xref: /llvm-project-15.0.7/mlir/lib/Dialect/Affine/Transforms/SuperVectorize.cpp (revision fa1b602e)

//===- SuperVectorize.cpp - Vectorize Pass Impl ---------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
//
// This file implements vectorization of loops, operations and data types to
// a target-independent, n-D super-vector abstraction.
//
//===----------------------------------------------------------------------===//

#include "mlir/Analysis/LoopAnalysis.h"
#include "mlir/Analysis/NestedMatcher.h"
#include "mlir/Analysis/SliceAnalysis.h"
#include "mlir/Analysis/Utils.h"
#include "mlir/Dialect/Affine/IR/AffineOps.h"
#include "mlir/Dialect/Affine/Passes.h"
#include "mlir/Dialect/StandardOps/IR/Ops.h"
#include "mlir/Dialect/Vector/VectorOps.h"
#include "mlir/Dialect/Vector/VectorUtils.h"
#include "mlir/IR/AffineExpr.h"
#include "mlir/IR/Builders.h"
#include "mlir/IR/Location.h"
#include "mlir/IR/Types.h"
#include "mlir/Pass/Pass.h"
#include "mlir/Support/Functional.h"
#include "mlir/Support/LLVM.h"
#include "mlir/Transforms/FoldUtils.h"

#include "llvm/ADT/DenseMap.h"
#include "llvm/ADT/DenseSet.h"
#include "llvm/ADT/SetVector.h"
#include "llvm/ADT/SmallString.h"
#include "llvm/ADT/SmallVector.h"
#include "llvm/Support/CommandLine.h"
#include "llvm/Support/Debug.h"

using namespace mlir;

///
/// Implements a high-level vectorization strategy on a Function.
/// The abstraction used is that of super-vectors, which provide a single,
/// compact, representation in the vector types, information that is expected
/// to reduce the impact of the phase ordering problem
///
/// Vector granularity:
/// ===================
/// This pass is designed to perform vectorization at a super-vector
/// granularity. A super-vector is loosely defined as a vector type that is a
/// multiple of a "good" vector size so the HW can efficiently implement a set
/// of high-level primitives. Multiple is understood along any dimension; e.g.
/// both vector<16xf32> and vector<2x8xf32> are valid super-vectors for a
/// vector<8xf32> HW vector. Note that a "good vector size so the HW can
/// efficiently implement a set of high-level primitives" is not necessarily an
/// integer multiple of actual hardware registers. We leave details of this
/// distinction unspecified for now.
///
/// Some may prefer the terminology a "tile of HW vectors". In this case, one
/// should note that super-vectors implement an "always full tile" abstraction.
/// They guarantee no partial-tile separation is necessary by relying on a
/// high-level copy-reshape abstraction that we call vector.transfer. This
/// copy-reshape operations is also responsible for performing layout
/// transposition if necessary. In the general case this will require a scoped
/// allocation in some notional local memory.
///
/// Whatever the mental model one prefers to use for this abstraction, the key
/// point is that we burn into a single, compact, representation in the vector
/// types, information that is expected to reduce the impact of the phase
/// ordering problem. Indeed, a vector type conveys information that:
///   1. the associated loops have dependency semantics that do not prevent
///      vectorization;
///   2. the associate loops have been sliced in chunks of static sizes that are
///      compatible with vector sizes (i.e. similar to unroll-and-jam);
///   3. the inner loops, in the unroll-and-jam analogy of 2, are captured by
///   the
///      vector type and no vectorization hampering transformations can be
///      applied to them anymore;
///   4. the underlying memrefs are accessed in some notional contiguous way
///      that allows loading into vectors with some amount of spatial locality;
/// In other words, super-vectorization provides a level of separation of
/// concern by way of opacity to subsequent passes. This has the effect of
/// encapsulating and propagating vectorization constraints down the list of
/// passes until we are ready to lower further.
///
/// For a particular target, a notion of minimal n-d vector size will be
/// specified and vectorization targets a multiple of those. In the following
/// paragraph, let "k ." represent "a multiple of", to be understood as a
/// multiple in the same dimension (e.g. vector<16 x k . 128> summarizes
/// vector<16 x 128>, vector<16 x 256>, vector<16 x 1024>, etc).
///
/// Some non-exhaustive notable super-vector sizes of interest include:
///   - CPU: vector<k . HW_vector_size>,
///          vector<k' . core_count x k . HW_vector_size>,
///          vector<socket_count x k' . core_count x k . HW_vector_size>;
///   - GPU: vector<k . warp_size>,
///          vector<k . warp_size x float2>,
///          vector<k . warp_size x float4>,
///          vector<k . warp_size x 4 x 4x 4> (for tensor_core sizes).
///
/// Loops and operations are emitted that operate on those super-vector shapes.
/// Subsequent lowering passes will materialize to actual HW vector sizes. These
/// passes are expected to be (gradually) more target-specific.
///
/// At a high level, a vectorized load in a loop will resemble:
/// ```mlir
///   affine.for %i = ? to ? step ? {
///     %v_a = vector.transfer_read A[%i] : memref<?xf32>, vector<128xf32>
///   }
/// ```
/// It is the responsibility of the implementation of vector.transfer_read to
/// materialize vector registers from the original scalar memrefs. A later (more
/// target-dependent) lowering pass will materialize to actual HW vector sizes.
/// This lowering may be occur at different times:
///   1. at the MLIR level into a combination of loops, unrolling, DmaStartOp +
///      DmaWaitOp + vectorized operations for data transformations and shuffle;
///      thus opening opportunities for unrolling and pipelining. This is an
///      instance of library call "whiteboxing"; or
///   2. later in the a target-specific lowering pass or hand-written library
///      call; achieving full separation of concerns. This is an instance of
///      library call; or
///   3. a mix of both, e.g. based on a model.
/// In the future, these operations will expose a contract to constrain the
/// search on vectorization patterns and sizes.
///
/// Occurrence of super-vectorization in the compiler flow:
/// =======================================================
/// This is an active area of investigation. We start with 2 remarks to position
/// super-vectorization in the context of existing ongoing work: LLVM VPLAN
/// and LLVM SLP Vectorizer.
///
/// LLVM VPLAN:
/// -----------
/// The astute reader may have noticed that in the limit, super-vectorization
/// can be applied at a similar time and with similar objectives than VPLAN.
/// For instance, in the case of a traditional, polyhedral compilation-flow (for
/// instance, the PPCG project uses ISL to provide dependence analysis,
/// multi-level(scheduling + tiling), lifting footprint to fast memory,
/// communication synthesis, mapping, register optimizations) and before
/// unrolling. When vectorization is applied at this *late* level in a typical
/// polyhedral flow, and is instantiated with actual hardware vector sizes,
/// super-vectorization is expected to match (or subsume) the type of patterns
/// that LLVM's VPLAN aims at targeting. The main difference here is that MLIR
/// is higher level and our implementation should be significantly simpler. Also
/// note that in this mode, recursive patterns are probably a bit of an overkill
/// although it is reasonable to expect that mixing a bit of outer loop and
/// inner loop vectorization + unrolling will provide interesting choices to
/// MLIR.
///
/// LLVM SLP Vectorizer:
/// --------------------
/// Super-vectorization however is not meant to be usable in a similar fashion
/// to the SLP vectorizer. The main difference lies in the information that
/// both vectorizers use: super-vectorization examines contiguity of memory
/// references along fastest varying dimensions and loops with recursive nested
/// patterns capturing imperfectly-nested loop nests; the SLP vectorizer, on
/// the other hand, performs flat pattern matching inside a single unrolled loop
/// body and stitches together pieces of load and store operations into full
/// 1-D vectors. We envision that the SLP vectorizer is a good way to capture
/// innermost loop, control-flow dependent patterns that super-vectorization may
/// not be able to capture easily. In other words, super-vectorization does not
/// aim at replacing the SLP vectorizer and the two solutions are complementary.
///
/// Ongoing investigations:
/// -----------------------
/// We discuss the following *early* places where super-vectorization is
/// applicable and touch on the expected benefits and risks . We list the
/// opportunities in the context of the traditional polyhedral compiler flow
/// described in PPCG. There are essentially 6 places in the MLIR pass pipeline
/// we expect to experiment with super-vectorization:
/// 1. Right after language lowering to MLIR: this is the earliest time where
///    super-vectorization is expected to be applied. At this level, all the
///    language/user/library-level annotations are available and can be fully
///    exploited. Examples include loop-type annotations (such as parallel,
///    reduction, scan, dependence distance vector, vectorizable) as well as
///    memory access annotations (such as non-aliasing writes guaranteed,
///    indirect accesses that are permutations by construction) accesses or
///    that a particular operation is prescribed atomic by the user. At this
///    level, anything that enriches what dependence analysis can do should be
///    aggressively exploited. At this level we are close to having explicit
///    vector types in the language, except we do not impose that burden on the
///    programmer/library: we derive information from scalar code + annotations.
/// 2. After dependence analysis and before polyhedral scheduling: the
///    information that supports vectorization does not need to be supplied by a
///    higher level of abstraction. Traditional dependence analysis is available
///    in MLIR and will be used to drive vectorization and cost models.
///
/// Let's pause here and remark that applying super-vectorization as described
/// in 1. and 2. presents clear opportunities and risks:
///   - the opportunity is that vectorization is burned in the type system and
///   is protected from the adverse effect of loop scheduling, tiling, loop
///   interchange and all passes downstream. Provided that subsequent passes are
///   able to operate on vector types; the vector shapes, associated loop
///   iterator properties, alignment, and contiguity of fastest varying
///   dimensions are preserved until we lower the super-vector types. We expect
///   this to significantly rein in on the adverse effects of phase ordering.
///   - the risks are that a. all passes after super-vectorization have to work
///   on elemental vector types (not that this is always true, wherever
///   vectorization is applied) and b. that imposing vectorization constraints
///   too early may be overall detrimental to loop fusion, tiling and other
///   transformations because the dependence distances are coarsened when
///   operating on elemental vector types. For this reason, the pattern
///   profitability analysis should include a component that also captures the
///   maximal amount of fusion available under a particular pattern. This is
///   still at the stage of rough ideas but in this context, search is our
///   friend as the Tensor Comprehensions and auto-TVM contributions
///   demonstrated previously.
/// Bottom-line is we do not yet have good answers for the above but aim at
/// making it easy to answer such questions.
///
/// Back to our listing, the last places where early super-vectorization makes
/// sense are:
/// 3. right after polyhedral-style scheduling: PLUTO-style algorithms are known
///    to improve locality, parallelism and be configurable (e.g. max-fuse,
///    smart-fuse etc). They can also have adverse effects on contiguity
///    properties that are required for vectorization but the vector.transfer
///    copy-reshape-pad-transpose abstraction is expected to help recapture
///    these properties.
/// 4. right after polyhedral-style scheduling+tiling;
/// 5. right after scheduling+tiling+rescheduling: points 4 and 5 represent
///    probably the most promising places because applying tiling achieves a
///    separation of concerns that allows rescheduling to worry less about
///    locality and more about parallelism and distribution (e.g. min-fuse).
///
/// At these levels the risk-reward looks different: on one hand we probably
/// lost a good deal of language/user/library-level annotation; on the other
/// hand we gained parallelism and locality through scheduling and tiling.
/// However we probably want to ensure tiling is compatible with the
/// full-tile-only abstraction used in super-vectorization or suffer the
/// consequences. It is too early to place bets on what will win but we expect
/// super-vectorization to be the right abstraction to allow exploring at all
/// these levels. And again, search is our friend.
///
/// Lastly, we mention it again here:
/// 6. as a MLIR-based alternative to VPLAN.
///
/// Lowering, unrolling, pipelining:
/// ================================
/// TODO(ntv): point to the proper places.
///
/// Algorithm:
/// ==========
/// The algorithm proceeds in a few steps:
///  1. defining super-vectorization patterns and matching them on the tree of
///     AffineForOp. A super-vectorization pattern is defined as a recursive
///     data structures that matches and captures nested, imperfectly-nested
///     loops that have a. conformable loop annotations attached (e.g. parallel,
///     reduction, vectorizable, ...) as well as b. all contiguous load/store
///     operations along a specified minor dimension (not necessarily the
///     fastest varying) ;
///  2. analyzing those patterns for profitability (TODO(ntv): and
///     interference);
///  3. Then, for each pattern in order:
///    a. applying iterative rewriting of the loop and the load operations in
///       DFS postorder. Rewriting is implemented by coarsening the loops and
///       turning load operations into opaque vector.transfer_read ops;
///    b. keeping track of the load operations encountered as "roots" and the
///       store operations as "terminals";
///    c. traversing the use-def chains starting from the roots and iteratively
///       propagating vectorized values. Scalar values that are encountered
///       during this process must come from outside the scope of the current
///       pattern (TODO(ntv): enforce this and generalize). Such a scalar value
///       is vectorized only if it is a constant (into a vector splat). The
///       non-constant case is not supported for now and results in the pattern
///       failing to vectorize;
///    d. performing a second traversal on the terminals (store ops) to
///       rewriting the scalar value they write to memory into vector form.
///       If the scalar value has been vectorized previously, we simply replace
///       it by its vector form. Otherwise, if the scalar value is a constant,
///       it is vectorized into a splat. In all other cases, vectorization for
///       the pattern currently fails.
///    e. if everything under the root AffineForOp in the current pattern
///       vectorizes properly, we commit that loop to the IR. Otherwise we
///       discard it and restore a previously cloned version of the loop. Thanks
///       to the recursive scoping nature of matchers and captured patterns,
///       this is transparently achieved by a simple RAII implementation.
///    f. vectorization is applied on the next pattern in the list. Because
///       pattern interference avoidance is not yet implemented and that we do
///       not support further vectorizing an already vector load we need to
///       re-verify that the pattern is still vectorizable. This is expected to
///       make cost models more difficult to write and is subject to improvement
///       in the future.
///
/// Points c. and d. above are worth additional comment. In most passes that
/// do not change the type of operands, it is usually preferred to eagerly
/// `replaceAllUsesWith`. Unfortunately this does not work for vectorization
/// because during the use-def chain traversal, all the operands of an operation
/// must be available in vector form. Trying to propagate eagerly makes the IR
/// temporarily invalid and results in errors such as:
///   `vectorize.mlir:308:13: error: 'addf' op requires the same type for all
///   operands and results
///      %s5 = addf %a5, %b5 : f32`
///
/// Lastly, we show a minimal example for which use-def chains rooted in load /
/// vector.transfer_read are not enough. This is what motivated splitting
/// terminal processing out of the use-def chains starting from loads. In the
/// following snippet, there is simply no load::
/// ```mlir
/// func @fill(%A : memref<128xf32>) -> () {
///   %f1 = constant 1.0 : f32
///   affine.for %i0 = 0 to 32 {
///     affine.store %f1, %A[%i0] : memref<128xf32, 0>
///   }
///   return
/// }
/// ```
///
/// Choice of loop transformation to support the algorithm:
/// =======================================================
/// The choice of loop transformation to apply for coarsening vectorized loops
/// is still subject to exploratory tradeoffs. In particular, say we want to
/// vectorize by a factor 128, we want to transform the following input:
/// ```mlir
///   affine.for %i = %M to %N {
///     %a = affine.load %A[%i] : memref<?xf32>
///   }
/// ```
///
/// Traditionally, one would vectorize late (after scheduling, tiling,
/// memory promotion etc) say after stripmining (and potentially unrolling in
/// the case of LLVM's SLP vectorizer):
/// ```mlir
///   affine.for %i = floor(%M, 128) to ceil(%N, 128) {
///     affine.for %ii = max(%M, 128 * %i) to min(%N, 128*%i + 127) {
///       %a = affine.load %A[%ii] : memref<?xf32>
///     }
///   }
/// ```
///
/// Instead, we seek to vectorize early and freeze vector types before
/// scheduling, so we want to generate a pattern that resembles:
/// ```mlir
///   affine.for %i = ? to ? step ? {
///     %v_a = vector.transfer_read %A[%i] : memref<?xf32>, vector<128xf32>
///   }
/// ```
///
/// i. simply dividing the lower / upper bounds by 128 creates issues
///    when representing expressions such as ii + 1 because now we only
///    have access to original values that have been divided. Additional
///    information is needed to specify accesses at below-128 granularity;
/// ii. another alternative is to coarsen the loop step but this may have
///    consequences on dependence analysis and fusability of loops: fusable
///    loops probably need to have the same step (because we don't want to
///    stripmine/unroll to enable fusion).
/// As a consequence, we choose to represent the coarsening using the loop
/// step for now and reevaluate in the future. Note that we can renormalize
/// loop steps later if/when we have evidence that they are problematic.
///
/// For the simple strawman example above, vectorizing for a 1-D vector
/// abstraction of size 128 returns code similar to:
/// ```mlir
///   affine.for %i = %M to %N step 128 {
///     %v_a = vector.transfer_read %A[%i] : memref<?xf32>, vector<128xf32>
///   }
/// ```
///
/// Unsupported cases, extensions, and work in progress (help welcome :-) ):
/// ========================================================================
///   1. lowering to concrete vector types for various HW;
///   2. reduction support;
///   3. non-effecting padding during vector.transfer_read and filter during
///      vector.transfer_write;
///   4. misalignment support vector.transfer_read / vector.transfer_write
///      (hopefully without read-modify-writes);
///   5. control-flow support;
///   6. cost-models, heuristics and search;
///   7. Op implementation, extensions and implication on memref views;
///   8. many TODOs left around.
///
/// Examples:
/// =========
/// Consider the following Function:
/// ```mlir
/// func @vector_add_2d(%M : index, %N : index) -> f32 {
///   %A = alloc (%M, %N) : memref<?x?xf32, 0>
///   %B = alloc (%M, %N) : memref<?x?xf32, 0>
///   %C = alloc (%M, %N) : memref<?x?xf32, 0>
///   %f1 = constant 1.0 : f32
///   %f2 = constant 2.0 : f32
///   affine.for %i0 = 0 to %M {
///     affine.for %i1 = 0 to %N {
///       // non-scoped %f1
///       affine.store %f1, %A[%i0, %i1] : memref<?x?xf32, 0>
///     }
///   }
///   affine.for %i2 = 0 to %M {
///     affine.for %i3 = 0 to %N {
///       // non-scoped %f2
///       affine.store %f2, %B[%i2, %i3] : memref<?x?xf32, 0>
///     }
///   }
///   affine.for %i4 = 0 to %M {
///     affine.for %i5 = 0 to %N {
///       %a5 = affine.load %A[%i4, %i5] : memref<?x?xf32, 0>
///       %b5 = affine.load %B[%i4, %i5] : memref<?x?xf32, 0>
///       %s5 = addf %a5, %b5 : f32
///       // non-scoped %f1
///       %s6 = addf %s5, %f1 : f32
///       // non-scoped %f2
///       %s7 = addf %s5, %f2 : f32
///       // diamond dependency.
///       %s8 = addf %s7, %s6 : f32
///       affine.store %s8, %C[%i4, %i5] : memref<?x?xf32, 0>
///     }
///   }
///   %c7 = constant 7 : index
///   %c42 = constant 42 : index
///   %res = load %C[%c7, %c42] : memref<?x?xf32, 0>
///   return %res : f32
/// }
/// ```
///
/// The -affine-vectorize pass with the following arguments:
/// ```
/// -affine-vectorize="virtual-vector-size=256 test-fastest-varying=0"
/// ```
///
/// produces this standard innermost-loop vectorized code:
/// ```mlir
/// func @vector_add_2d(%arg0 : index, %arg1 : index) -> f32 {
///   %0 = alloc(%arg0, %arg1) : memref<?x?xf32>
///   %1 = alloc(%arg0, %arg1) : memref<?x?xf32>
///   %2 = alloc(%arg0, %arg1) : memref<?x?xf32>
///   %cst = constant 1.0 : f32
///   %cst_0 = constant 2.0 : f32
///   affine.for %i0 = 0 to %arg0 {
///     affine.for %i1 = 0 to %arg1 step 256 {
///       %cst_1 = constant dense<vector<256xf32>, 1.0> :
///                vector<256xf32>
///       vector.transfer_write %cst_1, %0[%i0, %i1] :
///                vector<256xf32>, memref<?x?xf32>
///     }
///   }
///   affine.for %i2 = 0 to %arg0 {
///     affine.for %i3 = 0 to %arg1 step 256 {
///       %cst_2 = constant dense<vector<256xf32>, 2.0> :
///                vector<256xf32>
///       vector.transfer_write %cst_2, %1[%i2, %i3] :
///                vector<256xf32>, memref<?x?xf32>
///     }
///   }
///   affine.for %i4 = 0 to %arg0 {
///     affine.for %i5 = 0 to %arg1 step 256 {
///       %3 = vector.transfer_read %0[%i4, %i5] :
///            memref<?x?xf32>, vector<256xf32>
///       %4 = vector.transfer_read %1[%i4, %i5] :
///            memref<?x?xf32>, vector<256xf32>
///       %5 = addf %3, %4 : vector<256xf32>
///       %cst_3 = constant dense<vector<256xf32>, 1.0> :
///                vector<256xf32>
///       %6 = addf %5, %cst_3 : vector<256xf32>
///       %cst_4 = constant dense<vector<256xf32>, 2.0> :
///                vector<256xf32>
///       %7 = addf %5, %cst_4 : vector<256xf32>
///       %8 = addf %7, %6 : vector<256xf32>
///       vector.transfer_write %8, %2[%i4, %i5] :
///                vector<256xf32>, memref<?x?xf32>
///     }
///   }
///   %c7 = constant 7 : index
///   %c42 = constant 42 : index
///   %9 = load %2[%c7, %c42] : memref<?x?xf32>
///   return %9 : f32
/// }
/// ```
///
/// The -affine-vectorize pass with the following arguments:
/// ```
/// -affine-vectorize="virtual-vector-size=32,256 test-fastest-varying=1,0"
/// ```
///
/// produces this more interesting mixed outer-innermost-loop vectorized code:
/// ```mlir
/// func @vector_add_2d(%arg0 : index, %arg1 : index) -> f32 {
///   %0 = alloc(%arg0, %arg1) : memref<?x?xf32>
///   %1 = alloc(%arg0, %arg1) : memref<?x?xf32>
///   %2 = alloc(%arg0, %arg1) : memref<?x?xf32>
///   %cst = constant 1.0 : f32
///   %cst_0 = constant 2.0 : f32
///   affine.for %i0 = 0 to %arg0 step 32 {
///     affine.for %i1 = 0 to %arg1 step 256 {
///       %cst_1 = constant dense<vector<32x256xf32>, 1.0> :
///                vector<32x256xf32>
///       vector.transfer_write %cst_1, %0[%i0, %i1] :
///                vector<32x256xf32>, memref<?x?xf32>
///     }
///   }
///   affine.for %i2 = 0 to %arg0 step 32 {
///     affine.for %i3 = 0 to %arg1 step 256 {
///       %cst_2 = constant dense<vector<32x256xf32>, 2.0> :
///                vector<32x256xf32>
///       vector.transfer_write %cst_2, %1[%i2, %i3] :
///                vector<32x256xf32>, memref<?x?xf32>
///     }
///   }
///   affine.for %i4 = 0 to %arg0 step 32 {
///     affine.for %i5 = 0 to %arg1 step 256 {
///       %3 = vector.transfer_read %0[%i4, %i5] :
///                memref<?x?xf32> vector<32x256xf32>
///       %4 = vector.transfer_read %1[%i4, %i5] :
///                memref<?x?xf32>, vector<32x256xf32>
///       %5 = addf %3, %4 : vector<32x256xf32>
///       %cst_3 = constant dense<vector<32x256xf32>, 1.0> :
///                vector<32x256xf32>
///       %6 = addf %5, %cst_3 : vector<32x256xf32>
///       %cst_4 = constant dense<vector<32x256xf32>, 2.0> :
///                vector<32x256xf32>
///       %7 = addf %5, %cst_4 : vector<32x256xf32>
///       %8 = addf %7, %6 : vector<32x256xf32>
///       vector.transfer_write %8, %2[%i4, %i5] :
///                vector<32x256xf32>, memref<?x?xf32>
///     }
///   }
///   %c7 = constant 7 : index
///   %c42 = constant 42 : index
///   %9 = load %2[%c7, %c42] : memref<?x?xf32>
///   return %9 : f32
/// }
/// ```
///
/// Of course, much more intricate n-D imperfectly-nested patterns can be
/// vectorized too and specified in a fully declarative fashion.

#define DEBUG_TYPE "early-vect"

using functional::makePtrDynCaster;
using functional::map;
using llvm::dbgs;
using llvm::SetVector;

/// Forward declaration.
static FilterFunctionType
isVectorizableLoopPtrFactory(const DenseSet<Operation *> &parallelLoops,
                             int fastestVaryingMemRefDimension);

/// Creates a vectorization pattern from the command line arguments.
/// Up to 3-D patterns are supported.
/// If the command line argument requests a pattern of higher order, returns an
/// empty pattern list which will conservatively result in no vectorization.
static std::vector<NestedPattern>
makePatterns(const DenseSet<Operation *> &parallelLoops, int vectorRank,
             ArrayRef<int64_t> fastestVaryingPattern) {
  using matcher::For;
  int64_t d0 = fastestVaryingPattern.empty() ? -1 : fastestVaryingPattern[0];
  int64_t d1 = fastestVaryingPattern.size() < 2 ? -1 : fastestVaryingPattern[1];
  int64_t d2 = fastestVaryingPattern.size() < 3 ? -1 : fastestVaryingPattern[2];
  switch (vectorRank) {
  case 1:
    return {For(isVectorizableLoopPtrFactory(parallelLoops, d0))};
  case 2:
    return {For(isVectorizableLoopPtrFactory(parallelLoops, d0),
                For(isVectorizableLoopPtrFactory(parallelLoops, d1)))};
  case 3:
    return {For(isVectorizableLoopPtrFactory(parallelLoops, d0),
                For(isVectorizableLoopPtrFactory(parallelLoops, d1),
                    For(isVectorizableLoopPtrFactory(parallelLoops, d2))))};
  default: {
    return std::vector<NestedPattern>();
  }
  }
}

static NestedPattern &vectorTransferPattern() {
  static auto pattern = matcher::Op([](Operation &op) {
    return isa<vector::TransferReadOp>(op) || isa<vector::TransferWriteOp>(op);
  });
  return pattern;
}

namespace {

/// Base state for the vectorize pass.
/// Command line arguments are preempted by non-empty pass arguments.
struct Vectorize : public FunctionPass<Vectorize> {
  Vectorize() = default;
  Vectorize(const Vectorize &) {}
  Vectorize(ArrayRef<int64_t> virtualVectorSize);
  void runOnFunction() override;

  /// The virtual vector size that we vectorize to.
  ListOption<int64_t> vectorSizes{
      *this, "virtual-vector-size",
      llvm::cl::desc("Specify an n-D virtual vector size for vectorization"),
      llvm::cl::ZeroOrMore, llvm::cl::CommaSeparated};
  /// Optionally, the fixed mapping from loop to fastest varying MemRef
  /// dimension for all the MemRefs within a loop pattern:
  ///   the index represents the loop depth, the value represents the k^th
  ///   fastest varying memory dimension.
  /// This is voluntarily restrictive and is meant to precisely target a
  /// particular loop/op pair, for testing purposes.
  ListOption<int64_t> fastestVaryingPattern{
      *this, "test-fastest-varying",
      llvm::cl::desc(
          "Specify a 1-D, 2-D or 3-D pattern of fastest varying memory"
          " dimensions to match. See defaultPatterns in Vectorize.cpp for a"
          " description and examples. This is used for testing purposes"),
      llvm::cl::ZeroOrMore, llvm::cl::CommaSeparated};
};

} // end anonymous namespace

Vectorize::Vectorize(ArrayRef<int64_t> virtualVectorSize) {
  vectorSizes->assign(virtualVectorSize.begin(), virtualVectorSize.end());
}

/////// TODO(ntv): Hoist to a VectorizationStrategy.cpp when appropriate.
/////////
namespace {

struct VectorizationStrategy {
  SmallVector<int64_t, 8> vectorSizes;
  DenseMap<Operation *, unsigned> loopToVectorDim;
};

} // end anonymous namespace

static void vectorizeLoopIfProfitable(Operation *loop, unsigned depthInPattern,
                                      unsigned patternDepth,
                                      VectorizationStrategy *strategy) {
  assert(patternDepth > depthInPattern &&
         "patternDepth is greater than depthInPattern");
  if (patternDepth - depthInPattern > strategy->vectorSizes.size()) {
    // Don't vectorize this loop
    return;
  }
  strategy->loopToVectorDim[loop] =
      strategy->vectorSizes.size() - (patternDepth - depthInPattern);
}

/// Implements a simple strawman strategy for vectorization.
/// Given a matched pattern `matches` of depth `patternDepth`, this strategy
/// greedily assigns the fastest varying dimension ** of the vector ** to the
/// innermost loop in the pattern.
/// When coupled with a pattern that looks for the fastest varying dimension in
/// load/store MemRefs, this creates a generic vectorization strategy that works
/// for any loop in a hierarchy (outermost, innermost or intermediate).
///
/// TODO(ntv): In the future we should additionally increase the power of the
/// profitability analysis along 3 directions:
///   1. account for loop extents (both static and parametric + annotations);
///   2. account for data layout permutations;
///   3. account for impact of vectorization on maximal loop fusion.
/// Then we can quantify the above to build a cost model and search over
/// strategies.
static LogicalResult analyzeProfitability(ArrayRef<NestedMatch> matches,
                                          unsigned depthInPattern,
                                          unsigned patternDepth,
                                          VectorizationStrategy *strategy) {
  for (auto m : matches) {
    if (failed(analyzeProfitability(m.getMatchedChildren(), depthInPattern + 1,
                                    patternDepth, strategy))) {
      return failure();
    }
    vectorizeLoopIfProfitable(m.getMatchedOperation(), depthInPattern,
                              patternDepth, strategy);
  }
  return success();
}

///// end TODO(ntv): Hoist to a VectorizationStrategy.cpp when appropriate /////

namespace {

struct VectorizationState {
  /// Adds an entry of pre/post vectorization operations in the state.
  void registerReplacement(Operation *key, Operation *value);
  /// When the current vectorization pattern is successful, this erases the
  /// operations that were marked for erasure in the proper order and resets
  /// the internal state for the next pattern.
  void finishVectorizationPattern();

  // In-order tracking of original Operation that have been vectorized.
  // Erase in reverse order.
  SmallVector<Operation *, 16> toErase;
  // Set of Operation that have been vectorized (the values in the
  // vectorizationMap for hashed access). The vectorizedSet is used in
  // particular to filter the operations that have already been vectorized by
  // this pattern, when iterating over nested loops in this pattern.
  DenseSet<Operation *> vectorizedSet;
  // Map of old scalar Operation to new vectorized Operation.
  DenseMap<Operation *, Operation *> vectorizationMap;
  // Map of old scalar Value to new vectorized Value.
  DenseMap<Value, Value> replacementMap;
  // The strategy drives which loop to vectorize by which amount.
  const VectorizationStrategy *strategy;
  // Use-def roots. These represent the starting points for the worklist in the
  // vectorizeNonTerminals function. They consist of the subset of load
  // operations that have been vectorized. They can be retrieved from
  // `vectorizationMap` but it is convenient to keep track of them in a separate
  // data structure.
  DenseSet<Operation *> roots;
  // Terminal operations for the worklist in the vectorizeNonTerminals
  // function. They consist of the subset of store operations that have been
  // vectorized. They can be retrieved from `vectorizationMap` but it is
  // convenient to keep track of them in a separate data structure. Since they
  // do not necessarily belong to use-def chains starting from loads (e.g
  // storing a constant), we need to handle them in a post-pass.
  DenseSet<Operation *> terminals;
  // Checks that the type of `op` is AffineStoreOp and adds it to the terminals
  // set.
  void registerTerminal(Operation *op);
  // Folder used to factor out constant creation.
  OperationFolder *folder;

private:
  void registerReplacement(Value key, Value value);
};

} // end namespace

void VectorizationState::registerReplacement(Operation *key, Operation *value) {
  LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ commit vectorized op: ");
  LLVM_DEBUG(key->print(dbgs()));
  LLVM_DEBUG(dbgs() << "  into  ");
  LLVM_DEBUG(value->print(dbgs()));
  assert(key->getNumResults() == 1 && "already registered");
  assert(value->getNumResults() == 1 && "already registered");
  assert(vectorizedSet.count(value) == 0 && "already registered");
  assert(vectorizationMap.count(key) == 0 && "already registered");
  toErase.push_back(key);
  vectorizedSet.insert(value);
  vectorizationMap.insert(std::make_pair(key, value));
  registerReplacement(key->getResult(0), value->getResult(0));
  if (isa<AffineLoadOp>(key)) {
    assert(roots.count(key) == 0 && "root was already inserted previously");
    roots.insert(key);
  }
}

void VectorizationState::registerTerminal(Operation *op) {
  assert(isa<AffineStoreOp>(op) && "terminal must be a AffineStoreOp");
  assert(terminals.count(op) == 0 &&
         "terminal was already inserted previously");
  terminals.insert(op);
}

void VectorizationState::finishVectorizationPattern() {
  while (!toErase.empty()) {
    auto *op = toErase.pop_back_val();
    LLVM_DEBUG(dbgs() << "\n[early-vect] finishVectorizationPattern erase: ");
    LLVM_DEBUG(op->print(dbgs()));
    op->erase();
  }
}

void VectorizationState::registerReplacement(Value key, Value value) {
  assert(replacementMap.count(key) == 0 && "replacement already registered");
  replacementMap.insert(std::make_pair(key, value));
}

// Apply 'map' with 'mapOperands' returning resulting values in 'results'.
static void computeMemoryOpIndices(Operation *op, AffineMap map,
                                   ValueRange mapOperands,
                                   SmallVectorImpl<Value> &results) {
  OpBuilder builder(op);
  for (auto resultExpr : map.getResults()) {
    auto singleResMap =
        AffineMap::get(map.getNumDims(), map.getNumSymbols(), resultExpr);
    auto afOp =
        builder.create<AffineApplyOp>(op->getLoc(), singleResMap, mapOperands);
    results.push_back(afOp);
  }
}

////// TODO(ntv): Hoist to a VectorizationMaterialize.cpp when appropriate. ////

/// Handles the vectorization of load and store MLIR operations.
///
/// AffineLoadOp operations are the roots of the vectorizeNonTerminals call.
/// They are vectorized immediately. The resulting vector.transfer_read is
/// immediately registered to replace all uses of the AffineLoadOp in this
/// pattern's scope.
///
/// AffineStoreOp are the terminals of the vectorizeNonTerminals call. They
/// need to be vectorized late once all the use-def chains have been traversed.
/// Additionally, they may have ssa-values operands which come from outside the
/// scope of the current pattern.
/// Such special cases force us to delay the vectorization of the stores until
/// the last step. Here we merely register the store operation.
template <typename LoadOrStoreOpPointer>
static LogicalResult vectorizeRootOrTerminal(Value iv,
                                             LoadOrStoreOpPointer memoryOp,
                                             VectorizationState *state) {
  auto memRefType = memoryOp.getMemRef().getType().template cast<MemRefType>();

  auto elementType = memRefType.getElementType();
  // TODO(ntv): ponder whether we want to further vectorize a vector value.
  assert(VectorType::isValidElementType(elementType) &&
         "Not a valid vector element type");
  auto vectorType = VectorType::get(state->strategy->vectorSizes, elementType);

  // Materialize a MemRef with 1 vector.
  auto *opInst = memoryOp.getOperation();
  // For now, vector.transfers must be aligned, operate only on indices with an
  // identity subset of AffineMap and do not change layout.
  // TODO(ntv): increase the expressiveness power of vector.transfer operations
  // as needed by various targets.
  if (auto load = dyn_cast<AffineLoadOp>(opInst)) {
    OpBuilder b(opInst);
    ValueRange mapOperands = load.getMapOperands();
    SmallVector<Value, 8> indices;
    indices.reserve(load.getMemRefType().getRank());
    if (load.getAffineMap() !=
        b.getMultiDimIdentityMap(load.getMemRefType().getRank())) {
      computeMemoryOpIndices(opInst, load.getAffineMap(), mapOperands, indices);
    } else {
      indices.append(mapOperands.begin(), mapOperands.end());
    }
    auto permutationMap =
        makePermutationMap(opInst, indices, state->strategy->loopToVectorDim);
    if (!permutationMap)
      return LogicalResult::Failure;
    LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ permutationMap: ");
    LLVM_DEBUG(permutationMap.print(dbgs()));
    auto transfer = b.create<vector::TransferReadOp>(
        opInst->getLoc(), vectorType, memoryOp.getMemRef(), indices,
        AffineMapAttr::get(permutationMap),
        // TODO(b/144455320) add a proper padding value, not just 0.0 : f32
        state->folder->create<ConstantFloatOp>(b, opInst->getLoc(),
                                               APFloat(0.0f), b.getF32Type()));
    state->registerReplacement(opInst, transfer.getOperation());
  } else {
    state->registerTerminal(opInst);
  }
  return success();
}
/// end TODO(ntv): Hoist to a VectorizationMaterialize.cpp when appropriate. ///

/// Coarsens the loops bounds and transforms all remaining load and store
/// operations into the appropriate vector.transfer.
static LogicalResult vectorizeAffineForOp(AffineForOp loop, int64_t step,
                                          VectorizationState *state) {
  using namespace functional;
  loop.setStep(step);

  FilterFunctionType notVectorizedThisPattern = [state](Operation &op) {
    if (!matcher::isLoadOrStore(op)) {
      return false;
    }
    return state->vectorizationMap.count(&op) == 0 &&
           state->vectorizedSet.count(&op) == 0 &&
           state->roots.count(&op) == 0 && state->terminals.count(&op) == 0;
  };
  auto loadAndStores = matcher::Op(notVectorizedThisPattern);
  SmallVector<NestedMatch, 8> loadAndStoresMatches;
  loadAndStores.match(loop.getOperation(), &loadAndStoresMatches);
  for (auto ls : loadAndStoresMatches) {
    auto *opInst = ls.getMatchedOperation();
    auto load = dyn_cast<AffineLoadOp>(opInst);
    auto store = dyn_cast<AffineStoreOp>(opInst);
    LLVM_DEBUG(opInst->print(dbgs()));
    LogicalResult result =
        load ? vectorizeRootOrTerminal(loop.getInductionVar(), load, state)
             : vectorizeRootOrTerminal(loop.getInductionVar(), store, state);
    if (failed(result)) {
      return failure();
    }
  }
  return success();
}

/// Returns a FilterFunctionType that can be used in NestedPattern to match a
/// loop whose underlying load/store accesses are either invariant or all
// varying along the `fastestVaryingMemRefDimension`.
static FilterFunctionType
isVectorizableLoopPtrFactory(const DenseSet<Operation *> &parallelLoops,
                             int fastestVaryingMemRefDimension) {
  return [&parallelLoops, fastestVaryingMemRefDimension](Operation &forOp) {
    auto loop = cast<AffineForOp>(forOp);
    auto parallelIt = parallelLoops.find(loop);
    if (parallelIt == parallelLoops.end())
      return false;
    int memRefDim = -1;
    auto vectorizableBody =
        isVectorizableLoopBody(loop, &memRefDim, vectorTransferPattern());
    if (!vectorizableBody)
      return false;
    return memRefDim == -1 || fastestVaryingMemRefDimension == -1 ||
           memRefDim == fastestVaryingMemRefDimension;
  };
}

/// Apply vectorization of `loop` according to `state`. This is only triggered
/// if all vectorizations in `childrenMatches` have already succeeded
/// recursively in DFS post-order.
static LogicalResult
vectorizeLoopsAndLoadsRecursively(NestedMatch oneMatch,
                                  VectorizationState *state) {
  auto *loopInst = oneMatch.getMatchedOperation();
  auto loop = cast<AffineForOp>(loopInst);
  auto childrenMatches = oneMatch.getMatchedChildren();

  // 1. DFS postorder recursion, if any of my children fails, I fail too.
  for (auto m : childrenMatches) {
    if (failed(vectorizeLoopsAndLoadsRecursively(m, state))) {
      return failure();
    }
  }

  // 2. This loop may have been omitted from vectorization for various reasons
  // (e.g. due to the performance model or pattern depth > vector size).
  auto it = state->strategy->loopToVectorDim.find(loopInst);
  if (it == state->strategy->loopToVectorDim.end()) {
    return success();
  }

  // 3. Actual post-order transformation.
  auto vectorDim = it->second;
  assert(vectorDim < state->strategy->vectorSizes.size() &&
         "vector dim overflow");
  //   a. get actual vector size
  auto vectorSize = state->strategy->vectorSizes[vectorDim];
  //   b. loop transformation for early vectorization is still subject to
  //     exploratory tradeoffs (see top of the file). Apply coarsening, i.e.:
  //        | ub -> ub
  //        | step -> step * vectorSize
  LLVM_DEBUG(dbgs() << "\n[early-vect] vectorizeForOp by " << vectorSize
                    << " : ");
  LLVM_DEBUG(loopInst->print(dbgs()));
  return vectorizeAffineForOp(loop, loop.getStep() * vectorSize, state);
}

/// Tries to transform a scalar constant into a vector splat of that constant.
/// Returns the vectorized splat operation if the constant is a valid vector
/// element type.
/// If `type` is not a valid vector type or if the scalar constant is not a
/// valid vector element type, returns nullptr.
static Value vectorizeConstant(Operation *op, ConstantOp constant, Type type) {
  if (!type || !type.isa<VectorType>() ||
      !VectorType::isValidElementType(constant.getType())) {
    return nullptr;
  }
  OpBuilder b(op);
  Location loc = op->getLoc();
  auto vectorType = type.cast<VectorType>();
  auto attr = DenseElementsAttr::get(vectorType, constant.getValue());
  auto *constantOpInst = constant.getOperation();

  OperationState state(loc, constantOpInst->getName().getStringRef(), {},
                       {vectorType}, {b.getNamedAttr("value", attr)});

  return b.createOperation(state)->getResult(0);
}

/// Tries to vectorize a given operand `op` of Operation `op` during
/// def-chain propagation or during terminal vectorization, by applying the
/// following logic:
/// 1. if the defining operation is part of the vectorizedSet (i.e. vectorized
///    useby -def propagation), `op` is already in the proper vector form;
/// 2. otherwise, the `op` may be in some other vector form that fails to
///    vectorize atm (i.e. broadcasting required), returns nullptr to indicate
///    failure;
/// 3. if the `op` is a constant, returns the vectorized form of the constant;
/// 4. non-constant scalars are currently non-vectorizable, in particular to
///    guard against vectorizing an index which may be loop-variant and needs
///    special handling.
///
/// In particular this logic captures some of the use cases where definitions
/// that are not scoped under the current pattern are needed to vectorize.
/// One such example is top level function constants that need to be splatted.
///
/// Returns an operand that has been vectorized to match `state`'s strategy if
/// vectorization is possible with the above logic. Returns nullptr otherwise.
///
/// TODO(ntv): handle more complex cases.
static Value vectorizeOperand(Value operand, Operation *op,
                              VectorizationState *state) {
  LLVM_DEBUG(dbgs() << "\n[early-vect]vectorize operand: " << operand);
  // 1. If this value has already been vectorized this round, we are done.
  if (state->vectorizedSet.count(operand.getDefiningOp()) > 0) {
    LLVM_DEBUG(dbgs() << " -> already vector operand");
    return operand;
  }
  // 1.b. Delayed on-demand replacement of a use.
  //    Note that we cannot just call replaceAllUsesWith because it may result
  //    in ops with mixed types, for ops whose operands have not all yet
  //    been vectorized. This would be invalid IR.
  auto it = state->replacementMap.find(operand);
  if (it != state->replacementMap.end()) {
    auto res = it->second;
    LLVM_DEBUG(dbgs() << "-> delayed replacement by: " << res);
    return res;
  }
  // 2. TODO(ntv): broadcast needed.
  if (operand.getType().isa<VectorType>()) {
    LLVM_DEBUG(dbgs() << "-> non-vectorizable");
    return nullptr;
  }
  // 3. vectorize constant.
  if (auto constant = dyn_cast_or_null<ConstantOp>(operand.getDefiningOp())) {
    return vectorizeConstant(
        op, constant,
        VectorType::get(state->strategy->vectorSizes, operand.getType()));
  }
  // 4. currently non-vectorizable.
  LLVM_DEBUG(dbgs() << "-> non-vectorizable: " << operand);
  return nullptr;
}

/// Encodes Operation-specific behavior for vectorization. In general we assume
/// that all operands of an op must be vectorized but this is not always true.
/// In the future, it would be nice to have a trait that describes how a
/// particular operation vectorizes. For now we implement the case distinction
/// here.
/// Returns a vectorized form of an operation or nullptr if vectorization fails.
// TODO(ntv): consider adding a trait to Op to describe how it gets vectorized.
// Maybe some Ops are not vectorizable or require some tricky logic, we cannot
// do one-off logic here; ideally it would be TableGen'd.
static Operation *vectorizeOneOperation(Operation *opInst,
                                        VectorizationState *state) {
  // Sanity checks.
  assert(!isa<AffineLoadOp>(opInst) &&
         "all loads must have already been fully vectorized independently");
  assert(!isa<vector::TransferReadOp>(opInst) &&
         "vector.transfer_read cannot be further vectorized");
  assert(!isa<vector::TransferWriteOp>(opInst) &&
         "vector.transfer_write cannot be further vectorized");

  if (auto store = dyn_cast<AffineStoreOp>(opInst)) {
    OpBuilder b(opInst);
    auto memRef = store.getMemRef();
    auto value = store.getValueToStore();
    auto vectorValue = vectorizeOperand(value, opInst, state);
    if (!vectorValue)
      return nullptr;

    ValueRange mapOperands = store.getMapOperands();
    SmallVector<Value, 8> indices;
    indices.reserve(store.getMemRefType().getRank());
    if (store.getAffineMap() !=
        b.getMultiDimIdentityMap(store.getMemRefType().getRank())) {
      computeMemoryOpIndices(opInst, store.getAffineMap(), mapOperands,
                             indices);
    } else {
      indices.append(mapOperands.begin(), mapOperands.end());
    }

    auto permutationMap =
        makePermutationMap(opInst, indices, state->strategy->loopToVectorDim);
    if (!permutationMap)
      return nullptr;
    LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ permutationMap: ");
    LLVM_DEBUG(permutationMap.print(dbgs()));
    auto transfer = b.create<vector::TransferWriteOp>(
        opInst->getLoc(), vectorValue, memRef, indices,
        AffineMapAttr::get(permutationMap));
    auto *res = transfer.getOperation();
    LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ vectorized store: " << *res);
    // "Terminals" (i.e. AffineStoreOps) are erased on the spot.
    opInst->erase();
    return res;
  }
  if (opInst->getNumRegions() != 0)
    return nullptr;

  SmallVector<Type, 8> vectorTypes;
  for (auto v : opInst->getResults()) {
    vectorTypes.push_back(
        VectorType::get(state->strategy->vectorSizes, v.getType()));
  }
  SmallVector<Value, 8> vectorOperands;
  for (auto v : opInst->getOperands()) {
    vectorOperands.push_back(vectorizeOperand(v, opInst, state));
  }
  // Check whether a single operand is null. If so, vectorization failed.
  bool success = llvm::all_of(vectorOperands, [](Value op) { return op; });
  if (!success) {
    LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ an operand failed vectorize");
    return nullptr;
  }

  // Create a clone of the op with the proper operands and return types.
  // TODO(ntv): The following assumes there is always an op with a fixed
  // name that works both in scalar mode and vector mode.
  // TODO(ntv): Is it worth considering an Operation.clone operation which
  // changes the type so we can promote an Operation with less boilerplate?
  OpBuilder b(opInst);
  OperationState newOp(opInst->getLoc(), opInst->getName().getStringRef(),
                       vectorOperands, vectorTypes, opInst->getAttrs(),
                       /*successors=*/{},
                       /*regions=*/{}, opInst->hasResizableOperandsList());
  return b.createOperation(newOp);
}

/// Iterates over the forward slice from the loads in the vectorization pattern
/// and rewrites them using their vectorized counterpart by:
///   1. Create the forward slice starting from the loads in the vectorization
///   pattern.
///   2. Topologically sorts the forward slice.
///   3. For each operation in the slice, create the vector form of this
///   operation, replacing each operand by a replacement operands retrieved from
///   replacementMap. If any such replacement is missing, vectorization fails.
static LogicalResult vectorizeNonTerminals(VectorizationState *state) {
  // 1. create initial worklist with the uses of the roots.
  SetVector<Operation *> worklist;
  // Note: state->roots have already been vectorized and must not be vectorized
  // again. This fits `getForwardSlice` which does not insert `op` in the
  // result.
  // Note: we have to exclude terminals because some of their defs may not be
  // nested under the vectorization pattern (e.g. constants defined in an
  // encompassing scope).
  // TODO(ntv): Use a backward slice for terminals, avoid special casing and
  // merge implementations.
  for (auto *op : state->roots) {
    getForwardSlice(op, &worklist, [state](Operation *op) {
      return state->terminals.count(op) == 0; // propagate if not terminal
    });
  }
  // We merged multiple slices, topological order may not hold anymore.
  worklist = topologicalSort(worklist);

  for (unsigned i = 0; i < worklist.size(); ++i) {
    auto *op = worklist[i];
    LLVM_DEBUG(dbgs() << "\n[early-vect] vectorize use: ");
    LLVM_DEBUG(op->print(dbgs()));

    // Create vector form of the operation.
    // Insert it just before op, on success register op as replaced.
    auto *vectorizedInst = vectorizeOneOperation(op, state);
    if (!vectorizedInst) {
      return failure();
    }

    // 3. Register replacement for future uses in the scope.
    //    Note that we cannot just call replaceAllUsesWith because it may
    //    result in ops with mixed types, for ops whose operands have not all
    //    yet been vectorized. This would be invalid IR.
    state->registerReplacement(op, vectorizedInst);
  }
  return success();
}

/// Vectorization is a recursive procedure where anything below can fail.
/// The root match thus needs to maintain a clone for handling failure.
/// Each root may succeed independently but will otherwise clean after itself if
/// anything below it fails.
static LogicalResult vectorizeRootMatch(NestedMatch m,
                                        VectorizationStrategy *strategy) {
  auto loop = cast<AffineForOp>(m.getMatchedOperation());
  OperationFolder folder(loop.getContext());
  VectorizationState state;
  state.strategy = strategy;
  state.folder = &folder;

  // Since patterns are recursive, they can very well intersect.
  // Since we do not want a fully greedy strategy in general, we decouple
  // pattern matching, from profitability analysis, from application.
  // As a consequence we must check that each root pattern is still
  // vectorizable. If a pattern is not vectorizable anymore, we just skip it.
  // TODO(ntv): implement a non-greedy profitability analysis that keeps only
  // non-intersecting patterns.
  if (!isVectorizableLoopBody(loop, vectorTransferPattern())) {
    LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ loop is not vectorizable");
    return failure();
  }

  /// Sets up error handling for this root loop. This is how the root match
  /// maintains a clone for handling failure and restores the proper state via
  /// RAII.
  auto *loopInst = loop.getOperation();
  OpBuilder builder(loopInst);
  auto clonedLoop = cast<AffineForOp>(builder.clone(*loopInst));
  struct Guard {
    LogicalResult failure() {
      loop.getInductionVar().replaceAllUsesWith(clonedLoop.getInductionVar());
      loop.erase();
      return mlir::failure();
    }
    LogicalResult success() {
      clonedLoop.erase();
      return mlir::success();
    }
    AffineForOp loop;
    AffineForOp clonedLoop;
  } guard{loop, clonedLoop};

  //////////////////////////////////////////////////////////////////////////////
  // Start vectorizing.
  // From now on, any error triggers the scope guard above.
  //////////////////////////////////////////////////////////////////////////////
  // 1. Vectorize all the loops matched by the pattern, recursively.
  // This also vectorizes the roots (AffineLoadOp) as well as registers the
  // terminals (AffineStoreOp) for post-processing vectorization (we need to
  // wait for all use-def chains into them to be vectorized first).
  if (failed(vectorizeLoopsAndLoadsRecursively(m, &state))) {
    LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ failed root vectorizeLoop");
    return guard.failure();
  }

  // 2. Vectorize operations reached by use-def chains from root except the
  // terminals (store operations) that need to be post-processed separately.
  // TODO(ntv): add more as we expand.
  if (failed(vectorizeNonTerminals(&state))) {
    LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ failed vectorizeNonTerminals");
    return guard.failure();
  }

  // 3. Post-process terminals.
  // Note: we have to post-process terminals because some of their defs may not
  // be nested under the vectorization pattern (e.g. constants defined in an
  // encompassing scope).
  // TODO(ntv): Use a backward slice for terminals, avoid special casing and
  // merge implementations.
  for (auto *op : state.terminals) {
    if (!vectorizeOneOperation(op, &state)) { // nullptr == failure
      LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ failed to vectorize terminals");
      return guard.failure();
    }
  }

  // 4. Finish this vectorization pattern.
  LLVM_DEBUG(dbgs() << "\n[early-vect]+++++ success vectorizing pattern");
  state.finishVectorizationPattern();
  return guard.success();
}

/// Applies vectorization to the current Function by searching over a bunch of
/// predetermined patterns.
void Vectorize::runOnFunction() {
  FuncOp f = getFunction();
  if (!fastestVaryingPattern.empty() &&
      fastestVaryingPattern.size() != vectorSizes.size()) {
    f.emitRemark("Fastest varying pattern specified with different size than "
                 "the vector size.");
    return signalPassFailure();
  }

  // Thread-safe RAII local context, BumpPtrAllocator freed on exit.
  NestedPatternContext mlContext;

  DenseSet<Operation *> parallelLoops;
  f.walk([&parallelLoops](AffineForOp loop) {
    if (isLoopParallel(loop))
      parallelLoops.insert(loop);
  });

  for (auto &pat :
       makePatterns(parallelLoops, vectorSizes.size(), fastestVaryingPattern)) {
    LLVM_DEBUG(dbgs() << "\n******************************************");
    LLVM_DEBUG(dbgs() << "\n******************************************");
    LLVM_DEBUG(dbgs() << "\n[early-vect] new pattern on Function\n");
    LLVM_DEBUG(f.print(dbgs()));
    unsigned patternDepth = pat.getDepth();

    SmallVector<NestedMatch, 8> matches;
    pat.match(f, &matches);
    // Iterate over all the top-level matches and vectorize eagerly.
    // This automatically prunes intersecting matches.
    for (auto m : matches) {
      VectorizationStrategy strategy;
      // TODO(ntv): depending on profitability, elect to reduce the vector size.
      strategy.vectorSizes.assign(vectorSizes.begin(), vectorSizes.end());
      if (failed(analyzeProfitability(m.getMatchedChildren(), 1, patternDepth,
                                      &strategy))) {
        continue;
      }
      vectorizeLoopIfProfitable(m.getMatchedOperation(), 0, patternDepth,
                                &strategy);
      // TODO(ntv): if pattern does not apply, report it; alter the
      // cost/benefit.
      vectorizeRootMatch(m, &strategy);
      // TODO(ntv): some diagnostics if failure to vectorize occurs.
    }
  }
  LLVM_DEBUG(dbgs() << "\n");
}

std::unique_ptr<OpPassBase<FuncOp>>
mlir::createSuperVectorizePass(ArrayRef<int64_t> virtualVectorSize) {
  return std::make_unique<Vectorize>(virtualVectorSize);
}

static PassRegistration<Vectorize>
    pass("affine-super-vectorize",
         "Vectorize to a target independent n-D vector abstraction");