Fast calculation of inverse square root with the use of magic constant – analytical approach

Abstract

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.

A Modification of the Fast Inverse Square Root Algorithm

“…Comparing ∆ (2) N,max with the analogical numerical bound for InvSqrt (see Eq. (4.23) in [45]) we conclude that in the case of two iterations the code InvSqrt2 is about 7 times more accurate than InvSqrt. We point out that round-off errors significantly decrease the theoretical improvement which is given by the factor 8.…”

“…It is worthwhile to point out that in this case the amplitude of the error oscillations is about 40% greater than the amplitude of oscillations of (ỹ 00 −ỹ 0 )/ỹ 0 (i.e., in the case of InvSqrt), see the right part of Fig. 2 in [45]. The errors of numerical values returned by InvSqrt2 belong (for e x = −126) to the following interval ∆ N,min = −6.21 · 10 −7 , ∆…”

“…Therefore, it is sufficient to find t = t I 0 and t = t II 0 such that 19) and to choose the greater of these two values. In [45] we have shown that…”

“…We point out that in general the optimum value of the magic constant can depend on the number of Newton-Raphson corrections. Calculations carried out in [45] gave the following results:…”

“…In this section we shortly present main results of [45]. We confine ourselves to positive floating-point numbers…”