EEM 486: Computer Architecture

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Toby Leonard
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1 EEM 486: Compuer Archecure Lecure 4 ALU EEM 486 MIPS Arhmec Insrucons R-ype I-ype Insrucon Exmpe Menng Commen dd dd $,$2,$3 $ = $2 + $3 sub sub $,$2,$3 $ = $2 - $3 3 opernds; overfow deeced 3 opernds; overfow deeced dd unsgned ddu $,$2,$3 $ = $2 + $3 3 opernds; no excepon sub unsgned subu $,$2,$3 $ = $2 - $3 3 opernds; no excepon dd mmede dd mmede unsgned dd $,$2, $ = $2 + ddu $,$2, $ = $2 + + consn; overfow deeced; sgn exenson + consn; no excepon; sgn exenson Arhmec operons: dd, subrc Lec 4.2

2 MIPS Logc Insrucons R-ype I-ype Insrucon Exmpe Menng Commen nd nd $,$2,$3 $ = $2 & $3 3 opernds; Logc AND or or $,$2,$3 $ = $2 $3 3 opernds; Logc OR xor xor $,$2,$3 $ = $2 Å $3 3 opernds; Logc XOR nor nor $,$2,$3 $ = ~($2 $3) 3 opernds; Logc NOR nd mmede or mmede xor mmede nd $,$2, $ = $2 & or $,$2, $ = $2 xor $,$2, $ = $2 Å Logc AND reg, consn Logc OR reg, consn Logc XOR reg, consn Logc operons: nd, or, xor, nor Lec 4.3 Numbers Bnry numbers (bse 2)... Bs re jus bs (no nheren menng) convenons defne reonshp beween bs nd numbers Of course ges more compced: numbers re fne (overfow) frcons nd re numbers negve numbers How do we represen negve numbers?.e., whch b perns w represen whch numbers? Lec 4.4

3 Possbe Represenons Sgn Mgnude: One's Compemen Two's Compemen = + = + = + = + = + = + = +2 = +2 = +2 = +3 = +3 = +3 = - = -3 = -4 = - = -2 = -3 = -2 = - = -2 = -3 = - = - Issues: bnce, number of zeros, ese of operons Whch one s bes? Why? Lec 4.5 Two s compemen 32 b sgned numbers: wo = en wo = + en wo = + 2 en mxn... wo = + 2,47,483,646 en wo = + 2,47,483,647 en wo = 2,47,483,648 en wo = 2,47,483,647 en wo = 2,47,483,646 en... mnn wo = 3 en wo = 2 en wo = en Lec 4.6

4 Two's Compemen Operons Negng wo's compemen number: nver bs nd dd remember: nege nd nver re que dfferen! Converng n b numbers no numbers wh more hn n bs: MIPS 6 b mmede ges convered o 32 bs for rhmec copy he mos sgnfcn b (he sgn b) no he oher bs -> -> Lec 4.7 Addon & Subrcon Jus ke n grde schoo (crry/borrow s) Two's compemen operons esy subrcon usng ddon of negve numbers + Overfow (resu oo rge for fne compuer word): e.g., ddng wo n-b numbers does no yed n n-b number + noe h overfow erm s somewh msedng, does no men crry overfowed Lec 4.8

5 Deecng Overfow No overfow when sgns re he oppose for ddon No overfow when sgns re he sme for subrcon Overfow occurs when he vue ffecs he sgn: overfow when ddng wo posves yeds negve or, ddng wo negves gves posve or, subrc negve from posve nd ge negve or, subrc posve from negve nd ge posve Consder he operons A + B nd A B Cn overfow occur f B s? Cn overfow occur f A s? Lec 4.9 Effecs of Overfow An excepon (nerrup) occurs Conro jumps o predefned ddress for excepon Inerruped ddress s sved for possbe resumpon Des bsed on sofwre sysem / nguge exmpe: fgh conro vs. homework ssgnmen Don' wys wn o deec overfow MIPS nsrucons: ddu, ddu, subu noe: ddu s sgn-exends! su, su for unsgned comprsons Lec 4.

6 -b ALU Bnver Operon Bnver Operon Resu Resu X nd b X or b b 2 + b - b CrryOu -b fu-dder: b Sum Cou = b + Cn + b Cn nd Sum = xor b xor Cn Crryou b seec c 2 o Mupexer c = f seec= ; c = b f seec = Lec 4. Torng he ALU for MIPS Need o suppor he se-on-ess-hn nsrucon (s) remember: s s n rhmec nsrucon produces f rs < r nd oherwse use subrcon: (-b) < mpes < b Need o suppor es for equy (beq $5, $6, be) use subrcon: (-b) = mpes = b Lec 4.2

7 Supporng s B n v e r O p e r o n B n v e r O p e r o n b 2 R e s u R e s u b 2 3 S e 3 O v e r f o w d e e c o n O v e r f o w C r r y O u - b ALU - b ALU for MSB Afer 32-b subrc operon: < b => Se = nd Resu =... o.w. => Se = nd Resu =... Lec 4.3 A 32-b ALU Benege Operon operon b A L U C r r y O u R e s u 32 ALU 32 resu b A L U C r r y O u R e s u Z e r o b zero overfow 2 b 2 A L U 2 C r r y O u R e s u 2 32 CrryOu 3 b 3 A L U 3 R e s u 3 S e O v e r f o w Conro nes Funcon And Or Add Subrc Se on ess hn Lec 4.4

8 Rppe Crry Adder s Sow! A 4-b Adder: Cn you see he rppe? c = bc + c + b c2 = bc + c + b c2 = bbc + bc + bb + bc + c + b + b c3 = b2c2 + 2c2 + 2b2 c3 =... c4 = b3c3 + 3c3 + 3b3 c4 =... Grows exponeny wh he number of bs c hs 2^ erms Rppe crry dder s no fesbe for wde dders Lec 4.5 Crry-Lookhed Adder Observe h c(+) = g() + p()*c() where g() = ()*b() c = g + pc p() = () + b() c2 = g + pc c3 = g2 + p2c2 c4 = g3 + p3c3 c2 = g + pg + ppc c3 = g2 + p2g + p2pg + p2ppc c4 = g3 + p3g2 + p3p2g + p3p2pg + p3p2ppc Q: Dd we ge rd of rppe? Q: Fesbe. Why? Lec 4.6

9 Crry-Lookhed Adder b b 2 b2 3 b3 4 b4 5 b5 6 b6 7 b7 ALU P G ALU P G C C2 p g c + p + g + c + 2 Resu--3 Crry-ookhed un Resu4--7 C(+) = G() + P()*C() where C = c P = p3p2pp, P = p7p6p5p4 P2 = ppp9p8, P3 = p5p4p3p2 G = g3 + p3g2 + p3p2g + p3p2pg 8 b8 9 b9 b b 2 b2 3 b3 4 b4 5 b5 ALU2 P2 G2 ALU3 P3 G3 C3 C4 p + 2 g + 2 c + 3 p + 3 g + 3 c + 4 Resu8-- Resu2--5 G =... G2 =... G3 =... CrryOu Lec 4.7 Summry We cn bud n ALU o suppor he MIPS nsrucon se key de: use mupexor o seec he oupu we wn we cn effceny perform subrcon usng wo s compemen we cn repce -b ALU o produce 32-b ALU Imporn pons bou hrdwre of he ges re wys workng he speed of ge s ffeced by he number of npus o he ge he speed of crcu s ffeced by he number of ges n seres (on he crc ph or he deepes eve of ogc ) Crry ookhed offers fser ph hn wng for he crres rppe hrough 32 -b dders Lec 4.8

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecure 0 Canoncal Transformaons (Chaper 9) Wha We Dd Las Tme Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple δi δ Addonal end-pon consrans pq H( q, p, ) d 0 δ q ( ) δq ( ) δ