A Naive question. Have you tried being flexible switch between quad and double precision? Will that give your further speedup?
No because we need quad for the precision. Doing any part of the calculation in double precision will lose precision and force us to take smaller steps and thus slow us down.
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J. Langou, J. Langou, P. Luszczek, J. Kurzak, A. Buttari and J. Dongarra, “Exploiting the Performance of 32 bit Floating Point Arithmetic in Obtaining 64 bit Accuracy (Revisiting Iterative Refinement for Linear Systems),” SC '06: Proceedings of the 2006 ACM/IEEE Conference on Supercomputing, 2006, pp. 50-50, doi: 10.1109/SC.2006.30.
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