lambda a: a.score >= m
aliases.filter(key)
# Project the candidate aliases into `exprs` to derive the final working set
mapper = {
k: v
for k, v in mapper.items() if v.free_symbols & set(aliases.aliaseds)
}
templated = [uxreplace(e, mapper) for e in exprs]
owset = wset(templated)
# Filter off the aliases with a weak flop-reduction / working-set tradeoff
key = lambda a: \
a.score > M or \
m <= a.score <= M and max(len(wset(a.pivot)), 1) > len(wset(a.pivot) & owset)
aliases.filter(key)
if not aliases:
return exprs, aliases
# Substitute the chosen aliasing sub-expressions
mapper = {
k: v
for k, v in mapper.items() if v.free_symbols & set(aliases.aliaseds)
}
exprs = [uxreplace(e, mapper) for e in exprs]
return exprs, aliases
def extract(cls, n, context, min_cost, max_alias, cluster, sregistry):
make = lambda: Scalar(name=sregistry.make_name(), dtype=cluster.dtype
).indexify()
# The `depth` determines "how big" the extracted sum-of-products will be.
# We observe that in typical FD codes:
# add(mul, mul, ...) -> stems from first order derivative
# add(mul(add(mul, mul, ...), ...), ...) -> stems from second order derivative
# To search the muls in the former case, we need `depth=0`; to search the outer
# muls in the latter case, we need `depth=2`
depth = n
exclude = {
i.source.indexed
for i in cluster.scope.d_flow.independent()
}
rule0 = lambda e: not e.free_symbols & exclude
rule1 = lambda e: e.is_Mul and q_terminalop(e, depth)
rule = lambda e: rule0(e) and rule1(e)
extracted = OrderedDict()
mapper = {}
for e in cluster.exprs:
for i in search(e, rule, 'all', 'bfs_first_hit'):
if i in mapper:
continue
key = lambda a: a.is_Add
terms, others = split(list(i.args), key)
if max_alias:
# Treat `e` as an FD expression and pull out the derivative
# coefficient from `i`
# Note: typically derivative coefficients are numbers, but
# sometimes they could be provided in symbolic form through an
# arbitrary Function. In the latter case, we rely on the
# heuristic that such Function's basically never span the whole
# grid, but rather a single Grid dimension (e.g., `c[z, n]` for a
# stencil of diameter `n` along `z`)
if e.grid is not None and terms:
key = partial(maybe_coeff_key, e.grid)
others, more_terms = split(others, key)
terms.extend(more_terms)
if terms:
k = i.func(*terms)
try:
symbol, _ = extracted[k]
except KeyError:
symbol, _ = extracted.setdefault(k, (make(), e))
mapper[i] = i.func(symbol, *others)
if mapper:
extracted = [e.func(v, k) for k, (v, e) in extracted.items()]
processed = [uxreplace(e, mapper) for e in cluster.exprs]
return extracted + processed, extracted
else:
return cluster.exprs, []
def callback(self, clusters, prefix, blk_size_gen=None):
if not prefix:
return clusters
d = prefix[-1].dim
if not any(TILABLE in c.properties[d] for c in clusters):
return clusters
# Create the block Dimensions (in total `self.levels` Dimensions)
base = self.sregistry.make_name(prefix=d.name)
if blk_size_gen is not None:
# By passing a suitable key to `next` we ensure that we pull the
# next par-tile entry iff we're now blocking an unseen TILABLE nest
step = sympify(blk_size_gen.next(clusters))
else:
# This will result in a parametric step, e.g. `x0_blk0_size`
step = None
name = self.sregistry.make_name(prefix="%s_blk" % base)
bd = BlockDimension(name, d, d.symbolic_min, d.symbolic_max, step)
step = bd.step
block_dims = [bd]
for _ in range(1, self.levels):
name = self.sregistry.make_name(prefix="%s_blk" % base)
bd = BlockDimension(name, bd, bd, bd + bd.step - 1, size=step)
block_dims.append(bd)
bd = BlockDimension(d.name, bd, bd, bd + bd.step - 1, 1, size=step)
block_dims.append(bd)
processed = []
for c in clusters:
if TILABLE in c.properties[d]:
ispace = decompose(c.ispace, d, block_dims)
# Use the innermost BlockDimension in place of `d`
exprs = [uxreplace(e, {d: bd}) for e in c.exprs]
# The new Cluster properties
# TILABLE property is dropped after the blocking.
properties = dict(c.properties)
properties.pop(d)
properties.update(
{bd: c.properties[d] - {TILABLE}
for bd in block_dims})
processed.append(
c.rebuild(exprs=exprs,
ispace=ispace,
properties=properties))
else:
processed.append(c)
return processed
def lower_exprs(expressions, **kwargs):
"""
Lowering an expression consists of the following passes:
* Indexify functions;
* Align Indexeds with the computational domain;
* Apply user-provided substitution;
Examples
--------
f(x - 2*h_x, y) -> f[xi + 2, yi + 4] (assuming halo_size=4)
"""
processed = []
for expr in as_tuple(expressions):
try:
dimension_map = expr.subdomain.dimension_map
except AttributeError:
# Some Relationals may be pure SymPy objects, thus lacking the subdomain
dimension_map = {}
# Handle Functions (typical case)
mapper = {
f: f.indexify(lshift=True, subs=dimension_map)
for f in retrieve_functions(expr)
}
# Handle Indexeds (from index notation)
for i in retrieve_indexed(expr):
f = i.function
# Introduce shifting to align with the computational domain
indices = [(lower_exprs(a) + o)
for a, o in zip(i.indices, f._size_nodomain.left)]
# Apply substitutions, if necessary
if dimension_map:
indices = [j.xreplace(dimension_map) for j in indices]
mapper[i] = f.indexed[indices]
subs = kwargs.get('subs')
# Add dimensions map to the mapper in case dimensions are used
# as an expression, i.e. Eq(u, x, subdomain=xleft)
mapper.update(dimension_map)
if subs:
# Include the user-supplied substitutions, and use
# `xreplace` for constant folding
processed.append(expr.xreplace({**mapper, **subs}))
else:
processed.append(uxreplace(expr, mapper))
if isinstance(expressions, Iterable):
return processed
else:
assert len(processed) == 1
return processed.pop()
def scalarize(clusters, template):
"""
Turn local "isolated" Arrays, that is Arrays appearing only in one Cluster,
into Scalars.
"""
processed = []
for c in clusters:
# Get any Arrays appearing only in `c`
impacted = set(clusters) - {c}
arrays = {i for i in c.scope.writes if i.is_Array}
arrays -= set().union(*[i.scope.reads for i in impacted])
# Turn them into scalars
#
# r[x,y,z] = g(b[x,y,z]) t0 = g(b[x,y,z])
# ... = r[x,y,z] + r[x,y,z+1]` ----> t1 = g(b[x,y,z+1])
# ... = t0 + t1
mapper = {}
exprs = []
for n, e in enumerate(c.exprs):
f = e.lhs.function
if f in arrays:
indexeds = [i.indexed for i in c.scope[f] if i.timestamp > n]
for i in filter_ordered(indexeds):
mapper[i] = Scalar(name=template(), dtype=f.dtype)
assert len(f.indices) == len(e.lhs.indices) == len(
i.indices)
shifting = {
idx: idx + (o2 - o1)
for idx, o1, o2 in zip(f.indices, e.lhs.indices,
i.indices)
}
handle = e.func(mapper[i], uxreplace(e.rhs, mapper))
handle = xreplace_indices(handle, shifting)
exprs.append(handle)
else:
exprs.append(e.func(e.lhs, uxreplace(e.rhs, mapper)))
processed.append(c.rebuild(exprs))
return processed
def rebuild(cluster, others, aliases, subs):
# Rebuild the non-aliasing expressions
exprs = [uxreplace(e, subs) for e in others]
# Add any new ShiftedDimension to the IterationSpace
ispace = cluster.ispace.augment(aliases.index_mapper)
# Rebuild the DataSpace to include the new symbols
accesses = detect_accesses(exprs)
parts = {k: IntervalGroup(build_intervals(v)).relaxed
for k, v in accesses.items() if k}
dspace = DataSpace(cluster.dspace.intervals, parts)
return cluster.rebuild(exprs=exprs, ispace=ispace, dspace=dspace)
def rebuild(cluster, others, schedule, subs):
"""
Plug the optimized aliases into the input Cluster. This leads to creating
a new Cluster with suitable IterationSpace and DataSpace.
"""
exprs = [uxreplace(e, subs) for e in others]
ispace = cluster.ispace.augment(schedule.dmapper)
ispace = ispace.augment(schedule.rmapper)
accesses = detect_accesses(exprs)
parts = {k: IntervalGroup(build_intervals(v)).relaxed
for k, v in accesses.items() if k}
dspace = DataSpace(cluster.dspace.intervals, parts)
return cluster.rebuild(exprs=exprs, ispace=ispace, dspace=dspace)
def callback(self, clusters, prefix):
if not prefix:
return clusters
d = prefix[-1].dim
# Create the block Dimensions (in total `self.levels` Dimensions)
name = self.template % (d.name, self.nblocked[d], '%d')
bd = IncrDimension(name % 0, d, d.symbolic_min, d.symbolic_max)
size = bd.step
block_dims = [bd]
for i in range(1, self.levels):
bd = IncrDimension(name % i, bd, bd, bd + bd.step - 1, size=size)
block_dims.append(bd)
bd = IncrDimension(d.name, bd, bd, bd + bd.step - 1, 1, size=size)
block_dims.append(bd)
processed = []
for c in clusters:
if TILABLE in c.properties[d]:
ispace = decompose(c.ispace, d, block_dims)
# Use the innermost IncrDimension in place of `d`
exprs = [uxreplace(e, {d: bd}) for e in c.exprs]
# The new Cluster properties
# TILABLE property is dropped after the blocking.
# SKEWABLE is dropped as well, but only from the new
# block dimensions.
properties = dict(c.properties)
properties.pop(d)
properties.update({bd: c.properties[d] - {TILABLE} for bd in block_dims})
properties.update({bd: c.properties[d] - {SKEWABLE}
for bd in block_dims[:-1]})
processed.append(c.rebuild(exprs=exprs, ispace=ispace,
properties=properties))
else:
processed.append(c)
# Make sure to use unique IncrDimensions
self.nblocked[d] += int(any(TILABLE in c.properties[d] for c in clusters))
return processed
def extract(cls, n, context, min_cost, max_alias, cluster, sregistry):
make = lambda: Scalar(name=sregistry.make_name(), dtype=cluster.dtype
).indexify()
rule = cls._extract_rule(context, min_cost, cluster)
extracted = []
mapper = OrderedDict()
for e in cluster.exprs:
for i in search(e, rule, 'all', 'dfs_first_hit'):
if i not in mapper:
symbol = make()
mapper[i] = symbol
extracted.append(e.func(symbol, i))
processed = [uxreplace(e, mapper) for e in cluster.exprs]
return extracted + processed, extracted
def eliminate_arrays(clusters):
"""
Eliminate redundant expressions stored in Arrays.
"""
mapper = {}
processed = []
for c in clusters:
if not c.is_dense:
processed.append(c)
continue
# Search for any redundant RHSs
seen = {}
for e in c.exprs:
f = e.lhs.function
if not f.is_Array:
continue
v = seen.get(e.rhs)
if v is not None:
# Found a redundant RHS
mapper[f] = v
else:
seen[e.rhs] = f
if not mapper:
# Do not waste time
processed.append(c)
continue
# Replace redundancies
subs = {}
for f, v in mapper.items():
for i in filter_ordered(i.indexed for i in c.scope[f]):
subs[i] = v[i.indices]
exprs = []
for e in c.exprs:
if e.lhs.function in mapper:
# Drop the write
continue
exprs.append(uxreplace(e, subs))
processed.append(c.rebuild(exprs))
return processed
def extract(cls, n, context, min_cost, cluster, sregistry):
make = lambda: Scalar(name=sregistry.make_name(), dtype=cluster.dtype
).indexify()
# The `depth` determines "how big" the extracted sum-of-products will be.
# We observe that in typical FD codes:
# add(mul, mul, ...) -> stems from first order derivative
# add(mul(add(mul, mul, ...), ...), ...) -> stems from second order derivative
# To search the muls in the former case, we need `depth=0`; to search the outer
# muls in the latter case, we need `depth=2`
depth = n
exclude = {
i.source.indexed
for i in cluster.scope.d_flow.independent()
}
rule0 = lambda e: not e.free_symbols & exclude
rule1 = lambda e: e.is_Mul and q_terminalop(e, depth)
rule = lambda e: rule0(e) and rule1(e)
extracted = OrderedDict()
mapper = {}
for e in cluster.exprs:
for i in search(e, rule, 'all', 'bfs_first_hit'):
if i in mapper:
continue
# Separate numbers and Functions, as they could be a derivative coeff
terms, others = split(i.args, lambda a: a.is_Add)
if terms:
k = i.func(*terms)
try:
symbol, _ = extracted[k]
except KeyError:
symbol, _ = extracted.setdefault(k, (make(), e))
mapper[i] = i.func(symbol, *others)
if mapper:
extracted = [e.func(v, k) for k, (v, e) in extracted.items()]
processed = [uxreplace(e, mapper) for e in cluster.exprs]
return extracted + processed, extracted
else:
return cluster.exprs, []
def optimize_clusters_msds(clusters):
"""
Relax the clusters by letting the expressions defined over MultiSubDomains to
rather be computed over the entire domain. This increases the likelihood of
code lifting by later passes.
"""
processed = []
for c in clusters:
msds = [
d for d in c.ispace.itdimensions
if isinstance(d, MultiSubDimension)
]
if msds:
mapper = {d: d.root for d in msds}
exprs = [uxreplace(e, mapper) for e in c.exprs]
ispace = c.ispace.relaxed(msds)
accesses = detect_accesses(exprs)
parts = {
k: IntervalGroup(build_intervals(v)).relaxed
for k, v in accesses.items() if k
}
intervals = [i for i in c.dspace if i.dim not in msds]
dspace = DataSpace(intervals, parts)
guards = {mapper.get(d, d): v for d, v in c.guards.items()}
properties = {mapper.get(d, d): v for d, v in c.properties.items()}
syncs = {mapper.get(d, d): v for d, v in c.syncs.items()}
processed.append(
c.rebuild(exprs=exprs,
ispace=ispace,
dspace=dspace,
guards=guards,
properties=properties,
syncs=syncs))
else:
processed.append(c)
return processed
def __new__(cls, *args, **kwargs):
if len(args) == 1 and isinstance(args[0], LoweredEq):
# origin: LoweredEq(devito.LoweredEq, **kwargs)
input_expr = args[0]
expr = sympy.Eq.__new__(cls, *input_expr.args, evaluate=False)
for i in cls._state:
setattr(expr, '_%s' % i,
kwargs.get(i) or getattr(input_expr, i))
return expr
elif len(args) == 1 and isinstance(args[0], Eq):
# origin: LoweredEq(devito.Eq)
input_expr = expr = args[0]
elif len(args) == 2:
expr = sympy.Eq.__new__(cls, *args, evaluate=False)
for i in cls._state:
setattr(expr, '_%s' % i, kwargs.pop(i))
return expr
else:
raise ValueError("Cannot construct LoweredEq from args=%s "
"and kwargs=%s" % (str(args), str(kwargs)))
# Well-defined dimension ordering
ordering = dimension_sort(expr)
# Analyze the expression
accesses = detect_accesses(expr)
dimensions = Stencil.union(*accesses.values())
# Separate out the SubIterators from the main iteration Dimensions, that
# is those which define an actual iteration space
iterators = {}
for d in dimensions:
if d.is_SubIterator:
iterators.setdefault(d.root, set()).add(d)
elif d.is_Conditional:
# Use `parent`, and `root`, because a ConditionalDimension may
# have a SubDimension as parent
iterators.setdefault(d.parent, set())
else:
iterators.setdefault(d, set())
# Construct the IterationSpace
intervals = IntervalGroup([Interval(d, 0, 0) for d in iterators],
relations=ordering.relations)
ispace = IterationSpace(intervals, iterators)
# Construct the conditionals and replace the ConditionalDimensions in `expr`
conditionals = {}
for d in ordering:
if not d.is_Conditional:
continue
if d.condition is None:
conditionals[d] = GuardFactor(d)
else:
conditionals[d] = diff2sympy(lower_exprs(d.condition))
if d.factor is not None:
expr = uxreplace(expr, {d: IntDiv(d.index, d.factor)})
conditionals = frozendict(conditionals)
# Lower all Differentiable operations into SymPy operations
rhs = diff2sympy(expr.rhs)
# Finally create the LoweredEq with all metadata attached
expr = super(LoweredEq, cls).__new__(cls,
expr.lhs,
rhs,
evaluate=False)
expr._ispace = ispace
expr._conditionals = conditionals
expr._reads, expr._writes = detect_io(expr)
expr._is_Increment = input_expr.is_Increment
expr._implicit_dims = input_expr.implicit_dims
return expr
def process(cluster, chosen, aliases, sregistry, platform):
clusters = []
subs = {}
for alias, writeto, aliaseds, distances in aliases.iter(cluster.ispace):
if all(i not in chosen for i in aliaseds):
continue
# The Dimensions defining the shape of Array
# Note: with SubDimensions, we may have the following situation:
#
# for zi = z_m + zi_ltkn; zi <= z_M - zi_rtkn; ...
# r[zi] = ...
#
# Instead of `r[zi - z_m - zi_ltkn]` we have just `r[zi]`, so we'll need
# as much room as in `zi`'s parent to avoid going OOB
# Aside from ugly generated code, the reason we do not rather shift the
# indices is that it prevents future passes to transform the loop bounds
# (e.g., MPI's comp/comm overlap does that)
dimensions = [d.parent if d.is_Sub else d for d in writeto.dimensions]
# The halo of the Array
halo = [(abs(i.lower), abs(i.upper)) for i in writeto]
# The data sharing mode of the Array
sharing = 'local' if any(d.is_Incr for d in writeto.dimensions) else 'shared'
# Finally create the temporary Array that will store `alias`
array = Array(name=sregistry.make_name(), dimensions=dimensions, halo=halo,
dtype=cluster.dtype, sharing=sharing)
# The access Dimensions may differ from `writeto.dimensions`. This may
# happen e.g. if ShiftedDimensions are introduced (`a[x,y]` -> `a[xs,y]`)
adims = [aliases.index_mapper.get(d, d) for d in writeto.dimensions]
# The expression computing `alias`
adims = [aliases.index_mapper.get(d, d) for d in writeto.dimensions] # x -> xs
indices = [d - (0 if writeto[d].is_Null else writeto[d].lower) for d in adims]
expression = Eq(array[indices], uxreplace(alias, subs))
# Create the substitution rules so that we can use the newly created
# temporary in place of the aliasing expressions
for aliased, distance in zip(aliaseds, distances):
assert all(i.dim in distance.labels for i in writeto)
indices = [d - i.lower + distance[i.dim] for d, i in zip(adims, writeto)]
subs[aliased] = array[indices]
if aliased in chosen:
subs[chosen[aliased]] = array[indices]
else:
# Perhaps part of a composite alias ?
pass
# Construct the `alias` IterationSpace
ispace = cluster.ispace.add(writeto).augment(aliases.index_mapper)
# Optimization: if possible, the innermost IterationInterval is
# rounded up to a multiple of the vector length
try:
it = ispace.itintervals[-1]
if ROUNDABLE in cluster.properties[it.dim]:
vl = platform.simd_items_per_reg(cluster.dtype)
ispace = ispace.add(Interval(it.dim, 0, it.interval.size % vl))
except (TypeError, KeyError):
pass
# Construct the `alias` DataSpace
accesses = detect_accesses(expression)
parts = {k: IntervalGroup(build_intervals(v)).add(ispace.intervals).relaxed
for k, v in accesses.items() if k}
dspace = DataSpace(cluster.dspace.intervals, parts)
# Finally, build a new Cluster for `alias`
built = cluster.rebuild(exprs=expression, ispace=ispace, dspace=dspace)
clusters.insert(0, built)
return clusters, subs
def collect(exprs, min_storage, ignore_collected):
"""
Find groups of aliasing expressions.
We shall introduce the following (loose) terminology:
* A ``terminal`` is the leaf of a mathematical operation. Terminals
can be numbers (n), literals (l), or Indexeds (I).
* ``R`` is the relaxation operator := ``R(n) = n``, ``R(l) = l``,
``R(I) = J``, where ``J`` has the same base as ``I`` but with all
offsets stripped away. For example, ``R(a[i+2,j-1]) = a[i,j]``.
* A ``relaxed expression`` is an expression in which all of the
terminals are relaxed.
Now we define the concept of aliasing. We say that an expression A
aliases an expression B if:
* ``R(A) == R(B)``
* all pairwise Indexeds in A and B access memory locations at a
fixed constant distance along each Dimension.
For example, consider the following expressions:
* a[i+1] + b[i+1]
* a[i+1] + b[j+1]
* a[i] + c[i]
* a[i+2] - b[i+2]
* a[i+2] + b[i]
* a[i-1] + b[i-1]
Out of the expressions above, the following alias to `a[i] + b[i]`:
* a[i+1] + b[i+1] : same operands and operations, distance along i: 1
* a[i-1] + b[i-1] : same operands and operations, distance along i: -1
Whereas the following do not:
* a[i+1] + b[j+1] : because at least one index differs
* a[i] + c[i] : because at least one of the operands differs
* a[i+2] - b[i+2] : because at least one operation differs
* a[i+2] + b[i] : because the distances along ``i`` differ (+2 and +0)
"""
# Find the potential aliases
found = []
for expr in exprs:
if expr.lhs.is_Indexed or expr.is_Increment:
continue
indexeds = retrieve_indexed(expr.rhs)
bases = []
offsets = []
for i in indexeds:
ii = IterationInstance(i)
if ii.is_irregular:
break
base = []
offset = []
for e, ai in zip(ii, ii.aindices):
if q_constant(e):
base.append(e)
else:
base.append(ai)
offset.append((ai, e - ai))
bases.append(tuple(base))
offsets.append(LabeledVector(offset))
if indexeds and len(bases) == len(indexeds):
found.append(Candidate(expr, indexeds, bases, offsets))
# Create groups of aliasing expressions
mapper = OrderedDict()
unseen = list(found)
while unseen:
c = unseen.pop(0)
group = [c]
for u in list(unseen):
# Is the arithmetic structure of `c` and `u` equivalent ?
if not compare_ops(c.expr, u.expr):
continue
# Is `c` translated w.r.t. `u` ?
if not c.translated(u):
continue
group.append(u)
unseen.remove(u)
group = Group(group)
# Apply callback to heuristically discard groups
if ignore_collected(group):
continue
if min_storage:
k = group.dimensions_translated
else:
k = group.dimensions
mapper.setdefault(k, []).append(group)
aliases = Aliases()
for _groups in list(mapper.values()):
groups = list(_groups)
while groups:
# For each Dimension, determine the Minimum Intervals (MI) spanning
# all of the Groups diameters
# Example: x's largest_diameter=2 => [x[-2,0], x[-1,1], x[0,2]]
# Note: Groups that cannot evaluate their diameter are dropped
mapper = defaultdict(int)
for g in list(groups):
try:
mapper.update({d: max(mapper[d], v) for d, v in g.diameter.items()})
except ValueError:
groups.remove(g)
intervalss = {d: make_rotations_table(d, v) for d, v in mapper.items()}
# For each Group, find a rotation that is compatible with a given MI
mapper = {}
for d, intervals in intervalss.items():
for interval in list(intervals):
found = {g: g.find_rotation_distance(d, interval) for g in groups}
if all(distance is not None for distance in found.values()):
# `interval` is OK !
mapper[interval] = found
break
if len(mapper) == len(intervalss):
break
# Try again with fewer groups
smallest = len(min(groups, key=len))
groups = [g for g in groups if len(g) > smallest]
for g in groups:
c = g.pivot
distances = defaultdict(int, [(i.dim, v[g]) for i, v in mapper.items()])
# Create the basis alias
offsets = [LabeledVector([(l, v[l] + distances[l]) for l in v.labels])
for v in c.offsets]
subs = {i: i.function[[l + v.fromlabel(l, 0) for l in b]]
for i, b, v in zip(c.indexeds, c.bases, offsets)}
alias = uxreplace(c.expr, subs)
# All aliased expressions
aliaseds = [i.expr for i in g]
# Distance of each aliased expression from the basis alias
distances = []
for i in g:
distance = [o.distance(v) for o, v in zip(i.offsets, offsets)]
distance = [(d, set(v)) for d, v in LabeledVector.transpose(*distance)]
distances.append(LabeledVector([(d, v.pop()) for d, v in distance]))
aliases.add(alias, list(mapper), aliaseds, distances)
return aliases
def __init__(self, mapper=None, replacer=None):
super(XSubs, self).__init__()
self.replacer = replacer or (lambda i: uxreplace(i, mapper))
def callback(self, clusters, prefix):
if not prefix:
return clusters
d = prefix[-1].dim
subiters = flatten(
[c.ispace.sub_iterators.get(d, []) for c in clusters])
subiters = {i for i in subiters if i.is_Stepping}
if not subiters:
return clusters
# Collect the index access functions along `d`, e.g., `t + 1` where `t` is
# a SteppingDimension for `d = time`
mapper = DefaultOrderedDict(lambda: DefaultOrderedDict(set))
for c in clusters:
indexeds = [
a.indexed for a in c.scope.accesses if a.function.is_Tensor
]
for i in indexeds:
try:
iaf = i.indices[d]
except KeyError:
continue
# Sanity checks
sis = iaf.free_symbols & subiters
if len(sis) == 0:
continue
elif len(sis) == 1:
si = sis.pop()
else:
raise InvalidOperator(
"Cannot use multiple SteppingDimensions "
"to index into a Function")
size = i.function.shape_allocated[d]
assert is_integer(size)
mapper[size][si].add(iaf)
# Construct the ModuloDimensions
mds = []
for size, v in mapper.items():
for si, iafs in list(v.items()):
# Offsets are sorted so that the semantic order (t0, t1, t2) follows
# SymPy's index ordering (t, t-1, t+1) afer modulo replacement so
# that associativity errors are consistent. This corresponds to
# sorting offsets {-1, 0, 1} as {0, -1, 1} assigning -inf to 0
siafs = sorted(iafs,
key=lambda i: -np.inf
if i - si == 0 else (i - si))
for iaf in siafs:
name = '%s%d' % (si.name, len(mds))
offset = uxreplace(iaf, {si: d.root})
mds.append(
ModuloDimension(name, si, offset, size, origin=iaf))
# Replacement rule for ModuloDimensions
def rule(size, e):
try:
return e.function.shape_allocated[d] == size
except (AttributeError, KeyError):
return False
# Reconstruct the Clusters
processed = []
for c in clusters:
# Apply substitutions to expressions
# Note: In an expression, there could be `u[t+1, ...]` and `v[t+1,
# ...]`, where `u` and `v` are TimeFunction with circular time
# buffers (save=None) *but* different modulo extent. The `t+1`
# indices above are therefore conceptually different, so they will
# be replaced with the proper ModuloDimension through two different
# calls to `xreplace_indices`
exprs = c.exprs
groups = as_mapper(mds, lambda d: d.modulo)
for size, v in groups.items():
mapper = {md.origin: md for md in v}
func = partial(xreplace_indices,
mapper=mapper,
key=partial(rule, size))
exprs = [e.apply(func) for e in exprs]
# Augment IterationSpace
ispace = IterationSpace(c.ispace.intervals, {
**c.ispace.sub_iterators,
**{
d: tuple(mds)
}
}, c.ispace.directions)
processed.append(c.rebuild(exprs=exprs, ispace=ispace))
return processed
def _cse(maybe_exprs, make, mode='default'):
"""
Main common sub-expressions elimination routine.
Note: the output is guaranteed to be topologically sorted.
Parameters
----------
maybe_exprs : expr-like or list of expr-like or Cluster
One or more expressions to which CSE is applied.
make : callable
Build symbols to store temporary, redundant values.
mode : str, optional
The CSE algorithm applied. Accepted: ['default'].
"""
# Note: not defaulting to SymPy's CSE() function for three reasons:
# - it also captures array index access functions (eg, i+1 in A[i+1] and B[i+1]);
# - it sometimes "captures too much", losing factorization opportunities;
# - very slow
# TODO: a second "sympy" mode will be provided, relying on SymPy's CSE() but
# also ensuring some form of post-processing
assert mode == 'default' # Only supported mode ATM
# Just for flexibility, accept either Clusters or exprs
if isinstance(maybe_exprs, Cluster):
cluster = maybe_exprs
processed = list(cluster.exprs)
scope = cluster.scope
else:
processed = list(maybe_exprs)
scope = Scope(maybe_exprs)
# Some sub-expressions aren't really "common" -- that's the case of Dimension-
# independent data dependences. For example:
#
# ... = ... a[i] + 1 ...
# a[i] = ...
# ... = ... a[i] + 1 ...
#
# `a[i] + 1` will be excluded, as there's a flow Dimension-independent data
# dependence involving `a`
exclude = {i.source.access for i in scope.d_flow.independent()}
mapped = []
while True:
# Detect redundancies
counted = count(mapped + processed, q_xop).items()
targets = OrderedDict([(k, estimate_cost(k, True)) for k, v in counted
if v > 1])
# Rule out Dimension-independent data dependencies
targets = OrderedDict([(k, v) for k, v in targets.items()
if not k.free_symbols & exclude])
if not targets:
break
# Create temporaries
hit = max(targets.values())
picked = [k for k, v in targets.items() if v == hit]
mapper = OrderedDict([(e, make()) for i, e in enumerate(picked)])
# Apply replacements
processed = [uxreplace(e, mapper) for e in processed]
mapped = [uxreplace(e, mapper) for e in mapped]
mapped = [Eq(v, k) for k, v in reversed(list(mapper.items()))] + mapped
# Update `exclude` for the same reasons as above -- to rule out CSE across
# Dimension-independent data dependences
exclude.update({i for i in mapper.values()})
# Prepare for the next round
for k in picked:
targets.pop(k)
processed = mapped + processed
# At this point we may have useless temporaries (e.g., r0=r1). Let's drop them
processed = _compact_temporaries(processed)
return processed
def visit_Expression(self, o):
return o._rebuild(expr=uxreplace(o.expr, self.mapper))
def visit_Call(self, o):
arguments = [uxreplace(i, self.mapper) for i in o.arguments]
return o._rebuild(arguments=arguments)
def actions_from_update_memcpy(cluster, clusters, prefix, actions):
it = prefix[-1]
d = it.dim
direction = it.direction
# Prepare the data to instantiate a PrefetchUpdate SyncOp
e = cluster.exprs[0]
size = 1
function = e.rhs.function
fetch = e.rhs.indices[d]
ifetch = fetch.subs(d, d.symbolic_min)
if direction is Forward:
fcond = make_cond(cluster.guards.get(d), d, direction, d.symbolic_min)
else:
fcond = make_cond(cluster.guards.get(d), d, direction, d.symbolic_max)
if direction is Forward:
pfetch = fetch + 1
pcond = make_cond(cluster.guards.get(d), d, direction, d + 1)
else:
pfetch = fetch - 1
pcond = make_cond(cluster.guards.get(d), d, direction, d - 1)
target = e.lhs.function
tstore0 = e.lhs.indices[d]
# If fetching into e.g., `ub[sb1]`, we'll need to prefetch into e.g. `ub[sb0]`
if is_integer(tstore0):
tstore = tstore0
else:
assert tstore0.is_Modulo
subiters = [md for md in cluster.sub_iterators[d] if md.parent is tstore0.parent]
osubiters = sorted(subiters, key=lambda i: Vector(i.offset))
n = osubiters.index(tstore0)
if direction is Forward:
tstore = osubiters[(n + 1) % len(osubiters)]
else:
tstore = osubiters[(n - 1) % len(osubiters)]
# Turn `cluster` into a prefetch Cluster
expr = uxreplace(e, {tstore0: tstore, fetch: pfetch})
guards = {d: And(*([pcond] + as_list(cluster.guards.get(d))))}
syncs = {d: [PrefetchUpdate(
d, size,
function, fetch, ifetch, fcond,
pfetch, pcond,
target, tstore
)]}
pcluster = cluster.rebuild(exprs=expr, guards=guards, syncs=syncs)
# Since we're turning `e` into a prefetch, we need to:
# 1) attach a WaitPrefetch SyncOp to the first Cluster accessing `target`
# 2) insert the prefetch Cluster right after the last Cluster accessing `target`
# 3) drop the original Cluster performing a memcpy-based fetch
n = clusters.index(cluster)
first = None
last = None
for c in clusters[n+1:]:
if target in c.scope.reads:
if first is None:
first = c
last = c
assert first is not None
assert last is not None
actions[first].syncs[d].append(WaitPrefetch(
d, size,
function, fetch, ifetch, fcond,
pfetch, pcond,
target, tstore
))
actions[last].insert.append(pcluster)
actions[cluster].drop = True