Test failure in `LinearAlgebra/addmul` for 5-way `mul!` with `BigFloat` and `Bidiagonal`

[giordano]

On 4633607ce9b9f077f32f89f09a136e04389bbac2 with Linux (either x86_64 or aarch64) I get

julia> Base.runtests("LinearAlgebra/addmul"; seed=0xca081a1e9adc6829030ff209636c84f4)
[...]
The global RNG seed was 0xca081a1e9adc6829030ff209636c84f4.

Error in testset LinearAlgebra/addmul:
Test Failed at ~/repo/julia/usr/share/julia/stdlib/v1.12/LinearAlgebra/test/addmul.jl:192
  Expression: ≈(collect(returned_mat), α * Ac * Bc + β * Cc, rtol = rtol)
   Evaluated: BigFloat[-0.0001400037216701636132108987549395176373104782747840190352000128617941687774417292;;] ≈ BigFloat[-0.0001400633263149390038358987549395176373104782747840190352000128617941687774417292;;] (rtol=0.00034526697709225118160247802734375)

Reduced to

julia> using LinearAlgebra

julia> α, β = true, big"1.4598638281754612311402752311551012098789215087890625";

julia> C = [big"-2.432162727109652065274123009949748377821320641416554145536270759853570050001279";;];

julia> Cc = collect(C);

julia> Af = Float32[0.50721234;;];

julia> Ac = collect(Af);

julia> Bf = Bidiagonal([7], Int64[], :U);

julia> Bc = [7;;];

julia> returned_mat = mul!(C, Af, Bf, α, β)
1×1 Matrix{BigFloat}:
 -0.0001400037216701636132108987549395176373104782747840190352000128617941687774417292

julia> α * Ac * Bc + β * Cc
1×1 Matrix{BigFloat}:
 -0.0001400633263149390038358987549395176373104782747840190352000128617941687774417292

Somewhat distressfully, this doesn't reproduce on aarch64-darwin, in the sense that the above reproducer gives

julia> returned_mat = mul!(C, Af, Bf, α, β)
1×1 Matrix{BigFloat}:
 -0.0001400633263149390038358987549395176373104782747840190352000128617941687774417292

julia> α * Ac * Bc + β * Cc
1×1 Matrix{BigFloat}:
 -0.0001400633263149390038358987549395176373104782747840190352000128617941687774417292

[jishnub]

The structured arrays seem to be a red herring in this one, and the issue may be reproduced using only Matrixes. Reduced further:

julia> α * Ac * Bc + β * C
1×1 Matrix{BigFloat}:
 -0.0001400633263149390038358987549395176373104782747840190352000128617941687774417292

julia> mul!(copy(C), Ac, Bc, α, β)
1×1 Matrix{BigFloat}:
 -0.0001400037216701636132108987549395176373104782747840190352000128617941687774417292

In the out-of-place case, the first multiplication α * Ac * Bc is carried out in Float32 precision, whereas in the second case, all the numbers are promoted to BigFloat first before the multiplication is carried out. This seems to explain the differences in the numbers:

julia> mul!(Float32.(zero(C)), Ac, Bc, α, false) + C * β
1×1 Matrix{BigFloat}:
 -0.0001400633263149390038358987549395176373104782747840190352000128617941687774417292

julia> mul!(zero(C), Ac, Bc, α, false) + C * β
1×1 Matrix{BigFloat}:
 -0.0001400037216701636132108987549395176373104782747840190352000128617941687774417292

Worth noting that the two terms being added are similar in magnitude and with opposite signs, so the rounding differences play a part.

julia> (α * Ac * Bc)[1], (β * C)[1]
(3.5504863f0, -3.550626389543966306191335898754939517637310478274784019035200012861794168777442)

Since we end up comparing numbers close to zero, perhaps we need an atol here as well.