diff options
| -rwxr-xr-x | booth_multiplier.py | 24 | ||||
| -rw-r--r-- | report/performance.pgf | 867 | ||||
| -rw-r--r-- | report/report.aux | 2 | ||||
| -rw-r--r-- | report/report.log | 388 | ||||
| -rw-r--r-- | report/report.pdf | bin | 110007 -> 183824 bytes | |||
| -rw-r--r-- | report/report.tex | 58 | ||||
| -rw-r--r-- | report/result_table.tex | 23 | ||||
| -rw-r--r-- | report/speed_table.tex | 23 | ||||
| -rw-r--r-- | texput.log | 21 |
9 files changed, 1362 insertions, 44 deletions
diff --git a/booth_multiplier.py b/booth_multiplier.py index dc32263..c17c2f8 100755 --- a/booth_multiplier.py +++ b/booth_multiplier.py @@ -1,7 +1,16 @@ #!/usr/bin/env python3 from tabulate import tabulate +import matplotlib import matplotlib.pyplot as plt +matplotlib.use("pgf") +matplotlib.rcParams.update({ + "pgf.texsystem": "pdflatex", + 'font.family': 'serif', + 'text.usetex': True, + 'pgf.rcfonts': False, +}) + with open('input.txt') as f: input_string = f.read().split('\n') @@ -60,6 +69,7 @@ def booth(multiplier, multiplicand, length): result = result >> 1 return (result, operations) +# TODO clean up def booth_mod(multiplier, multiplicand, length): operations = 0 multiplicand |= ((1 << length - 1) & multiplicand) << 1 # extend multiplicand sign to prevent overflow when mult/sub by 2 @@ -129,16 +139,24 @@ if __name__ == "__main__": opcount_table.append([bin(multiplicand), bin(multiplier), length, result_booth[1], result_mod_booth[1]]) debug(debug_results) - print(tabulate(result_table, result_headers)) + print(tabulate(result_table, result_headers, tablefmt="latex")) print(tabulate(opcount_table, opcount_headers)) + # output + with open("report/result_table.tex", 'w') as f: + f.write(tabulate(result_table, result_headers, tablefmt="latex_booktabs")) + + with open("report/speed_table.tex", "w") as f: + f.write(tabulate(opcount_table, opcount_headers, tablefmt="latex_booktabs")) + - # generate graph + + plt.title("Operations vs Operand Length") plt.plot(lengths, ops_booth, '^--m', label='booths algorithim') plt.plot(lengths, ops_mod_booth, 'v--c', label='modified booths algorithim') plt.gca().set_xlabel("Length of Operands") plt.gca().set_ylabel("Number of Additions and Subtractions") plt.legend(loc='upper left') - plt.show() + plt.savefig('report/performance.pgf') diff --git a/report/performance.pgf b/report/performance.pgf new file mode 100644 index 0000000..3c144ed --- /dev/null +++ b/report/performance.pgf @@ -0,0 +1,867 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% \def\mathdefault#1{#1} +%% \everymath=\expandafter{\the\everymath\displaystyle} +%% +%% \makeatletter\@ifpackageloaded{underscore}{}{\usepackage[strings]{underscore}}\makeatother +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{6.400000in}{4.800000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% +\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{6.400000in}{0.000000in}}% 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PDF statistics: - 25 PDF objects out of 1000 (max. 8388607) - 15 compressed objects within 1 object stream + 57 PDF objects out of 1000 (max. 8388607) + 36 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 1 words of extra memory for PDF output out of 10000 (max. 10000000) + 13 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/report/report.pdf b/report/report.pdf Binary files differindex 899037f..7f67724 100644 --- a/report/report.pdf +++ b/report/report.pdf diff --git a/report/report.tex b/report/report.tex index 581d2ba..646bca8 100644 --- a/report/report.tex +++ b/report/report.tex @@ -3,6 +3,13 @@ \usepackage{titling} \usepackage{helvet} \usepackage{tgpagella} % text only +\usepackage[utf8]{inputenc} +\usepackage{booktabs} +\usepackage{float} + +\usepackage{pgf} + +\newcommand{\mathdefault}[1][]{} \geometry{ @@ -24,13 +31,60 @@ Analyzing Performance of Booth’s Algorithm and Modified Booth’s Algorithm} \begin{document} \maketitle \begin{abstract} -In this paper, the performance of Booth’s Algorithm is compared to modified Booth's Algorithm. Each multiplier is simulated in Python, and performance is observed by counting the number of add and subtract operations for various inputs. Results are analyzed and discussed to highlight the tradeoffs between the two multipliers. +In this paper, the performance of Booth’s Algorithm is compared to modified Booth's Algorithm. Each multiplier is simulated in Python, and performance is observed by counting the number of add and subtract operations for various inputs. Results are analyzed and discussed to highlight the potential tradeoffs to consider when implimenting hardware multipliers. \end{abstract} \section*{Introduction} Multiplication is among the most time consuming mathematical operations for processors. In many applications, the time it takes to multiply dramatically influences the speed of the program. Applications of digital signal processing (such as audio modification and image processing) require constant multiply and accumulate operations for functions such as fast fourier transformations and convolutions. Other applications are heavily dependent on multiplying large matrices, such as machine learning, 3D graphics and data analysis. In such scenarios, the speed of multiplication is vital. Consequently, most modern processors implement hardware multiplication. However, not all multiplication circuits are equal; there is often a stark contrast between performance and hardware complexity. To further complicate things, multiplication circuits perform differently depending on what numbers are being multiplied. -\section*{Implementation} +\section*{Algorithm Description and Implementation} +Booth's algorithim computes the multiplication of two signed two's compliment numbers. To avoid overflow, the result is placed into a register two times the size of the operands (or two registers the size of a single operand). Additionally, the algorithim must work with a space that is exended one bit more then the result. For the purpose of brevity, the result register and extra bit will be refered to as the workspace. First, the multiplier is placed into the workspace and shifted left by 1. From there, an operation is performed based off the last two bits, as shown by the following table: +\begin{table}[H] + \centering +\begin{tabular}{lll} +\toprule +Bit 1 & Bit 0 & Action \\ +\midrule +0 & 0 & None \\ +0 & 1 & Add \\ +1 & 0 & Subtract \\ +1 & 1 & None \\ +\bottomrule +\end{tabular} +\end{table} +After all iterations are complete, the result is arithmaticlly shifted once to the left, and the process repeats for the number of bits in an operand.\\ + +Modified booth's algorithim functions similar to Booth's algorithim, but checks the last \textit{three} bits instead. As such, there are a larger selection of actions for each iteration: +\begin{table}[H] + \centering +\begin{tabular}{llll} +\toprule +Bit 2 & Bit 1 & Bit 0 & Action \\ +\midrule +0 & 0 & 0 & None \\ +0 & 0 & 1 & Add \\ +0 & 1 & 0 & Add \\ +0 & 1 & 1 & Add $\times 2$ \\ +1 & 0 & 0 & Sub $\times 2$ \\ +1 & 0 & 1 & Sub \\ +1 & 1 & 0 & Sub \\ +1 & 1 & 1 & None \\ +\bottomrule +\end{tabular} +\end{table} +Because some operations require multiplying the multiplicand by 2, an extra bit is added to the most significant side of the workspace to avoid overflow. After each iteration, the result is arithmaticlly shifted right twice. The number of iterations is only half of the length of the operands. After all iterations, the workspace is shifted right once, and the second most significant bit is set to the first most significant bit as the result register does not include the extra bit.\\ +The purpose of this paper is to analyze and compare the peformance of these two algorithms for various operand lengths and values. As such, all arithmatic bitwise operations had to account for the length of operand sizes. Take for example, the arithmatic shift right functions: +\begin{center} + put phseudo code here +\end{center} +Additionally, after each iteration, the bits more significant then the workspace length had to be erased (the bitwise functions purposefully do not account for this). \section*{Results} +\begin{center} + \input{performance.pgf} +\end{center} +\input{speed_table.tex}\\ +\input{result_table.tex}\\ + \section*{Analysis} \section*{Conclusion} \end{document} + diff --git a/report/result_table.tex b/report/result_table.tex new file mode 100644 index 0000000..9ec6507 --- /dev/null +++ b/report/result_table.tex @@ -0,0 +1,23 @@ +\begin{tabular}{llll} +\toprule + multiplicand & multiplier & result (bin) & result (hex) \\ +\midrule + 0b1110 & 0b1111 & 0b10 & 0x2 \\ + 0b101 & 0b0 & 0b0 & 0x0 \\ + 0b111111 & 0b111111 & 0b1 & 0x1 \\ + 0b101110 & 0b110111 & 0b10100010 & 0xa2 \\ + 0b111011 & 0b100011 & 0b10010001 & 0x91 \\ + 0b11111 & 0b1010101 & 0b101001001011 & 0xa4b \\ + 0b11010111 & 0b1010101 & 0b1111001001100011 & 0xf263 \\ + 0b1010101 & 0b11010111 & 0b1111001001100011 & 0xf263 \\ + 0b1110111 & 0b110011 & 0b1011110110101 & 0x17b5 \\ + 0b0 & 0b1110111 & 0b0 & 0x0 \\ + 0b101010101 & 0b101010101 & 0b11100011000111001 & 0x1c639 \\ + 0b1100111011 & 0b1001110000 & 0b10011001111010000 & 0x133d0 \\ + 0b1001101110 & 0b101111010 & 0b11011010111001101100 & 0xdae6c \\ + 0b10101010101 & 0b10101010101 & 0b111000110111000111001 & 0x1c6e39 \\ + 0b1111100111 & 0b0 & 0b0 & 0x0 \\ + 0b101010101010 & 0b101010101010 & 0b111000111100011100100 & 0x1c78e4 \\ + 0b111001110000 & 0b11111111 & 0b111111100111000110010000 & 0xfe7190 \\ +\bottomrule +\end{tabular}
\ No newline at end of file diff --git a/report/speed_table.tex b/report/speed_table.tex new file mode 100644 index 0000000..32c8d03 --- /dev/null +++ b/report/speed_table.tex @@ -0,0 +1,23 @@ +\begin{tabular}{llrrr} +\toprule + multiplicand & multiplier & length & booth & modified booth \\ +\midrule + 0b1110 & 0b1111 & 4 & 1 & 1 \\ + 0b101 & 0b0 & 4 & 0 & 0 \\ + 0b111111 & 0b111111 & 6 & 1 & 1 \\ + 0b101110 & 0b110111 & 6 & 3 & 3 \\ + 0b111011 & 0b100011 & 6 & 3 & 3 \\ + 0b11111 & 0b1010101 & 8 & 8 & 4 \\ + 0b11010111 & 0b1010101 & 8 & 8 & 4 \\ + 0b1010101 & 0b11010111 & 8 & 5 & 4 \\ + 0b1110111 & 0b110011 & 8 & 4 & 4 \\ + 0b0 & 0b1110111 & 8 & 4 & 4 \\ + 0b101010101 & 0b101010101 & 10 & 10 & 5 \\ + 0b1100111011 & 0b1001110000 & 10 & 3 & 3 \\ + 0b1001101110 & 0b101111010 & 10 & 6 & 4 \\ + 0b10101010101 & 0b10101010101 & 12 & 12 & 6 \\ + 0b1111100111 & 0b0 & 12 & 0 & 0 \\ + 0b101010101010 & 0b101010101010 & 12 & 11 & 6 \\ + 0b111001110000 & 0b11111111 & 12 & 2 & 2 \\ +\bottomrule +\end{tabular}
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