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by theophrastus 4501 days ago
It ought to be pointed out that this method (and many others even more cunning) are detailed in the delightful tome "Hacker's Delight" (ISBN 978-0-201-91465-8) by Henry S. Warren

on about page 481 or so:

"The Newton step is a standard Newton-Raphson calculation for the reciprocal square root function (see Appendix B). Simply repeating this step reduces the relative error to the range 0 to -0.0000047. The optimal constant for this is 0x5F37599E."