Fast calculation of inverse square root with the use of magic constant   $-$ analytical approach
  We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm. 
Leonid V. Moroz, Cezary J. Walczyk, Andriy Hrynchyshyn, Vijay Holimath   and Jan L. Cie\'sli\'nski
cs.MS