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	On bit-shifting - libzahl - big integer library
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	---
	commit 4ff3671b42f011862ab7a8c6b8ddf66780a2054b
	parent 79484235aae6729a2489c8477695c72ba77e9b58
	Author: Mattias Andrée <[email protected]>
	Date: Sat, 14 May 2016 16:40:46 +0200

	On bit-shifting

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/bit-operations.tex \| 58 +++++++++++++++++++++++++++++…

	1 file changed, 57 insertions(+), 1 deletion(-)
	---
	diff --git a/doc/bit-operations.tex b/doc/bit-operations.tex
	@@ -25,7 +25,63 @@ TODO % zand zor zxor znot
	\section{Shift}
	\label{sec:Shift}

	-TODO % zlsh zrsh
	+There are two functions for shifting bits
	+in integers:
	+
	+\begin{alltt}
	+ void zlsh(z_t r, z_t a, size_t b);
	+ void zrsh(z_t r, z_t a, size_t b);
	+\end{alltt}
	+
	+\noindent
	+{\tt zlsh} performs a left-shift, and {\tt zrsh}
	+performs a right-shift. That is, {\tt zlsh} adds
	+{\tt b} trailing binary zeroes, and {\tt zrsh}
	+removes the lowest {\tt b} binary digits. So if
	+
	+$a = \phantom{00}10000101_2$ then
	+
	+$r = 1000010100_2$ after calling {\tt zlsh(r, a, 2)}, and
	+
	+$r = \phantom{0100}100001_2$ after calling {\tt zrsh(r, a, 2)}.
	+\vspace{1em}
	+
	+{\tt zlsh(r, a, b)} is equivalent to $r \gets a \cdot 2^b$,
	+and {\tt zrsh(r, a, b)} is equivalent to
	+$r \gets \lfloor a \div 2^b \rfloor$, {\tt zlsh} and
	+{\tt zrsh} are significantly faster than {\tt zpowu}
	+and should be used whenever possible. {\tt zpowu}
	+does not check if it is possible for it to use {\tt zlsh}
	+instead, even if it would, {\tt zlsh} and {\tt zrsh}
	+would still be preferable in most cases because it
	+removes the need for {\tt zmul} and {\tt zdiv},
	+respectively.
	+
	+{\tt zlsh} and {\tt zrsh} are implemented in two steps:
	+(1) shift whole characters, that is, groups of aligned
	+64 bits, and (2) shift on a bit-level between characters.
	+
	+If you are implementing a calculator, you may want to
	+create a wrapper for {\tt zpow} that uses {\tt zlsh}
	+whenever possible. One such wrapper could be
	+
	+\begin{alltt}
	+ void
	+ pow(z_t r, z_t a, z_t b)
	+ \{
	+ size_t s1, s2;
	+ if ((s1 = zlsb(a)) + 1 == zbits(a) &&
	+ zbits(b) <= 8 * sizeof(SIZE_MAX)) \{
	+ s2 = zzero(b) ? 0 : b->chars[0];
	+ if (s1 <= SIZE_MAX / s2) \{
	+ zsetu(r, 1);
	+ zlsh(r, r, s1 * s2);
	+ return;
	+ \}
	+ \}
	+ zpow(r, a, b);
	+ \}
	+\end{alltt}


	\newpage