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A compiler-based leakage reduction technique by power-gating functional units in embedded microprocessors

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Material Information

Title:
A compiler-based leakage reduction technique by power-gating functional units in embedded microprocessors
Physical Description:
Book
Language:
English
Creator:
Roy, Soumyaroop
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Supply-control
Sleep instruction
Low power design
Power-aware ARM processor
SimpleScalar ARM
Dissertations, Academic -- Computer Engineering -- Masters -- USF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: Power-gating is a technique investigated widely for reducing leakage energy in the functional units of microprocessors at the architectural level. Effective power-gating involves deactivating idle functional units for sustained periods incurring little or no performance degradation. Accurate prediction of long idle periods is essential, which, in turn, depends on the application program characteristics.In this thesis, we propose a compiler-based leakage reduction technique for embedded architectures by exploiting the well-known attributes of embedded applications, namely, small code size and intensive loops. From the control flow graph (CFG) representation of the source program, we construct a forest of loop hierarchy trees (LHTs), which capture the nesting loop properties of the program. As an LHT satisfies the partial ordering on the loop nesting, we exploit this property to identify maximal subgraphs (of functional unit idleness) in the original program. For each subgraph so found, a sleep instruction is introduced at the entry point of the corresponding code segement, thus optimizing the number of sleep instructions. The sleep instruction has one operand, a bit-vector comprised of ON/OFF controlbits for all functional units in the data path. Our target architecture is a modified ARM processor model comprising of functional units with power-gating ability. We obtained an average leakage energy reduction of 34.1% for 12 benchmarks chosen from the MiBench suite, with range of 19.5% and standard deviation of 6.5%.
Thesis:
Thesis (M.S.Cp.E.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Soumyaroop Roy.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 52 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001920056
oclc - 187072338
usfldc doi - E14-SFE0001832
usfldc handle - e14.1832
System ID:
SFS0026150:00001


This item is only available as the following downloads:


Full Text
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A compiler-based leakage reduction technique by power-gating functional units in embedded microprocessors
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ABSTRACT: Power-gating is a technique investigated widely for reducing leakage energy in the functional units of microprocessors at the architectural level. Effective power-gating involves deactivating idle functional units for sustained periods incurring little or no performance degradation. Accurate prediction of long idle periods is essential, which, in turn, depends on the application program characteristics.In this thesis, we propose a compiler-based leakage reduction technique for embedded architectures by exploiting the well-known attributes of embedded applications, namely, small code size and intensive loops. From the control flow graph (CFG) representation of the source program, we construct a forest of loop hierarchy trees (LHTs), which capture the nesting loop properties of the program. As an LHT satisfies the partial ordering on the loop nesting, we exploit this property to identify maximal subgraphs (of functional unit idleness) in the original program. For each subgraph so found, a sleep instruction is introduced at the entry point of the corresponding code segement, thus optimizing the number of sleep instructions. The sleep instruction has one operand, a bit-vector comprised of ON/OFF controlbits for all functional units in the data path. Our target architecture is a modified ARM processor model comprising of functional units with power-gating ability. We obtained an average leakage energy reduction of 34.1% for 12 benchmarks chosen from the MiBench suite, with range of 19.5% and standard deviation of 6.5%.
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PAGE 19

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y#Hz6Y{#vw‡b “$z6†PvA‚|Q€“$|V£#|M‚„vw…z3{Mx}‹… ‰ +(*-&`'K-&&(rOcW8rA1r*').2mr(*-&r$Ri.r8:Kr%?rV+r(*-&I[8:K#RT`&n40I+1!wE98:r*!(*+1!MR R%&+&')(*+&E%&+&(*:+rD2mr&1r>&*J')r*Keh=Œ.(cJ8:rw&*r*r+W(B#2&KrRSr)ŠJ?+?(*+R(*-1rwr+X(**r RTKr8*neh,N-1r+ScJ8:r31rK')K2mrA(*-&rwV0&U&r>\"AF£K'.-&(r')(*%1*r3%&*r+Q(*-?8:*>8N-&'.-6-& (*-1r6*%1&m*(3RT#E98:r*!(*+1!kR$(*-1rQR%&+&')(*+&`%&+?(*M+§(*M?(*?(*-Seg=Œ+?4c78:r&*rr+X( (*-1r>(r'K-&+&I/X%1rQr?94rRTMU?+?&+1!%1(D(*-1rQ?r*r!+&D+(*-&rQ&*!.+&§+&r*(*I+1!(*-1r .K<+?(K%&')(*+?e rYI1r*')K2mr\r*2?%&(V(*-1rQ(*'S+1RTK(K%?')(*%1*r\8N-&'.-Qr+&2nrB%& (Q1r]Jr3(*-?($(r'K-&+?/X%&re bmd yD#ˆ‰Y7>WTXOŠWTtsnuWƒ:X?0??Q07~9TE “9#ˆX#n9nq?6N0rXO…1 T~ ,$-&rh')?r*r]rr!rh*r&%&')(*+wRiKVr8:*31?(rY+w(*-&m8*3S*-&O8N+YI+3=7!%1*rYn0e–e 2&KrR~r)Š0?I+&(*+R7(*-1rY]K%&BI+1?%1(*(V(*-?Ri.r8:K$!]r+2mrO8Ma g A&?I'(*+ L %1K')r601rQ\,$-&r>%1.')r>')J1r>RT#r'K-&?I'(*+D')?r+& Kr&*rr+X(rg\QI($Rh**rM2?4Wr]r~+&(.%&')(*+&e ‰ +G(*-&N8:Kncn8:rr%&rY2mr+&'.-&V*J RT*„(*-&rQFHr+&'.-*%?(rG œ9eQ,$-1r&Kr')?rkr)ŠJr'%1(*2?rMRiM(*-1rr2mr+&'K-?V*JY*r O] I2?rhRT~I((*r)r+&&I+3V'K-&I+1rfK%?+&+&+1!+0%JŠw+Y(*-1rGe7,$-&rh*rD2?4Xr]rW+?(K%&' (*+§I(MRTD(*-&r>2mr+&'.-&V*JM8r*r2&(*+1r§204(*-1r2J_r')(r&%&VR$(*-1r&*r')V?r r)ŠJr'%1(*2?re g 5798:r<;3(rNK'.-&(r')(*%1KrM,$-1rmO8r!(r &*J')r*>K'K-&(r')(*%1*rHQ0&rr+ (*-&r‹r)ŠJ*(*+1!A"\FG]&§&*J')r* n9e’ ?(*+1!m+X(6%?+&(6-&62mrr+ ?1r¢((*-1r ?(*?(*-‹RT3(*-1rVJ&U&rkA"\FK'.-&(r')(*%1KrV%&r‹+k(*-&\8*H2mr'%&*r(*-1rA"\FG]& ?*0')rK:')*rw-&S_%&*(`+X(r!r:R%&+&')(*+?&%&+&(*e7,$-&r$&(*+1!m+X(`%&+?(h`V01rr+ –)™

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(*-&r\Žw=Œ5:Ÿ9 L ]r')($?(*+1!m+X(')&K0')r**M nŸ9c18N-&'.-'+62mr\')+&+1r')(rr)ŠJ(rK+&4 8A(*-V+1r8r$]rK*+RŒ(*-1rYA"\F£&*J')r*c&\"AFkŸ C e g =?%&+&')(*+&m@\+&(5~98:r*<;w(*+1! Z r(*: L +?')rN(*-1r*rw+O]rK-1r6r+&r*!4>')m+1r+X( K0'I(rH8N(*-6(*-1rYmO8r!(*+1!>(r'K-&+?/X%&rc1(*-1rŒ‹)"**"}Ž‹mrKJRiAr'K-gRT%?+&')(*+& %?+&(AI\'I'%&(rSe3,$-1r‹)"**"}Ž§Er.0fcn'rH(*-1r W"*.0lz‹mrKJScEA&rU?+1r :(*-1r\V+&#%&PmrKJrRS(*r$RT`8A-&'K-QYR%&+&')(*+&?%&+&(h*-&%&r2mr$r&(1r')(*](r K%&'K-(*-&r$r !rNr+1r*!4*]J+1!`&%1K+&!N(*-&(hmrK0rŒr/0%&J(Y(*-1r$K%&‡Rm(*-1rBO]rK-&r r+&r*!r$**0'(rH8N(*-(*N')(*] (*+H+&G&r')(*] (*+Se Identification Idle FU Subgraph Analysis Static Code Application Source Profiling Dynamic Instructions Insert Sleep Simulation Cycle-accurate FU Powergating Details Power-gated Architecture Compiler Extensions =Œ!%&*rMn0e–M=1.r8:K6\1&(rGRiN5798:r<;3(*+1!1e P2&KrR1r*')K&(*+6R7(*-1rY]K%&(*JNI+6(*-1rYRTKr8*I$BRTO8Aa g L (*(*'kJ1rkA+&40*Ir Z %1KI+1!‹(*-&>(*!rc$(*-1rk&?'(*+%1.')rH')0&rkQ(*(*'4 +?4X¡rRTRT%&+?')(*+&?%?+&(h*r/0%&Krr+W(*eŒq')+W(*m?O8 !K?-‹=~;Y`RT(*-1r\*%1K')r –›

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?*0')rKA')*reB,$-&r L ?r*'$\"AF&(.2?%1(*+6N%&rG+G(*-&B8*68N-&'.-G*#%J I(rw(*-1r ‰ +X(r L (K+1!N. ?*0')rK\')Kr œJ–eY 1r(*rk1r*')K&(*+GR:(*-&A(*!rV !]r+H+ L r')(*+Hœ0ešn0ešœ0e bmpb † SŒX~0z6>† z97I9X k>i9A0#0…619~qn“~~~9TE3W7i9 Incrementer Address Register Bank Write Data Register Multiplier Booth's Shifter Barrel Instruction Decode & Control Logic Address Register Instruction Pipeline & Read Data Register ALU =Œ!%1KrMn0ešn L (*+&&.A"\F£AK'.-&(r')(*%1*r n9e =Œ!%1Kr\n0ešn#*-198N(*-1rA2n0'KV&!K—R#!r+1rKI'A\"AF„.'K-&(r')(*%&*rr\"AFH]&:*r&*J&%&')r RT* n9eYAw'+k2mr#rr+kRT* (*-1r=7!%1*rcE(*-1rD?*0')rK3')*r1Xrw+1(w-&O]rr+X4H?(*+1! m+X(\R%&+&')(*+&[%&+&(*er,N-&3K'.-&(r')(*%1*r>3J&U&rk(Gr+&2nrmO8r!(*+1!GR%&+&')(*+&(^4 RT$(*-1rYR%&+&')(*+?E%?+&(*eh,$-&rwV0&U&rK'K-?(r')(*%1*rMIB*-1O8A+G+=Œ!%1*rMn0ešœ0e ’?(*+1!m+X(M%&+&(D-&M2mrr+&1r§Ri#(*-1r&(*+1!m+X(M+?(K%&')(*+?Y*%1&m*(r2X4 (*-1rM\"AF ‰ +&(K%?')(*+ L r(e:,N-1r3RT%?+&')(*+&S%&+?(*B*rM1rK!+1rG*%?'K-(*-&(A(*-1rM(r+?')46I+6(*-1r ')(*](*+ (*r+1r‹'J'* ')40'r j e,N-1rk&r(*RY(*-1rR%&+&')(*+&\%&+&(*>*r‹&I*'%&*r + D ¨§?§pPp9{¨9""f~~D\?'e~/eeDI~eD\I\‚e‚~'e’)‚‚ŠŠ)?eŠ}Š7Peef/w‚ wŠfw>eŠZ1ee~D< w~'fw"\"e}D/e}?+e~?ePeŠ?e~/?KDŠ ~Šf7fPw1f}f91f‚I9 )e ?1D'f –O

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Incrementer Address Register Bank Write Data Register Instruction Decode & Control Logic Multiplier Booth's Shifter Barrel Floating Point Unit Address Register Instruction Pipeline & Read Data Register AU MUL DS ALU SCR X 1 1 1 X =Œ!%1*rMn0ešœ>FGJ&U?r\"AFAK'K-?(r')(*%1*rY8A(*-57O8r<;w(*+1!>Ri\=&%&+&')(*+&S@A+?(*e L r')(*+‹œ0e–e3,$-1rr+&*(K%&')(*+H?mr+1r3\')+&*I1r*rg(2mr#>!r+&rK'DU&]r)(*!r?mrI+1rw8N(*zXŽ"K ?zr)ŠJr'%1(*+‹?Er6 œn9eV4k&]rK*U&rgr)ŠJr'%1(*+‹?Ercm8rVr+‹(*-&(3(*-&r*rV*r ?KIrm*%121?ErI+1rhrV?O4J+1!V?ErKr+W(BR%&+&')(*+?E%?+&(*B+6(*-1rMr)Š0r'%1(r#(*!rW..e Z r')(*](*+1!§(*-1rGR%&+&')(*+&%&+?(*V'*.r%1(V%&K+1!(*-1r‹.^+&(K%?')(*+Se L rr +X(*S"$r!I(rM L "w`I?1r((*-1r3+&(K%?')(*+>&r')01r3!I'e C 'K-6R%&+&')(*+&m%&+&(:(*-&( +1rr&>(2mrg1r')(*](r>*%1&?Ir6+mrK+?((*-1r*rr¢+&(K%?')(*+Se L I+&')rHr'.R%&+&')(*+&m%&+?(B'+2mr3*r/X%?*r(V2mrwr(*-1r$')(*](rGN&r')(*] (rfc?+1r32?(I*r/X%?*r( mr'RT4>r'.-kEr.+&Se^I–¨n+&?'(r$(*-&(A(*-1rMR%&+&')(*+&f%&+&($-&$(>2ErD1r')(*](rSe$^šoJ¨ +&?'(rh(*-1rw(*(*%&:RS(*-1r\RT%?+&')(*+&&%?+&(`*-&%&n 2mrA'.-&+1!rSe -1r+M*rr6+&*(K%&')(*+ A1r')0&rScn^I–¨EN8$K((r+g+X(Q(*-1r#')**rm+&&I+1!2?($J'(*+k+(*-1rr*rrk')+W(K*r!I(r +&G^šoJ¨?B!+1KrSe ‰ +(*-&rHRTO8A+1!r)Š1?rcB(*-&rgmrK+&?V?**r((*-1r.^ +?(K%&')(*+*rH+X(r!r #%&(*?IrcW?(*+1!m+X($K(*-&r(*I'\%?+&(c&+&6&(*I+1!m+W($#%&(*?re ‘)št›u›}– {•”'Ÿ#šu'•u~Ÿu'•f”aŸ#š –

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-1r+(*-1rY2m9]rM+&(.%&')(*+6B1r')J1rScS¨–¨?I8$K((r++(*-1r L "r+X(Kr$')*Krm+&&+1! (3+X(r!rŒ#%&(*?rc =Œ5‹K(*-&r(*I'`%&+&(c +&r=Œ5‹#%&(*?IrcO+?&'(*+1!N(*-&(Œ(*-1rrRT%?+&')(*+& %&+&(*$*rMn%1($(>Krrfe,$-1rYr+X(Kr\')**rm+&?+1!r(Q(*-1rD2?**r*-&RT(rA+&H(*-1rM=Œ5¢&]J*+ +&*/0%&*rw*0(B%&+&(1+1($'.-&+1!rc?+?&'(r2X4g¨ ¦ ¨e&,N-&*')r+&KrIK-1O8N+I+=Œ!%&*rMn0ešœJe ,$-&rN')(*](*+QRm(*-1rART%?+&')(*+&&%?+&(*h-&?Er+?[(:(*-1r"zW*. zW3*(*!r\(**rR^e[,$-&r$RT%?+&')(*+& %&+&(\!r(*#')(*](r§?%1K+1!6(*-1r>')40'rQRiO8N+&!(*-1rV+1rQI+‹8N-&'.-(*-1rQ+?(K%&')(*+r+X(rKD(*-1r mrK+& )(*!r‹G(*-&rzIJ" )q*(*!r+ (*-1r‹?mr+&r§ œn9e’,$-1r*rRT*rc\2X4 (*-1r(*r (*-1r+&(.%&')(*+r+X(rK6(*-1rP0* | ‹(*!rcA(*-&rgR%&+&')(*+?A%&+?(Q')(*]r(ErKRiK %&rR%& ')?%&(*(*+Se`,$-?]&N(*-1rM+1rrGRiN*r?K(r`"~|9+&(K%?')(*+&e bmck yD#ˆ‰Y7>WTXHˆfKYYh?9 ‰ +k(*-&wr')(*+ScS8r+&40¡rr(*-1rr*r/X%?*rr+X(*\Ri3&?40+&!QmO8r!(*+1!G(w(*-1r')nr r]rAV(*-1rV(*'I4+&?*mrH(r'.-&+&/0%1rgRi1r+X(*Ri4J+1!Q ŠJQ$*%12&!.?-&V+(*-1r &*!.mr*(*+&+&!A(#&r+1rK~RE(*-1r$R%&+&')(*+&1%&+&(*e rA&r*')K2mr(*-1r$&(*D(K%&')(*%1KrŒ(*-&( 8rG%&r6(?rr+W(#(*-&r(r'.-&+&/0%1r+&&Krr+W(!.(*-&VrRTV1r+X(*RT40+&!g+&+?r*(*+1! .K<‹I+&(K%&')(*+&e bmckf 3 ?? 3ˆ z63WmV ,$-&r`')+W(KW&98§rQ+W(*'~R1$&K!Kq*r:*r&*r*r+W(rD+3(*-1r:RiKqR1N')+X(*W?O8!.?=~;M.cn+68N-?'K-6(*-1r3]rK(*')r$*r?*rr+X(B(*-1r32?*'w2n0'K0B+?(*-1r3r1!rN*r&*rr+X(B(K+&*Rir$R ')+X(*&O82Er(^8:rr+(*-&r2n*'2?J'*J œn9e=Œ!%1Krn0e ™I%&(.(rD=7; 8N(*-Ri%&2?*' 2?J'*JD&r+1(rk2X46*r')(*+1!rA8A(*-G&*-&rH+&r.r'K-')+X(*+&+&!Q>+0%&D2mr$Rh+&(.%&')(*+& 1r+1(rk2046O]..eA,N-1rD&*r')(rGr1!r\*r?*rr+X(N&98 Rh')+X(*72mr(8rr+g(*-&rM2?*I'32?J'*Je ,$-1r*rNr1!rB*rw+&?%&')r204V(*-1r\2&K+&'.->+?(K%&')(*+?N1r+1(r6204K-1D2?%&r.eŒ,N-1rNK%?+J(*r r)ŠJr'%1(*+R`&*!.„r&\(6(*-1r140+?V'Y(.]rK*ŒR:(*-1rr]r*(*')r3+&k(*-1rrr1!rwR(* =~;reX,$-1rN2&K+&'.-V+&(.%&')(*+&[+&V(*-1rh2&.+&'K->')+&&(*+?$+V'rARf')+&&(*+?J2&K+&'.-1r. &')(*(rM(*-1rM')(*%&Sn(*-RŒ(K]rKK&%1.+1!#&*!K’r)ŠJr'%1(*+fe –Ÿ

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=Œ!%1*rMn0e ™‡+X(*f=Œ98‡;\.?-=~;YM œn9e ,$-&r#VI+H(*HRh(*-1rr')?rwA(+&40¡rD(*-1r#&*!. 2mr-&O]0\+&k&Kr&')($*r!+& +(*-1rQ&*!K8N-1r*rV')r*(*I+R%&+&')(*+&[%?+&(*3*rQ+&(P)1* Pz(G2mr*r/X%?*r‹&%1.+1!6(*-1r r)ŠJr'%1(*+R(*-1rQ&*!K£G(*-&(DE98:r*!(*+1!g'+2mrV&?rfeV,$-X%?c!]r+§g=~;¤RTD &*!.c&(*-1r32J_r')(*]r#B(VU?+?6%1(NV Š1VS*%&2&!K?-&BI+(Bmr*(*+&+&!D(>(*-1rY&r+&r* +1+J%?*!r:R?r'K-#RT%?+&')(*+&X%&+&(e7,$-1r+ScX.K<+&(.%&')(*+&fK-1%&M2Er:+&r*(r#(7(*-1r:r+X(*4YR (*-1rr*%12&!.?-&SA(*-&((*-1r:RT%?+&')(*+&%&+?(*f*r1r')(*](rM8N-&r+Y(*-1rh?*!KP')+W(*Wr+W(r. (*-1rrGK%12&!K?-?#?%1K+1!g&*!Kr)ŠJr'%1(*+fe Z %1r(‹(*-1r?*rr+&')r6Rw')+&&(*+?:(r.(*]r (K%?')(*%1*rDŒXi0?$+?"Q0?B8N(*-H14J+&V'3(rK(*+H')%?+W(..cm(*-1rDK%&+J(*Vr3(K')rDR (*-1rM=~; R7M?*!K 8NI&2mrA&Er*r+X(Ri$&Er*r+X(K%&+J(*VrA+1n%1(*eŒ,$-1r\&*!K c0(*-0%&c 14J+&V'I4‹&*U?r§8A(*-r(^4X?I'B+1?%1(r&(*‹('.-&K')(rK¡rG(*-1r&*!.2mr-&]J (*-&($B+&(Br]0&r+W(RT* (*-1rY(*(*'3+?40KR~(*-1rDB=~;#e&rG+6(*-&rrY'K-&.')(rK(*'c&(*-1r ')?Ir$+&rK(*(*-1rM&&K&K(r3*rrg+&(K%?')(*+&e no

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p p v v v v basic block requiring functional unit r r basic block not requiring functional unit u k 1 2 k-1 p k-1 k 2 1 p =Œ!%1*rMn0eš›Q L %12&!K?-RŒ(*-1rY"A%&+J,$Vrw,~K')rYR[>+W(*=7O8P;\.?-Se r+X]r(*!(r‹(*-&rk?*2?rRD&r+W(*RT4J+1!(*-1rm(r+X(*\*r!+&6*%12?!K?-&+ (*-1r =~;‡8N-1r*rwmO8r!(*+1!V'+62mr3&?IrQrIH>'r+W(*4e rYr?94>*I?rA(*rrY&(*(K%?')(*%1*r *r&Krr+W(*I+1!0-&rK.'K-X4I+&*!.((*-&Ir+&Se ‰ +(*-&rG*rV+?1rVRw(*-&V*r')(*+Sc 8rD+&40(*'46V01r(*-1rM?*2?r’RhI+&r*(*+&!(*-1rD*rrk+?(K%&')(*+?B+&H?*rr+X(AQK?r &&K'K-G(>]rM(e bmckfcb x~JE9>qwV~§A&OJE9T3WAXr0…619>XEiE~>9~‹yr“B… -&r!r+1rK(*I+1!‹(*-1rg=~;Ri(*-1rG*%1K')r&*!.chr'K-]r*(r)Šr+&+1(*(r8N(*-(*-1r R%&+&')(*+&J%&+?(*r/0%&Krr+W(ŒRn(*-1rB2?*':2?J'*#')**rm+&?+1!N(3(*-?(h+10&re7,$-?Œ+1RTKV(*+ [Kr/X%&*r+rU?+&?+1!w*%12&!Kn-&[+(*-1rN&*!K—B=~;8A-&'K-KrN*%&(*2nrBRi:&?4J+1!wmO8r !(*+1!1e3=Œ!%1Kr#n0eš›%?(K(rN*%12?!K?-gRh(*-1r#K%&+J(*VrY(K')rrRh(*-1rV=~;—RT3Q&K!Ke Ž[r*(r)Šd *r&Krr+W(*‹2?K'2?J'*(*-&(10r>+1(>%&rH?*(*'%&IVRT%?+&')(*+&$%?+&(c*4Je Ž[r*(*')r B ` ` ` *r&*rr+X(r2?*'>2?J'*Jr(*-&(%&r Jer( *r&*rr+X(r(*-1rr(RNn(*-& B ` ` ` 8N-&r*r Br?(*-RT* H( c&*%&'.-6(*-&($(*-1r3+?4Q]r*(r)Š8N-?'K-%&*rrIn e jA+1rYR7(*-1rM+X(rKr&I(r3]r*(*')rNI+ %?r Je nJ–

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r( N 2ErD(*-1rr'0'KHmrKJH+? $2Er#(*-1rr+X%?M2mr\R`')40'rAKr/X%&*rH(r)ŠJr'%1(r(*-1r +&*(K%&')(*+&BI+?(*r eB,$-1r+G(*-&rY((*7(*rMmr+X(N+r)ŠJr'%1(*+&!Q(*-1rMI+&(K%&')(*+&$+ !]r+204c N dn0e–O r(A .c?(*-1rVux GRŒ(*-1rMn(*c&2mrM&r*')K2mr+(r.V$RŒ(*-1r#+X%&D2mr$R[+&(.%&')(*+& + e ‰ R7(*-1rYV Š1D%? ‰ 5‡+&(K%?')(*+&mr$')40'r9')%&+X(c&8N-&'.-B1r(rKQ+1r6204>(*-1r +&‰zX*KzW#8N&(*->R[?*0')rKc&fc&(*-1r+ A dn0ešn ,$-&rrI+1r/X%?(4G-&&\+ C /0%&(*+dn0ešn\2mr'%&rr(*-1r ‰ 5P')%&+X(wRiY6')J1rVr!r+X(Y'+ 2mrY*VrB(*-&+H+'')%&+X(NR[''K-1rDV*rc12&K+&'.-GV?*r&')(*+?cW?ErI+1rNn%&*-1rcJr(*'e C V+?(*+1! RT* C /X%?(*+dn0e–OB+? C /0%&(*+dn0ešn.cE8:rM2&(*+Sc – A N dn0ešœ =1 –c \ N ek,N-&#DI+W(*%&(*]r>+(*-&(chRTkQ Š ‰ 5')%?+W(((6–c (*-1rVVI+&#%&¤(*Vrr*Er+X(3+(*-1rrr)ŠJr'%1(*+‹R:(*-1rV+&(.%&')(*+&\+ '+2mrr2E%?+&1rg2X4k(*-1r +0%&M2mrBR[+&*(K%&')(*+&+ e r(h%&Œ+1O8*K%&r(*-&(Œ(*-1r$r!rBr+1r*!4#*]rmr[%&+&(~(*Vr204Yrr?I+1!w(*-1rBRT%?+&')(*+& %&+&(KN1r')(*] (r "!`+&D(*-&(Œ(*-1r14J+&V'`r+1r*!4Y9]rK-1r+#')(*](*+r+&r1r')(*](*+ $# eA,$-1r+G(*-&rDV+?D%?’(*rc N&% RTA8A-&'K-œQ*-1%?2mrM1r')(*](rk2mrRi*r#(AI$8r+ %1G$!]r+204c N&% # "! dn0e ™X L 1c0RTM?(*(#!r+1rK(rwr+1rK!4Q*]J+1!cX(*-1rA(*rwmr+W(+Vr)ŠJr'%1(*+&!M(*-1rw+&(.%&')(*+& + K-1%&2mrY*rY(*-?+ N % c?de rec N % eh,$-&BBK(*U&rGR^c – \ N N'% # ! dn0eš› nn

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8N-?'K-V?rc \ (#! N dn0ešp ,$-&rV/X%&+X(*(^4)(# !+* N 3'Ir‹(*-1r W"*.0lzr+1!(*-+&3+&&I'(r2X4, I e ‰ R(*-1r *r(*+Sc&mr'U&r6204 C /X%?(*+dn0ešp.cEIB*(*U&rGRiNI r c&(*-&r+ N&% `n.r dn0e 8N-?'K-Dr+&*%1Kr~(*-?(7(*-&rr !rBr+1rK!4D*O]rV+D(*-1rB*%12&!Kn-D*-&O8N+r+#=Œ!%1*rBn0eš›\r)ŠJ')rr? (*-1rM&40+&Q'\r+1r*!4O]rK-1rk+&'%&**rSc0(*-1r*r204c?!]J+1!r9]rKf*O]0I+1!e \(*-1%1!-fcW(*-1rw2mO]r3&&*'.-Qr+?*%1*r:r+1r*!4V*O]0I+1!:I+Qr]r*4Q?(*->I+V(*-1r\*%12&!Kn-0/G c (K(*-&rV')+&r*](*]rH&?*'K-Se Z %1KI+1!g(*-&rr)Š0r'%1(*+RA(*-&r&*!KcrR\(*-1r ?(*-&+ V4>2mrA!*r(rA+>r+1!(*-6+&n9(K]r.rG*r\RT*r/X%&r+W(*4Q(*-&+(*-1rw*r(eh,$-0%&c (*-1r*r#$QEK*2?I(4(*-&(A(*-1rD')D2?+&rr+1r*!4G*O]0I+1!N'.-&r]rk+1!V(*-1*rM?(*-&Nr)ŠJ')rr? (*-1r6')M2n+1rr+1r*!4*rr+&'%1K*r+&!g(*-1r*r(#R$(*-1r6?(*-?ch(*`*r*%&(*+1!H+§O]r. r+1r*!4*O]0I+1!e ‰ R1F `(*-1r\+0%&D2mr`RS(*r:(*-1rA?(*.r Ih(KO]rKrScJ(*-1r+ScJ%&*I+1! C /0%&(*+dn0ešp.c1(*-1r r!rMr+1rK!4*O]0I+1!$'+G2mr3r)ŠJ&*r**rGc 1 32 9 5456 F \ 7 N .8# dn0eš ‰ R G:9 2mr\(*-1r3r(BR7?(*-&8N-1r3r+&!(*-!*r(rN(*-&+; I 1rU?+1r6c G =<>r +&?A I c~e recf(*-1rQr(3R?(*-&Y8N-1rVr+&!(*-‹w!*r(rM(*-?+) I (*-1r+ C /0%&(*+ dn0ešB'+H2mr3*r8$K((r+c 1 32 9 „ 5456 F \ 7 N .# . 5456 6 F @# \ 7 N dn0ešŸ =1K (*-&rD*rK%&(AI+ C /X%&(*+dn0ešŸ.cS8r#'+k2?*r*]rr(*-&(\R[8:rr1r+X(*RT46K%12&!K?-?N+g(*-1r &*!.”=~;q8N-?'K-V'+>*r*%?(h+&K!K r)ŠJr'%1(*+Q?(*-&:RSr+1!(*-Q(r(A I K%&'K-Q(*-&(a nœ

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1 l 6 l 7 l 4 l 5 l 2 l 3 l 2 l 3 l 5 l 4 l 7 l 6 l 1 l (b) Loop hierarchy forest for the CFG (a) CFG of a sample function =Œ!%1Kr3n0ešpQf0H•\rKK'.-W4,~*rre iMHR%&+&')(*+&`%&+?(D#+1(#*r/0%&*r§+(*-1*rn(*-&Iv`+&¢i3(*-1r?(*-&r*r6k(K]rK*r *r(*]r4G*rr+X%?M2mrNR`(*rcE8:r#U?+&Hm(r+W(*~*r!+?A+g(*-1rD&*!K =~; 8N-?'K-k*r !00'+&&I&(rrRirmO8r!(*+&!1ejN(r(*-?(c:+§&K')(*')rc`(*-1r]rK!rG?*!K ‰ 5—'+ 2mrg*QrQ(*-?+Bfe‡,$-&Q8NBKr*%&(+VI**+1!%1(>]K%?VmrKJ&Q?%1K+1!‹(*-&rg&K!K r)ŠJr'%1(*+>8N-1r*rAE98:r*O]0+&!:')%&IV2mrA'.-&r]rSeh•AO8r]rc1r]r+(*-1%1!->(*-1r ‰ 5V4Q] *4 RTN&mr*r+X(B?(*-&$+6(*-&r3K%&+J(*Vr\(.')rYRŒ(*-1rD=~;#c?%1N&&K'K-G!]r$V')+&*I(r+W(AO8r 2m%&+&+G(*-&r3r+1r*!4*]J+1!$+(*-1rY?(*-?B(*-&($*rK%&(B&%1KI+1!D?*!K’r)ŠJr'%1(*+Se bmckflk ˆSDC7? Y77Qvw~%WTn0 >i97T ƒ`S#Y7 =1IO8NI+1!(*-1r+&4J*r+(*-1r6&*r]J%&r*%12nr')(*+Sch8rG%&r6(*-1r0?r+(*-1rG%&K')r6&* !K(1r+X(*RT4G*%12&!.?-&A8N-&'.-g8r-1mrD8%&g*r*%?(\I+H?(*-?\K(*RT40+&!>(*-1rr')+&&(*+& r+X(*+1r+(*-1r3?*r]0%&r')(*+Se rD+W(K0&%?')rA(*-1rY+&(*+6R:.Xd""")X' "*K)>•\,$. (>'?(*%1*rM(*-&rD+1r(*I+1!Q(K%?')(*%1*rMRŒ(*-1rD&*!K r!r+X(*e Z %1.+1!(*-1r#(*(*'M')J1r#+&4J* ?-&*rcWRT`r'K-R%&+&')(*+Q+(*-1rN*%1K')rN?*!KcX8:r\')*r(r\YRi*r*(`Rm(*-1rA~•A,Ne C 'K->]r*(r)Š R•\,1r+1(rY0+k(*-1rV%&K')r&*!K„+?‹(*31r')r+&&+X(*Y1r+&(r(*-1rV0?\(*-&( *rw+1r(r8N(*-?+V(*-&(0fe C '.-6]r*(r)Š+&+1(*(r68A(*-Q(*-&rAR%&+&')(*+??%&+&(`*r/0%&*rVr+W( R7(*-1r3XG')K*rm+&&+&!M(V(*-&(B]rK(r)ŠEe`=Œ!%&*rYn0ešp1WB*-198N(*-1rY')+X(*S&98!K?-6R7(*-1r n ™

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\!K(*-? œ§1r*')K2ErQ(*-1r!K(*-&(')*r(r‹§%?+&/0%1rH*rr+?(K%&')(*+RiA(*-1r R%&+&')(*+&S%?+&(*8N-&I'K--&O]rM(*-1rMKrMJ'(*+G+6(*-&rM')0&re ‰ +G+1r–c&(*-&rYrrr+X(*N+?R*r *(r+K')r+&&+1!gK1rD2X4‹(*-1rDJ'(*+&e ‰ +I+1rMn .™1chH%?+&/0%1r6.KˆyD+ˆYWT“iV L I+&')rw(*-1r3*%1(*+&r ‰ j L 5 C B,?rVr+W(*')+&&(*+& Z = L (K]rKK+ c&(*8K('*r (*r')?r)ŠJ(^4D~!]r+r204”Q < < [ < 1 <  œœ9eh•N98:r]rc0*+&')rRTŒ\*0(rr(*rrc < 1 < < < .–c (*-1rw8K('r3(*Vr\')nr)ŠJ(4VR ‰ j L 5 C I”Q < < .e:jN(r3(*-&( < < 1r+1(r(*-&(B+X%?M2mr Rh0?NI+HVR%&+&')(*+SeB,$-1r*rRT*rcE(*-1rM8K('rr(*VrM')?r)Š1(^4R ‰ j L 5 C ,qIN+&rN+ (*-1rM+0%&D2mrBR70?$I+6(*-1r3RT%?+&')(*+')**r*E+?&+1!r(V(*-1rY~•A, e ,$-&r3(*rY')?r)ŠJ(^4>RŒ*%&(*+1r3@Aj ‰3Š\‰ =Zh”Q < R < < R < .e ,$-&rM*%1(*I+1r ‰ j L C "Œ, L C:C 5 Vrw'$( ‰ j L 5 C ,Riwf(*-&rD•\,$ARiAr'K-R%&+&' (*+&7%&+&(e L +?')rM(*-1r*rrA+1rr•\,qRTwr'K-R%&+&')(*+g+H(*-&rD&K!Kcm(*-1r#8:K*('rr(*r ')?r)ŠJ(^4Rh+1rM–<.§p>+ ‰ j L C "Œ, L C`C 5 •”VXF;– < ƒQ< .cn8A-1r*r’F$(*-&rM((*Œ+X%?M2mrNR 0?`I+V(*-1r\r+W(**rN&K!K ')**&R%&+&')(*+&`+& < ƒQ< I`(*-1rA((*m+X%&D2mr:RSm98:r!(*I+1! r+&2?rHR%&+&')(*+?~%&+?(*e3•N98:r]rOc < ƒQ< A')+?(*+W(YK+&')r#(*-1r#+0%&D2mr\R`RT%&+?')(*+&7%&+&(*\ ')+&(*+X(eA,N-X%&c&(*-1rM8K('*r#(*IrM')?r)ŠJ(^4Rh+1rM–<.p>+ ‰ j L C "Œ, L C`C 5 •”VXF~.c 8N-&I'K-#~I+1rŒ+M(*-&r+0%&D2Er7R?0?Œ+D(*-1r&*!KeŒ•AO8r]rcW(*-1r+0%&M2mr7Rn*rr#+?(K%&' (*+&h(*-&(:'+Q2Er$I+&r*(rV'+Q2mrNYV Š1#%&Rm(*-1rN+0%&D2Er`RmX?h&*rr+X(`+&ScW(*-1r*rRT*rc < R < dF—– < ƒQ< e:,N-X%&c1(*-1rY8K('rD(*rM')nr)ŠJ(4R7(*-&rM'S(>(*-1rY*%1(*I+1r3@Aj ‰Šw‰ =Zh+ +&rAR ‰ j L C "Œ, L C`C 5kI”VXF!\F~.eŒ,$-1r*rRT*rc(*-1rB8K('*rB(*rB')?r)Š1(^4MR?K%1(*+1r ‰ j L C "Œ, L C:C 5I”VXF!\F~.e bmckf™ G?~WT~ˆmO?~7&9 ƒ:dC79?V“~~~9Tn7 ,$-&r*(*+&&K¢2?K*4*%1(*I+1r(8A-&'K-¢'I*r‹V1rRiK (*-1r‹%1K')r‹&K!K '+ r(*-1rB2mr\(*r(rH2n'*02mŠJr$r(*-&($+1*rrGI+&(K%&')(*+&:KrY+&r*(r6+X(r(*-1r $(*-1r4 '+k2mrD+&40¡rkRi\(*-1rr+&rK(*+GR`KrrkI+&(K%&')(*+&e ‰ +GRTKr3'rcm(*-1r#RT%?+&')(*+&~%&+&( nŸ

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*r/0%&*rVr+W(SRT(*-1r:(*-1rhr+X(**r`I2&K*4A*%1(*+&rh'+M2mrhRT%&+&w%1(~2X43+&4X¡+&!B(*-1r:&I**rD2?r (*-1rr2J_r')(3')01r#RT\(*-&rrI2&K*4ew•AO8r]rc+g(*-1r((r3'rcf(*-1rV=~;—RT3(*-1r#RT%?+&')(*+k-& (‹2mr+?4X¡rg_*%&(r+X4§(*-&rRT%&+?')(*++(*-1r%1K')r')J1re L I+&')r‹*(*(*'4+1r 2?K*4DI`YKr0'(*2nrN2J_r')(')01rc0!r+1rK(*+QRfr=~;qRiK(*-1r\&***rM2?r2J_r')(')J1r `EK*2?re7%1(c0I+&r*(*+VRf*rr>I+&(K%&')(*+8:%?V*r/0%&Kr$*r!r+1rK(*+QRS(*-1rA2J_r')(')J1r +&6(*-1r+I+1J+1!Y(*-1r3+1r821_r')($')J1rY+&*(r>RŒ&I+1!D(*-&rw*(*+&&K2&K*4e ‰ +(*-&I8:*Ec 8rM1V+1(NI+&r*(B+X46KrrG+&(.%&')(*+&8N(*-&+>(*-1rM(*+&?KG2?K*4VR%&+&')(*+?e bmckfLš ˆSˆ‰ˆ&fV ‰ +(*-&r'K-&?(rc`8r1r*')K2mr§(*-1r6Ri.r8:K(*-&(r8r6%?r6Ri&?40+&!HmO8r!(*+1!‹R R%&+&')(*+&B%?+&(*V+rM2mr&1r?*0')rKKe rH&Krr+W(r(*-1rHJ&U?rA"\F .'K-&(r')(*%&*r %&r +(*-?8*8N-&'.--&Q(*-1rgK%1&m*(RT>mO8r!(*+&!§R3(*-1rgRT%?+&')(*+&$%?+&(*V+ (* &(*?(*-feM=Œ+&4cm8r1r*')K2mrgV(*-1rV(*'[I+1Ri.(K%&')(*%1KrM8N-&I'K-gr+&2?rw%&\(G1r]Jr (*-1rV(r'.-&+&/0%1rrnO4rRi#1r+X(*Ri4J+1!6V Š1VhK%12&!K?-?3+&‹&*r*r+W(Y(*-&rQ!K(*-?Vw( +&*r*(\.K^+&*(K%&')(*+&e œo

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y#Hz6Y{#vw k v$›YvwQx†‡vw}H{z6ƒP>vMˆfƒE{Vˆ ‰ +(*-?D'.-&&(rc[8r61r*')K2mrV(*-&r>r)ŠJmrKr+X(*:*r(*%1+?(*-1r*r*%?(*eg=Œ.(cŒ8:r&*rr+X( (*-1rrmr'U?'(*+?BRh(*-1rDRT%&+?')(*+&%?+&(*N')Kr(rkI+G(*-&N8:Kne\,$-1r+G8rr1r*')K2mrY(*-&rDJJ U?'(*+&YV1rQ(H(*-1r L ?r L '\"AFKD%?(YRTM(*-&Y8:*Ee>jAr)ŠJ(c78rQ1rK')K2mr(*-1r ''%&I(*+&#RN]K%&r')m+1r+X(*rR$r+1rK!4§8N-?'K-*r6*r/0%&*r§2X4§(*-1r')?rDRT(*-1r +&*r*(*+gR*rrI+&(K%&')(*+&eM=Œ+&I4cn8rV*-198(*-1rVr!rr+1r*!4*]J+1!w2?(*+1rk+ r($R[Fgr+&'K-HrM2mr&1r62mr+&'.-&V*Je kSd u60TE u6XO?>Wr#q\AXr0…61O0 “7~9Tn~3WB7I rM')Kr(rHR%&+&')(*+?S%&+&(*+')n+&')r38N(*-6(*-1rYR%&+&')(*+&mmr'Un'(*+&RŒ(*-1rYR%&+&' (*+&%&+&(*RTN(*-1rDA"\F?*0')rKe:=&N(*-1rYn%1*mr3RŒr(*V(*I+1!mO8rc&(*-1rrDRT%?+&')(*+& %&+&(*V*r1r*')K2mr*(K%&')(*%1KI4%?*+1!O8rr]r\')m+1r+X(*>IrG™2?('**40-1r ~wNJ&%&rcm92?(NKr!(rKcS92?(w*-&Ri($Kr!(rKcS92?(\D%?(*?r)ŠJrKc?r(*'e\8A-&'K-g*rr')+J (K%?')(r +'K'%&(Vr]rA+& *rg'.-&K')(rK¡rRT>m98:r%&*+1! o+&F lYL ‰ L (r'.-&+1!4 J1rJU?re7,$-&rBE98:r*!(rQ]rKK+RSJ(*-1rrA')m+1r+W(*`r?94#+>&&*?K(r4#*¡r RTX(r*rr>(K+&**(e[=1:r)Š1?rcX(*-1rwœn92?(&1r:I+(*-1rAI+W(r!rA~@`')+?(K%&')(r>%1( R7r*(*!rBR~(*-&r\™2n(N 0?%&rA+?6(*-1r3:0(*-G#%&(*nr`B')V&Kr&.VK4DR[œn9 2?(\&1rcfœn92?(\#%&(*nr)Š0r.N+&‹œn92?(w*-&Ri(A*r!(rKew,N-1rDm98:r!(rRT%&+?')(*+&7%&+&(* ')+&*I($R[(*-1rDE98:r*!(rk')VE+&r+W(*A8N-&r3(*-1rY*r!%&INR%&+&')(*+&%&+&(*$KrD')&.rGR ')m+1r+X(*$8N(*-1%1(B+W4*rr(K+&**(Ke œJ–

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kSd? z i9ˆX9THƒ:?nT#WB~i ,$-&r\+X(r!rN\@qErKRiKQ:+X(r!rNK(*-&r(*'AmrK(*+&r3&?(*+Sc0*%12&(K')(*+Sc1')r ?KI+mrK(*+&c?+&G!I'fmrK(*+&NrM+?'%&*]rwcE+&Sc?+&(N+&Gr)Š1'%&K]r3e,$-1r \@%?+&(D&Xrr+1(-&O]r(*-1r6E98:r*!(*+1!Rir(*%&*r62Er'%?r6(*-1r+W(r!rV?1r#(*-1r( RT*r/X%&r+W(*46%&r6R%&+&')(*+&S%&+&(B+&!Qm(*-1r\%&+&(*e kSd?cb ‘#&9 aWAˆS7>qdO0>7i @\+&1r(*-1rwA"\F +&*(K%&')(*+>r(cJ(*-1rw2?**rn*-&RT(rI:%&*r>8A(*-V(*-1r3K(*-&r(*'A+&>!' +&*(K%&')(*+&7+&Q(*r$+&*(K%&')(*+&e~,$-1rB+?(K%&')(*+?8N-?'K-r%&rB(*-&r$A~@+&r(*-1rB2n**r *-&Ri(r[(*rA(8D')4J'r`+V(*-1rBr)Š0r'%&(rA(*!re`,N-1rB2?**r1K-&RT(rhmr*Ri.VŒ(*-1r$K(*-?r(*'K!-X( *-&Ri(c1!'rRi($+?K!-X($*-&RT(RT%?+&')(*+&B+Q*+1!r3')4J're kSd?lk ‘#S&9œQ†‡ZWi9XYWi06~i ,$-&rr#%&(*?4H+?(K%&')(*+?AVr>%&rRG%&+&(\(*-&(MI?rr+X(*3n92?(Y0(*-S¨M!.(*-&e ,$-1rKr*%&(Yw(*-&rQr(MK!+&U?'+X(Yœn2?(*wR(*-&rV&*J&%&')(wR(*-1rV(^8:œn92?(3mrK+&&eV,$-1r ')K(*I'f?(*-HR[(*-1rD%&+?(N-&N(*-1rD1rI4JNRh›9+1?%1(AD%&(*?r)ŠJrN+&H(*-&(NRhœn92?(w&1r 1rO4e Z %1K+&!#(*-&r3U&K($')4J'rc&(*-1rY''%?D%&I(N*r!*(rN2&*%&!-W(B(V(*-&rMA~@Mc?8N-&'.->r(*-1r (K+&KV(*(>6&K0&%?')rQ¡r*1rmr+&&+&!‹+ (*-&rI+&(K%&')(*+2mr+1!§FgGFH@\7V( +&(*¡r3(*-1r#1r(*+?(*+HKr!(re Z %&K+1!V(*-&rD*rr')4J'rcm(*-1r##%&(*?rBmrK+&gN1r +X(D(*-&r\0(*-S¨*-&RT(`*r!I(re[,$-1rwD%?(*?r[Er.+&Q*-&Ri(r>K!-X(nM2?(*:mr:')4J'rc1+&Sc (*-1r*rRT*rY(*r#–p')4J'rB(')?%1(r3(*-1rMU?+&m*r*%?(e ,$-&rNX(*-S¨:D%?(*?rŒ%?rV+\"AF ]rKK+>\rnO4JhrK4#(r.V+&(*+fe7•AO8r]rcJ(*-1r ]K2?I(4r+(*-1rAU?+&K-&+1!w(*rARS(*-1rA#%&(*?'(*+rmrK(*+Q-&+1(`2mrr+>0&rrQ+Q(*-1r ')4J'rM''%1K(r#*#%&($+&Hr8:K*(N'r–pV')40'r9B(r+&')4-&B2mrr+G%&*rSe œn

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kSd?„ “/WT?1 T~§AnE6~i9 ,$-&rV&(*+1!m+X(3R%&+&')(*+&`%&+&(*3-&O]rQ2mrr+J1rrRi#*%1&m*(*+&!>(*-1r ‰zXr"}#zQc |1""#" Z L ?mr+&r g "{zH"}#zQ "* L :?mr+&r ,$-&rr3?mr+&r:-&O]rD')VV+Rir(*'.-H+&H1r')0&rM(*!re •AO8r]rcRTw%1w8:Kncm(*-1rr&(*+&!>m+X(NR%&+&')(*+?~%&+?(*A-?]rr2mrr+k*r?K(r4k?r) r+X(rHa g ?(*+1!m+X(M"". nQ d |a& g ?(*+1!m+X(Jn"|J E'Œ|a& g ?(*+1!m+X(’zmŽ9>zXr"}#zQc |1""#" |a& r')+&(K%&')(~*+1!r)?*r'*+\R%&+&')(*+?%&+&(*S%&K+1!(*-1rh?.r(rK¡r3?(*+1!$m+X(S%&+&( 1r*!+&N?*rr+X(rg+ œ ™1cœ›9eM•N98:r]rOc*+&')rD8:rr')+&(.%&')(A(*-&rr1r*!+&A%&*+&!>')m+1r+X(* RT* %1N2&K*4c0(*-1rY(r+&'rBR~(*-&rMD%?(*?4V%?+&(+&(*-1rY&]0&r\+&K/X%&Kr3*X($%?+&(B*r &mr*r+X(BRiK’(*-1rMI+6(*-1r3K!+?m8:*Ee kSd?c… “/WT?1 T~§AnE6u60~?}ˆ3WX†W~§?~¢}G?}ˆ#WTX†W~7i9 ,$-&r+&KV¡r RTKV(RD&(*I+1!EI+W(] I%1r61rU?+1r¢6(*-1rRTKV(+ 8N-&I'Kr)Š1')(*4+1rQ+1+J¡rKg&!(\RTKVY(*-1rQ+X(r!rM?K(3R(*-&rQV+X(***68N-?'K-I+‹'rQR2?+?*4 *r&Krr+W(*(*+G^I–¨e ,$-&r$*%&+&?+1!w+&Q+1.V¡(*+Q%&+&([*%1?EK(*Œ(8DRm(*-1r$RT%1hK%&+&&+&!wV01r:mr'U&r 204 ‰C`C:C (*+&?KSaNi:*%?+&6(V¡r*Q+?dn*%?+&(>+1r*r*(e =Œ!%1KrBœ0e–BK-1O8NŒ(*-&rK'K-1rV(*I'B&!KVŒRn(*-1rB1r+1KQ¡+1!\+&+1KQ¡+1!A0?%&re œœ

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Norm Rnd_Add rnd e_out f_out ex_in done s_out Bitwise OR rdy e f ex_in f_int e_int s e s f s_out f_out e_out (a) (b) =Œ!%&*rMœ0e– L '.-1rV(*' Z I!K’R Z r+1KV¡+1!+&GjAKV¡+1!VFGJ&%&re kSd?™ “/WT?1 T~§AnE6z>i9ˆX9TH~i ,$-&rD&(*I+1!6EI+W(wK(*-&Vr(*'D%?+&(Nmr*RTKVA&(*+&!m+X(w&&(*+k+&*%12?(K')(*+k1 rK(*+?ek,N-1r')+X]rK*+V0&%?r3Y+X(r!rGT2E(*-*!+1r§+?§%&+?*!+1rnw(k?(*+1!kEI+W( ')+X]rK*++&?(*+1!Gm+X(3(kI+W(r!r6T2E(*-*!+&r+&%?+&*!+1rEN')+X]rK*+Sc[-&]r6+1( 2mrr+V')+&(.%&')(rQ+r(*-?Œ8:Kneh=Œ!%1KrNœ0ešn3*-198Nh(*-1rA*'K-1rQ(*'$&!.Rm(*-1rB&(*+1!MEI+W( K(*-?r(*'3%&+&(e kSd?Lš “/WT?1 T~§AnE6†PWi9XYWmV7i ,$-&r>&(*I+1!gm+X(rD%&(*?4‹%&+&(Mmr*RTKVD&(*+1!km+W(#D%?(*?4mrK(*+fe,$-1rU&K( (*!rR#(*-1r‹J&%&r#%&(*')4J'rdp')4J'r.>2mr'%&rR#(*-1r(r+&')4¢')+&(KI+W(*6R#(*-1r ]2nr##%&(*?r\1r*!+&\+k%132?K*4e3=Œ!%1*rœ0ešœ6*-198Nw(*-1r*'K-&rV(*'&!. R:(*-1r &(*+&!EI+W($#%&(*n4V%&+&(e kSd? “/WT?1 T~§AnE6umsfT7gnqˆmf~& .>S&~i ,$-&r\&]0&rA+&*/0%&*r\*0($%&+&(:mr*Ri.”K+1!rA&*r'*+>&]0K+Q+&K/X%&KrA*0(BmrK (*+&:%&*+&!MY(*2?rA00%&>*'.-1rr# œ›9e:=7!%1*r\œ0e ™r*-198N:(*-1rA*'.-1rV(*'A&!K—RS(*-1r\%&+&(e œ ™

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Shift Adjust Swap Add/Sub Corrector Comparator en ex clear f s e1 e2 f2 s1 f1 sm lg op1 op1 ex_in rdy s2 s1 e ex_out res done =7!%1*rMœ0ešn L '.-1rV(*' Z I!K¤RŒ=Œ(*+1!Q57+X($AK(*-&r(*I'3@A+&(e œ›

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e2 e1 s2 s1 Multiplier (multicycled) f1 f2 Zero Detection Exception Detection Exception Detection bias 1 Adder rdy ex_in op1 op2 done ex_out e1,f1 e2,f2 Concatenation Subtractor sign exp man 0 res Mux =Œ!%&*rMœ0ešœ L '.-1rV(*' Z !.¤R[=Œ(*+1!V57+X(AFg%&(*?4V@A+&(e œp

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Multiplier Multiplier Multiplier Multiplier 4 Operand Adder (CSA) Lookup Table Divisor (2m) Lookup Table Lookup Table Sqrt Op (4j) Dividend (2m) Mux Mux Mux Mux Multiplier (2m+2) (m) (j) (j+1) (4j) (2m) (j) (j) (j) (j) (2j) (j) (3j) (2j) (2j) (3j) (4j) (2m+2) (4j) (2m+2) (4j / 2m+2) (4j) (2m) =7!%1*rMœ0e ™ L '.-1rV(*' Z I!K¤RŒ=Œ(*+1!V57+X( Z ]J1r3+& L /0%&*r)"NX(A@A+&(e œ

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Denorm FP Module precision module IEEE single single precision Operand in IEEE single precision Operand in IEEE single precision Result in IEEE Denorm Rnd_Norm =7!%1*rMœ0eš› ‰C`C:C L +&!r35h*r'K+GFGJ&%&re =1 ‰C`C:C *+1!rA&*r'K+VRTKV(cŸ D+&0  –n0e L (*!r3–c&n0c&+&6œ#-&]rY(r+?'rR~0c XcS+&Q'J'*g')40'rcn*r*Er')(*]r4cn2mr'%&*rDRh(*-&rD#%&(*')4J'rH#%&(*?rKe=&wK/X%&KrD*0( mrK(*+6n(*-1rw(*!rBRf(*-1rw?ErI+1r$*rw%&rSeh=&$&]J*+mrK(*+ScX(*-&rw*r')+&6(*!rY 204X?KrG+&+?4Q(*-1rDD%&(*?r.`+G*(*!rQ–Y+&G*(*!r#œr*rM%?rSe kSd?I¡ z6aˆCWmV#q$9~k“/WTn19NEE>†PS7WT0 =Œ!%1Krœ0eš›w*-&O8NŒ(*-1rB**rM2?4YR?(*-1r&(*I+1!wEI+W(~RT%&+?')(*+&0%&+&(*78N(*-D(*-1rB+1KQ¡+1! +&G&r+1KV¡+1!r0?%&re ,72?rYœ0e–c?1rK')K2mr`(*-&rY(r+&')4>*Er'U?'(*+&R7(*-1rr3')m+1r+X(*BRi$r'J'*mrKJQR –o6+&ew,$-1rr#(r+&')4G]%1rw*rr%&rk&%1.+1!Q')4J'r)''%1K(rVKD%?(*+G8A(*-H(*-&rDrD2mr&1r 2mr+&'K-?V*Je œ

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,~2nrMœ0e–M~(r+&'rBR[=&%&+?')(*+&f@\+&(*e =?%&+&')(*+& 5:mr+1r ~(r+&')4 %?+&( (*!r ')4J'r. ‰ +X(r!rA\@ o – **r L -&RT(r o – ‰ +W(r!rwFH%&(*?r – –p =Œ5 NK(*-&r(*'w@\+&( ™ p =Œ5 Fg%&(*?4Q@\+&( œ –o =Œ5 Z ]X L /X*($@\+&( ™ –Ÿ kSpb † SŒX~W19Tn7OˆS>ˆ‰YWT?ˆm#Wi?.z6† ˆS>ˆZWT19? ‰ +(*-&IYr')(*+8:r1r*')K2mr(*-1r>0?U?'(*+&Y(*-&(M8r*r>&+1rV(H(*-&r L V?r*'Y\"AF +1RTK(.%&')(*%1*r œo ~(*-&(N8A%&r(QI?rr+X((*-1r3&*mr6(r'.-&+&/0%1re kSpbS †PS7X719Tn79§z6>† x ˆfzu6tŒ~i9TE “NXWTJ ,~&*9]0I1rG(*-1rH'n2?(^4§R3''%1.(r4r*%1.+1!(*-1rH%&K!rHRw(*-1rH]K%?RT%?+&')(*+& %&+&(*B')+&(K%&')(rGrKr$(*-&rYA"\F ‰ L P1rU?+&(*+U?r8r*rMJ&U&rfe ,$-&r§1rU?+?(*+R*r*%1K')r'*rgRik(*-1r§(*-1r\"AF .'K-&(r')(*%&*rI+(*-1r-1r1rkU?rc ”"£¢¤+“¦3›§a¤mc[8VJ&U?rSe,$-1r'K+¦' M¨k8V&&r(‹&*O]J1r6(*-1rG'n2?(^4(mr RTK')4J'r)''%1K(rVKD%?(*+G')+&K1rK+&!(*-1rM2?K*r*-?Ri(rA\>r?K(r#R%&+&')(*+&%&+&(e ,$-1rkr)Š1(*+1!r?K(r‹'Kr 'šta) +5+& 'štu{ †M8r*r‹r*!r¢+X(§*I+1!rg'I* š%a) " MY£EeŒ,N-1rG'*r+¦' "5nc'šta) †aEc7+&n'štu{ M8r*r*r9]r*+?')r(*-&rr '**rB8:rKrM%&+0%&r+6(*-1r ‰ L P1rU?+?(*+U?rc%”"£¢¤"“>¦3›§¡a›%™ne ,$-&r\"AF ‰ L 1rU?+1rr*%12?*r(MRN(*-&r6?(*g?*0')rK*+1!k+&(K%?')(*+&D(k2mr62?r(‹%&r (*-1r>2?**r*-&RT(rMRTr*-&RT(*+1!+1r>RN(*M*r!(rrmrK+&&M2mrRi*r>mr*Ri.V+1!(*-1r+X(r+&1r mrK(*+Se ‰ (QI1rU?+1rV(*-1rH+?(*rk+&(K%?')(*+&#(2mrG2?r(§ Er')(>(*-1rH+&&r)Š ]%1rM8A(*-(*-1rr-1rGR`(*-1rD*-?Ri(r\&%1KI+1!+&&Kr')(N&1*rK*+1!1eB•AO8r]rc L nr*'N\"AF +&'I%&1r7(*-1rN*-?Ri(*I+1!wRT%&+?')(*+&(^4D+X(3(*-1rNI+W(r!rA~@ +?V10rh+1(h(*r(:(*-1r$2?K*r1*-&RT(r œŸ

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