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# Towards a Library of Shufflevector Patterns

The LLVM IR contains a very powerful vector operation
called shufflevector. Unless the target architecture
supports a similarly powerful operation directly, good
code generation for `shufflevector` instructions with arbitrary
shuffle patterns is very difficult.

However, there are many `shufflevector` patterns that
map to simpler operations, both at the LLVM IR level and
at the target architecture level. This document introduces
a library of patterns as well as a systematic approach to
code generation for `shufflevector` operations.

## Pattern Recognition

With a potentially large library of shuffle vector patterns and corresponding implementations, how do we systematically search for patterns without incurring high pattern matching costs for brute-force sequential search? One answer is to abstract the patterns and to define quick, parallel methods of pattern recognition.

### Shuffle Vector Canonical Form

In order to facilitate recognition of shuffle vector patterns, we first perform some transformations to put shuffle vectors in canonical form.

#### An Initial Example

For example, consider the following three `shufflevector` patterns
are all equivalent.

shufflevector <4 x i32> zeroinitializer, <4 x i32> %x, <4 x i32> <i32 4, i32 2, i32 5, i32 1> shufflevector <4 x i32> %x, <4 x i32> zeroinitializer, <4 x i32> <i32 0, i32 6, i32 1, i32 5> shufflevector <4 x i32> %x, <4 x i32> zeroinitializer, <4 x i32> <i32 0, i32 4, i32 1, i32 4>

This example illustrates two points about equivalent forms.
First, it is always possible to produce equivalent results
by exchanging the order of the two input vectors and modifying
the shuffle selection mask, accordingly. Second, when
one of the input vectors is a vector of all zeroes (`zeroinitializer`), then
any selection mask index in this vector is equivalent to any other.

This example also illustrates two rules for the conversion to canonical form
in this case. As described below, if `zeroinitializer` occurs
as an input vector, then it should normally occur as the second
input vector. Secondly, the selection mask values for `zeroinitializer` in canonical form
always refer to the lowest element. In the example, the
selection mask values 5 and 6 are replaced by 4.

#### Canonical Selection from Constant Splats

In general, in discussing canonical forms, we refer to the
three consecutive arguments of `shufflevector` as
vector 1 or `v1`, vector 2 or `v2` and `mask`.

A constant splat is a constant vector having the same
value in every field. When used as a vector, `zeroinitializer`
is a constant splat of the value `0`.

Whenever one of the input vectors is a constant splat vector,
then the canonical selection mask values for selecting a
value from this vector must alwayws specify
selection from the first field in the vector.
That is, if the size of the input vectors is `n`,
and vector 2 is a constant splat, then the index `n`
is the canonical index for selection from this vector.
Conversion to constant indexes in this case is illustrated
in the second step of the example above. If vector 1
is a constant splat, then the canonical index
for selection from this vector is `0`.

#### Canonical Order of Vectors

Consider a shufflevector invocation of the following form.

shufflevector <n x <ty>> v1, <n x <ty>> v2, <m x i32> mask

Given such a shufflevector is always possible to reverse the order of the vectors with a suitable transformation of the mask.

shufflevector <n x <ty>> v2, <n x <ty>> v1, <m x i32> mask'

The mask transformation replaces each mask element `mask`[i]
with the value which is either `mask`[i] + n,
if `mask`[i] < n, or `mask`[i] - n, if `mask`[i] >= n.
This may be expressed by the following equation.

`mask'`[i] = (`mask`[i] + n) mod (2n)

The following rules define canonical order for the vector arguments.

- If one of the vectors is
`undef`, then vector 2 must be`undef`in canonical order. - If one of the vectors is
`zeroinitializer`(or the equivalent constant), and the other vector is not`undef`, then vector 2 must be`zeroinitializer`in canonical order. - If one of the vectors is a constant vector and the other vector is neither constant nor
`undef`, then vector 2 must be a constant vector in canonical order. - If neither vector is a constant splat or
`undef`, then the first selector value must be < n, (i.e., selecting from vector 1).

## Lane Detection in Patterns

Lane detection determines whether the shufflevector operation is
a natural SIMD operation, i.e., whether it may be viewed as
a set of parallel operations in 2^{n} lanes for some value of n.

Lane detection is a recursive binary process. The first step is to determine if the overall operation may be viewed as two separate instances of the same operation applied independently in two lanes. If so, then the operation within the lanes are analyzed to determine whether a further division by 2 is possible. When no further binary division into lanes is possible, the process terminates.

To determine whether a shuffle vector pattern may be divided into lanes, the following algorithm is applied.

- Given a shuffle vector mask of 2
^{n}elements labelled m_{0}, ... m_{2n-1}, form the two submask vectors m_{0}, mm_{2n-1-1}and m_{2n-1}, ... m_{2n-1}. - Compare the two vectors, element by element. The mask is ruled a laned operation if the following conditions hold in each case.
- The value of m
_{2n-1+i}- m_{i}= 2^{n-1}, or - The value of m
_{2n-1+i}= m_{i}= n, and vector 2 is a constant splat.

- The value of m

## Useful Functions

def pattern_diff(P, m, k): return [P[i + k] - P[i] for i in range(0, m-k)] def pattern_signature(P): return (P[0], pattern_diff(P, len(P), 1))

## Standard Pattern Examples

In the following examples, the patterns are defined assuming that the shufflevector arguments have been first placed in canonical form as described above and lanes have been determined.

### The Rotate Pattern

Rotations may easily be identified as patterns which have a pattern unit difference of 1 in all cases, except a single instance of a 1-n difference, where n is the vector (or double vector) size.

>>> pattern_diff([1,2,3,4,5,6,7,0], 8,1) [1, 1, 1, 1, 1, 1, -7] >>> pattern_diff([2,3,4,5,6,7,0,1], 8,1) [1, 1, 1, 1, 1, -7, 1] >>> pattern_diff([3,4,5,6,7,0,1,2], 8,1) [1, 1, 1, 1, -7, 1, 1]

A rotation may occur within lanes. For example, the pattern [1,2,3,0,5,6,7,4] is a rotate right 1 within each of 2 lanes. The lane detection is accomplished by noting that the n/2 pattern diff is a constant vector of n/2 values. Then the rotation pattern is exposed by examining a single lane.

>>> pattern_diff([1,2,3,0,5,6,7,4], 8, 4) [4, 4, 4, 4] >>> pattern_diff([1,2,3,0,5,6,7,4], 4, 1) [1, 1, -3] >>>

### The Shift Left Pattern

Shift left patterns require that the second shufflevector argument be zeroinitializer so that 0s can be inserted on the right. Note the little-endianness issue.

>>> pattern_signature([8,8,1,2,3,4,5,6]) (8, [0, -7, 1, 1, 1, 1, 1]) >>> pattern_signature([8,8,8,1,2,3,4,5]) (8, [0, 0, -7, 1, 1, 1, 1])

### The Shift Right Logical Pattern

Shift right logical patterns require that the second shufflevector argument be zeroinitializer so that 0s can be inserted on the left. Note Little-endianness. The shift constant is conveniently the value of the first element times the field width.

(2, [1, 1, 1, 1, 1, 1, 0]) >>> pattern_signature([1,2,3,4,5,6,7,8]) (1, [1, 1, 1, 1, 1, 1, 1]) >>> pattern_signature([0,1,2,3,4,5,6,7]) (0, [1, 1, 1, 1, 1, 1, 1]) >>> pattern_signature([5,6,7,8,8,8,8,8]) (5, [1, 1, 1, 0, 0, 0, 0])

### The Zero-Extend Pattern

### The Merge Pattern

Merge patterns have alternation n and 1-n values.

>>> pattern_signature([0,8,1,9,2,10,3,11]) (0, [8, -7, 8, -7, 8, -7, 8])

### The Pack Pattern

Pack patterns are characterized by a constant pattern difference of 2.

>>> pattern_signature([0,2,4,6,8,10,12,14]) (0, [2, 2, 2, 2, 2, 2, 2]) >>> pattern_signature([1,3,5,7,9,11,13,15]) (1, [2, 2, 2, 2, 2, 2, 2])

### The Parallel Extract Pattern

The Parallel Extract Pattern is essentially a vector filter operation,
which preserves some elements from a given input vector, deleting
others. Argument #2 must be `zeroinitializer`.

The pattern must consist of a set of increasing selectors from
the first vector, after which all remaining selectors are
zero selectors (i.e., selecting from `zeroinitializer`.

In the case of `i1` vectors, this corresponds to the parallel bit
extract of the
BitShuffle project.

### The Parallel Deposit Pattern

The Parallel Deposit Pattern is essentially a vector expand operation,
which preserves some elements from a given input vector, inserting
zeroes inbtween. Argument #2 must be `zeroinitializer`.

The pattern must consist of a set of consecutive selectors from
the first vector (starting with 0), interspersed with
zero selectors (i.e., selecting from `zeroinitializer`.

In the case of `i1` vectors, this corresponds to the parallel bit
deposit of the
BitShuffle project.