Arithmetic: Past Revisited Milo s D. Ercegovac Computer Science - - PowerPoint PPT Presentation

SLIDE 1

Arithmetic: Past Revisited

Miloˇ s D. Ercegovac Computer Science Department University of California Los Angeles Ambitious beginnings Two pillars of digital arithmetic: signed-digit and carry-save DCL and Illiacs at the University of Illinois Urbana-Champaign Antonin Svoboda and ARITH-4, Santa Monica, 1978.

SLIDE 2

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Two specimens from the past

Early attempt at arithmetic chaining - A. J. Thompson integrating and differencing machine made of four connected calculators. Unfortunately, Moore’s Law did not kick in here.

1

SLIDE 3

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Enthusiasm for fast arithmetic rising - mathematicians in peril! Eventually rounding problems will save them. New York World, March 1, 1920.

2

SLIDE 4

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Amusing and, perhaps, silly today?

That has been the fate of most ideas and their implementations – after years passed by. But there are some ”old” ideas in arithmetic that don’t share such a fate ....

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SLIDE 5

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Amazingly modern work on signed-digit arithmetic: Early 18th century

4

SLIDE 6

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Reverend John Colson, first Vicar of Chalk, Kent, then Lucasian Professor of Mathematics at Cambridge (1739-1760) - translated Newton’s works - Fellow of Royal Society. Philosophical Transactions of the Royal Society, Vol.34 (1726-27) [In May 1726, the London newspapers announced the arrival of Francois Marie Arouet, the renowned French dramatist and poet, known as Voltaire. Surely not to practice signed-digit arithmetic - after a brief free stay in Bastille, he was exiled for behavior that irritated nobility.]

5

SLIDE 7

ENS Medal Talk - 22 June 2015 ARITH-22 2015 6

SLIDE 8

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Colson proposes a radix-10 left-to-right conversion algorithm of SD number into conventional form. Let pi and ni denote positive and negative digits (”Figures”). The conversion rules are based on pairs of digits:

• 1. pini−1 ⇒ (pi − 1)
• 2. nipi−1 ⇒ (10 − |ni|)
• 3. nini−1 ⇒ (9 − |ni|)
• 4. All other digits must remain unchanged.

Sign of zero (”Cypher”) is the sign of the next non-zero digit - requires propagation. Pairs of non-zero digits are converted in parallel.

7

SLIDE 9

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Example: Signed-digit input Conventional output Rules: pini−1 ⇒ (pi − 1); nipi−1 ⇒ (10 − |ni|); nini−1 ⇒ (9 − |ni|)

8

SLIDE 10

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Sign Propagation Input digit: xi = (σi, mi), sign σi =(0 if positive, 1 if negative), Magnitude mi ∈ {0, 1, . . . , r − 1}, zero signal zi = (mi = 0) Switch network for sign propagation

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SLIDE 11

ENS Medal Talk - 22 June 2015 ARITH-22 2015

SD–to–Conventional Radix-2 Converter Circuit

10

SLIDE 12

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Throwing out all the large digits {9, 8, 7, 6} → {1¯ 1, 1¯ 2, 1¯ 3, 1¯ 4} 5 ”esteemed” to be large or small depending on context Rules: 1. SL ⇒ (S + 1); 2. LL ⇒ −(9 − L); 3. LS ⇒ −(10 − L)

( 163)

• r

iall, according as the Ftgure following is ciher

large

• r Emall. Some

Examples

• f this

Redudion l)!alE Ilere follow, both in whole NumlJers and Decitnat Fr$- d;tionsz

_ a

37O68259764 43 I 3234o:++

7295289607 39957

I 3 3 I S3 IO 4I 340043

g26087239S,87294 = I 1 34

I I32404,133

I4

Or (9)92So87239S87294

(to) 1; I 34I13240413 3 tR

(m)3879I64O7953>&C

• (m) AI2I244I21$3,

&c. It ts to be obServed that in this lal Exampfe tl> Numbers are what I call

infermixate,

• r Approximatiff

Qns

• nly; that

is, the {;rR and moR valuable Figures

are exprefs'd) and all the reR (whether finite or infi- 11ite in S7umber, whether known

• r unknown)

are o mitted as inconliderable, and intinuated by the Marlx &cZ Alfo the Index m before the Number flands for fome Integer, expreSlng the Diflance

• f the firlt

Figure 3 or q froln the Place

• f Units which Integer

is ei ther affirmative

• r negatine,

according as the faid firft Figure {lands in integral or fradional

• Places. Thz

Example immediately before is a particular Ini:lance

• f

ttliS

And thus

much by way of Notation: To procecd

ttlerefore to the

Operations to be perfornzed witll tl<>fe Numbers, wllether reduced to Inzall Figuros

• r nor;

an(l tirS

• f Addition.

131ace

tlac Numbers to be added juR. under

• ne

ano tller, obferving the Homogeneity

• f Places $S in

ec3m

mon

• Nulnbers. Then beginning

at the Right Hand, clledr the Figures in the firEt llow or Codumn, ac- aording to tlleir

Signs,

and place the Refult under

\' 2

neatlz:

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• Index m ”stands for some Integer, expressing the Distance of the

first Figure from the Place of Units; which Integer is either affirmative or negative;” &c. denotes ”interminate” (approximate) number, ”... and all the rest (whether finite or infinite in Number, whether known or unknown)”

11

SLIDE 13

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Multioperand SD Addition

( 164 ) neath: And Io fuccelElvely

• f all

the other Colutnns, as in Example

I.

But if at any time this refult canno£ be exprefsd by

a f1ngle

Figure)

it may

be writ down with two

• r

more Figures)

• bServing

the Homegeneity

• f

Places, and tllen the Sum may be colleded

• ver
• again. But

to fave this trouble, it will be fuflicient to referve the Figure

in rnind which belongs to the next Column, and col- ed it wtth tlle Figures

• f that Column; as in Ex--

amplcs ^, . If the Numbers to be added are reduced to fmall Figures, as in Example 3) their Additiotl will be very sllrlple and the Sum may alfo be exhibited in fmall Fi

gures) by an cafie Subrtitution

• f Equivalentsa

where ttscre is occaflon.

(

)

(2&)

_ _ _ _ _ _ _

25? 384263 647O396 82

__ __ .____ _

t70982 I3 70

49 827365t

_ _

580 7305

sI94Q3 76 $

_ . t_ ,*

_ _ q _ _ _ _ _ _ _ _ _

953 3642IX 5864643894. ( 3* )

._ * _ * _ _

(m) 2IS3I4043 J2I32 &ca

_ _

(S) Jo420 3I4255I22

&cb

(g-I) 43 I023 X 024I 3>

&c

(g-2) sI342II032Iy

&c

.rtn-3)

2I 3042I032

&G

_ _

(94 4)

1 3 202 I 2248 &cv

Cev

5)

X

3:243

T5* &cv

.

(gX 3[) 3333

2X41 343 I2 7 &it err v- z s__ __

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12

SLIDE 14

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Subtraction by adding negated subtrahend:

( ses ) SubtradEtion in this Arithmetick is reduced to Add tion by changing a10 the Signs

• f the Number

tQ be fubtraded

abusiffrom (7W3 7zg384XyG37}

• Ac. we are to

fiubtraA (X-t) 8 xo73 S9z6 &cw the Remainder will be found as in Exatuple 4. (4 )

• .
• S
• (X)

729 98 4z963 7 &cll

_ w _

(s-z) 8 Io73S926i &c*

t b S

_ _ _ _. _ _

) :3747

I 54343,

8co

.

Thus in a11 Cafes wtill Addition and SulJtradliorl beE eafily perfortned: But the chief ure

• f ttlis Metlled

will bes to eafe the trouble

• f prolix

Multipli>tions. And here9 as wrell as in Dierifson, the firll and moll ra luable Figpres may be firA found,, and conSequently the Produd may be continued to as many Places as Ihali be reqllired, withollt finding any unneceXry Figures

^

which is a convenience not to be had in the ordinary way

• f Multiplication,

Let it be propefed to Multiply together the Numbers 86O5yz93987 1 5 and 389175836438> which reduced to imall Flgures wilJ be tl4t433X4oX3rS and 4ttt wiz4444X S$trite down thefe two Numbers

• ne

un der the other upon a ilip of Paper) with the Figures at eqrlal difl:ances, and then cut them aEunderw Take either

• f the Numbers

for a Multipliers and place it

• urer

the other in an inverted pof1tion,

b as its firR

4lgur; This content downloaded from 128.97.245.208 on Tue, 26 May 2015 05:12:49 UTC All use subject to

13

SLIDE 15

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Multiplication - Left to Right

( tti6 )

• figrc. lway

be

• l0
• ver

the firR Figure

• f

tlle Multis

plicand hfoveab m#MSer

xtfkbz+zz

X X

1^

,* # _ t vr

Slkiplicand

* t |F .

I4t.433t40t315

* t _ w *+. *, . _ *

s6 1 79

X

8

X

S4gS 6qoG

l 606080

_ _ _

IIIX3 1124

td3^

X t3t1

_ _ . w, _ _ _ _

Pfaak2 = 4Sio86tg37Og6ot707z6X3I70 Then Multip3y thetE twto 15rR Figures togetber, and their Produdt (4 x I-47 place underneath. Then move your Multiplier a place fbrwarder, fb that two

• f

its firR Figures may be

• ver

two

• f the

firR Figures

• f

tlle Multiplicand; and colleding their

tWO Produds

(4xtElxI=s) put their Refult underneath inthe

next

• place. Move the tMultiplier

a place forwarder. and colled:ting the ehree Produds

(4X4

+ t X t + r X X X

c) put the Reilt underneatbs as in tlle Example lQove tlle Multiplier, and colleA tlle four Produds

_ _ . . *

(4X I + I X4+ I X I + 2 X I - It ) , liCll write under

neath as before. And fO proceed by Qne

rtOp at a

eime, as ong as any Figures

• f the Multiplier

can be

• ver any Figures
• f the Multiplicand. Lalily col-

leEt the ProduEt into one Line, wIlich being reduced to a common Number will be 3-34gt4I93dpo3gg69 a7377X7OJ FroLn this ProceX it may be obfirv-ed, rllat at > srery nesv S>ituation

• f the moveable

Multiplier ehofe

Figures

• nly

are to be Multiply'd tgether each by cacll

as arv iXnd

• ver
• ne an(9tterN

Ared

ttliS NSulBk

tip.0it.

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14

SLIDE 16

ENS Medal Talk - 22 June 2015 ARITH-22 2015

• Convert operands to ”small” signed-digit form
• Perform pairwise multiplication of overlapped digits and add:

4 × 1 = 4 4 × ¯ 1 + ¯ 1 × 1 = ¯ 5 4 × ¯ 4 + ¯ 1 × ¯ 1 + ¯ 1 × 1 = ¯ 1¯ 6 etc.

• Shift the multiplier one position right after obtaining sum of digit

products

• Repeat for all digit positions
• To obtain an approximate product, compute one more place

15

SLIDE 17

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Multiplication - Right to Left

Moveable Me41tiplier

xbEbfzb%r

X

1f

_ _ _ _ _ _ _

Mkiplicawd

,ffl, _

I 4I43

3 t4O1 3iS

*. 1. w _ e , ^ ,

• :

3349X419369o39969z7377X7o

• Prodd

Place the 1noveable

Multiplier

inverted in fuch a nanner, as rhat its laft Figure

2 maY be

ju&

• ver S

tile laR Figure

• f the Multiplicand. Multiply

thelie to gether (; x

s-IO

) and

fet down the laft Figure

• f

the Produd;t

• jal under,

referving the firR Figure 1

for the next place Then move the Multiplier a place

iorwarder

fo that two of its laft Figure§

may be qrer two

• f the

laR Figures

• f the

Multiplicand, and then Multiplying and colled;ting you will have

^I + 2 X I + 4 X 5 = I 7.

Set down 7 in the next place

• f the EroduEt,

and reServe

• t. At the

next remove you willhave

I + n-X ^t+4XI+4X 5 = 31.

Setdown

w

X anc carry

• 3. T zen 3+1

xI +

4 x3

+4X I++X

s

2 3 - 37* Set down

7 and carry

3

Then 3 +-t X

_ _ _

• +4Xt+4X3+4Xt+4X5

= 3 = I7,

bet down

7 and

carry

I

And fo proceed as long as there can he any F}gures

• ver
• ne

another, and the

Produd wiSl

be found as before. This way of i!wlultiplication is fo eafy, and may be

made fiQ fsamiliar

by a little Pradice, that it will be but little Ihort

• f M8ltiWlication

by Zn>Spediot,

and eJill

doubtleSs feem very {urprizing to thole who are

• nly

acquainted with the common tediolus

AFaat

• f Multipli-

eavion : efpecially if we content

• ur

felves svith a rnen- ta1 preparation

• f ttie

N;umbers gixren,

• r
• nly

mark shole Figures rhat ate to be changed) uthich by fotne Pradice is eaElly atrained

170 e

( 168 )

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• Shift the multiplier one position left after obtaining a product digit
• Perform pairwise multiplication of overlapped digits, form their sum

and add carry immediately from the position to the right

16

SLIDE 18

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Signed-Digit Division

• Operands first reduced to small digit form; form divisor multiples
• Quotient digit selection by comparing residual with divisor multiples

(1 × d, . . . , 5 × d):

( t76

tO Emall

Figures and form a Tarifls

• r able of a13

the Multiples

• f the Divifor

as far as 5* Compare thefe Multiples with the Dividend, and with the Weveral .Remainders after the Mllltiples have been Subtrad:t-

cd) by tvhich

means you will dificover tlle feveral finall

Figures and their Signs, to be put fuccelfively in the

QOtiente

• C3

forrn the Table

• f Multiples,

Set down the Disrifor above, draving a ltnea under which Set doxvn the Disri for

• veragain,

putting

X

• ver

againI}

• it. Add

thefe

tWO

togetheraccording to the Rule for Additionin Snall Figures, anel put 2 over againft vlleir

• Sum. Add tllis

laIl and tlle DiviSor together, and put 3 oxrer again their

• Sum. Then

add this

laft and the

Divifor togetller, and put 4 over againft tlletr

• Sum. La(tlyX

add this arld the Divifor together, putting 5 over againll their

• Susn. Tflus

will you have a Table

• f all the

N5Ulti

ples of the Disrifors as fir as will be necefEary Tllus for Example, if (^n

+ n) 5639567S8t22 &c.

_ _ _ _ _ _ _

=(mEn+X) 444o43z4zzzzX &c. is to l)e di vided by (n) 1836097Xw699, &c. - (n)

£2-4+

_ _ _

103E345I,

• aC. the

Procefs will be as here follows, by which the Quotient will be found (m) 3

I 3

t SO3

4t3 S4 &c. _ (m) 307X497+1746,&C

tH

B L E of llIteltiplese

_ _ _ _

(n) 2244l03

134$I &c.

_ _ _ _ _

f

(n) 2244I 03 I 345

I, &cs 2

(n) 43 3

2XT4253 02n &c.

_ _ _ _ _ 3 (n) .SS

X

23

X

t42I 5 X, 8gc 4 (t1+1) Xg

g44412$1+04n&c.

_ _ _ _

5 (n

+

I ) 1 I 22o5 I 443 24&) &c

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17

SLIDE 19

ENS Medal Talk - 22 June 2015 ARITH-22 2015

( 171 )

t8otiett f) 3 I 3 I So bA23 S4) &ce

_ _____

Diqvi4 (2tZ 4 1) + I) 144043

2422

22)

&c.

_ _ _ _ _ _

55

I 2 3 T

I42 I 53v &G.

• .

* s

I 3 1 3 2 42 0 3 3 5v &Ce 224JI

• ; I g45v

&c

__ . * _ _ _ _ _

XS23 345 lot o) &c

S S t ^ g X I42 I ) Ac,

• x
• -r

3 g5

X4O4E I) 8(Cw

_ _ _ _ _

22449to3I3X&

".. ,_

t I I 3 3 I 3

24) &cv t2205144)

&c

X S g4220> &Cs 55 I ^ 3 I , &Cf I 24451

&c. 13X4+4X&

__

320f &G.

• 3332)

&cB

F

tS;gx &Ct 55 I g

aG #

s24) &C.

*__

I2n&c .t 4

1 lle This content downloaded from 128.97.245.208 on Tue, 26 May 2015 05:12:49 UTC All use subject to 18

SLIDE 20

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Colson concludes:

£ a73

)

tnoured by a little

J£tention

to tlleir reltedive IndexesS In the Divifion

• f interluinate

Numbers, tlat fime

refiridtions are to obrain}

as are aIready mention'd in Multiplication. +4nd rllts tuay fiSfli¢@ for a fllort Stlmtnary

• f Nega

t;vo-atErmative ArithnetickX as to the ordinary Ope rations

• f AdditionX

Subtraftion} Multiplication and

:)iYtfion. 97hat

improvements anay be had iom 11ence in the Extradion nf Roots, wilether

• f pure or aX

ded Equations) I tizall

leave to fu£llre illtuirF

BUt Il th;ic OperationsX

vvllether in the common

].)eciInal

Aritlllnetick, nr n tiliX here compendioully de(Pribed; and not only in llecimal Arithmeticl) l)ut in tilt fevcral

• ttler

Specxes}

as Duodecinnal

Sexages finalX Centefimal}

• Sc. all theSe Vperations

(Is^ay) may be much more eafily alld readily perforln'ds and

withequalaccuracy5by anlnRrument I llave contrived caJld

ibacss or theCo8atit*lCASe}

wthichI hope {llort- ly to comulunicate to the inquifitive in theSe rnatters} and by whtcll all

lQNg Calculations

may be very nuci

• facilitated. In the mean

time tllis Dlort account 0f Stew gativbafErlnative Arithtnetick, in Denary

• r Decimal

NumbersX may be preluisd} by way of Introdudlorx

to

the knos>}edge atld ttI;e

• f tbe fAid

Infirument

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”I have contrived an Instrument, call’d Abacus, or the Counting Table, which I hope shortly to communicate to the inquisitive in these matters and by which all long Calculations may be very much facilitated.”

19

SLIDE 21

ENS Medal Talk - 22 June 2015 ARITH-22 2015

1840 - Signed-Digit French Way: Cauchy

20

SLIDE 22

ENS Medal Talk - 22 June 2015 ARITH-22 2015

In Extrait No. 105 Calculus Num´

• erique. - Sur le moyens d’eviter les

erreurs dans les calculs num´ eriques, Comptes Rendus de L ’Academie, 16 Nov. 1840: Cauchy describes signed-digit representation and recoding to limit the decimal digit set to {−4, . . . , 5} 1¯ 1 = 9 1¯ 21 = 81 10¯ 2¯ 45¯ 312¯ 4¯ 2 = 976471158 and comments that such a signed-digit set simplifies all operations. In particular, multiples of 2, 3, 4 = 2 × 2 and 5 = 10/2 suffice in multiplication: that is, doubling, tripling and taking half.

21

SLIDE 23

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Example of multiplication: 8 2 5 6 = 1 ¯ 2 3 ¯ 4 ¯ 4 9 9 7 8 = 1 0 0 ¯ 2 ¯ 2 Partial products with simple multiples ¯ 2 4 ¯ 5 ¯ 1 ¯ 2 ¯ 2 4 ¯ 5 ¯ 1 ¯ 2 1 ¯ 2 3 ¯ 4 ¯ 4 1 ¯ 2 2 4 ¯ 2 ¯ 2 4 ¯ 3 ¯ 2

22

SLIDE 24

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Simple fractions to decimal fractions 1 7 =

• 0. 1 4 2 8 5 7 1 4 2 8 5 7 . . .

=

• 0. 1 4 3 ¯

1 ¯ 4 ¯ 3 1 4 3 ¯ 1 ¯ 4 ¯ 3 . . . 1 11 =

• 0. 0 9 0 9 0 9 0 9 0 9 0 9 . . .

=

• 0. 1 ¯

1 1 ¯ 1 1 ¯ 1 1 ¯ 1 1 ¯ 1 1 ¯ 1 . . . 1 13 =

• 0. 0 7 6 9 2 3 0 7 6 9 2 3 . . .

=

• 0. 1 ¯

2 ¯ 3 ¯ 1 2 3 1 ¯ 2 ¯ 3 ¯ 1 2 3 . . .

23

SLIDE 25

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Applies binomial formula for squares and cubes: ((1 0) (1 3))2 = (1 0)2∗104+2∗(1 0)∗(1 3)+(1 3)2 = (1 0 2 6 1 6 9) ((1 0) (¯ 1 ¯ 3))2 = (1 0 ¯ 2 ¯ 6 1 6 9) (987)2 = 9 7 4 1 6 9 ((1 0) (0 6))3 = 1 0 1 8 1 0 8 2 1 6 ((1 0) (0 ¯ 6))3 = 1 0 ¯ 1 ¯ 8 1 0 8 ¯ 2 ¯ 1 ¯ 6 (994)3 = 9 8 2 1 0 7 7 8 4

24

SLIDE 26

ENS Medal Talk - 22 June 2015 ARITH-22 2015

What in the world is this?

25

SLIDE 27

ENS Medal Talk - 22 June 2015 ARITH-22 2015

It’s a 5-bit multiplier! Project Whirlwind 1946. Norman H.Taylor The Five-Digit Multiplier R-134 Servomechanisms Laboratory, MIT, December 3, 1948 (R-134 Vol.2 Schematics)

26

SLIDE 28

ENS Medal Talk - 22 June 2015 ARITH-22 2015

One bit of the accumulator: all parts accessible at all times.

27

SLIDE 29

ENS Medal Talk - 22 June 2015 ARITH-22 2015

It is a carefully designed prototype of a 5-bit multiplier.

• Built as a proof of concept, to learn about technology, to check

circuitry, and to understand errors.

• 2MHz clock; 5µsec multiplication time
• 10 cycles: in odd cycles add, in even cycles perform ”shift and

carry” operation (carry-save)

• probably the first electronic implementation of the carry-save

concept

• 400 vacuum tubes
• a few weeks between errors

28

SLIDE 30

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Man who helped me begin this journey

1968: Norman Scott, University of Michigan, Ann Arbor, visited Belgrade. We met at the Institute Mihailo Pupin where I was an engineer involved in the design of fixed/floating-point 32-bit unit for HRS-100 hybrid computer for the USSR Academy in Moscow. He strongly recommended that I should go to Urbana for graduate studies under Jim Robertson - which I did in 1970.

29

SLIDE 31

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Illinois Days 1970-75 with Jim Robertson

30

SLIDE 32

ENS Medal Talk - 22 June 2015 ARITH-22 2015

HAL 9000 - the deceptive, clever, and sinister computer in Arthur C. Clarke’s and Stanley Kubrick’s 2001: A Space Odyssey, who ”became operational at the HAL Plant in Urbana, Illinois”

31

SLIDE 33

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Spreading Arithmetic Know-How

32

SLIDE 34

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Early PhD Work in Arithmetic Guided by JER

• G. Metze, A Study of Parallel One’s Complement Arithmetic Units

with Separate Carry or Borrow Storage, 1958.

• A. Avizienis, A Study of Redundant Number Representations for

Parallel Digital Computers, 1960. J.O. Penhollow, A Study of Arithmetic Recoding with Applications to Multiplication and Division, 1962.

• R. Shively, Stationary Distributions of Partial Remainders in SRT

Digital Division, 1963. F. Rohatch, A Study

• f

Transformations Applicable to the Development of Limited Carry-Borrow Propagation Adders, 1967.

33

SLIDE 35

ENS Medal Talk - 22 June 2015 ARITH-22 2015

Digital Computer Laboratory (DCL): home of ILLIACS and CS Dept.

34

SLIDE 36

ENS Medal Talk - 22 June 2015 ARITH-22 2015

1949 - 1951: ORDVAC U.S.Army, 40-bit, add 92 µs, multiply 700 µs, 1952 - 1962: ILLIAC I (similar to ORDVAC design), 2,800 vacuum tubes, 5 tons

35

SLIDE 37

ENS Medal Talk - 22 June 2015 ARITH-22 2015

1957 - 1967: ILLIAC II: a breakthrough into a new generation of machines On the Design of Very High- Speed Computer, DCL Report

• No. 80, Dec. 1957, widely read

as a computer design Bible. Prepared by D.B. Gillies, R. E.Meagher, D.E. Muller, R.W Kay, J.P . Nash, J.E Robertson and A.H Taub. Initial 300 copies exhausted quickly.

36

SLIDE 38

ENS Medal Talk - 22 June 2015 ARITH-22 2015

• 100-200 times faster than ILLIAC I: add 0.3 µs; multiply 3.5-4 µs;

division 7 - 20µs Many ”firsts”:

• Fully transistorized, 15,400 transistors, 34,000 diodes and 42,000

resistors; 8K 52-bit words, 1.5 µs access time - CAD?

• Separate carry storage (radix 4) and sum storage (radix 2).
• SRT radix 2 division unit designed by Jim Robertson.
• IBM Stretch project borrowed many of its ideas from ILLIAC II
• One of the first pipelined computers: Advanced Control, Delayed

Control, and Interplay stages (D. Gillies)

• The first computer to incorporate speed-independent circuitry in

control unit (David E. Muller).

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• In 1963 Donald Gillies discovers 3 new Mersenne primes (3000

digits) on Illiac-II in spare time. US Postal Service notices a bit later (1965).

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1966 - 1970: ILLIAC III Pattern recognition - never finished. Good ideas, too early. 32 by 32 arithmetic elements. Anyway, some good arithmetic developed by Dan Atkins.

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1965 - 1981: ILLIAC IV 64-processor SIMD supercomputer; 200 MFLOPs. Many ”firsts”, genesis

• f

parallel compilers and parallel programming, interconnection networks, conflict-free access to parallel memories, ECL logic, 12-layer boards ...

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Old DCL, still alive, inside a new building. CSD moved to Siebel Center, reaching new heights.

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What was DCL in these days?

• Prof. Daniel Slotnick, designer of Westinghouse SOLOMON parallel

computer and Illiac IV supercomputer said it well in 1969: ”Computer Science means nothing. Attempts to give the word meaning yielded nothing but a greasy, stinking, intellectual soup. The Computer Science Department at Illinois is, on the other hand, a group of guys almost all of whom have done something during their lifetime and many of whom swing. Academic ”weltschmerz” and phony respectability are still rare here. Big machines are in the middle of life and most of us are in the middle of big machines.

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The good people here do things, not just write about things, worry about things, obsess about things, criticize things. Good fights can still be seen here - people are deeply involved in their work - they care if people spit on it - they care if it gets done or not. The kids are great. They’re smart; no project could live without them; they’re are irritating - sometimes I could live without them.”

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Nostalgia Notes: ARITH-4, Santa Monica, 1978.

Antonin Svoboda, keynote speaker From Prague, via Paris to MIT Radiation Lab - designing linkage computers for war efforts (Mark 56), back to Prague, proposing RNS arithmetic with Miroslav Valach and designing first fault-tolerant computers SAPO and EPOS, and, finally, to UCLA.

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SAPO, a relay computer with stored-program capabilities, had three arithmetic units comparing the results to ensure correctness. EPOS was a vacuum tube computer, utilizing RNS. At UCLA he continued to work on arithmetic, logic design methods, and switching theory.

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1950-1952 Svoboda helped design the M1 special-purpose relay computer for a 3D Fourier synthesis. Uses probably the first pipelined arithmetic unit to evaluate producing one result per one relay activation!

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A man of many talents:

• a mathematician and a physicist
• an engineer
• authored New Theory of Bridge - a scientific theory of bidding

strategies

• a pianist for the Prague Wind Quintet (Smet´

aˇ cek), a percussionist for the Czech Philharmonic Orchestra

• (Bohumil Martinu dedicated several pieces to him)
• an accomplished photographer and lens designer
• an expert APL programmer (his PRESTO, one of the first

minimization programs, leading to IBM MIN, was written in APL)

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Best known for his pioneering work on the RNS theory and algorithms Solved multiple-output minimization problem, disliked Karnaugh maps and loved Marquand charts (maps with binary labels); used punched cards to find prime implicants and minimal covers, ..., Proposed a decimal division with simple selection, decimal arithmetic, arithmetic circuits with fault detection, .... A witty, noble, exceptionally talented man with a charming ”akzent”, admired for his enthusiastic lectures

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Dan Atkins and Luigi Dadda watching Svoboda speak at Arith-4 ”I did not develop RNS primarily for speed. I also looked for a reliable design using unreliable components because I did not want to end up in Siberia. A truly modular design, inherent to RNS, was a key to increasing reliability.”

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During the WWII, Svoboda developed methods for designing bar-linkage computers, for antiaircraft fire control (Mark 56). These are mechanical analog computers, small, fast, reliable but difficult to design - Toni develped fundamentals.

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MIT Radiation Lab Linkage Computer of Mark 56: Toni on the right

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Examples of elementary bar linkage ”computers”: Multiplier Xi = Xk × Xj

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Adder (A1 + A2)X3 = A1X1 + A2X2

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The ideal harmonic transformer: input an angle Xi and output a displacement Xk = Rsin(Xi)

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A double three-bar linkage is a logarithm generator with evenly spaced inputs 1 ≤ Xi ≤ 50, and outputs with a maximum error of 0.003.

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Jan Oblonsky on RNS origin: ”The original impulse came from Svoboda in 1950. While explaining the theory of linkage multipliers, he noted how in the analog world there is no structural difference between an adder and a multiplier (the difference being only in applying proper scales at input and

• utput), whereas in the digital implementations adder and multiplier are

completely different structures. He challenged his students to try to find a digital implementation that would perform both multiplication and addition with comparable ease”. Miroslav Valach suggested the idea of residue encoding .... which led to Svoboda’s Numerical System of Residue Classes ...

Annals of the History of Computing, Vol. 2. No. 4, 1980

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Arith-4, 1978, Santa Monica: with Jim Robertson and Robert Gregory

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Arith-4, 1978, Santa Monica: with Algirdas Aviˇ zenis

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36 years later, at the 2014 UCLA CSD Retreat

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Thank you again for this honor and for being part of this humbling event. Let me also thank my many colleagues who, with talent and dedication, kept arithmetic field alive, interesting, contentious, and relevant all these many years. THANK YOU

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