From 878cac1217cc49df321cb5c49d7d3438f2681a10 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 23 Jan 2026 11:57:27 +0100 Subject: [PATCH] [SPCA] Additions to floating point --- .../parts/01_c/06_floating-point/00_intro.tex | 2 +- .../01_c/06_floating-point/04_rounding.tex | 22 +++++++++++------- semester3/spca/spca-summary.pdf | Bin 702697 -> 702837 bytes 3 files changed, 14 insertions(+), 10 deletions(-) diff --git a/semester3/spca/parts/01_c/06_floating-point/00_intro.tex b/semester3/spca/parts/01_c/06_floating-point/00_intro.tex index 6b35a5b..5bc7c83 100644 --- a/semester3/spca/parts/01_c/06_floating-point/00_intro.tex +++ b/semester3/spca/parts/01_c/06_floating-point/00_intro.tex @@ -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. diff --git a/semester3/spca/parts/01_c/06_floating-point/04_rounding.tex b/semester3/spca/parts/01_c/06_floating-point/04_rounding.tex index addcf3b..84956ba 100644 --- a/semester3/spca/parts/01_c/06_floating-point/04_rounding.tex +++ b/semester3/spca/parts/01_c/06_floating-point/04_rounding.tex @@ -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} diff --git a/semester3/spca/spca-summary.pdf b/semester3/spca/spca-summary.pdf index bc306ed39da4e9e4bf52d22ff74ab48618708070..63423a6584cb1d59fef4b995b36ae59a65f4c01e 100644 GIT binary patch delta 22446 zcmV)IK)k={@hbK5DuA>B6MmP$69FlI&0AZO+qM;c-(SH;?o5Q>PSTk?#EsJ?9cSFw zGwpQa2TI~fOmuNY%JwGBfA5D2B|-wZ7ptD^%ZdbWaBy%g-#GwjvPfw1*Q@7muD<+E z@}wYz<~n(Eo5+G_s*|jcq{vkAW}e)nukV_+S~u4dAw>FWw_WVY^}60H+>3I5GdDkE z`j_qc-M+1^S-NvCzS+%|bz9Ba{jR#6unYv#pSII_;|4x2XUl5$h3or6xxv7?aZi8S zx4&J#ee=WBi#J#QxnkgzChW)?BP=hH+4}0{EluWV`yru36nXN&_FSWRGp=-t_4fC^`>dct!z(mrqY>#2QJ?YvmaYdje-ndO)6-L87)-Zs^y1+}dg-1fF>FIR4W zr6(IZs6+{s^X@byP=tq5lP-mWR^QsK%ae6`QA<# z<*cn|+#x9zHba=+KxqiAXBgf=VN%8zW9<;FGL+C^I{dr26PB&M422AJMrclBqF2q8XtXJGha6L~clA#ubS% 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