Added algebraic transformations to report.

report/report.tex

@@ -112,11 +112,11 @@ We now add the instruction above the first use, and write the result in a new
 variable. Then all occurrences of this expression can be replaced by a move of
 from new variable into the original destination variable of the instruction.
 
-This is a less efficient method then the dag, but because the basic blocks are
+This is a less efficient method then the DAG, but because the basic blocks are
 in general not very large and the execution time of the optimizer is not a
 primary concern, this is not a big problem.
 
-\subsubsection*{Fold constants}
+\subsubsection*{Constant folding}
 
 
 
@@ -158,7 +158,18 @@ removed by the dead code elimination.
 
 \subsubsection*{Algebraic transformations}
 
+Some expression can easily be replaced with more simple once if you look at
+what they are saying algebraically. An example is the statement $x = y + 0$, or
+in Assembly \texttt{addu \$1, \$2, 0}. This can easily be changed into $x = y$
+or \texttt{move \$1, \$2}.
 
+Another case is the multiplication with a power of two. This can be done way
+more efficiently by shifting left a number of times. An example:
+\texttt{mult \$regA, \$regB, 4    ->  sll  \$regA, \$regB, 2}. We perform this
+optimization for any multiplication with a power of two.
+
+There are a number of such cases, all of which are once again stated in
+appendix \ref{opt}. 
 
 \section{Implementation}
 
@@ -195,7 +206,7 @@ The optimizations are done in two different steps. First the global
 optimizations are performed, which are only the optimizations on branch-jump
 constructions. This is done repeatedly until there are no more changes.
 
-After all possible global optimizations are done, the program is seperated into
+After all possible global optimizations are done, the program is separated into
 basic blocks. The algorithm to do this is described earlier, and means all
 jump and branch instructions are called leaders, as are their targets. A basic
 block then goes from leader to leader.
@@ -207,7 +218,7 @@ steps can be done to optimize something.
 \subsection{Writing}
 
 Once all the optimizations have been done, the IR needs to be rewritten into
-Assembly code, so the xgcc crosscompiler can make binary code out of it.
+Assembly code, so the xgcc cross compiler can make binary code out of it.
 
 The writer expects a list of statements, so first the blocks have to be
 concatenated again into a list. After this is done, the list is passed on to