71 lines
1.9 KiB
TeX
71 lines
1.9 KiB
TeX
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\section{The Fast Inverse Square Root}
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The following code is present in \textit{Quake III Arena} (1999):
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\lstset{
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breaklines=false,
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numbersep=5pt,
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xrightmargin=0in
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}
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\begin{lstlisting}[language=C]
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float Q_rsqrt( float number ) {
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long i = * ( long * ) &number;
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i = 0x5f3759df - ( i >> 1 );
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return * ( float * ) &i;
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}
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\end{lstlisting}
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It defines a method \texttt{Q\_rsqrt} that consumes a float named \texttt{number} and quickly approximates its inverse
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square root (in other words, \texttt{Q\_rsqrt} computes $1/\sqrt{\texttt{number}}$).
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\vspace{8mm}
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If we rewrite this using the notation we're familiar with, we get the following:
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\begin{equation*}
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\frac{1}{\sqrt{n_f}} \approx \kappa - (n_i \div 2)
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\end{equation*}
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Where $\kappa$ is the magic constant $6240089$ (which is \texttt{0x5f3759df} in hexadecimal)
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\problem{}
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Find the exact value of $\kappa$ in terms of $\varepsilon$. \par
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\hint{Remember that $\varepsilon$ is the correction term in the approximation $\log_2(1 + a) = a + \varepsilon$.}
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\problem{}
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Rewrite $\log_2(1 / \sqrt{x})$ in terms of $\log_2{x}$.
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\begin{solution}
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Say $g_f = \frac{1}{\sqrt{n_f}}$---that is, $g_f$ is the value we want to compute. \par
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Then:
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\begin{align*}
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\log_2(g_f)
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&=\log_2(\frac{1}{\sqrt{n_f}}) \\
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&=\frac{-1}{2}\log_2(n_f) \\
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&=\frac{-1}{2}\left( \frac{n_i}{2^{23}} + \varepsilon - 127 \right)
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\end{align*}
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But we also know that
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\begin{align*}
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\log_2(g_f)
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&=\frac{g_i}{2^{23}} + \varepsilon - 127
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\end{align*}
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So,
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\begin{align*}
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\frac{g_i}{2^{23}} + \varepsilon - 127
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&=\frac{-1}{2}\left( \frac{n_i}{2^{23}} + \varepsilon - 127 \right) \\
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\frac{g_i}{2^{23}}
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&=\frac{-1}{2}\left( \frac{n_i}{2^{23}} \right) + \frac{3}{2}(\varepsilon - 127) \\
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g_i
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&=\frac{-1}{2}\left(n_i \right) + 2^{23}\frac{3}{2}(\varepsilon - 127) \\
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&=2^{23}\frac{3}{2}(\varepsilon - 127) - \frac{n_i}{2}
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\end{align*}
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thus, $\kappa = 2^{23}\frac{3}{2}(\varepsilon - 127)$.
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\end{solution}
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\pagebreak
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