A common operation performed in graphics rendering is computing

the inverse square root of a number, $\frac{1}{\sqrt[\phantom{2}]{n}}$

This is used when normalizing vectors to calculate angles of incidence

and reflection$--$an operation so common modern $3$D graphics

programs typically perform milions of these calculations every second

to simulate lighting.

In the early $1990$ s$,$ when $3$D graphics rendering started to become

widespread in the video game industry, the floating point processing

power of most CPUs was too slow to perform the inverse square root

at the speed required, necessitating certain speedups. This question

investigates the tradeoffs between these speedups.

Suppose the application QUAKE spends $20\%$ of its execution time

computing the inverse square root of a floating point number, and $15\%$

of its time computing the inverse square root of an integer. Consider

each of the following speedups:

$(2)$ On existing hardware $($without the speedup in $($a$)),$ computing the inverse

square root of an integer is $30\%$ faster than that of a floating point number.

However, by utilizing a trick involving a magic number, $0$x$5$F$3759$DF$,$ a floating

point number can be converted to an integer $($for the purposes of this

computation$).$ This allows all inverse square root calculations to be as fast as

those for integers.

What is the overall speedup for the QUAKE application considering this

optimization alone? Show your work and round your result to the nearest

thousandth.