Wow thank you I do not understand everything written and didn't take time to do so (and can't even find reference 23 paper on Brent stability), but I was thinki

Wow thank you I do not understand everything written and didn't take time to do so (and can't even find reference 23 paper on Brent stability), but I was thinking we don't necessarily need a new algorithm, we could tweak existing ones. For example swapping columns and rows before applying a product, so that accuracy is best kept (then reswapping back at the end of the operation). It seems to me the Strassen-like algorithm choose arbitrarily some elements of the matrices over others to make their temporary sum-products variables, so we could exploit this asymmetry (not sure if it's the correct word here) to choose which elements are to be chosen