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\title{Basics of Information Systems}
\author{Winter Semester 2022--23}
\date{For discussion on Wednesday, November 16, 2022}
\maketitle

\begin{enumerate}
\item Convert the following single-precision floating point number to
decimal:
\begin{equation*}
  1 \, 10001010 \, 11000010000000000000000
\end{equation*}

\item Show that floating point division has moderate growth of
relative error for all numbers.

\item Does the associative law hold for floating point computations?

\item Identify values of $x$ for which there is a substantial growth
of relative error (due to subtraction of almost equal numbers), and
suggest an alternate formula that improves accuracy for the
problematic range of $x$.
\begin{enumerate}
\item $\displaystyle \frac{1-(1-x)^3}{x}$
\item $\displaystyle \frac{1 - \sqrt{1-x^2}}x$
\item $\displaystyle \frac{1 - \sec x}{\tan^2 x}$
\end{enumerate}

\emph{Hint:} Recall that $\sec x = (\cos x)^{-1}$; use the well-known
trigonometric identity $\sec^2x = \tan^2x+1$.

\end{enumerate}
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