Merge pull request #36 from level2fast/wes237-sdaniels-cordic-errors

Fixed grammatical erros in CORDIC chapter

cordic.tex

@@ -375,9 +375,9 @@ \section{Cartesian to Polar Conversion}
 \section{Number Representation}
 \label{sec:number_representation}
 
-The \lstinline{cordic} function uses currently uses common types for the variables. For example, the variable \lstinline{sigma} is defined as an \lstinline{int} and other variables are defined to use custom data types (e.g., \lstinline{THETA_TYPE} and \lstinline{COS_SIN_TYPE}).  It is sometimes convenient to think that such types represent arbitrary numbers, when in actuality they don't.  In practice, it digital representation of numbers has a huge impact on the complexity of the logic to compute with those numbers.   In many cases, HLS tools are able to optimize the representation of each value to simplify the generated hardware.  For instance, in Figure \ref{fig:cordic_code}, the variable \lstinline{sigma} is restricted to be either \lstinline{1} or \lstinline{-1}.  Even though the variable is declared as an \lstinline{int} type of at least 32 bits, many fewer bits can be used to implement the variable without changing the behavior of the program. In other cases, particularly function inputs, memories, and variables that appear in recurrences, the representation cannot be automatically optimized.  In these cases, modifying the code to use smaller datatypes is a key optimization to avoid unnecessary resource usage.
+The \lstinline{cordic} function uses currently uses common types for the variables. For example, the variable \lstinline{sigma} is defined as an \lstinline{int} and other variables are defined to use custom data types (e.g., \lstinline{THETA_TYPE} and \lstinline{COS_SIN_TYPE}).  It is sometimes convenient to think that such types represent arbitrary numbers, when in actuality they don't.  In practice, it's digital representation of numbers has a huge impact on the complexity of the logic to compute with those numbers.   In many cases, HLS tools are able to optimize the representation of each value to simplify the generated hardware.  For instance, in Figure \ref{fig:cordic_code}, the variable \lstinline{sigma} is restricted to be either \lstinline{1} or \lstinline{-1}.  Even though the variable is declared as an \lstinline{int} type of at least 32 bits, many fewer bits can be used to implement the variable without changing the behavior of the program. In other cases, particularly function inputs, memories, and variables that appear in recurrences, the representation cannot be automatically optimized.  In these cases, modifying the code to use smaller datatypes is a key optimization to avoid unnecessary resource usage.
 
-Although reducing the size of variables is generally a good idea, this optimization can change the behavior of the program. A data type with fewer number of bits will not be able to express as much information as a data type with more bits and no finite binary representation can represent all real numbers with infinite accuracy. Fortunately, as designers we can pick numeric representations that are tuned to accuracy requirements of particular applications and tradeoff between accuracy, resource usage, and performance.
+Although reducing the size of variables is generally a good idea, this optimization can change the behavior of the program. A data type with fewer number of bits will not be able to express as much information as a data type with more bits and no finite binary representation which can represent all real numbers with infinite accuracy. Fortunately, as designers we can pick numeric representations that are tuned to accuracy requirements of particular applications and tradeoff between accuracy, resource usage, and performance.
 
 Before discussing these number representation optimizations further using our \lstinline{cordic} function, we first give a background on number representation. We provide the basics, as this is important in understand the data type specific representations provided by \VHLS. The next section starts with a fundamental background on number representation, and then proceeds to discuss the arbitrary precision variables available in \VHLS.