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[SPCA] Additions to floating point

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Janis Hutz
2026-01-23 11:57:27 +01:00
parent 25b3381294
commit 878cac1217
3 changed files with 14 additions and 10 deletions
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2
semester3/spca/parts/01_c/06_floating-point/00_intro.tex
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@@ -5,4 +5,4 @@ Floating point numbers are a representation of real numbers.
Though there are many ways to accomplish this, \textit{IEEE Standard 754} is used practically everywhere, also in \verb|x86|. This standard is a little more complicated than fractional binary numbers, but has a few numeric advantages, especially for representing very large (very small) numbers.
\hlurl{float.exposed}\ is an excellent website to understand floating point by example.
\hlhref{https://float.exposed}{float.exposed}\ is an excellent website to understand floating point by example.

22
semester3/spca/parts/01_c/06_floating-point/04_rounding.tex
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@@ -13,22 +13,26 @@ The default used is \textit{Nearest Even}\footnote{Changing the rounding mode is
Rounding can be defined using 3 different bits from the \textit{exact} number: $G, R, S$
$$
a = 1.BB\ldots BB\underbrace{G}_\text{Guard}\underbrace{R}_\text{Round}\underbrace{XX\ldots XX}_\text{Sticky}
a = 1.B_1B_2\ldots B_{n - 2}B_{n - 1}\underbrace{G}_\text{Guard}\underbrace{R}_\text{Round}
\underbrace{X_1X_2\ldots X_{k - 1}X_k}_\text{Sticky}
$$
where $n$ is the number of bits in the mantissa of the format (e.g. $3$ as in the above example of an $8$bit floating point number).
\begin{enumerate}
\item \textbf{Guard Bit} $G$ is the least significant bit of the (rounded) result
\item \textbf{Guard Bit} $G$ is the least significant bit of the (rounded) result (i.e. it is $B_n$)
\item \textbf{Round Bit} $R$ is the $1$st bit cut off after rounding
\item \textbf{Sticky Bit} $S$ is the logical OR of all remaining cut off bits.
\item \textbf{Sticky Bit} $S$ is the logical OR of all remaining cut off bits $X_i$.
\end{enumerate}
Based on these bits the rounding can be decided:
$$
R \land S \implies \text{ Round up} \qquad\qquad
G \land R \land \lnot S \implies \text{ Round to even}
$$
Based on these bits the rounding can be decided (we increment the rounded part if the expression evaluates to true):
\hrmvspace
\begin{align*}
\text{Round up: } R \land S
& &
\text{Round to even: } G \land R \land \lnot S
\end{align*}
\drmvspace
\content{Example} Rounding $8$b precise results to $8$b precision floating point ($4$b mantissa):
\renewcommand{\arraystretch}{1.2}