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We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Schönhage's efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but novel idea, and yield significant practical acceleration of the known near optimal subdivision algorithms for complex and real root-finding of user's choice. We achieve this by means of significant saving of exclusion tests and Taylor's shifts, which are thearXiv:2102.10821v2 fatcat:z522mpedgnaa7hbnq7hd7m5owm