Branch on Bool alpha in bidiag matmul

[jishnub]

Similar to #1256, we may reduce branches in @stable_muladdmul for a non-zero Bool alpha in Bidiagonal matmul. The idea is that if alpha::Bool is non-zero, it must be true. We may therefore hardcode this value and reduce branches in @stable_muladdmul. In addition, if beta is unused in a method, we may hardcode beta = false as well, which further helps with compilation.

julia> using LinearAlgebra

julia> B = Bidiagonal(1:4, 1:3, :U); D = Diagonal(1:4); v = (1:4);

With this,

julia> @time B * B;
  0.406036 seconds (2.34 M allocations: 105.696 MiB, 4.31% gc time, 99.99% compilation time) # nightly
  0.141480 seconds (588.18 k allocations: 26.049 MiB, 99.95% compilation time) # This PR

The rest are mainly reductions in allocation:

julia> @time B * D;
  0.141903 seconds (487.43 k allocations: 24.557 MiB, 99.96% compilation time) # nightly
  0.147749 seconds (382.60 k allocations: 19.385 MiB, 99.96% compilation time) # this PR

julia> @time D * B;
  0.136308 seconds (491.83 k allocations: 24.782 MiB, 99.95% compilation time) # nightly
  0.136909 seconds (386.35 k allocations: 19.591 MiB, 99.94% compilation time) # this PR

julia> @time B * v;
  0.087207 seconds (428.64 k allocations: 21.620 MiB, 99.93% compilation time) # master
  0.089002 seconds (342.46 k allocations: 17.306 MiB, 99.92% compilation time) # this PR

This also improves performance for small Bidiagonal multiplication, as there are fewer operations to carry out.

julia> n = 1; T = Bidiagonal(ones(n), ones(max(n-1,0)), :U); C = Matrix(T);

julia> @btime mul!($C, $T, $T);
  33.360 ns (0 allocations: 0 bytes) # nightly
  23.773 ns (0 allocations: 0 bytes) # this PR

julia> n = 2; T = Bidiagonal(ones(n), ones(max(n-1,0)), :U); C = Matrix(T));

julia> @btime mul!($C, $T, $T);
  78.685 ns (0 allocations: 0 bytes) # nightly
  31.388 ns (0 allocations: 0 bytes) # this PR

julia> n = 3; T = Bidiagonal(ones(n), ones(max(n-1,0)), :U); C = Matrix(T);

julia> @btime mul!($C, $T, $T);
  161.577 ns (0 allocations: 0 bytes) # nightly
  41.256 ns (0 allocations: 0 bytes) # this PR

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 92.15%. Comparing base (27de9e9) to head (cb026ae).
Report is 18 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1257      +/-   ##
==========================================
- Coverage   92.15%   92.15%   -0.01%     
==========================================
  Files          34       34              
  Lines       15604    15645      +41     
==========================================
+ Hits        14380    14417      +37     
- Misses       1224     1228       +4     

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