Reduce generic matrix*vector latency (#56289)

```julia
julia> using LinearAlgebra

julia> A = rand(Int,4,4); x = rand(Int,4); y = similar(x);

julia> @time mul!(y, A, x, 2, 2);
  0.330489 seconds (792.22 k allocations: 41.519 MiB, 8.75% gc time, 99.99% compilation time) # master
  0.134212 seconds (339.89 k allocations: 17.103 MiB, 15.23% gc time, 99.98% compilation time) # This PR
```
Main changes:
- `generic_matvecmul!` and `_generic_matvecmul!` now accept `alpha` and
`beta` arguments instead of `MulAddMul(alpha, beta)`. The methods that
accept a `MulAddMul(alpha, beta)` are also retained for backward
compatibility, but these now forward `alpha` and `beta`, instead of the
other way around.
- Narrow the scope of the `@stable_muladdmul` applications. We now
construct the `MulAddMul(alpha, beta)` object only where it is needed in
a function call, and we annotate the call site with `@stable_muladdmul`.
This leads to smaller branches.
- Create a new internal function with methods for the `'N'`, `'T'` and
`'C'` cases, so that firstly, there's less code duplication, and
secondly, the `_generic_matvecmul!` method is now simple enough to
enable constant propagation. This eliminates the unnecessary branches,
and only the one that is taken is compiled.

Together, this reduces the TTFX substantially.

stdlib/LinearAlgebra/src/matmul.jl

@@ -78,27 +78,20 @@ _mul!(y::AbstractVector, A::AbstractVecOrMat, x::AbstractVector,
 generic_matvecmul!(y::StridedVector{T}, tA, A::StridedVecOrMat{T}, x::StridedVector{T},
                 alpha::Number, beta::Number) where {T<:BlasFloat} =
     gemv!(y, tA, A, x, alpha, beta)
-generic_matvecmul!(y::StridedVector{T}, tA, A::StridedVecOrMat{T}, x::StridedVector{T},
-                _add::MulAddMul = MulAddMul()) where {T<:BlasFloat} =
-    gemv!(y, tA, A, x, _add.alpha, _add.beta)
+
 # Real (possibly transposed) matrix times complex vector.
 # Multiply the matrix with the real and imaginary parts separately
 generic_matvecmul!(y::StridedVector{Complex{T}}, tA, A::StridedVecOrMat{T}, x::StridedVector{Complex{T}},
                 alpha::Number, beta::Number) where {T<:BlasReal} =
     gemv!(y, tA, A, x, alpha, beta)
-generic_matvecmul!(y::StridedVector{Complex{T}}, tA, A::StridedVecOrMat{T}, x::StridedVector{Complex{T}},
-                _add::MulAddMul = MulAddMul()) where {T<:BlasReal} =
-    gemv!(y, tA, A, x, _add.alpha, _add.beta)
+
 # Complex matrix times real vector.
 # Reinterpret the matrix as a real matrix and do real matvec computation.
 # works only in cooperation with BLAS when A is untransposed (tA == 'N')
 # but that check is included in gemv! anyway
 generic_matvecmul!(y::StridedVector{Complex{T}}, tA, A::StridedVecOrMat{Complex{T}}, x::StridedVector{T},
                 alpha::Number, beta::Number) where {T<:BlasReal} =
     gemv!(y, tA, A, x, alpha, beta)
-generic_matvecmul!(y::StridedVector{Complex{T}}, tA, A::StridedVecOrMat{Complex{T}}, x::StridedVector{T},
-                _add::MulAddMul = MulAddMul()) where {T<:BlasReal} =
-    gemv!(y, tA, A, x, _add.alpha, _add.beta)
 
 # Vector-Matrix multiplication
 (*)(x::AdjointAbsVec,   A::AbstractMatrix) = (A'*x')'
@@ -539,9 +532,9 @@ Base.@constprop :aggressive function gemv!(y::StridedVector{T}, tA::AbstractChar
     if tA_uc in ('S', 'H')
         # re-wrap again and use plain ('N') matvec mul algorithm,
         # because _generic_matvecmul! can't handle the HermOrSym cases specifically
-        return @stable_muladdmul _generic_matvecmul!(y, 'N', wrap(A, tA), x, MulAddMul(α, β))
+        return _generic_matvecmul!(y, 'N', wrap(A, tA), x, α, β)
     else
-        return @stable_muladdmul _generic_matvecmul!(y, tA, A, x, MulAddMul(α, β))
+        return _generic_matvecmul!(y, tA, A, x, α, β)
     end
 end
 
@@ -564,7 +557,7 @@ Base.@constprop :aggressive function gemv!(y::StridedVector{Complex{T}}, tA::Abs
         return y
     else
         Anew, ta = tA_uc in ('S', 'H') ? (wrap(A, tA), oftype(tA, 'N')) : (A, tA)
-        return @stable_muladdmul _generic_matvecmul!(y, ta, Anew, x, MulAddMul(α, β))
+        return _generic_matvecmul!(y, ta, Anew, x, α, β)
     end
 end
 
@@ -591,9 +584,9 @@ Base.@constprop :aggressive function gemv!(y::StridedVector{Complex{T}}, tA::Abs
     elseif tA_uc in ('S', 'H')
         # re-wrap again and use plain ('N') matvec mul algorithm,
         # because _generic_matvecmul! can't handle the HermOrSym cases specifically
-        return @stable_muladdmul _generic_matvecmul!(y, 'N', wrap(A, tA), x, MulAddMul(α, β))
+        return _generic_matvecmul!(y, 'N', wrap(A, tA), x, α, β)
     else
-        return @stable_muladdmul _generic_matvecmul!(y, tA, A, x, MulAddMul(α, β))
+        return _generic_matvecmul!(y, tA, A, x, α, β)
     end
 end
 
@@ -825,82 +818,83 @@ end
 # NOTE: the generic version is also called as fallback for
 #       strides != 1 cases
 
-Base.@constprop :aggressive generic_matvecmul!(C::AbstractVector, tA, A::AbstractVecOrMat, B::AbstractVector, alpha::Number, beta::Number) =
-    @stable_muladdmul generic_matvecmul!(C, tA, A, B, MulAddMul(alpha, beta))
+# legacy method, retained for backward compatibility
+generic_matvecmul!(C::AbstractVector, tA, A::AbstractVecOrMat, B::AbstractVector, _add::MulAddMul = MulAddMul()) =
+    generic_matvecmul!(C, tA, A, B, _add.alpha, _add.beta)
 @inline function generic_matvecmul!(C::AbstractVector, tA, A::AbstractVecOrMat, B::AbstractVector,
-                                    _add::MulAddMul = MulAddMul())
+                                    alpha::Number, beta::Number)
     tA_uc = uppercase(tA) # potentially convert a WrapperChar to a Char
     Anew, ta = tA_uc in ('S', 'H') ? (wrap(A, tA), oftype(tA, 'N')) : (A, tA)
-    return _generic_matvecmul!(C, ta, Anew, B, _add)
+    return _generic_matvecmul!(C, ta, Anew, B, alpha, beta)
 end
 
-function _generic_matvecmul!(C::AbstractVector, tA, A::AbstractVecOrMat, B::AbstractVector,
-                            _add::MulAddMul = MulAddMul())
-    require_one_based_indexing(C, A, B)
-    @assert tA in ('N', 'T', 'C')
-    mB = length(B)
-    mA, nA = lapack_size(tA, A)
-    if mB != nA
-        throw(DimensionMismatch(lazy"matrix A has dimensions ($mA,$nA), vector B has length $mB"))
-    end
-    if mA != length(C)
-        throw(DimensionMismatch(lazy"result C has length $(length(C)), needs length $mA"))
-    end
-
+# legacy method, retained for backward compatibility
+_generic_matvecmul!(C::AbstractVector, tA, A::AbstractVecOrMat, B::AbstractVector, _add::MulAddMul = MulAddMul()) =
+    _generic_matvecmul!(C, tA, A, B, _add.alpha, _add.beta)
+function __generic_matvecmul!(f::F, C::AbstractVector, A::AbstractVecOrMat, B::AbstractVector,
+                            alpha::Number, beta::Number) where {F}
     Astride = size(A, 1)
-
     @inbounds begin
-    if tA == 'T'  # fastest case
-        if nA == 0
-            for k = 1:mA
-                _modify!(_add, false, C, k)
-            end
-        else
-            for k = 1:mA
-                aoffs = (k-1)*Astride
-                firstterm = transpose(A[aoffs + 1])*B[1]
-                s = zero(firstterm + firstterm)
-                for i = 1:nA
-                    s += transpose(A[aoffs+i]) * B[i]
-                end
-                _modify!(_add, s, C, k)
-            end
-        end
-    elseif tA == 'C'
-        if nA == 0
-            for k = 1:mA
-                _modify!(_add, false, C, k)
+        if length(B) == 0
+            for k = eachindex(C)
+                @stable_muladdmul _modify!(MulAddMul(alpha,beta), false, C, k)
             end
         else
-            for k = 1:mA
+            for k = eachindex(C)
                 aoffs = (k-1)*Astride
-                firstterm = A[aoffs + 1]'B[1]
+                firstterm = f(A[aoffs + 1]) * B[1]
                 s = zero(firstterm + firstterm)
-                for i = 1:nA
-                    s += A[aoffs + i]'B[i]
+                for i = eachindex(B)
+                    s += f(A[aoffs+i]) * B[i]
                 end
-                _modify!(_add, s, C, k)
+                @stable_muladdmul _modify!(MulAddMul(alpha,beta), s, C, k)
             end
         end
-    else # tA == 'N'
-        for i = 1:mA
-            if !iszero(_add.beta)
-                C[i] *= _add.beta
-            elseif mB == 0
+    end
+end
+function __generic_matvecmul!(::typeof(identity), C::AbstractVector, A::AbstractVecOrMat, B::AbstractVector,
+                            alpha::Number, beta::Number)
+    Astride = size(A, 1)
+    @inbounds begin
+        for i = eachindex(C)
+            if !iszero(beta)
+                C[i] *= beta
+            elseif length(B) == 0
                 C[i] = false
             else
                 C[i] = zero(A[i]*B[1] + A[i]*B[1])
             end
         end
-        for k = 1:mB
+        for k = eachindex(B)
             aoffs = (k-1)*Astride
-            b = _add(B[k])
-            for i = 1:mA
+            b = @stable_muladdmul MulAddMul(alpha,beta)(B[k])
+            for i = eachindex(C)
                 C[i] += A[aoffs + i] * b
             end
         end
     end
-    end # @inbounds
+    return C
+end
+function _generic_matvecmul!(C::AbstractVector, tA, A::AbstractVecOrMat, B::AbstractVector,
+                            alpha::Number, beta::Number)
+    require_one_based_indexing(C, A, B)
+    @assert tA in ('N', 'T', 'C')
+    mB = length(B)
+    mA, nA = lapack_size(tA, A)
+    if mB != nA
+        throw(DimensionMismatch(lazy"matrix A has dimensions ($mA,$nA), vector B has length $mB"))
+    end
+    if mA != length(C)
+        throw(DimensionMismatch(lazy"result C has length $(length(C)), needs length $mA"))
+    end
+
+    if tA == 'T'  # fastest case
+        __generic_matvecmul!(transpose, C, A, B, alpha, beta)
+    elseif tA == 'C'
+        __generic_matvecmul!(adjoint, C, A, B, alpha, beta)
+    else # tA == 'N'
+        __generic_matvecmul!(identity, C, A, B, alpha, beta)
+    end
     C
 end
 

stdlib/LinearAlgebra/src/triangular.jl

@@ -1066,7 +1066,7 @@ for TC in (:AbstractVector, :AbstractMatrix)
         if isone(alpha) && iszero(beta)
             return _trimul!(C, A, B)
         else
-            return @stable_muladdmul generic_matvecmul!(C, 'N', A, B, MulAddMul(alpha, beta))
+            return _generic_matvecmul!(C, 'N', A, B, alpha, beta)
         end
     end
 end