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2$|ҖyQsr40 2$f#E7JE 0AA 3@8?@w ʚ;ʚ;g4ddddF0&ppp@ <4dddd@0Pl3 <4KdKd(A0P80___PPT10 ?-May. 2009 vWu Jinyuan, (Fermilab jywu168@fnal.gov), Huang Yifei (IMSA)O  = 7An FPGA Computing Demo Core for Space Charge Simulation4Wu, Jinyuan (Fermilab) Huang, Yifei (IMSA) May, 2009> Outline*Electronic Aspect of FPGA: LED Flashing Logic Elements in a Nutshell TDC and ADC FPGA as a Computing Fabric: Moore s Law Forever? Space Charge Computing with FPGA Cores Doublet Matching & Hash Sorter Triplet Matching & Tiny Triplet Finder Enclosed Loop Micro-Sequencer (ELMS)T88The Space Charge ComputingThe FPGA Board|The 16-bit Demo Core~The Lookup Tablej!Two Electrons with Natural Scales|]"256 Charged Particles, Iteration 0##}^"256 Charged Particles, Iteration 5##~_#256 Charged Particles, Iteration 10$$`#256 Charged Particles, Iteration 15$$a#256 Charged Particles, Iteration 20$$b#256 Charged Particles, Iteration 25$$c#256 Charged Particles, Iteration 30$$d#256 Charged Particles, Iteration 35$$e#256 Charged Particles, Iteration 40$$}!Speed Comparison with Regular CPUOne Board: 8 FPGA Coresh$The Execution & Non-Execution CyclesOutline*Electronic Aspect of FPGA: LED Flashing Logic Elements in a Nutshell TDC and ADC FPGA as a Computing Fabric: Moore s Law Forever? 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___f3̙;/f9___PPT10i.p̥D+D=' b= @B + ?0N0 `%$ ^(  $ x $ c $Zb `p  b  $  6obfS" `P  b In space charge simulation computing task, in an low cost FPGA, a 16-bit demo core is developed: One FPGA = 5 Intel Core 2 Duo 2.2GHz CPU (0.5 W) vs. (5 x 75 W)Fa@l  3   3H $ 0޽h ? ___f3̙;/f9___PPT10i.p̥D+D=' b= @B +r4 # ư106(   wEquation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.05Equation Equation.30,Microsoft Equation 3.0J/ 0|DArialgsRomanPP3ؖF0ؖDGaramondRomanPP3ؖF0ؖ DTimes New RomanPP3ؖF0ؖ0DWingdingsRomanPP3ؖF0ؖ@DSymbolgsRomanPP3ؖF0ؖ@0.  @n?" dd@  @@`` <%4%X$ }PxPUa&%b>>>    ()  5 ()  21 5 ` a b[UK\ceuw5n#%$4J[ 01c)6}a  NEJD P xlD (*%2"NO 2*\ Fv^O=-$)U`(s`DUD P  1c w2\ *)N7w *}Dkz&8(BA$ Bqe[1AL4us\ w^?X?&s:<4## a wo8GA: nNhLDQ  )T[=A &$M# ,ehn;6 _B, >ZJ.{) 3W5m'5/o?EQQPXJ&/ab.Y 7B7 e!\3kV30B E}" Ua)/A8+&F:6#BA4J[ 1cOt RVT$$R$pf!@;A܋$$$$$$$$$$$$$2$[HWo|6܋2$IlH¤q\Q ;$$$$$$$$$$$$$$2$pMIIGbo$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$2$Oc^[$ɚAi{7;v<m2$pM$t# ]=2$wjGHh2_ >\2$a#Ýd>h%2$nHn0nЧpy>c2$aJ0 >B>2$ELxC(>2$  f}(8ns2$Nɕ=!$$2$4ٓpQa0g5W:2$U.3ͼ3U !e<$R$X`QVR8l3=2$Y&'޿! 2$|ҖyQsr40 2$f#E7JE 0AA f3@8?@w ʚ;ʚ;g4ddddsF0Vppp@ <4dddd@0Pl3 <4KdKd(A0P80___PPT10 ?-May. 2009 vWu Jinyuan, (Fermilab jywu168@fnal.gov), Huang Yifei (IMSA)O  =7An FPGA Computing Demo Core for Space Charge SimulationOWu, Jinyuan (Fermilab) Huang, Yifei (Illinois Math & Science Academy) May, 2009,-%About Illinois Math & Science AcademyWhat?The Space Charge ComputingThe FPGA Board|The 16-bit Demo Core~The Lookup Tablej!Two Electrons with Natural Scales|]"256 Charged Particles, Iteration 0##}^"256 Charged Particles, Iteration 5##~_#256 Charged Particles, Iteration 10$$`#256 Charged Particles, Iteration 15$$a#256 Charged Particles, Iteration 20$$b#256 Charged Particles, Iteration 25$$c#256 Charged Particles, Iteration 30$$d#256 Charged Particles, Iteration 35$$e#256 Charged Particles, Iteration 40$$}!Speed Comparison with Regular CPUOne Board: 8 FPGA Coresh$The Execution & Non-Execution CyclesThe EndThanks@0N0  (     0DsS"       0sAS"    H  0޽h ? 33___PPT10i.'n+D=' ~= @B + ?0N0 $ (   x  c $ظ p      60S"   <$ 0  One coauthor (Huang Yifei) is with Illinois Math & Science Academy (IMSA). IMSA enrolls grade 10-12 academically talented Illinois students. Nobel Laureate Dr. Leon Lederman is an IMSA Founder and Resident Scholar at IMSA. The work is done through the Student Inquiry and Research program. (The SIR consists 22 Wednesdays in 08-09 academic year. The work is done in Fermilab. Transportation to Fermilab is provided by IMSA).H  0޽h ? ___f3̙;/f9  ___PPT10z .p̥D+pCD ' ~= @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* K%(D' =-s6Bwipe(left)*<3<* KD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* K%(D' =-s6Bwipe(left)*<3<* KD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* +8+0+ 0 + ?0N0 `%$ (  $ x $ c $Zb `p  b  $  6obfS" `P <$ 0 b In space charge simulation computing task, in an low cost FPGA, a 16-bit demo core is developed: One FPGA = 5 Intel Core 2 Duo 2.2GHz CPU (0.5 W) vs. (5 x 75 W)Fa@l  3   3H $ 0޽h ? ___f3̙;/f9  ___PPT10~ .p̥D+pCD ' 7~= @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$ a%(D' =-s6Bwipe(left)*<3<*$ aD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$ a%(D' =-s6Bwipe(left)*<3<*$ aD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$ %(D' =-s6Bwipe(left)*<3<*$ +8+0+$ 0 +r &^%k(z:# H106(   wEquation Equation.30,Microsoft Equation 3.0%Ueڄ4CDP)R%R6V,E޳{O@9a=@V.2XؿcS +oѷGm y-',8K nG s ϫ>7}8++x1U*-7xʙMu̻`I^l:CS$덴Be qۉߧ{/sp>-̤ᣮjhxznV†$1䆉^};yH?ē;v(y[)Eqxak?;IC)tl'ɼ=7B2ri37R{[(K}|=j&lfu?e<GH5 iÖOm+iv^߅࠾A;'m(Bɹ@W_܍V(ohm#Tʮuܪ {6>UΆ`A e?,HPu1_*J~1))o4@LByni>u/x}՘ =^\ E}6S/>g/IPwKk&]^g6W 3іO].1RcP4}G,

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(   XB   0D P XB  0DP PP XB  0DP^  6f XB  0DP P XB  0D` P P @  <0f   DLUT 10b in 16b outXB  0D  XB 0 0Dp  pL P 0   #  P 2   66f   Fx2(K2   6;f   Fx2(K2   6?f  Fx2(KZB   s *D p ZB   s *D 0 ZB  s *D TB   c $D  TB   c $Df   6P 0 2   NEf"`p0 0  5+x  c $|H@      2AU ??c"$`?? 0 RB  s *D P f  0A^?"`p  2  6o `p,$D0   8A_ ??c"$`??P ~@8 $D 0H  0޽h ? 33  ___PPT10 .F u+iQzD ' = @B D| ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D1' =4@BBBB%()D' =,54 *3>Bfillcolor=@BPB<* D' =1:B solid*a3>Bfill.type<* D' =1:B true*]3>Bfill.on<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* + ?0N0   T (  x  c $TF-Q  - ^  6  B11AV?"`_ ,$D0l 0?P 0?P,$D 0TB  c $D0?0TB  c $DP?P  <\I-`G :256 nm0l      ,$D 0TB  c $D  TB  c $D    <O- l  828ps0  <2gA?"` ,$D 0l @`  ` ,$D 02  ZC~fd" 0P 7e0`B  0DԔ`P`` 2  ZQ~fd" Pp 7e0`B  0DԔ`@`PH  0޽h ? 33tl___PPT10L.F u+-_kD ' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D`' =4@BB#BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?HCB4DBCBB*k3>'Bstyle.rotation<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D1' =%(D' =%(DC' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#slide(fromRight)*<3<*D' =%(D?' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bslide(fromTop)*<3<*D' =%(D?' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bslide(fromTop)*<3<*+D> ?0 !:\ (   x  c $^[Q  [   0b[  `,$  0 One board has a calculation capacity as 40 dual core CPUs. The power consumption of one board is < 4.5 W. Newer FPGAs capable of hosting 4 cores/FPGA are available..({r  s <A\BoardPic"`@T   B4f[G`HT  @  ,$D 0 X One Core/FPGA = 5 Dual Core CPUs!0!  Bk[GlH= @  ,$D 0 X One Core/FPGA = 5 Dual Core CPUs!0!  6fpp ,$D 0"  6Ԕ 0 ,$D0l P@    ,$D 0l   <G2HP@f   6G2HP f   6G2H`Pf   6G2H3Pf   6G2HP}l  p  \ p  ,$D 0|N P@    `  f   6G2HP@f   6G2HP f   6G2H`Pf   6G2H3Pf   6G2HP}|N P@    @ f   6G2HP@f !  6G2HP f "  6G2H`Pf #  6G2H3Pf $  6G2HP}|N P@ %    ` f &  6G2HP@f '  6G2HP f (  6G2H`Pf )  6G2H3Pf *  6G2HP}|N P@ C   f D  6G2HP@f E  6G2HP f F  6G2H`Pf G  6G2H3Pf H  6G2HP}|N P@ I   f J  6G2HP@f K  6G2HP f L  6G2H`Pf M  6G2H3Pf N  6G2HP}|N P@ O   f P  6G2HP@f Q  6G2HP f R  6G2H`Pf S  6G2H3Pf T  6G2HP}|N P@ U   pf V  6G2HP@f W  6G2HP f X  6G2H`Pf Y  6G2H3Pf Z  6G2HP} [ B@GuHX  ,$@ 0 w!8 Cores/Board = 40 Dual Core CPUs:"0H  0޽h ??0  [  33_!W!___PPT107!.F u+f?,D+ ' = @B D' = @BA?%,( < +O%,( < +D5' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*[ %(D' =-u6Bwipe(right)*<3<*[ D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\ %(D' =-s6Bwipe(down)*<3<*\ D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ;%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* ;D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* ;D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ;j%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* ;jD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* ;jD#' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* j%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* jD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* j++0+ 0 ++0+ 0 ++0+ 0 ++0+[ 0 +<, ?0 (  r  S h[Q  [ n  0[ ,$ 0 fIn current micro-processors: Each instruction takes one clock cycle to execute. It takes many clock cycles to prepare for executing an instruction. Pipelined? Yes. But the non-execution pipeline stages consume silicon area, power etc. To execute an instruction != to do useful calculation. Can we do something different? 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`    8AY ??c"$`??@8 $D 0   8AZ ??c"$`?? 8 $D042  # ZBqCENGkH[ILQ1 q(xq(xL`T(xL`T ,$D0"  6jJp0`,$@0"  6jJ p,$D042  # ZBqCENGkH[ILQ1 q(xq(xL`T(xL`T ,$D042  # ZBqCENGkH[ILQ1 q(xq(xL`T(xL`T ,$D0"  6jJ`P ,$@0"  6jJp,$D0"  6jJ`` ,$D 0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T`,$@ 0.2   ZCWENG.Hv `T'W`T'W&T'W&TP @`,$@  0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T0 ,$@ 0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T 0P ,$@0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T 0,$@0H  0޽h ?         33..___PPT10-.F u+)D-' = @B D~-' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D ' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D' =%(D3' =4@BBBB%(D'  =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* Dk' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =4@BB1BB%()?)@D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+4 8?HCBCBCBB*k3>'Bstyle.rotation<* D' =-g6B fade*<3<* DR' =%(D' =%(Di' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =4@BB1BB%()?)@D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+4 8?HCBCBCBB*k3>'Bstyle.rotation<* D' =-g6B fade*<3<* ++ ?0 N0 **%], P)(  ,  , <ަ pPp Bxi$IXB , 0Dp^ , 6p@p^ , 6p@2 , T f"`@P 5-XB , 0Dp0pXB  , 0DPpPXB  , 0DXB  , 0DpXB  , 0DppXB  , 0D @ XB , 0D @ XB , 0D @ RB , s *D RB , s *D  RB , s *D ^ , 6f  XB , 0D  XB , 0D @@ 2 , <i f@ ` 5X2 , <Lr f  5X2 , <% f@ `  5X , <4 f 0  DLUT 10b in 16b outXB , 0D   XB , 0D@ XB , 0D XB , 0D@ XB , 0D  XB , 0D @ XB  , 0D RB !, s *D@ @  RB ", s *D @  XB #, 0D XB $, 0D XB %, 0Dp pXB &, 0DP PXB ', 0D XB (, 0D RB ), s *D RB *, s *D RB +, s *D RB ,, s *D RB -, s *D ^ ., 6f` XB /, 0D XB 0, 0D  RB 1, s *D  5, <P Ppp Byi$I^ 6, 6@pp^ 7, 6@p 8, <(> pPp  Bzi$I^ 9, 6p@p ^ :, 6p@  ;, Z "`   D16-bit Coordinates <, Z "` V  ? 32-bit Forces^ =, 6f@p XB >, 0D09 ^ ?, 6fp XB @, 0D 09 !^ A, 6f@p XB B, 0D09 XB C, 0DPpXB D, 0D Pp XB E, 0DPpXB U, 0D  ^ \, 6f0 0 XB ], 0D @@ 2 ^, <U f  @  GS*CKXB _, 0D  XB d, 0DP P^ k, 6f XB l, 0D`@@@2 m, < f  GS*CKXB n, 0DPPXB s, 0D ^ z, 6fp pXB {, 0D@@2 |, <$ f`  GS*CKXB }, 0D2 , <~ f @`  Fx2(K2 , <Dm f @`0  Fx2(K2 , <(w f @`  Fx2(KXB , 0D ` XB , 0D  @ XB , 0D   RB , s *D `  RB , s *D `  d , <` @ 2 , T f"`@   5+2 , T f"`P 5-2 , T f"`@P  5-x , c $J P@   H , 0޽h ? 33___PPT10i.F u+D='  = @B +g ?0 ^ZVZp%( Y(  ( X ( 0A?px ( c $  `p   U  > ( # #"ZN@ ,$D 0 (  H ?"` K255M (  H ?"` I0M (  H, ?"`B  K253M (  HH$ ?"`B  K255M  (  H0 ?"` B  @M  (  H44 ?"` B  @M  (  H= ?"` F  I0M  (  HPL ?"` F  I1M  (  HlU ?"`0   K255M (  H^ ?"` 0 @M (  Hg ?"`  I4M (  Hp ?"`   I3M (  Hhs ?"``   I2M (  H ?"` ` I1M (  HH ?"`  I0M (  H4 ?"`  K255M (  HP ?"`0J   @M (  Hl ?"`J0  @M (  Hx ?"`J  @M (  H0 ?"` J  @M (  H ?"``J   @M (  HD ?"`J`  @M (  H ?"`J  @M (  Hh ?"`J  @M (  H ?"`0 J K255M (  H ?"`0J @M (  H ?"`J I4M  (  H ?"` J I3M !(  H ?"`` J I2M "(  H ?"``J I1M #(  H ?"`J I0M $(  H| ?"`J I4M %(  H* ?"`0  K255M &(  H 3 ?"`0 @M '(  H* ?"` I4M ((  H; ?"`  I3M )(  HN ?"``  I2M *(  HW ?"`` I1M +(  Ha ?"` I0M ,(  Hi ?"` I3M -(  Hr ?"`0   K255M .(  H{ ?"` 0 @M /(  H ?"`  I4M 0(  H ?"`   I3M 1(  H ?"``   I2M 2(  HH ?"` ` I1M 3(  HT ?"`  I0M 4(  H ?"`  I2M 5(  H ?"`0N   K255M 6(  H ?"`N0  @M 7(  H ?"`N  I4M 8(  H ?"` N  I3M 9(  H` ?"``N   I2M :(  H< ?"`N`  I1M ;(  H| ?"`N  I0M <(  H< ?"`N  I1M =(  H  ?> I0M >(  Hp ?> I0M ?(  H ? K254M @(  H) ? I0M A(  HD, ?   I2M B(  H; ?   I4M C(  H= ?   I1M D(  HL ?   I3M E(  HTO ?F   I0M F(  H@^ ?F   I2M G(  H\g ? > 7A double-layer loop is followed by a single-layer loop.88fM H(  H(` ?  I0M I(  H|y ?  I0M J(  H ?0 N K255M K(  H ?0N @M L(  H, ?N I4M M(  H ? N I3M N(  HP ?` N I2M O(  H ?`N I1M P(  Ht ?N I0M Q(  H, ?N I0M R(  H ?  JAAM S(  HP ? JBAMlB T(  <o ? fB U(  61 ? fB V(  61 ?N NfB W(  61 ?  fB X(  61 ?  fB Y(  61 ?  fB Z(  61 ?  fB [(  61 ?lB \(  <o ?> >fB ](  61 ? ZB ^(  s *1 ?   ZB _(  s *1 ? ZB `(  s *1 ? ZB a(  s *1 ?J J`B b(  01 ?   `B c(  01 ? ZB d(  s *1 ?F F lB e(  <o ? lB f(  <o ? lB g(  <o ?>lB h(  <1 ? fB i(  61 ?>lB j(  <1 ? fB k(  61 ?>lB l(  <1 ?` `lB m(  <1 ?   lB n(  <1 ? lB o(  <1 ? lB p(  <1 ?0 0lB q(  <o ?   lB r(  <o ?   lB s(  <o ?  >ZB t(  s *1 ?B B ZB u(  s *1 ?ZB v(  s *1 ?fB w(  61 ? fB x(  61 ?`` fB y(  61 ?   fB z(  61 ? fB {(  61 ? fB |(  61 ?00 " }( H o"` 0,$D0 @ Inner Loop 0 " ~( H o"`P0 ,$D0 @ Outer Loop 0 " ( N o"`` ,$D0 C State Control0H ( 0޽h ? 33&  ___PPT10 .F u+(?D '  = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-o6Bwipe(up)*<3<*( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-o6Bwheel(1)*<3<*( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*}( %(D' =-o6Bwheel(1)*<3<*}( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*~( %(D' =-o6Bwheel(1)*<3<*~( +r$j0X  v# FN16(   Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.05Equation Equation.30,Microsoft Equation 3.0J/ 0|DArialgsRomanPP3ؖF0ؖDGaramondRomanPP3ؖF0ؖ DTimes New RomanPP3ؖF0ؖ0DWingdingsRomanPP3ؖF0ؖ@DSymbolgsRomanPP3ؖF0ؖ@0.  @n?" dd@  @@`` L%D%h, }PxPUa&%b>>>    ()  5 ()  21 5 ` a b[UK\ceuw5n#%$4J[ 01c)6}a  NEJD P xlD (*%2"NO 2*\ Fv^O=-$)U`(s`DUD P  1c w2\ *)N7w *}Dkz&8(BA$ Bqe[1AL4us\ w^?X?&s:<4## a wo8GA: nNhLDQ  )T[=A &$M# ,ehn;6 _B, >ZJ.{) 3W5m'5/o?EQQPXJ&/ab.Y 7B7 e!\3kV31]B E}" Ua)/A8+&F:6#BA4J[ 1cOt RVWXT2$xF膷>L0M&R 2$3w%;t_Mt zG R$pf!@;A܋$$$$$$$$$$$$$2$[HWo|6܋2$IlH¤q\Q ;$$$$$$$$$$$$$$2$pMIIGbo$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$2$Oc^[$ɚAi{7;v<m2$pM$t# ]=2$wjGHh2_ >\2$a#Ýd>h%2$nHn0nЧpy>c2$aJ0 >B>2$ELxC(>2$  f}(8ns2$Nɕ=!$$2$4ٓpQa0g5W:2$U.3ͼ3U !e<$R$X`QVR8l3=2$Y&'޿! 2$|ҖyQsr40 2$f#E7JE 0AA ̙f@8?@w ʚ;ʚ;g4ddddPuF0&ppp@ <4dddd@0Pl3 <4KdKd(A0P80___PPT10 ?-May. 2009 vWu Jinyuan, (Fermilab jywu168@fnal.gov), Huang Yifei (IMSA)O  =7An FPGA Computing Demo Core for Space Charge SimulationOWu, Jinyuan (Fermilab) Huang, Yifei (Illinois Math & Science Academy) May, 2009,-%About Illinois Math & Science AcademyWhat?The Space Charge ComputingThe FPGA Board|The 16-bit Demo Core~The Lookup TableNumber of Bits for Input to LUT'A Double-Layer + Single-Layer Sequencerj!Two Electrons with Natural Scales|]"256 Charged Particles, Iteration 0##}^"256 Charged Particles, Iteration 5##~_#256 Charged Particles, Iteration 10$$`#256 Charged Particles, Iteration 15$$a#256 Charged Particles, Iteration 20$$b#256 Charged Particles, Iteration 25$$c#256 Charged Particles, Iteration 30$$d#256 Charged Particles, Iteration 35$$e#256 Charged Particles, Iteration 40$$}!Speed Comparison with Regular CPUOne Board: 8 FPGA Coresh$The Execution & Non-Execution CyclesThe EndThanksr>>    ()  5 ()  21 5 ` a b[UK\ceuw5n#%$4J[ 01c)6}a  NEJD P xlD (*%2"NO 2*\ Fv^O=-$)U`(s`DUD P  1c w2\ *)N7w *}Dkz&8(BA$ Bqe[1AL4us\ w^?X?&s:<4## a wo8GA: nNhLDQ  )T[=A &$M# ,ehn;6 _B, >ZJ.{) 3W5m'5/o?EQQPXJ&/ab.Y 7B7 e!\3kV31]B E}" Ua)/A8+&F:6#BA4J[ 1cOt RVWXYT2$xF膷>L0M&R 2$3w%;t_Mt zG R$pf!@;A܋$$$$$$$$$$$$$2$[HWo|6܋2$IlH¤q\Q ;$$$$$$$$$$$$$$2$pMIIGbo$$$                         ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z | } ~  $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$2$Oc^[$ɚAi{7;v<m2$pM$t# ]=2$wjGHh2_ >\2$a#Ýd>h%2$nHn0nЧpy>c2$aJ0 >B>2$ELxC(>2$  f}(8ns2$Nɕ=!$$2$4ٓpQa0g5W:2$U.3ͼ3U !e<$R$X`QVR8l3=2$Y&'޿! 2$|ҖyQsr40 2$f#E7JE 0AA ̙f@8?@w ʚ;ʚ;g4ddddPuF0&ppp@ <4dddd@0Pl3 <4KdKd(A0P80___PPT10 ?-May. 2009 vWu Jinyuan, (Fermilab jywu168@fnal.gov), Huang Yifei (IMSA)O  =7An FPGA Computing Demo Core for Space Charge SimulationOWu, Jinyuan (Fermilab) Huang, Yifei (Illinois Math & Science Academy) May, 2009,-%About Illinois Math & Science AcademyWhat?The Space Charge ComputingThe FPGA Board|The 16-bit Demo Core~The Lookup TableNumber of Bits for Input to LUTa 'A Double-Layer + Single-Layer Sequencerj!Two Electrons with Natural Scales|]"256 Charged Particles, Iteration 0##}^"256 Charged Particles, Iteration 5##~_#256 Charged Particles, Iteration 10$$`#256 Charged Particles, Iteration 15$$a#256 Charged Particles, Iteration 20$$b#256 Charged Particles, Iteration 25$$c#256 Charged Particles, Iteration 30$$d#256 Charged Particles, Iteration 35$$e#256 Charged Particles, Iteration 40$$}!Speed Comparison with Regular CPUOne Board: 8 FPGA Coresh$The Execution & Non-Execution CyclesThe EndThanksx ?0 N0 [[@! Z(   JF P0      6W0 Bxi$IZB   s *D`   00`   002   Nt]f"` 5-ZB   s *D00ZB   s *DZB   s *DpZB   s *D  ZB   s *D@0 0 ZB   s *DP  ZB   s *D ` ` ZB   s *D  TB  c $DP`P TB   c $D  ` TB   c $D`  `   0fP  ZB   s *D ` P ` ZB  s *D p P 2   6cf@`  5X2   6gf@` 5X2   6kf@ `  5X   6ofp P  DLUT 10b in 16b outZB   s *D p ZB   s *D @ZB   s *DP `@`ZB   s *D @@@ZB   s *DP @ZB   s *D @ ZB   s *DP ` @` TB ! c $D  TB "  c $DP ZB #  s *DPZB $  s *DPZB %  s *D0 P0 ZB &  s *D`PZB '  s *D P ZB (  s *Dp Pp TB )  c $D`  TB *  c $D p TB +  c $D TB ,  c $D`TB - c $D` .  0fp ` ZB / s *D` @ ZB 0  s *D p TB 1  c $D`  ZB 2  s *DZB 3  s *DZB 4  s *D`   5  6({0  Byi$I` 6  00 ` 7  00  8  60   Bzi$I` 9  00  ` :  00   ;  T<"`  D16-bit Coordinates <  T"`pP  ? 32-bit Forces` =  0fP ZB > s *D` a` ?  0fP @ZB @ s *D ` A  0f P ZB B s *D` a ZB C  s *D``ZB D  s *DZB E  s *D` ` TB F  c $D@@TB G  c $DpTB H  c $D@@@ I  64 Bxj$I` J  0` K  0 L  6X Byj$I` M  0` N  0 O  6 Bzj$I` P  0` Q  0T    R # P   S  6p   Cvzj$I2 T  6fP   5+ZB U  s *Dp Pp ` V  0`  ` W  00 ` TB X c $D p TB Y B c $D  ZB Z  s *D  ZB [  s *Dp pp ` \  0f @ ZB ]  s *D ` 2 ^  6쪯f   GS*CKZB _  s *D@p p TB `  c $Dp p T    a # P`  b  6xp   Cvyj$I2 c  6$fP   5+ZB d  s *Dp Pp ` e  0`  ` f  00 ` TB g c $D p TB h B c $D  ZB i  s *D  ZB j  s *Dp pp ` k  0f @ ZB l  s *D ` 2 m  6\f   GS*CKZB n  s *D@p p TB o  c $Dp p T    p # P0 q  6p   Cvxj$I2 r  6ůfP   5+ZB s  s *Dp Pp ` t  0`  ` u  00 ` TB v c $D p TB w B c $D  ZB x  s *D  ZB y  s *Dp pp ` z  0f @ ZB {  s *D ` 2 |  6˯f   GS*CKZB }  s *D@p p TB ~  c $Dp p T P 0   #  2   6Яf   Fx2(K2   6կf   Fx2(K2   6lگf  Fx2(KZB   s *D p ZB   s *D 0 ZB  s *D TB   c $D  TB   c $Df   6P 0 2   Nf"`p0 0  5+2   Nf"`@ 5-2   Nf"`   5-2   Nxf"`   5+ZB  B s *D@ @@ 2   Nf"`P   5+2   Nf"`@   5+   T("`PP Y C16-bit VelocitiesTB   c $D`  TB   c $D TB   c $D`TB   c $DTB   c $D`@@TB   c $D@0TB   c $D@@@@ TB   c $Dpp` TB   c $D@P ZB  B s *D` p` ZB  B s *DP P TB  B c $D`TB  c $D`@` TB  c $D TB  c $D@ ZB   s *D@@ZB   s *DZB   s *D`@@TB  B c $DTB  B c $D00ZB  s *D` `ZB  s *D ZB  s *D 0x  c $x@      8AU ??c"$`?? `    8AY ??c"$`??@8 $D 0   8AZ ??c"$`?? 8 $D042  # ZBqCENGkH[ILQ1 q(xq(xL`T(xL`T ,$D0"  6jJp0`,$@0"  6jJ p,$D042  # ZBqCENGkH[ILQ1 q(xq(xL`T(xL`T ,$D042  # ZBqCENGkH[ILQ1 q(xq(xL`T(xL`T ,$D0"  6jJ`P ,$@0"  6jJp,$D0"  6jJ`` ,$D 0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T`,$@ 0.2   ZCWENG.Hv `T'W`T'W&T'W&TP @`,$@  0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T0 ,$@ 0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T 0P ,$@0:2  3 ZBqCENGkH[ILQԔ q(xq(xL`T(xL`T 0,$@0H  0޽h ?         33..___PPT10-.F u+)D-' = @B D~-' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D ' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* Dk' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =4@BB1BB%()?)@D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+4 8?HCBCBCBB*k3>'Bstyle.rotation<* D' =-g6B fade*<3<* DR' =%(D' =%(Di' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D' =4@BB1BB%()?)@D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+4 8?HCBCBCBB*k3>'Bstyle.rotation<* D' =-g6B fade*<3<* + ?0 N0 o g `!  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I0M >(  H|? I0M ?(  H|?q 0 K254M @(  H |?q 0 I0M A(  H(|?4   I2M B(  H0|?4   I4M C(  H8|?u 4  I1M D(  H@|?u 4  I3M E(  HH|? u  I0M F(  HP|? u  I2M G(  HX|?8  7A double-layer loop is followed by a single-layer loop.88fM H(  H|?8 I0M I(  H|?8 I0M J(  H|?0  K255M K(  H|?@0 @M L(  H}?@ I4M M(  H|?  I3M N(  H@}?  I2M O(  H}? I1M P(  Hh(}? I0M Q(  H}? I0M R(  H:}?@  JAAM S(  HLC}?@ JBAMlB T(  <o ?@ @fB U(  61 ? fB V(  61 ? fB W(  61 ?fB X(  61 ?u u fB Y(  61 ?4 4 fB Z(  61 ?  fB [(  61 ?00lB \(  <o ? fB ](  61 ?@yZB ^(  s *1 ?} }ZB _(  s *1 ?< <ZB `(  s *1 ? ZB a(  s *1 ? `B b(  01 ?y y`B c(  01 ?8 8ZB d(  s *1 ?  lB e(  <o ?y8lB f(  <o ?@ylB g(  <o ?8lB h(  <1 ?y8fB i(  61 ?8lB j(  <1 ?y8fB k(  61 ?8lB l(  <1 ?y8lB m(  <1 ? y 8lB n(  <1 ?y8lB o(  <  1 ?@y@8lB p(  <1 ?0y08lB q(  <o ? y 8lB r(  <o ? @ ylB s(  <o ? 8 ZB t(  s *1 ?  ZB u(  s *1 ?q q ZB v(  s *1 ?fB w(  61 ?yfB x(  61 ?yfB y(  61 ?  yfB z(  61 ?yfB {(  61 ?@@yfB |(  61 ?00y" }( HP}o"` P,$D0 @ Inner Loop 0 " ~( NU}o"`P0,$D0 @ Outer Loop 0 " ( N+B#style.visibility<*( %(D' =-o6Bwipe(up)*<3<*( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*}( %(D' =-o6Bwheel(1)*<3<*}( D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-s6Bwipe(left)*<3<*( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*~( %(D' =-o6Bwheel(1)*<3<*~( D' =%(D?' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-{6Bslide(fromTop)*<3<*( D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-o6Bwipe(up)*<3<*( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-o6Bwheel(1)*<3<*( D' =%(D?' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-{6Bslide(fromTop)*<3<*( D?' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-{6Bslide(fromTop)*<3<*( +5 ?0 N0 `! y(   XB   0D P XB  0DP PP XB  0DP^  6f XB  0DP P XB  0D` P P @  <0f   DLUT 10b in 16b outXB  0D  XB 0 0Dp  pL P 0   #  P 2   60f   Fx2(K2   60f   Fx2(K2   6آ0f  Fx2(KZB   s *D p ZB   s *D 0 ZB  s *D TB   c $D  TB   c $Df   6P 0 2   Nh0f"`p0 0  5+x  c $d0@  0    2AU ??c"$`?? 0 RB  s *D P f  0A^?"`ppg 2  6o `p,$D0   8A_ ??c"$`??P @8 $D 0~"  Z06 Ԕ"`` ,$D 0 NThe LUT replaces: A Square Rooting Two Multiplications A Reciprocal OperationsH2 2 H  0޽h ? 33___PPT10.F u+?9DD'  = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D1' =4@BBBB%()D' =,54*3>Bfillcolor=@BPB<* D' =1:B solid*a3>Bfill.type<* D' =1:B true*]3>Bfill.on<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwheel(1)*<3<* D{' =%(D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* +8+0+ 0 +Z ?0N0 **%Z, P *(  ,  , <0Zzr Bxi$IXB , 0DRzR^ , 6Jzr^ , 6Jr2 , T40f"`Z 5-XB , 0D:XB  , 0DzXB  , 0Dr rXB  , 0DRzRXB  , 0DXB  , 0D J XB , 0D" J" XB , 0D J RB , s *D" RB , s *D" RB , s *D" ^ , 6f  XB , 0D" " XB , 0D2 JJ 2 , <0f  5X2 , <X0fB b 5X2 , <00f  5X , <0fr   DLUT 10b in 16b outXB , 0D   XB , 0DJ XB , 0D" "XB , 0DJ XB , 0D XB , 0DJ XB  , 0D" "RB !, s *DJ J RB ", s *D J XB #, 0D XB $, 0Dr* rXB %, 0D XB &, 0D XB ', 0DR R XB (, 0D2 * 2 RB ), s *DR * RRB *, s *DR* * 2 RB +, s *D B RB ,, s *DR * RRB -, s *Dr* * R^ ., 6f  XB /, 0DB  B XB 0, 0DB  B RB 1, s *DB B  5, <0rZz Byi$I^ 6, 6rJz^ 7, 6rJ 8, <0Zzr  Bzi$I^ 9, 6Jzr ^ :, 6Jr  ;, Zx0"` $  D16-bit Coordinates <, Z0"`0 @d  ? 32-bit Forces^ =, 6fz XB >, 0D":C#^ ?, 6fBz XB @, 0D:C^ A, 6fz XB B, 0D":C#XB C, 0D"Zz"XB D, 0DZzXB E, 0D"Zz"^ \, 6f XB ], 0DB J J " XB _, 0D2 2 ^ k, 6fR RXB l, 0DJ J XB n, 0D ^ z, 6f XB {, 0DJ J bXB }, 0Dr r2 , <Dyf Jj2  Fx2(K2 , <yf Jj  Fx2(K2 , <` yf Jj2  Fx2(KXB , 0D" j" XB , 0D ** XB , 0D ** RB , s *D j* RB , s *D j* d , <Jb 2 , Tyf"`   5+2 , TPyf"`BZ 5-2 , Tyf"`Z 5-x , c $yQ  y  , T0"`b :0 G32-bit Sum of SquaresG" , ZyԔ"` PEr,$D 0 {-A 32-bit input LUT is too big. 232=4G words.2.!K 1 , B%y?"@   cShifters are used before and after the LUT. Leading zeros are eliminated: 00000001010110 01010110002JJ̙̙H , 0޽h ? 33w/o/___PPT10O/.F u+D.' = @B Dn.' = @BA?%,( < +O%,( < +D' =%(Dy' =%(D!' =4@BBBB%(D' =,54 f*3>Bfillcolor=@BPB<*;, D' =1:B solid*a3>Bfill.type<*;, D' =1:B true*]3>Bfill.on<*;, D' =%(Dy' =%(D!' =4@BBBB%(D' =,54 f*3>Bfillcolor=@BPB<*, D' =1:B solid*a3>Bfill.type<*, D' =1:B true*]3>Bfill.on<*, D{' =%(D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*, %(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<*, D' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<*, Dp!' =%(D' =%(D' =4@BBBB%()D' =,54 U*3>!Bstyle.color='`B@BPB<*, D' =,54 U*3>Bfillcolor=@BPB<*, D' =,54 U*3>#Bstroke.color=@BPB<*, D' =1:B solid*a3>Bfill.type<*, D' =4@BBBB%()D' =,54 U*3>!Bstyle.color='`B@BPB<*\, D' =,54 U*3>Bfillcolor=@BPB<*\, D' =,54 U*3>#Bstroke.color=@BPB<*\, D' =1:B solid*a3>Bfill.type<*\, D' =4@BBBB%()D' =,54 U*3>!Bstyle.color='`B@BPB<*k, D' =,54 U*3>Bfillcolor=@BPB<*k, D' =,54 U*3>#Bstroke.color=@BPB<*k, D' =1:B solid*a3>Bfill.type<*k, D' =4@BBBB%()D' =,54 U*3>!Bstyle.color='`B@BPB<*z, D' =,54 U*3>Bfillcolor=@BPB<*z, D' =,54 U*3>#Bstroke.color=@BPB<*z, D' =1:B solid*a3>Bfill.type<*z, D' =%(Dh' =A@BBBB0B@@%()))D' =1:B#000066*z3>!Bstyle.color= `B<*, ,D' =1:B#000066*a3>Bfillcolor<*, ,D' =1:B solid*a3>Bfill.type<*, ,D ' =%(Dh' =A@BBBB0B@@%()))D' =1:B#000066*z3>!Bstyle.color= `B<*, ,JD' =1:B#000066*a3>Bfillcolor<*, ,JD' =1:B solid*a3>Bfill.type<*, ,JDh' =A@BBBB0B@@%()))D' =1:B#000066*z3>!Bstyle.color= `B<*, JYD' =1:B#000066*a3>Bfillcolor<*, JYD' =1:B solid*a3>Bfill.type<*, JYDh' =A@BBBB0B@@%()))D' =1:B#000066*z3>!Bstyle.color= `B<*, YdD' =1:B#000066*a3>Bfillcolor<*, YdD' =1:B solid*a3>Bfill.type<*, Yd+p+0+, 0 ++0+, 0 +< ?0N0 **%`a8 *(  8 l  pV `8 p V,$D 0` *8  0f @` +8  0f @ ` ,8  0f  ` -8  0f   ` .8  0   ` /8  0 @ ` 08  0f0 ` 18  0f0  ` 28  0f0  ` 38  0fP ` 48  0fP  ` 58  0fP ` 68  0fPTB 78 B c $D ` 88  0f p` 98  0f p ` :8  0fp ` ;8  0fp` <8  0f 0` =8  0f 0 ` >8  0f0 ` ?8  0f0` @8  0f  ` A8  0f   ` B8  0f  ` C8  0f ZB D8  s *Dp  ZB E8  s *D  ZB F8  s *D`   G8  TM"` j  4x1 H8  TQ"``j  4x2 I8  TU"`p@ 9(x1-x2) J8  TZ"`0-j ; (x1-x2)^2  K8  T]"``pV CSum of 3 Squares` W8  0P` X8  0p` Y8  00sl  a8 ,$D 0` 8  0ff @  ` 8  0ff  ` 8  0ff ` ` 8  0ff `  ` 8  0ff  ` 8  0ff  ` 8  0ff  `  8  0ff @TB  8  c $D p ZB  8  s *DԔ p ZB  8  s *DfԔ p `  8  0ffp @` 8  0ffp @ ` 8  0ffp @ ZB 8  s *DԔ @ TB 8  c $D  ZB 8  s *D̙Ԕ @ @ ZB 8  s *DԔ @ ` 8  0ff P` ` 8  0ff` P TB 8  c $DZB 8  s *D̙Ԕ  ZB 8  s *DԔZB 8  s *DԔ@  ` 8  0ff @` 8  0ff@ ` 8  0ff @ ` 8  0ff@ ` 8  0ff @` 8  0ff@ `  8  0ff @` !8  0ff@ ` "8  0ffp @ ` #8  0ff @` $8  0ff ` %8  0ff `` &8  0ff `ZB '8  s *D̙Ԕ p ZB (8  s *DԔ p ZB )8  s *DfԔ p ` L8  0` ` M8  0`  TB N8 B c $DP` O8  0`  ` P8  0`  ZB Q8  s *D   R8  Tc"` 5LUTTB S8 c $DP ` TB T8 c $DP ` ` V8  0`  ` Z8  0ff P@` [8  0ff@Pr \8 S Pm `p    ]8 0A^?"`3$  ,$D 0 ^8  8A_ ??c"$`??t"8 $D 07 _8 TTt"`  ,$D 0 q?If ((High Bits) != 0) Choose (High Bits) Else Choose (Low Bits)@@H 8 0޽h ? 33'___PPT10.F u+XdD' d = @B DV' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`8 %(D' =-s6Bwipe(left)*<3<*`8 D~ ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a8 %(D' =-s6Bwipe(left)*<3<*a8 D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*_8 %(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<*_8 D' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<*_8 D' =%(Du' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*]8 %(D' =-6B%slide(fromBottom)*<3<*]8 D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*^8 %(D' =-s6Bwipe(left)*<3<*^8 +8+0+_8 0 +0 ?0N0 A 9 %Na< (  < X < 0ff@ ` X < 0ff ` X < 0ff `` X < 0ff` ` X < 0ff ` X < 0ff ` X < 0ff ` X  < 0ff `@X  < 0ff @` X  < 0ff` @ LB  < c $D`@LB  < c $D `@ LB < c $D`@LB < c $D@ `@ X < 0ff@`X < 0ff`X < 0ff``X < 0ff``X < 0f  X < 0f  X < 0f   < Tl=y"`@  9(x1-x2)X < 0  X < 0  LB < @ c $DX < 0  X  < 0  RB !< s *D 0 P  "< TXCy"`0 ( j 5LUTLB #< c $D  LB $< c $D  RB &< s *D @ ` X 4< 0  LB >< c $DPLB ?< c $D PLB @< c $D PX A< 0ff  @X B< 0ff@   X C< 0ff  @ LB D< c $D @ LB E< c $D LB F< c $D@ @ LB G< c $D@ X H< 0ff P@X I< 0ff@ P X J< 0ff P@ X K< 0ff@P X L< 0ffP@X M< 0ff@PX N< 0ffP@X O< 0ff@PX P< 0ff@ LB Q< c $D P LB R< c $DP LB S< c $DP X T< 0ff @@X U< 0ff@@r V< S _y `p  y l   `<  ,$D 0` <  0 P ` <  0 P ` <  0 P ` <  0 P ` (<  0 ` )<  0  ` *<  0 ` +<  0` <<  0f P ` =<  0f@` W<  0` ` Z<  0`` [<  0f` ZB \<  s *DP @ZB ]<  s *DP ` ) ^< T"`c I,$D 0 cShift 2n before LUT4 ( _< T"`c I,$D 0 bShift 3n after LUT4  a<  8A_ ??c"$`??p~`8 $D 0H < 0޽h ? 33___PPT10.F u+nDZ'  = @B D' = @BA?%                         ! 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