\documentclass[12pt]{article} \usepackage{amssymb, amsmath, amsfonts, amsthm, graphicx, tcolorbox} \usepackage{arydshln} \parskip = .2in \parindent = 0in \pagestyle{empty} %========== %========== \begin{document} X = 0.0111 0011 1001 (1849/4096) S = 0.1010 1100 0000 (2752/4096) {\small \begin{center} \begin{tabular}{ccccc} &$X $&$ 0000.0111\ 0011\ 1001 $& &$S_0={\color{blue}0001}.0000\ 0000\ 0000$\\ &$WS_0=2(X-1) $&$ 1110.1110\ 0111\ 0010 $& &$SM_0={\color{blue}0000}.0000\ 0000\ 0000\phantom{M}$\\ &$WC_0 $&$ 0000.0000\ 0000\ 0000 $& &$K_0=0001.0000\ 0000\ 0000\ $\\ & & & &$C_0=1111.0000\ 0000\ 0000\,$\\ \hdashline\\ Step 1: &$WS_0 $&$ 1110.1110\ 0111\ 0010 $& &\\ &$WC_0 $&$ 0000.0000\ 0000\ 0000 $& &($W_{msbs}=1110\ \text{so}\ s_1=-1$)\\ &$F_1=2S_0-K_1 $&${\color{blue}0001.1}000\ 0000\ 0000$& &$S_1={\color{blue}0000.1}000\ 0000\ 0000$\\ & & & &$SM_1={\color{blue}0000.0}000\ 0000\ 0000\phantom{M}$\\ \cline{2-3} &$sum $&$ 1111.0110\ 0111\ 0010 $&$\ll1 $&$K_1=0000.1000\ 0000\ 0000\ $\\ &$carry $&$ 0001.0000\ 0000\ 0000 $&$\ll1 $&$C_1=1111.1000\ 0000\ 0000\,$\\ \hdashline\\ Step 2: &$WS_1 $&$ 1110.1100\ 1110\ 0100 $& &\\ &$WC_1 $&$ 0010.0000\ 0000\ 0000 $& &($W_{msbs}=0000\ \text{so}\ s_2=1$)\\ &$F_2=-2S_1-K_2$&${\color{blue}1110.11}00\ 0000\ 0000$& &$S_2={\color{blue}0000.11}00\ 0000\ 0000$\\ & & & &$SM_2={\color{blue}0000.10}00\ 0000\ 0000\phantom{M}$\\ \cline{2-3} &$sum $&$ 0010.0000\ 1110\ 0100 $&$\ll1 $&$K_2=0000.0100\ 0000\ 0000\ $\\ &$carry $&$ 1101.1000\ 0000\ 0000 $&$\ll1 $&$C_2=1111.1100\ 0000\ 0000\,$\\ \hdashline\\ Step 3: &$WS_2 $&$ 0100.0001\ 1100\ 1000 $& &\\ &$WC_2 $&$ 1011.0000\ 0000\ 0000 $& &($W_{msbs}=1111\ \text{so}\ s_3=0$)\\ &$F_3=0 $&${\color{blue}0000.000}0\ 0000\ 0000$& &$S_3={\color{blue}0000.110}0\ 0000\ 0000$\\ & & & &$SM_3={\color{blue}0000.101}0\ 0000\ 0000\phantom{M}$\\ \cline{2-3} &$sum $&$ 1111.0001\ 1100\ 1000 $&$\ll1 $&$K_3=0000.0010\ 0000\ 0000\ $\\ &$carry $&$ 0000.0000\ 0000\ 0000 $&$\ll1 $&$C_3=1111.1110\ 0000\ 0000\,$\\ \hdashline\\ Step 4: &$WS_3 $&$ 1110.0011\ 1001\ 0000 $& &\\ &$WC_3 $&$ 0000.0000\ 0000\ 0000 $& &($W_{msbs}=1110\ \text{so}\ s_4=-1$)\\ &$F_4=2S_3-K_4 $&${\color{blue}0001.0111}\ 0000\ 0000$& &$S_4={\color{blue}0000.1011}\ 0000\ 0000$\\ & & & &$SM_4={\color{blue}0000.1010}\ 0000\ 0000\phantom{M}$\\ \cline{2-3} &$sum $&$ 1111.0100\ 1001\ 0000 $&$\ll1 $&$K_4=0000.0001\ 0000\ 0000\ $\\ &$carry $&$ 0000.0110\ 0000\ 0000 $&$\ll1 $&$C_4=1111.1111\ 0000\ 0000\,$\\ \hdashline\\ Step 5: &$WS_4 $&$ 1110.1001\ 0010\ 0000 $& &\\ &$WC_4 $&$ 0000.1100\ 0000\ 0000 $& &($W_{msbs}=1110\ \text{so}\ s_5=-1$)\\ &$F_5=2S_4-K_5 $&${\color{blue}0001.0101\ 1}000\ 0000$& &$S_5={\color{blue}0000.1010\ 1}000\ 0000$\\ & & & &$SM_5={\color{blue}0000.1010\ 0}000\ 0000\phantom{M}$\\ \cline{2-3} &$sum $&$ 1111.0000\ 1010\ 0000 $&$\ll1 $&$K_5=0000.0000\ 1000\ 0000\ $\\ &$carry $&$ 0001.1010\ 0000\ 0000 $&$\ll1 $&$C_5=1111.1111\ 1000\ 0000\,$\\ \hdashline\\ Step 6: &$WS_5 $&$ 1110.0001\ 0100\ 0000 $& &\\ &$WC_5 $&$ 0011.0100\ 0000\ 0000 $& &($W_{msbs}=0001\ \text{so}\ s_6=1$)\\ &$F_6=-2S_5-K_6$&${\color{blue}1110.1010\ 11}00\ 0000$& &$S_6={\color{blue}0000.1010\ 11}00\ 0000$\\ & & & &$SM_6={\color{blue}0000.1010\ 10}00\ 0000\phantom{M}$\\ \cline{2-3} &$sum $&$ 0011.1111\ 1000\ 0000 $&$\ll1 $&$K_6=0000.0000\ 0100\ 0000\ $\\ &$carry $&$ 1100.0000\ 1000\ 0000 $&$\ll1 $&$C_6=1111.1111\ 1100\ 0000\,$\\ & & & &$sum+carry=0$, terminate\\ \hdashline\\ Terminate&Square Root&0.101011 \end{tabular} \end{center} } {\small \begin{center} \begin{tabular}{cccc} \hdashline\\ Step 6: &$WS $&$0111.1111\ 0000\ 00$&\\ &$WC $&$1000.0001\ 0000\ 00$&($W_{msbs}=1111\ \text{so}\ s_6=0$)\\ &$F $&$0000.0000\ 0000\ 00$&$S_6=\mathbf{1.0101\ 10}00\ 00$\\ & & &$SM_6=\mathbf{1.0101\ 01}00\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$1111.1110\ 0000\ 00$&$\ll1$\\ &$WC $&$0000.0010\ 0000\ 00$&$\ll1$\\ \hdashline\\ Step 7: &$WS $&$1111.1100\ 0000\ 00$&\\ &$WC $&$0000.0100\ 0000\ 00$&($W_{msbs}=1111\ \text{so}\ s_7=0$)\\ &$F $&$0000.0000\ 0000\ 00$&$S_7=\mathbf{1.0101\ 100}0\ 00$\\ & & &$SM_7=\mathbf{1.0101\ 011}0\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$1111.1000\ 0000\ 00$&$\ll1$\\ &$WC $&$0000.1000\ 0000\ 00$&$\ll1$\\ \hdashline\\ Step 8: &$WS $&$1111.0000\ 0000\ 00$&\\ &$WC $&$0001.0000\ 0000\ 00$&($W_{msbs}=0000\ \text{so}\ s_8=1$)\\ &$F $&$1110.1010\ 0111\ 10$&$S_8=\mathbf{1.0101\ 1001}\ 00$\\ & & &$SM_8=\mathbf{1.0101\ 1000}\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$0000.1010\ 0111\ 10$&$\ll1$\\ &$WC $&$1110.0000\ 0000\ 00$&$\ll1$\\ \hdashline\\ Step 9: &$WS $&$0001.0100\ 1111\ 00$&\\ &$WC $&$1100.0000\ 0000\ 00$&($W_{msbs}=1101\ \text{so}\ s_9=-1$)\\ &$F $&$0001.0101\ 1000\ 11$&$S_9=\mathbf{1.0101\ 1000\ 1}0$\\ & & &$SM_9=\mathbf{1.0101\ 1000\ 0}0\phantom{M}$\\ \cline{2-3} &$WS $&$1100.0001\ 0111\ 11$&$\ll1$\\ &$WC $&$0010.1001\ 0000\ 00$&$\ll1$\\ \hdashline\\ Step 10:&$WS $&$1000.0010\ 1111\ 10$&\\ &$WC $&$0101.0010\ 0000\ 00$&($W_{msbs}=1101\ \text{so}\ s_{10}=-1$)\\ &$F $&$0001.0101\ 1000\ 01$&$S_{10}=\mathbf{1.0101\ 1000\ 01}$\\ & & &$SM_{10}=\mathbf{1.0101\ 1000\ 00}\phantom{M}$\\ \cline{2-3} &$WS $&$1100.0001\ 0111\ 11$&$\ll1$\\ &$WC $&$0010.0101\ 0000\ 00$&$\ll1$\\ \hdashline\\ Step 11:&$WS $&$1000.0010\ 1111\ 10$&\\ &$WC $&$0100.1010\ 0000\ 00$&($W_{msbs}=1101\ \text{so}\ s_{11}=-1$)\\ &$F $&$0001.0101\ 1000\ 00$&$S_{11}=\mathbf{1.0101\ 1000\ 00}$\\ & & &$SM_{11}=\mathbf{1.0101\ 1000\ 00}\phantom{M}$\\ \cline{2-3} &$WS $&$1101.1101\ 0111\ 10$&$\ll1$\\ &$WC $&$0000.0101\ 0000\ 00$&$\ll1$\\ \hdashline\\ Step 12:&$WS $&$1101.1000\ 0111\ 10$&\\ &$WC $&$0000.1010\ 0000\ 00$&($W_{msbs}=1101\ \text{so}\ s_{12}=-1$)\\ &$F $&$0001.0101\ 1000\ 00$&$S_{12}=\mathbf{1.0101\ 1000\ 00}$\\ & & &$SM_{12}=\mathbf{1.0101\ 1000\ 00}\phantom{M}$\\ \cline{2-3} &$WS $&$1100.0111\ 1111\ 10$&$\ll1$\\ &$WC $&$0011.0000\ 0000\ 00$&$\ll1$\\ \end{tabular} \end{center} } {\small \begin{center} \begin{tabular}{cccc} Step 13:&$WS $&$1000.1111\ 1111\ 00$&\\ &$WC $&$0110.0000\ 0000\ 00$&($W_{msbs}=1110\ \text{so}\ s_{13}=-1$)\\ &$F $&$0001.0101\ 1000\ 00$&$S_{13}=\mathbf{1.0101\ 1000\ 00}$\\ & & &$SM_{13}=\mathbf{1.0101\ 1000\ 00}\phantom{M}$\\ \cline{2-3} &$WS $&$1111.1010\ 0111\ 10$&$\ll1$\\ &$WC $&$0000.1011\ 0000\ 00$&$\ll1$\\ \end{tabular} \end{center} } \vfill \eject {\small \begin{center} \begin{tabular}{ccccc} &$X $&$ 0000.1100\ 0000 $& &$S_0={\color{blue}0001}.0000\ 0000\ 00$\\ &$WS=X-1 $&$ 1111.1100\ 0000 $& &$SM_0={\color{blue}0000}.0000\ 0000\ 00\phantom{M}$\\ &$WC $&$ 0000.0000\ 0000 $& &$K_0=0000.0100\ 0000\ 00\ $\\ & & & &$C_0=1111.1100\ 0000\ 00\,$\\ \hdashline\\ Step 1: &$WS $&$ 1111.1100\ 0000 $& &\\ &$WC $&$ 0000.0000\ 0000 $& &($W_{msbs}=1111\ \text{so}\ s_1=0$)\\ &$F_1=0 $&${\color{blue}0000.00}00\ 0000$& &$S_1={\color{blue}0001.0}000\ 0000\ 00$\\ & & & &$SM_1={\color{blue}0000.1}000\ 0000\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$ 1111.1100\ 0000 $&$\ll1 $&$K_1=0000.0010\ 0000\ 00\ $\\ &$WC $&$ 0000.0000\ 0000 $&$\ll1 $&$C_1=1111.1110\ 0000\ 00\,$\\ \hdashline\\ Step 2: &$WS $&$ 1111.1000\ 0000 $& &\\ &$WC $&$ 0000.0000\ 0000 $& &($W_{msbs}=1111\ \text{so}\ s_2=0$)\\ &$F_2=0 $&${\color{blue}0000.000}0\ 0000$& &$S_2={\color{blue}0001.00}00\ 0000\ 00$\\ & & & &$SM_2={\color{blue}0000.11}00\ 0000\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$ 1111.1000\ 0000 $&$\ll1 $&$K_2=0000.0001\ 0000\ 00\ $\\ &$WC $&$ 0000.0000\ 0000 $&$\ll1 $&$C_2=1111.1111\ 0000\ 00\,$\\ \hdashline\\ Step 3: &$WS $&$ 1111.0000\ 0000 $& &\\ &$WC $&$ 0000.0000\ 0000 $& &($W_{msbs}=1111\ \text{so}\ s_3=0$)\\ &$F_3=0 $&${\color{blue}0000.0000}\ 0000$& &$S_3={\color{blue}0001.000}0\ 0000\ 00$\\ & & & &$SM_3={\color{blue}0000.111}0\ 0000\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$1111.0000\ 0000 $&$\ll1 $&$K_3=0000.0000\ 1000\ 00\ $\\ &$WC $&$0000.0000\ 0000 $&$\ll1 $&$C_3=1111.1111\ 1000\ 00\,$\\ \hdashline\\ Step 4: &$WS $&$1110.0000\ 0000 $& &\\ &$WC $&$0000.0000\ 0000 $& &($W_{msbs}=1110\ \text{so}\ s_4=-1$)\\ &$F_4=S_3-K_3 $&${\color{blue}0000.1111\ 1}000$& &$S_4={\color{blue}0000.1111}\ 0000\ 00$\\ & & & &$SM_4={\color{blue}0000.1110}\ 0000\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$1110.1111\ 1000 $&$\ll1 $&$K_4=0000.0000\ 0100\ 00\ $\\ &$WC $&$0000.0000\ 0000 $&$\ll1 $&$C_4=1111.1111\ 1100\ 00\,$\\ \hdashline\\ Step 5: &$WS $&$1101.1111\ 0000 $& &\\ &$WC $&$0000.0000\ 0000 $& &($W_{msbs}=1101\ \text{so}\ s_5=-1$)\\ &$F_5=S_4-K_4 $&${\color{blue}0000.1110\ 11}00$& &$S_5={\color{blue}0000.1110\ 1}000\ 00$\\ & & & &$SM_5={\color{blue}0001.1110\ 0}000\ 00\phantom{M}$\\ \cline{2-3} &$WS $&$1101.0001\ 1100 $&$\ll1 $&$K_5=0000.0000\ 0010\ 00\ $\\ &$WC $&$0001.1100\ 0000 $&$\ll1 $&$C_5=1111.1111\ 1110\ 00\,$\\ Terminate \end{tabular} \end{center} } \vfill \eject X = 0.1010101101(685/1024) S = 0.1101000110(838/1024) once R4 sslc gets here i can fill this in \vfill \eject X = 1.1001 (25/16) S = 1.0100 (20/16) {\small \begin{center} \begin{tabular}{cccc} Attempt 1:& $X$ is normalized& to $1/2