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/dev/null and b/docs/divsqrt_tex/div2.pdf differ diff --git a/docs/divsqrt_tex/div2.tex b/docs/divsqrt_tex/div2.tex new file mode 100644 index 000000000..03daaf1a8 --- /dev/null +++ b/docs/divsqrt_tex/div2.tex @@ -0,0 +1,123 @@ +\documentclass[12pt]{article} + +\usepackage{amssymb, amsmath, amsfonts, amsthm, graphicx, tcolorbox} + +\usepackage{arydshln} + +\parindent = 0in + +\pagestyle{empty} + +%========== +%========== + +\begin{document} + +\begin{center} +\begin{tabular}{cccc} + Initialization&$D$&$0001.1010\ 000$&\\ + &$-D=\overline{D}+1$&$1110.0101\ 111$&(+ 1 ulp)\\ + &&&\\ + &$WS_{-1}=X$&$0001.0000\ 010$&\\ + &$WC_{-1}$&$0000.0000\ 000$&\\ + \hdashline\\ + Step 0:&$WS_{-1}$&$0001.0000\ 010$&\\ + &$WC_{-1}$&$0000.0000\ 00\mathbf{1}$&($W_{msbs}=0001\ \text{so}\ q_0=1$)\\ + &$-q_0D$&$1110.0101\ 111$&\\ + \cline{2-3} + &$sum$&$1111.0101\ 100$&$\ll1$\\ + &$carry$&$0000.0000\ 110$&$\ll1$\\ + \hdashline\\ + Step 1:&$WS_0$&$1110.1011\ 000$&\\ + &$WC_0$&$0000.0001\ 10\mathbf{0}$&($W_{msbs}=1110\ \text{so}\ q_1=-1$)\\ + &$-q_1D$&$0001.1010\ 000$&\\ + \cline{2-3} + &$sum$&$1111.0000\ 100$&$\ll1$\\ + &$carry$&$0001.0110\ 000$&$\ll1$\\ + \hdashline\\ + Step 2:&$WS_1$&$1110.0001\ 000$&\\ + &$WC_1$&$0010.1100\ 00\mathbf{1}$&($W_{msbs}=0000\ \text{so}\ q_2=1$)\\ + &$-q_2D$&$1110.0101\ 111$&\\ + \cline{2-3} + &$sum$&$0010.1000\ 110$&$\ll1$\\ + &$carry$&$1100.1010\ 010$&$\ll1$\\ + \hdashline\\ + Step 3:&$WS_2$&$0101.0001\ 100$&\\ + &$WC_2$&$1001.0100\ 10\mathbf{0}$&($W_{msbs}=1110\ \text{so}\ q_3=-1$)\\ + &$-q_3D$&$0001.1010\ 000$&\\ + \cline{2-3} + &$sum$&$1101.1111\ 000$&\\ + &$carry$&$0010.0001\ 000$&$sum+carry=0$, terminate.\\ + \hdashline\\ + Terminate&Quotient&0.101 +\end{tabular} +\end{center} + +\vfill +\eject + +X = 1.0110\ 011 (179/128) + +D = 1.0011\ 000 (152/128) + +Q = 1.0010\ 1101\ 0 + +D[1.3] = 1.001, so we use the ``"1.001" column of chart 13.X. This means we select a quotient bit of 2 if the partial remainder is greater than or equal to 3.5, a quotient bit of 1 if the partial is greater or equal to than 1.0, a zero if the partial is greater than or equal to -1.5, -1 if the partial is greater than or equal to -3.75, and a -2 otherwise. +\begin{center} +\begin{tabular}{cccc} + Initialization&$D$&$0001.0011\ 000$&\\ + &$2D$&$0010.0110\ 000$&\\ + &$-D=\overline{D}+1$&$1110.1100\ 111$&(+ 1 ulp)\\ + &$-2D=\overline{2D}+1$&$1101.1001\ 111$&(+ 1 ulp)\\ + &&&\\ + &$X=WS$&$0001.0110\ 011$&\\ + &$WC$&$0000.0000\ 000$&\\ + \hdashline\\ + Step 4:&$WS$&$0001.0110\ 011$&\\ + &$WC$&$0000.0000\ 00\mathbf{1}$&($RW_{msbs}=0001.010\ \text{so}\ q_4=1$)\\ + &$-q_7D$&$1110.1100\ 111$&\\ + \cline{2-3} + &$WS$&$1111.1010\ 101$&$\ll2$\\ + &$WC$&$0000.1000\ 110$&$\ll2$\\ + \hdashline\\ + Step 3:&$WS$&$1110.1010\ 100$&\\ + &$WC$&$0010.0011\ 000$&($RW_{msbs}=0000.110\ \text{so}\ q_3=1$)\\ + &$-q_6D$&$0000.0000\ 000$&\\ + \cline{2-3} + &$WS$&$1100.1001\ 100$&$\ll2$\\ + &$WC$&$0100.0100\ 000$&$\ll2$\\ + \hdashline\\ + Step 2:&$WS$&$0010.0110\ 000$&\\ + &$WC$&$0001.0000\ 00\mathbf{1}$&($RW_{msbs}=0011.010\ \text{so}\ q_2=-1$)\\ + &$-q_5D$&$1110.0101\ 111$&\\ + \cline{2-3} +\end{tabular} +\end{center} + +page 269 306 + +\vfill +\eject + +\large{\bf{ +Math for the recurrence relation}} + +**going to have to change notation for sure, change the subscripts for steps and might have to get rid of some exponents** +\begin{align*} + w[j+1] &= r^{j+1}\big(x-S[j+1]^2\big)\\ + &= r^{j+1}\big(x-(S[j]+s_{j+1}r^{-(j+1)})^2\big)\\ + &= r^{j+1}x-r^{j+1}\big(S[j]^2+2S[j]s_{j+1}r^{-(j+1)}+s^2_{j+1}r^{-2(j+1)}\big)\\ + &= r^{j+1}\big(x-S[j]^2\big)-\big(2S[j]s_{j+1}+s_{j+1}^2r^{-(j+1)}\big)\\ + &= rw[j]-\big(2S[j]s_{j+1}+s_{j+1}^2r^{-(j+1)}\big)\\ + &= rw[j]+F[j] +\end{align*} +where +\begin{align*} + F[j]=-\big(2S[j]s_{j+1}+s_{j+1}^2r^{-(j+1)}\big) +\end{align*} + +Since there is a term of $S$ in the expression of $F$, we must come up with a way to represent $S$ using only zeros and ones, rather than using the bit set $\{-a,\ldots,a\}$. This is done using on-the-fly conversion just as we did to compute the quotient for the divider. We keep a running copy of $S$, but we also keep the value $SM=S-1$. The logic is still the same for computing $S$ and $SM$ on the next step; see figure 13.15. + +Now that $S$ is in a form such that we can use it in a CSA, we need to compute $F$. 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/dev/null and b/docs/divsqrt_tex/div4.pdf differ diff --git a/docs/divsqrt_tex/div4.tex b/docs/divsqrt_tex/div4.tex new file mode 100644 index 000000000..a5b95dda1 --- /dev/null +++ b/docs/divsqrt_tex/div4.tex @@ -0,0 +1,84 @@ +\documentclass[12pt]{article} + +\usepackage{amssymb, amsmath, amsfonts, amsthm, graphicx, tcolorbox} + +\usepackage{arydshln} + +\parskip = .2in +\parindent = 0in + +\pagestyle{empty} + +%========== +%========== + +\begin{document} + + +X = 1.0000\ 1101 (269/256) + +D = 1.0011\ 0110 (310/256) + +Q = 0.1101\ 1110 (222/256) + +D[1.3] = 1.001, so we use the ``1.001" column of chart 13.X. This means we select a quotient bit of 2 if the partial remainder is greater than or equal to 3.5, a quotient bit of 1 if the partial is greater or equal to than 1.0, a zero if the partial is greater than or equal to -1.5, -1 if the partial is greater than or equal to -3.75, and a -2 otherwise. + +{\small +\begin{center} +\begin{tabular}{cccc} + Initialization&$D$&$0001.0011\ 0110\ 00$&\\ + &$2D$&$0010.0110\ 1100\ 00$&\\ + &$-D=\overline{D}+1$&$1110.1100\ 1001\ 11$&(+ 1 ulp)\\ + &$-2D=\overline{2D}+1$&$1101.1001\ 0011\ 11$&(+ 1 ulp)\\ + &&&\\ + &$WS_{-1}=X$&$0001.0000\ 1101\ 00$&\\ + &$WC_{-1}$&$0000.0000\ 0000\ 00$&\\ + \hdashline\\ + Step 0: &$WS_{-1} $&$0001.0000\ 1101\ 00$&\\ + &$WC_{-1} $&$0000.0000\ 0000\ 0\mathbf{1}$&($W_{msbs}=0001.000\ \text{so}\ q_0=1$)\\ + &$-q_0D $&$1110.1100\ 1001\ 11$&\\ + \cline{2-3} + &$sum $&$1111.1100\ 0100\ 10$&$\ll2$\\ + &$carry $&$0000.0001\ 0010\ 10$&$\ll2$\\ + \hdashline\\ + Step 1: &$WS_0 $&$1111.0001\ 0010\ 00$&\\ + &$WC_0 $&$0000.0100\ 1010\ 0\mathbf{0}$&($W_{msbs}=1111.010\ \text{so}\ q_1=-1$)\\ + &$-q_1D $&$0001.0011\ 0110\ 00$&\\ + \cline{2-3} + &$sum $&$1110.0110\ 1110\ 00$&$\ll2$\\ + &$carry $&$0010.0010\ 0100\ 00$&$\ll2$\\ + \hdashline\\ + Step 2: &$WS_1 $&$1001.1011\ 1000\ 00$&\\ + &$WC_1 $&$1000.1001\ 0000\ 0\mathbf{1}$&($W_{msbs}=0010.010\ \text{so}\ q_2=2$)\\ + &$-q_2D $&$1101.1001\ 0011\ 11$\\ + \cline{2-3} + &$sum $&$1100.1011\ 1011\ 10$&$\ll2$\\ + &$carry $&$0011.0010\ 0000\ 10$&$\ll2$\\ + \hdashline\\ + Step 3: &$WS_2 $&$0010.1110\ 1110\ 00$&\\ + &$WC_2 $&$1100.1000\ 0010\ 0\mathbf{0}$&($W_{msbs}=1111.011\ \text{so}\ q_3=-1$)\\ + &$-q_3D $&$0001.0011\ 0110\ 00$\\ + \cline{2-3} + &$sum $&$1111.0101\ 1010\ 00$&$\ll2$\\ + &$carry $&$0001.0100\ 0100\ 00$&$\ll2$\\ + \hdashline\\ + Step 4: &$WS_3 $&$1101.0110\ 1000\ 00$&\\ + &$WC_3 $&$0101.0001\ 0000\ 0\mathbf{1}$&($W_{msbs}=0010.011\ \text{so}\ q_4=2$)\\ + &$-q_4D $&$1101.1001\ 0011\ 11$\\ + \cline{2-3} + &$sum $&$0101.1110\ 1011\ 10$&$\ll2$\\ + &$carry $&$1010.0010\ 0000\ 10$&$\ll2$\\ + \hdashline\\ + Step 5: &$WS_4 $&$0111.1010\ 1110\ 00$&\\ + &$WC_4 $&$1000.1000\ 0010\ 0\mathbf{0}$&($W_{msbs}=0000.001\ \text{so}\ q_5=0$)\\ + &$-q_5D $&$0000.0000\ 0000\ 00$\\ + \cline{2-3} + &$sum $&$1111.0010\ 1100\ 00$&$\ll2$\\ + &$carry $&$0001.0000\ 0100\ 00$&$\ll2$\\ + \hdashline\\ + Terminate&Quotient&$00.11\ 01\ 11\ 10\ (00\ 1)$ +\end{tabular} +\end{center} +} + +\end{document} diff --git a/docs/divsqrt_tex/sqrt2.aux b/docs/divsqrt_tex/sqrt2.aux new file mode 100644 index 000000000..4fd86f1fe --- /dev/null +++ b/docs/divsqrt_tex/sqrt2.aux @@ -0,0 +1,2 @@ +\relax +\gdef \@abspage@last{11} diff --git a/docs/divsqrt_tex/sqrt2.log b/docs/divsqrt_tex/sqrt2.log new file mode 100644 index 000000000..1019ba309 --- /dev/null +++ b/docs/divsqrt_tex/sqrt2.log @@ -0,0 +1,557 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023) (preloaded format=pdflatex 2023.7.19) 20 AUG 2023 13:10 +entering extended mode + restricted \write18 enabled. + 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a/docs/divsqrt_tex/sqrt2.tex b/docs/divsqrt_tex/sqrt2.tex new file mode 100644 index 000000000..3c40b037c --- /dev/null +++ b/docs/divsqrt_tex/sqrt2.tex @@ -0,0 +1,385 @@ +\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 patch level 1 +L3 programming layer <2023-02-22> +(/usr/local/texlive/2023/texmf-dist/tex/latex/base/article.cls +Document Class: article 2022/07/02 v1.4n Standard LaTeX document class +(/usr/local/texlive/2023/texmf-dist/tex/latex/base/size12.clo +File: size12.clo 2022/07/02 v1.4n Standard LaTeX file (size option) +) +\c@part=\count185 +\c@section=\count186 +\c@subsection=\count187 +\c@subsubsection=\count188 +\c@paragraph=\count189 +\c@subparagraph=\count190 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+[3] +Overfull \vbox (101.09952pt too high) has occurred while \output is active [] + + +[4] (./sqrt4.aux) ) +Here is how much of TeX's memory you used: + 13098 strings out of 476025 + 263892 string characters out of 5790017 + 1878388 words of memory out of 5000000 + 33355 multiletter control sequences out of 15000+600000 + 517233 words of font info for 51 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 83i,12n,89p,418b,230s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on sqrt4.pdf (4 pages, 75105 bytes). +PDF statistics: + 50 PDF objects out of 1000 (max. 8388607) + 31 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 13 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/docs/divsqrt_tex/sqrt4.pdf b/docs/divsqrt_tex/sqrt4.pdf new file mode 100644 index 000000000..363d6eca3 Binary files /dev/null and b/docs/divsqrt_tex/sqrt4.pdf differ diff --git a/docs/divsqrt_tex/sqrt4.tex b/docs/divsqrt_tex/sqrt4.tex new file mode 100644 index 000000000..1d40864eb --- /dev/null +++ b/docs/divsqrt_tex/sqrt4.tex @@ -0,0 +1,147 @@ +\documentclass[12pt]{article} + +\usepackage{amssymb, amsmath, amsfonts, amsthm, graphicx, tcolorbox} + +\usepackage{arydshln} + +\parskip = .2in +\parindent = 0in + +\pagestyle{empty} + +%========== +%========== + +\begin{document} + + + +X = 0.1011 0111 10 (734/1024) + +S = 0.1101 1000 11\ 00 (3468/4096), negative sticky bit +{\small +\begin{center} +\begin{tabular}{ccccc} + &$X $&$ 0000.1011\ 0111\ 1000 $& &$S_0={\color{blue}0001}.0000\ 0000\ 0000$\\ + &$WS_0=4(X-1) $&$ 1110.1101\ 1110\ 0000 $& &$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 $&$ 1110.1101\ 1110\ 0000 $& &\\ + &$WC_0 $&$ 0000.0000\ 0000\ 0000 $& &($W_{msbs}=1110.110\ \text{so}\ s_1=-1$)\\ + &$F_1=2S_0-K_1$&${\color{blue}0001.11}00\ 0000\ 0000$& &$S_1={\color{blue}0000.11}00\ 0000\ 0000$\\ + & & & &$SM_1={\color{blue}0000.10}00\ 0000\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1111.0001\ 1110\ 0000 $&$\ll2 $&$K_1=0000.0100\ 0000\ 0000\ $\\ + &$carry $&$ 0001.1000\ 0000\ 0000 $&$\ll2 $&$C_1=1111.1100\ 0000\ 0000\,$\\ + \hdashline\\ + Step 2: &$WS_1 $&$ 1100.0111\ 1000\ 0000 $& &\\ + &$WC_1 $&$ 0110.0000\ 0000\ 0000 $& &($W_{msbs}=0010.011\ \text{so}\ s_2=2$)\\ + &$F_2=-4S_1-4K_2$&${\color{blue}1100.1100}\ 0000\ 0000$& &$S_2={\color{blue}0000.1110}\ 0000\ 0000$\\ + & & & &$SM_2={\color{blue}0000.1101}\ 0000\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 0110.1011\ 1000\ 0000 $&$\ll2 $&$K_2=0000.0001\ 0000\ 0000\ $\\ + &$carry $&$ 1000.1000\ 0000\ 0000 $&$\ll2 $&$C_2=1111.1111\ 0000\ 0000\,$\\ + \hdashline\\ + Step 3: &$WS_2 $&$ 1010.1110\ 0000\ 0000 $& &\\ + &$WC_2 $&$ 0010.0000\ 0000\ 0000 $& &($W_{msbs}=1100.111\ \text{so}\ s_3=-2$)\\ + &$F_3=4S_2-4K_3$&${\color{blue}0011.0111\ 00}00\ 0000$& &$S_3={\color{blue}0000.1101\ 10}00\ 0000$\\ + & & & &$SM_3={\color{blue}0000.1101\ 01}00\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1011.1001\ 0000\ 0000 $&$\ll2 $&$K_3=0000.0000\ 0100\ 0000\ $\\ + &$carry $&$ 0100.1100\ 0000\ 0000 $&$\ll2 $&$C_3=1111.1111\ 1100\ 0000\,$\\ + \hdashline\\ + Step 4: &$WS_3 $&$ 1110.0100\ 0000\ 0000 $& &\\ + &$WC_3 $&$ 0011.0000\ 0000\ 0000 $& &($W_{msbs}=0001.010\ \text{so}\ s_4=1$)\\ + &$F_4=-2S_3-K_4$&${\color{blue}1110.0100\ 1111}\ 0000$& &$S_4={\color{blue}0000.1101\ 1001}\ 0000$\\ + & & & &$SM_4={\color{blue}0000.1101\ 1000}\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 0011.0000\ 1111\ 0000 $&$\ll2 $&$K_4=0000.0000\ 0001\ 0000\ $\\ + &$carry $&$ 1100.1000\ 0000\ 0000 $&$\ll2 $&$C_4=1111.1111\ 1111\ 0000\,$\\ + \hdashline\\ + Step 5: &$WS_4 $&$ 1100.0011\ 1100\ 0000 $& &\\ + &$WC_4 $&$ 0010.0000\ 0000\ 0000 $& &($W_{msbs}=1110.001\ \text{so}\ s_5=-1$)\\ + &$F_5=2S_4-K_5 $&${\color{blue}0001.1011\ 0001\ 11}00$& &$S_5={\color{blue}0000.1101\ 1000\ 11}00$\\ + & & & &$SM_5={\color{blue}0000.1101\ 1000\ 10}00\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1111.1000\ 1101\ 1100 $&$\ll2 $&$K_5=0000.0000\ 0000\ 0100\ $\\ + &$carry $&$ 0000.0110\ 0000\ 0000 $&$\ll2 $&$C_5=1111.1111\ 1111\ 1100\,$\\ + \hdashline\\ + Step 6: &$WS_5 $&$ 1110.0011\ 0111\ 0000 $& &\\ + &$WC_5 $&$ 0001.1000\ 0000\ 0000 $& &($W_{msbs}=1111.101\ \text{so}\ s_6=0$)\\ + &$F_6=0 $&${\color{blue}0000.0000\ 0000\ 0000}$& &$S_6={\color{blue}0000.1101\ 1000\ 1100}$\\ + & & & &$SM_6={\color{blue}0000.1101\ 1000\ 1011}\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1111.1011\ 0111\ 0111 $&$\ll2 $&$K_6=0000.0000\ 0000\ 0001\ $\\ + &$carry $&$ 0000.0000\ 0000\ 0000 $&$\ll2 $&$C_6=1111.1111\ 1111\ 1111\,$\\ + \hdashline\\ + Terminate&Square Root&0000.11 01\ 10 00\ 10 (11) +\end{tabular} +\end{center} +} + +X = 0.1001\ 0101\ 00 (596/1024) + +S = 0.1100 0011 01\ 01 (3125/4096) +{\small +\begin{center} +\begin{tabular}{ccccc} + &$X $&$ 0000.1001\ 0101\ 0000 $& &$S_0={\color{blue}0001}.0000\ 0000\ 0000$\\ + &$WS_0=4(X-1) $&$ 1110.0101\ 0100\ 0000 $& &$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 $&$ 1110.0101\ 0100\ 0000 $& &\\ + &$WC_0 $&$ 0000.0000\ 0000\ 0000 $& &($W_{msbs}=1110.010\ \text{so}\ s_1=-1$)\\ + &$F_1=2S_0-K_1 $&${\color{blue}0001.11}00\ 0000\ 0000$& &$S_1={\color{blue}0000.11}00\ 0000\ 0000$\\ + & & & &$SM_1={\color{blue}0000.10}00\ 0000\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1111.1001\ 0100\ 0000 $&$\ll2 $&$K_1=0000.0100\ 0000\ 0000\ $\\ + &$carry $&$ 0000.1000\ 0000\ 0000 $&$\ll2 $&$C_1=1111.1100\ 0000\ 0000\,$\\ + \hdashline\\ + Step 2: &$WS_1 $&$ 1110.0101\ 0000\ 0000 $& &\\ + &$WC_2 $&$ 0010.0000\ 0000\ 0000 $& &($W_{msbs}=0000.010\ \text{so}\ s_2=0$)\\ + &$F_2=0 $&${\color{blue}0000.0000}\ 0000\ 0000$& &$S_2={\color{blue}0000.1100}\ 0000\ 0000$\\ + & & & &$SM_2={\color{blue}0000.1110}\ 0000\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1100.0101\ 0000\ 0000 $&$\ll2 $&$K_2=0000.0001\ 0000\ 0000\ $\\ + &$carry $&$ 0100.0000\ 0000\ 0000 $&$\ll2 $&$C_2=1111.1111\ 0000\ 0000\,$\\ + \hdashline\\ + Step 3: &$WS_2 $&$ 0001.0100\ 0000\ 0000 $& &\\ + &$WC_2 $&$ 0000.0000\ 0000\ 0000 $& &($W_{msbs}=0001.010\ \text{so}\ s_3=1$)\\ + &$F_3=-2S_2-K_3$&${\color{blue}1110.0111\ 11}00\ 0000$& &$S_3={\color{blue}0000.1100\ 01}00\ 0000$\\ + & & & &$SM_3={\color{blue}0000.1100\ 00}00\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1111.0011\ 1100\ 0000 $&$\ll2 $&$K_3=0000.0000\ 0100\ 0000\ $\\ + &$carry $&$ 1000.1000\ 0000\ 0000 $&$\ll2 $&$C_3=1111.1111\ 1100\ 0000\,$\\ + \hdashline\\ + Step 4: &$WS_3 $&$ 1100.1111\ 0000\ 0000 $& &\\ + &$WC_3 $&$ 0010.0000\ 0000\ 0000 $& &($W_{msbs}=1110.111\ \text{so}\ s_4=-1$)\\ + &$F_4=2S_3-K_3 $&${\color{blue}0001.1000\ 0111}\ 0000$& &$S_4={\color{blue}0000.1100\ 0011}\ 0000$\\ + & & & &$SM_4={\color{blue}0000.1100\ 0010}\ 0000\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 1111.0111\ 0111\ 0000 $&$\ll2 $&$K_4=0000.0000\ 0001\ 0000\ $\\ + &$carry $&$ 0001.0000\ 0000\ 0000 $&$\ll2 $&$C_4=1111.1111\ 1111\ 0000\,$\\ + \hdashline\\ + Step 5: &$WS_4 $&$ 1101.1101\ 1100\ 0000 $& &\\ + &$WC_4 $&$ 0100.0000\ 0000\ 0000 $& &($W_{msbs}=0001.110\ \text{so}\ s_5=1$)\\ + &$F_5=-2S_4-K_4 $&${\color{blue}1110.0111\ 1001\ 11}00$& &$S_5={\color{blue}0000.1100\ 0011\ 01}00$\\ + & & & &$SM_5={\color{blue}0000.1100\ 0011\ 00}00\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 0111.1010\ 0101\ 1100 $&$\ll1 $&$K_5=0000.0000\ 0000\ 0100\ $\\ + &$carry $&$ 1000.1011\ 0000\ 0000 $&$\ll1 $&$C_5=1111.1111\ 1111\ 1100\,$\\ + \hdashline\\ + Step 6: &$WS_5 $&$ 1110.1001\ 0111\ 0000 $& &\\ + &$WC_5 $&$ 0010.1100\ 0000\ 0000 $& &($W_{msbs}=0001.010\ \text{so}\ s_6=1$)\\ + &$F_6= $&$ $& &$S_5={\color{blue}0000.1100\ 0011\ 0101}$\\ + & & & &$SM_5={\color{blue}0000.1100\ 0011\ 0100}\phantom{M}$\\ + \cline{2-3} + &$sum $&$ 0001.1110\ 1001\ 0111 $&$\ll1 $&$K_5=0000.0000\ 0000\ 0001\ $\\ + &$carry $&$ 1110.0010\ 1100\ 0000 $&$\ll1 $&$C_5=1111.1111\ 1111\ 1111\,$\\ + \hdashline\\ + Terminate +\end{tabular} +\end{center} +} + +\end{document}