Hello,
This is a Clangfocused follow up to the original proposal on llvmdev (http://lists.llvm.org/pipermail/llvmdev/2019October/136240.html). On the LLVM side, we recently landed the first commit adding matrix intrinsics as proposed.
On the Clang side, we would like to propose adding support for matrix math operations to Clang. This includes adding a new matrix type (similar to ext_vector_type) and a set of builtins to operate on values of the matrix type.
Our main motivation for the matrix support in Clang is to give users a way to

Guarantee generation of highquality code for matrix operations. For isolated operations, we can guarantee vector code generation suitable for the target. For trees of operations, the proposed value type helps with eliminating temporary loads & stores.

Make use of specialized matrix ISA extensions, like the new matrix instructions in ARM v8.6 or various proprietary matrix accelerators, in their C/C++ code.

Move optimizations from matrix wrapper libraries into the compiler. We use it internally to simplify an Eigenstyle matrix library, by relying on LLVM for generating tiled & fused loops for matrix operations.
The rest of this RFC is structured as follows: First we propose a draft specification for the matrix type and accompanying builtins. Next we show an example of how matrix operations will be lowered by Clang, followed by a discussion of the contributing criteria for new extensions. We wrap up the RFC by discussing possible extensions to the matrix type.
Draft Specification### Matrix TYPE AttributeThe attributetoken matrix_type
is used to declare a matrix type. It shall appear at most once in each attributelist. The attribute shall only appertain to a typedefname of a typedef of a nonvolatile type that is a signed integer type, an unsigned integer type, or a floatingpoint type. An attributeargumentclause must be present and it shall have the form:
(*constantexpression*, *constantexpression*)
Both constantexpressions shall be a positive nonzero integral constant expressions. The maximum of the product of the constants is implementation defined. If that implementation defined limit is exceeded, the program is illformed.
An attribute of the form matrix_type(
R
,
C
)
forms a matrix type with an element type of the cvqualified type the attribute appertains to and R rows and C columns.
If a declaration of a typedefname has a matrix_type attribute, then all declaration of that typedefname shall have a matrix_type attribute with the same element type, number of rows, and number of columns.
Matrix TypeA matrix type has an underlying element type, a constant number of rows, and a constant number of columns. Matrix types with the same element type, rows, and columns are the same type. A value of a matrix type contains rows * columns
values of the element type laid out in columnmajor order without padding in a way compatible with an array of at least that many elements of the underlying element type.
A matrix type is a scalar type with the same alignment as its underlying element type, but objects of matrix type are not usable in constant expressions.
TODO: Allow reinterpret_cast from pointer to element type. Make aliasing work.
Future Work: Initialization syntax.
Future Work: Access syntax. m[col][row]
.
Future Work: Conversions between matrix types with const qualified and unqualified element types.
Future Work: Conversions between matrix types with different element types.
Matrix Type builtin OperationsEach matrix type supports a collection of builtin expressions that look like function calls but do not form an overload set. Here they are described as function declarations with rules for how to construct the argument list types and return type and the library description elements from [library.description.structure.specifications]/3 in the C++ standard.
Definitions:

M, M1, M2, M3  Matrix types

T  Element type

row, col  Row and column arguments respectively.
All operations on matrix types match the behavior of the underlying element type with respect to signed overflows.
Element Operations
Preconditions: row
and col
are in the ranges [0, rows in M) and [0, columns in M) respectively.
*M* __builtin_matrix_insert(*M* matrix, int row, int col, *T* elt)
Remarks: The return type and the type T are inferred from the cvunqualified type of the matrix
argument and its cvunqualified element type respectively.
Returns: a copy of matrix
with the element at the specified row and column set to elt
.
*T* __builtin_matrix_extract(*M* matrix, int row, int col)
The return type is inferred from the cvunqualified type of the matrix
argument’s element type.
Returns: a copy of the element at the specified row and column.
Simple Binary Operations
For the following binary operations matrix1
and matrix2
shall be matrix values of the same cvunqualified type, and the return type is the cvunqualified version of that type.
*M* __builtin_matrix_add(*M* matrix1, *M* matrix2)
Returns: A matrix Res
equivalent to the code below, where col
refers to the number of columns of M, row
to the number of rows of M and EltTy
to the element type of M.
*M* Res;
for (int C = 0; C < col; ++C) {
for (int R = 0; R < row; ++R) {
EltTy Elt = __builtin_matrix_extract(matrix1, R, C) +
__builtin_matrix_extract(matrix2, R, C)
Res = __builtin_matrix_insert(Res, R, C, Elt);
}
}
*M* __builtin_matrix_sub(*M* matrix1, *M* matrix2)
Returns: A matrix Res
equivalent to the code below, where col
refers to the number of columns of M, row
to the number of rows of M and EltTy
to the element type of M.
*M* Res;
for (int C = 0; C < col; ++C) {
for (int R = 0; R < row; ++R) {
EltTy Elt = __builtin_matrix_extract(matrix1, R, C) 
__builtin_matrix_extract(matrix2, R, C)
Res = __builtin_matrix_insert(Res, R, C, Elt);
}
}
Other Operations
*M3* __builtin_matrix_multiply(*M1* matrix1, *M2* matrix2)
Mandates: M1 and M2 shall be matrix types with the same cvunqualified element type and M1’s number of columns matching M2’s number of row.
Remarks: The return type is a cvunqualified matrix type with the same element type as M1 and M2 if both M1 and M2’s element type is const, or the cvunqualified element type otherwise, and with the same number of rows as M1 and the same number of columns as M2.
Returns: A matrix Res
equivalent to the code below, where col
refers to the number of columns of M, row
to the number of rows of M, EltTy
to the element type of M and inner
refers to the number of columns of M1.
*M* Res;
for (int C = 0; C < col; ++C) {
for (int R = 0; R < row; ++R) {
EltTy Elt = 0;
for (int K = 0; K < inner; ++K) {
Elt += __builtin_matrix_extract(matrix1, R, K) *
__builtin_matrix_extract(matrix2, K, C)
}
Res = __builtin_matrix_insert(Res, R, C, Elt);
}
Remark: With respect to rounding errors, the operation preserves the behavior of the separate multiply and add operations by default. We propose to provide a Clang option to override this behavior and allow contraction of those operations (e.g. ffpcontract=matrix).
*M2* __builtin_matrix_transpose(*M1* matrix)
Remarks: The return type is a cvunqualified matrix type that has the same element type as M1 and has the the same number of rows as M1 has columns and the same number of columns as M1 has rows.
Returns: A matrix Res
equivalent to the code below, where col
refers to the number of columns of M, and row
to the number of rows of M.
*M* Res;
for (int C = 0; C < col; ++C) {
for (int R = 0; R < row; ++R) {
EltTy Elt = __builtin_matrix_extract(matrix, R, C);
Res = __builtin_matrix_insert(Res, C, R, Elt);
}
}
M
__builtin_matrix_column_load(
T
*ptr, int row, int col, int stride)
Mandates: row
and col
shall be integral constants greater than 0.
Preconditions: stride >= ``row
.
Remarks: The return type is a cvunqualified matrix type with an element type of the cvunqualified version of T and a number of rows and columns equal to row
and col
respectively.
Returns: A matrix Res
equivalent to:
*M* Res;
for (int C = 0; C < col; ++C) {
for (int R = 0; R < row; ++K)
Res = __builtin_matrix_insert(Res, R, C, ptr[R]);
ptr += stride
}
void __builtin_matrix_column_store(*M* matrix, *T* *ptr, int stride)
Preconditions: stride
is greater than or equal to the number of rows in M.
Effects: Equivalent to:
for (int C = 0; C < *columns **in** M*; ++C) {
for (int R = 0; R < *rows **in M*; ++K)
ptr[R] = __builtin_matrix_extract(matrix, R, C);
ptr += stride
}
Remarks: The type T is the constunqualified version of the matrix
argument’s element type.
*M* __builtin_matrix_scalar_multiply(*M* matrix, *T* scalar)
Returns: A matrix Res
equivalent to the code below, where col
refers to the number of columns of M, and row
to the number of rows of M.
*M* Res;
for (int C = 0; C < col; ++C) {
for (int R = 0; R < row; ++R) {
EltTy Elt = __builtin_matrix_extract(matrix, R, C) * scalar;
Res = __builtin_matrix_insert(Res, R, C, Elt);
}
}
Remarks: The return type and the type T are the cvunqualified type of the matrix
argument and its cvunqualified element type respectively.
ExampleThis code performs a matrixmultiply of two 4x4 matrices followed by an matrix addition:
typedef float m4x4_t __attribute__((matrix_type(4, 4)));
void f(m4x4_t *a, m4x4_t *b, m4x4_t *c, m4x4_t *r) {
*r = __builtin_matrix_add(__builtin_matrix_multiply(*a, *b), *c);
}
This will get lowered by Clang to the LLVM IR below. In our current implementation, we use LLVM’s array type as storage type for the matrix data. Before accessing the data, we cast the array to a vector type. This allows us to use the element width as alignment, without running into issues with LLVM’s large default alignment for vector types, which is problematic in structs.
define void @f([16 x float]* %a, [16 x float]* %b, [16 x float]* %c, [16 x float]* %r) #0 {
entry:
%a.addr = alloca [16 x float]*, align 8
%b.addr = alloca [16 x float]*, align 8
%c.addr = alloca [16 x float]*, align 8
%r.addr = alloca [16 x float]*, align 8
store [16 x float]* %a, [16 x float]** %a.addr, align 8
store [16 x float]* %b, [16 x float]** %b.addr, align 8
store [16 x float]* %c, [16 x float]** %c.addr, align 8
store [16 x float]* %r, [16 x float]** %r.addr, align 8
%0 = load [16 x float]*, [16 x float]** %a.addr, align 8
%1 = bitcast [16 x float]* %0 to <16 x float>*
%2 = load <16 x float>, <16 x float>* %1, align 4
%3 = load [16 x float]*, [16 x float]** %b.addr, align 8
%4 = bitcast [16 x float]* %3 to <16 x float>*
%5 = load <16 x float>, <16 x float>* %4, align 4
%6 = call <16 x float> @llvm.matrix.multiply.v16f32.v16f32.v16f32(<16 x float> %2, <16 x float> %5, i32 4, i32 4, i32 4)
%7 = load [16 x float]*, [16 x float]** %c.addr, align 8
%8 = bitcast [16 x float]* %7 to <16 x float>*
%9 = load <16 x float>, <16 x float>* %8, align 4
%10 = fadd <16 x float> %6, %9
%11 = load [16 x float]*, [16 x float]** %r.addr, align 8
%12 = bitcast [16 x float]* %11 to <16 x float>*
store <16 x float> %10, <16 x float>* %12, align 4
ret void
}
declare <16 x float> @llvm.matrix.multiply.v16f32.v16f32.v16f32(<16 x float>, <16 x float>, i32 immarg, i32 immarg, i32 immarg)
Contributing CriteriaEvidence of a significant user community: This is based on a number of factors, including an existing user community, the perceived likelihood that users would adopt such a feature if it were available, and any secondary effects that come from, e.g., a library adopting the feature and providing benefits to its users.
Currently this is part of one of our compiler toolchains and used on a few large internal codebases. The matrix type can be used by matrix libraries like Eigen, to offload some of the optimization responsibility from the library to the compiler. It would also be suitable target for implementing a standard matrix library. It also provides functionality similar to various libraries for matrix math on small matrixes, like https://developer.apple.com/documentation/accelerate/working_with_matrices, with more flexibility (supports any combination of input dimensions).
A specific need to reside within the Clang tree: There are some extensions that would be better expressed as a separate tool, and should remain as separate tools even if they end up being hosted as part of the LLVM umbrella project.
We want to expose this feature at the C/C++ level. For that, it needs to be part of Clang.
A specification: The specification must be sufficient to understand the design of the feature as well as interpret the meaning of specific examples. The specification should be detailed enough that another compiler vendor could implement the feature.
We currently have the design above and will work on a more comprehensive spec.
Representation within the appropriate governing organization: For extensions to a language governed by a standards committee (C, C++, OpenCL), the extension itself must have an active proposal and proponent within that committee and have a reasonable chance of acceptance. Clang should drive the standard, not diverge from it. This criterion does not apply to all extensions, since some extensions fall outside of the realm of the standards bodies.
We think this extension would fall outside of the realm of the standards bodies. It is an implementation detail used to implement matrix math libraries and such, much like the vector extensions are an implementation detail for SIMD libraries.
A longterm support plan: increasingly large or complex extensions to Clang need matching commitments to supporting them over time, including improving their implementation and specification as Clang evolves. The capacity of the contributor to make that commitment is as important as the commitment itself.
We are using this internally and adding this feature to Clang upstream means we intend to support it as part of our ongoing Clang work.
A highquality implementation: The implementation must fit well into Clang’s architecture, follow LLVM’s coding conventions, and meet Clang’s quality standards, including diagnostics and complete AST representations. This is particularly important for language extensions, because users will learn how those extensions work through the behavior of the compiler.
Will we provide a series of patches to implement the extension soon and look forward to any feedback to make sure the patches meet the quality requirement.
A test suite: Extensive testing is crucial to ensure that the language extension is not broken by ongoing maintenance in Clang. The test suite should be complete enough that another compiler vendor could conceivably validate their implementation of the feature against it
We will provide this as part of Clang’s unit tests and testsuite.
ExtensionsInitially we want to focus on 2D matrixes without padding in columnmajor layout as a concrete use case. This is similar to the defaults for the Matrix type in Eigen, for example. But our proposed type can be extended naturally to

Support N (known constant) dimensions by turning matrix_type attribute into a variadic attribute.

Support column/rowwise padding, by adding a column_padding clause to the attribute.
Dealing with the padding could be exclusively handled on the frontend side, by emitting additional shufflevector instructions to extract the data. If there is a desire to exploit the padding more on the LLVM side, we can add a set of intrinsics for that. 
Support row & column major layouts, by adding a layout clause to the attribute.
Again, this naively could be handled while lowering to LLVM IR in Clang using shufflevector to produce flattened vectors with the required layout. For better optimisations, the LLVM intrinsics relying on shape/layout information can be extended to take the layout as additional argument. Through propagating the layout information similar to the dimensions, we should be able to optimise the points where we need to transform the layout of the underlying matrixes.
In all cases, we require known integer constants as dimensions and we do not plan to support dynamic dimensions for now, as the main optimization potential comes from the fact that we know the dimensions. Supporting dynamic dimensions should be fairly straight forward, but means we lose the ability to type check matrix expressions at compile time and we also have to rely on dynamic dimension during code generation.
Cheers,
Florian