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HH5}HJ6HHlGG HUV 5~I6 HUV AAJUUh  f1e HUV 5JH6HUV lII qHH_tKD HH  GxRL ` Sum-Roots Procedure &UT UT`   ;UR UT` (define (sum-roots from to) 5xPUP UT` / ! (define (sum-iter sum from to) eUN UT` -  (if (> from to) zUL UT` 2  sum UJ UT` 0 & (sum-iter (+ sum (sqrt from)) UH UT` 3  (+ 1 from) UF UT` 4  to))) UD UT`  (sum-iter 0 from to)) UB UT`  HH_vLD5}HHlTKK HUV 5M5 5~HUV NUUe !  HUV 5NPF5HUV lMM H$ 5O5 H$ PUUe "  H$ 5PN5urH$ lOO m-rHZmjQfe =UgUNUTddRTTTUJUTHHdSR troHHj from) T 4 ` S ))Register Machine for Sum-Roots ite) ` # UB T) {UN UT` (controller VUL UT`  sum-roots UJ UT` V  (assign sum (const 0)) !UH UT` sum-iter UF UT` ' (test (op >) (reg from) (reg to)) UD UT` W  (branch (label done)) UB UT` X ( (assign temp (op sqrt) (reg from)) U@ UT` Y . (assign sum (op +) (reg sum) (reg temp)) #U> UT` Z N/ (assign from (op +) (const 1) (reg from)) 8U< UT` [ O (goto (label sum-iter)) MU: UT` f done)) d ` K UN THHdTRTHHlLSS l3?jUVQVWl3?fro|;ZQ`32jVUWQUW S))|;ZQ`32G|G sum l3?jWVXQZnmal3?UV-?jXWYQYZ e) mB#5?K (l-?ZZ_Gl32 &jYXZQXZ B@4 &32LBL + -?jZY[QW]q??XY j[Z\Q\] o (]! UT j\[]Q[] (si]- (r fj]\^QZ^gem[\ V`32j^]_Q]_ efr V`32  sum<-adder`I32j_^`Q^b `I32`KC`KC  from<-inc?j`_aQab?Z_@32ja`bQ`b VZ_@32$ХХ to?jbacQ_g)?`a $-jcbdQdh  $-ma - 32jdceQch #5 32 1 jedfQfg XZ &32!! jfegQeg ! XY!**jgfhQbhef $-jhgiQgk(si $-cdl{?jihjQjkl{?^zZQ"32jjikQik zZQ"32z from32l{?jkjlQhn32l{?ij~$$jlkmQmnC~$$ &32jmlnQln  &32 >~$$jnmoQko~$$lm?jonpQnp? ??i$jpoqQoqai$-I"I"ii$)-jqprQpt$)-QQ$$3Q?jrqsQst Q?fgQ~cklZQ &32jsrtQrt klZQ &32pץklץ +Q?jtsuQquigQ?rs-6jutvQtvj-6{-cc~-jvuwQuw32~-fr~~u-ZjwvxQvxu-Z$uu jxwyQwyn   cjyxzQxzocH?jzy{Qy{ ]#6?HZum32j{z|Qz~ m32~zz sqrt)Q?j|{}Q}~)Q?6 XZ_#32j}|~Q|~ 6 XZ_#32Hd6 d temp)Q?j~}Q{)Q?|}deJ2HHeK HHQjQQ  h $ *Sum-Roots Data Flow and Control Flow:g HHeMHHlT ZhZj~Q~hZxyhZZc?jQc?yzc)cZ jQ Z z{ZZcc jQ c jcllZjQZHS jQ HS PÕPÕ  temp<-sqrt-$-jQ -$-~-Z6C7 32jQ 7 32DD 0-$-jQ -$-6D6jQQ6D6 6DlDdeHHe HH~ ` % %Register Machine Controller Language ) ` &  (reg )) QUR UT` ( -(assign (const )) fUP UT` ) 9(assign (op ) ...) {UN UT` *  UL UT` +  UJ UT` , /(perform (op ) ...) QUH UT` 1  UF UT` 5 Õ,(test (op ) ...) UD UT` 6 (branch (label )) UB UT` 7 ˴(goto (label )) U@ UT` 8  2#U> UT` 9  8U< UT` :  MU: UT` ;  bU8 UT` <  y ` = jNotes: QU6 UT` >  U4 UT` ? / is either (const ) or U2 UT` @ $ (reg ) e ` A -... thus no "nested" operations are allowed. U0 UT` B  HHeMhiHHl ` '?: jQ )?: (a?:Q:HC?C jQ aig?C -%;32-2-2 init-sum 5 3$jQ>.. 3$W32ZjQT32ZlabH!32jQ 2U>H!323232 sum<-addU:,32ZjQ,32ZNoq3I32jQ ?q3I32?q?  from<-inc>32jQ 32 e32323266jQ e 3266 32323232he jQ he  >3266jQ326632-jQ32-ig32 32o1 jQ o1 3232 yes)o! jQ u)o! )32)32 donein32jQ323$3232,32ZjQ2,32Z32QjQ!32Q32G32Y32Y32323232jQ 323232 no32ZjQc>32Zp0ٍO32jQ ep0ٍO32p0  temp<-sqrt32ZjQ32Z32jQ32e32,3232ZjQ32ZK32ZjQK32Z-R;32jQ R;32^^ init-sumK32ZjQjK32Zf32jQoinf32jf3232032jQ32032j032K32 G32jQ  G32,KC,KC  sum-roots2k32jQ 2k32x32x32   x32;32jQ o x32;32 ~u ~u sum-iterz8vԀjQ z8vԀzz DATAp0z:ԀjQ z:ԀzԀzԀ FLOWV eԀjQ V eԀV2V2 CONTROLZV5:ԀjQ 2V5:ԀVFԀVFԀ FLOWdkp32 HHkq sHHjK32  ` %Recursive Sum-Roots Register Machine ) ` u  2? `  U  h i HHks2HHl df32HHf HH ~u  ` C %Register Machine Controller Language ) ` D p0 (reg )) zQUR UT` F FL-(assign (const )) fUP UT` G Ԁ9(assign (op ) ...) {UN UT` H )(assign (label )) UL UT` I  pUJ UT` J /(perform (op ) ...) UH UT` L  UF UT` M ,(test (op ) ...) UD UT` N (branch (label )) UB UT` O M(goto (label )) 2U@ UT` P (goto (reg )) #U> UT` Q  8U< UT` R  2MU: UT` \  bU8 UT` ]  y ` ^ Notes: dU6 UT` _  U4 UT` ` / is either (const ) or U2 UT` a $ (reg )  ` b -... thus no "nested" operations are allowed. tU0 UT` c le aHHf EHHl ` FHAkvae>8  i(aigu(?kw2..u(?` HdfC> UJUTHHfD t <HHUT` Mp  ` 2>%Register Machine Controller Language l) ` )  B (reg )) ` PQUR UT` re-(assign (const )) fUP UT` UT9(assign (op ) ...) {UN UT` UT)(assign (label )) tUL UT` t  nUJ UT` /(perform (op ) ...) g-nUH UT`  bUF UT` "n,(test (op ) ...) cUD UT` (branch (label )) UB UT` (goto (label )) U@ UT` (goto (reg )) #U> UT`  8U< UT` (a(save ) MU: UT` (restore ) bU8 UT`  y ` Notes: U6 UT`  U4 UT` / is either (const ) or U2 UT` $ (reg )  ` -... thus no "nested" operations are allowed. U0 UT` B T00ZG32kx <g-00ZG32<0< val `32`K9`K9 "n?k ?&ZU@32k G&ZU@32-ť&ť to?k?$-k #+$-6 32k  32 1<  k UT    k l  k dfHHf HH ` er Subroutine Version of Sum-Roots &UT UT` 32 ;UR UT` (controller PUP UT` ` 9eUN UT`  ... zUL UT` U ' ;; Contract: input registers from, to 32UJ UT` % ;; output in register sum UH UT` / ;; Returns to label in the continue register. ?UF UT` %  sum-roots  ;; entry point UD UT`  (assign sum (const 0)) UB UT` k sum-iter U@ UT` ˴$ (test (op >) (reg from) (reg to))  U> UT`  (branch (label done)) "U< UT` % (assign temp (op sqrt) (reg from)) 7U: UT` + (assign sum (op +) (reg sum) (reg temp)) LU8 UT` , (assign from (op +) (const 1) (reg from)) aU6 UT`  (goto (label sum-iter)) vU4 UT`  done U2 UT`  (goto (reg continue)) U0 UT`  ... U. UT`  ) U, UT` f  HHfeerHHl dfNUTUTHHf mtoHH  n regist : :UT ` d toRecursive Sum-Roots r&UT UT` UT ;UR UT` -r(define (sum-roots from to) UDPUP UT` g  (if (> from to) ) eUN UT` h k' 0 ;; base case (ozUL UT` l eg4 (+ (sqrt from) ;; deferred operation UJ UT` m 5 (sum-roots (+ 1 from) to)))) ;; recursion ` i m   h e tem  `  HHfU6HHl dfcti .HHf fHH7eer  ` j %Recursive Sum-Roots (Failed Attempt) &UT UT` v  N;UR UT` (controller PUP UT`  - ;; On entry -- continue holds return label eUN UT` 6 ;; -- registers from, to hold input values zUL UT` m-+ ;; On return -- register val holds answer ` UJ UT` k -r sum-roots UDUH UT` n $ (test (op >) (reg from) (reg to)) hUF UT` o  (branch (label base-case)) LUD UT` p 2 ;; Need to recurse, so remember what well need UB UT` # ;; for the deferred operation... )) U@ UT` ' (assign old-continue (reg continue))  U> UT`  (assign old-from (reg from)) "U< UT` U63 (assign continue (label do-deferred-operations)) 7U: UT` s , (assign from (op +) (const 1) (reg from)) LU8 UT` t .& (goto (label sum-roots)) ; recurse aU6 UT` x base-case vU4 UT` z  (assign val (const 0)) U2 UT` { j (goto (reg continue)) (FU0 UT` w UT do-deferred-operations UTU. UT` q tr (assign from (reg old-from)) OnU, UT` r e % (assign temp (op sqrt) (reg from)) U* UT` ~ r+ (assign val (op +) (reg val) (reg temp)) U( UT`  et' (assign continue (reg old-continue)) UT U& UT` } -r (goto (reg continue)) U$ UT` | re) 3U" UT` y UF THHfLUDHHl dfrd  HHf )HHa om))    ` la!Recursive Sum-Roots (With Stack) :&UT UT`  s;UR UT`  on(define-machine sum-roots PUP UT`  ot# (registers continue from to temp) UTeUN UT`  e- (operations + sqrt >) zUL UT`  on (controller UJ UT`  ot. ;; On entry -- continue holds return label UH UT` UT7 ;; -- registers from, to hold input values U,UF UT` , ;; On return -- register val holds answer UD UT` n sum-roots g UB UT` ˪& (test (op >) (reg from) (reg to)) U@ UT` )) (branch (label base-case)) (r U> UT` U$4 ;; Need to recurse, so remember what well need "U< UT` % ;; for the deferred operation... 7U: UT`  (save from) LU8 UT`  (save continue) raU6 UT` 5 (assign continue (label do-deferred-operations)) vU4 UT` . (assign from (op +) (const 1) (reg from)) U2 UT` ( (goto (label sum-roots)) ; recurse (WU0 UT` UT base-case sU. UT` on (assign val (const 0)) PUPU, UT` ( (goto (reg continue)) U* UT` ˪ do-deferred-operations rU( UT` ˪5 (restore continue) ;; restore in reverse order! n U& UT` ol (restore from) UTU$ UT` ' (assign temp (op sqrt) (reg from)) va3U" UT` ˪- (assign val (op +) (reg val) (reg temp)) aHU UT` ˪ (goto (reg continue)))) UT]U UT` (t HHf))HHl  ;;$-ke$-up?k:UTup?xZG"32k ntxZG"32 fromcoup?k4UTup?r$$k2UTr$$$z &32k 0UT$z &32$$ >r$$k,UTr$$3k -d3s rU(33^$kee ^$ yI"yI"^^)-k)re)- UTEE(Z?k go Z? UTUZltlZG &32k tlZG &32y̥tl̥ +Z?kZ?66k66p6ll-k32-fr~-Zk~-Z$~~)k)$))ckc<?k dR#,?U({<Nib32k b32~oo sqrt2E?k2E?? MZU#32k ? MZU#32QY? Y temp2E?k2E?qNk2qN+qNNW?kW?kW2WN k N NNWW k W kW``NkN< k < EËEË !$-k !$-!N?8$, 32k U($, 32$8$8 0!$-k!$-?86k?86?8u8H. k H. 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