NASA Technical Reports Server (NTRS) 20040110810: Verification of IEEE Compliant Subtractive Division Algorithms

16 Pages·0.2 MB·English

by  NASA Technical Reports Server (NTRS)

Veri(cid:12)cation of IEEE Compliant Subtractive
Division Algorithms
1 2
Paul S. Miner and James F. Leathrum, Jr.
1
NASA Langley Research Center, Hampton VA23681-0001 USA,
E-mail: [email protected]
2
Old Dominion University, Department ofElectrical and Computer Engineering,
Norfolk VA23529 USA,E-mail: [email protected]
Abstract. A parameterized de(cid:12)nition of subtractive (cid:13)oating point di-
vision algorithms is presented and veri(cid:12)ed using PVS. The general al-
gorithm is proven to satisfy a formal de(cid:12)nition of an IEEE standard
for (cid:13)oating point arithmetic. The utility of the general speci(cid:12)cation is
illustrated usinganumberofdi(cid:11)erentinstancesofthegeneralalgorithm.
1 Introduction
As computing systems become more complex, it becomes increasingly di(cid:14)cult
toensure thattestingfullyexercises thedesign.Thiswasmadeabundantlyclear
by the infamousbug in the (cid:13)oatingpoint unitof the Intel Pentium(TM)micro-
processor. The bug consists of (cid:12)ve missing entries in a lookup table. Pratt [20]
providesathoroughanalysisofthiserror.Heprovidescompellingargumentsthat
athoroughmanualanalysisofadesign maystillallowerrors to evadedetection.
This is particularly true if the (cid:13)aw is in a region of the design that is thought
to be unreachable. Machine assisted reasoning is crucial to prevent such errors.
The SRT divider[19] veri(cid:12)ed by Rue(cid:25), Srivas, and Shankar[17] illustrates how
the Prototype Veri(cid:12)cation System (PVS) [16] can be used to prevent omissions
in lookup tables similar to that employed by the Pentium (TM). Their work
describes a formalrelationship froma veri(cid:12)ed algorithmto aformaldescription
of a hardware design. In order to complete the veri(cid:12)cation, it is necessary to
relate the algorithmto a formaldescription of the (cid:13)oating point operation.
Our work provides a formalrelationship fromthe IEEE (cid:13)oating point stan-
dards [6,7]toaclassofveri(cid:12)eddivisionalgorithms.Astrengthoftheoremprover
based veri(cid:12)cation is that it allows veri(cid:12)cation of classes of algorithms. Once a
class is veri(cid:12)ed with respect to the standard, itis trivialto instantiate it with a
speci(cid:12)c instance. The SRTdivider [19]veri(cid:12)ed in[17] is an instance of the class
wehaveveri(cid:12)ed.Otherinstances includetheclassicalrestoringandnonrestoring
division algorithms.Thus, our general theory provides a standard speci(cid:12)cation
for IEEE compliantsubtractive division.
The veri(cid:12)cation is structured in a hierarchical fashion. At the root is a ver-
i(cid:12)cation of the class of subtractive division algorithms.We illustrate the utility
ofthe general veri(cid:12)cation byexhibitinganumber ofinstances. Then it is shown
how these algorithms can be extended to provide IEEE compliant rounding.
These instances provideacollectionofformalspeci(cid:12)cations forimplementations
of (cid:13)oating point division that are known to meet the standard.
2 Related work
Much of the previous work in theorem prover based veri(cid:12)cation of (cid:13)oating
point algorithms has focused on verifying core algorithms and a correspond-
inghardware design.The (cid:12)rst e(cid:11)orts targeted veri(cid:12)ed implementationofbinary
nonrestoring algorithms.Leeser, O’Leary, and others present a veri(cid:12)cation (us-
ing Nuprl) of a binary nonrestoring square root algorithmand its implementa-
tion[12,9].Verkest,et alpresent asimilarveri(cid:12)cation(usingnqthm)ofabinary
nonrestoring divisionalgorithm[21].
In response to the (cid:13)aw in the Pentium[20], several researchers investigated
theorem prover based veri(cid:12)cations of SRT division hardware. Clarke, German,
and Zhao used the Analytica theorem prover to verify Taylor’s[19] radix-4
SRT division circuit [4]. Their veri(cid:12)cation includes an abstract representation
of the lookup table and a proof that it de(cid:12)nes all necessary values for the quo-
tient selection logic. Rue(cid:25), Srivas, and Shankar generalize this work using the
PVS theoremprover. They present a general veri(cid:12)cationofarbitrary radixSRT
division algorithms,instantiate their theory with Taylor’sradix-4 SRT division
circuit [19], and verify a description of the hardware. Included in their work is
a technique to verify a concrete representation of the lookup table. By virtue of
PVS’ type system,proof obligationsare automaticallygenerated to ensure that
blank entries in the table are inaccessible.
In all of the e(cid:11)orts above, the veri(cid:12)cations address the functional correct-
ness of (cid:13)oating point algorithms and an associated hardware design. They do
not address the issue ofrelatingthe algorithmstoa formalde(cid:12)nition of(cid:13)oating
point operations. Harrison has presented a veri(cid:12)cation of two (cid:13)oating point al-
gorithms;square root, and a CORDIC[22] natural logarithmalgorithm[8]. For
both algorithms, Harrison relates the proofs to a (cid:13)oating point interpretation.
Although he does not present hardware descriptions, he does address some of
the preliminaryerror analysis necessary to provide correct rounding. The IEEE
standards for (cid:13)oating point arithmetic unambiguouslystate that operations
shall be performedas if it (cid:12)rst produced an intermediate resultcorrect to
in(cid:12)niteprecisionandwithunbounded range,andthenthatresultrounded
according to one of the modes ::: [6, 7]
Barrett manuallyveri(cid:12)ed a general rounding algorithmwith respect to a Z for-
malization of IEEE 754[1, 2]. When Barrett performed his veri(cid:12)cation, there
was no machine assisted reasoning for the Z speci(cid:12)cation language. Some tools
for machine assisted application of Z have recently been developed [13]. Thus
far, these have not been applied to (cid:13)oating point veri(cid:12)cation.
Recently,the microcode for the (cid:13)oating point divisionand square root algo-
TM
rithmsoftheAMD5K86 microprocessorhasbeenmechanicallyveri(cid:12)edusing
the ACL2theoremprover[15,18].Bothalgorithmsassumecorrect hardware for
(cid:13)oating point multiplication, addition, and subtraction. Both veri(cid:12)cations in-
cludedetailedanalysisofroundingandproofthatthedeliveredresultisrounded
in accordance with the IEEE standard. In addition, the veri(cid:12)cations guarantee
that all intermediate results of the algorithms (cid:12)t the datapath of the existing
(cid:13)oating point hardware.
Our work is a generalization of the Rue(cid:25), Srivas, and Shankar veri(cid:12)cation.
In addition, to the SRT algorithms, our veri(cid:12)cation encompasses most of the
algorithmspresented in[5]. In addition, our veri(cid:12)cation includes a formal path
relating the algorithmto the IEEE standard.
2.1 Brief introduction to PVS
The Prototype Veri(cid:12)cation System (PVS) [16] is a veri(cid:12)cation system that pro-
vides support for general purpose theorem proving. The speci(cid:12)cation language
is a higher order logic augmented with dependent types. Theories can be pa-
rameterized, and the dependent type mechanism allows for stating arbitrary
constraints on theory parameters. The type system of PVS includes predicate
subtypes and is therefore undecidable. PVS frequently generates proof obliga-
tionstoensurethatexpressions arewelltyped.PVShaspowerfuldecisionproce-
dures,so manyproofsinvolvingsimplearithmeticexpressions canbe discharged
automatically.Inaddition,PVSprovidesacollectionofpre-provenresults inthe
prelude.Alsoincluded with PVS are libraries providingsupport for bit-vectors
and (cid:12)nite sets. In PVS, the real numbers are a base type, and other numeric
types are de(cid:12)ned as subtypes of the reals. This allows speci(cid:12)cations to freely
mixoperations on numeric types.
3 General veri(cid:12)cation of subtractive division algorithms
Therearetwoprincipleclassesof(cid:13)oatingpointdivisionalgorithms.Thesubtrac-
tivealgorithmsuseshiftingfollowedbyaddition/subtractiontogeneratequotient
digitsintimelinearwithrespect tothesize oftheoperands.Multiplicativealgo-
rithms,such as Goldschmidt’sAlgorithmorNewton-Raphsoniterations provide
fewer iterations,but the operations ineach iterationgrowincreasinglycomplex.
Ercegovac and Lang present a detailed study of subtractive algorithms algo-
rithms for both division and square root[5]. The general division algorithm is
presented inPVSprovidingaparameterizedclassofveri(cid:12)edsubtractive division
algorithms.
3.1 General algorithm de(cid:12)nition
Subtractive algorithms generate one quotient digit per iteration. They are de-
(cid:0)i
signed so that in iteration i the remainder is no larger than r for a radix-r
algorithm. Ercegovac and Lang present a series of interrelated factors which
di(cid:11)erentiate subtractive division algorithms: the radix (r), the quotient-digit
set (f(cid:0)a;(cid:0)a+1;:::;(cid:0)1;0;1;:::;a(cid:0)1;ag), the range of the divisor (b), and the
quotient-digit select function (qs) [5]. These factors are all parameters to the
general division theory. The formalparameters are:
r : {i : posint | i > 1},
a : {i : posint | ceiling((r-1)/2) <= i & i < r},
b : {i : posint | 1 < i & i <= r},
(IMPORTING divide_types[r, a, b])
qs : qs_type
The (cid:12)rst three parameters are de(cid:12)ned using predicate subtypes of the positive
integers. In addition, the types for a and b are constrained by the value of r.
The type signature for function qs depends on all of the previous parameters.
The declaration for type qs type is imported from theory divide types. Theory
a
divide types declares constant (cid:26)= r(cid:0)1 and the followingtypes:
D_type : TYPE = {d : real | 1 <= d & d < b}
dividend : TYPE = {x : posreal | 1/r <= x & x < rho}
p_type(D) : TYPE = {p : real | abs(p) <= rho*D}
qs_type : TYPE =
[D : D_type, p : p_type(D) ->
{q : subrange(-a,a) | abs(r*p - q*D) <= rho*D}]
Constant (cid:26) denotes the redundancy factor of the quotient-digit set. It repre-
sents a trade-o(cid:11) in design complexity between the quotient selection function
and generation of divisor multiples. The type of the divisor is constrained by
D type, which is de(cid:12)ned to include the numeric range of the signi(cid:12)cand of a
normalized (cid:13)oating point number. The type of the dividend is constrained to
ensure that it satis(cid:12)es constraints imposed on the partial remainder. Parame-
terized type p type(D) encodes an invariant, dependent on divisor D, that the
partial remainder must satisfy during execution of the algorithm. Finally, the
type of the quotient selection function, qs type, is restricted to functions that,
given a divisor D and a partial remainderp such that jpj(cid:20)(cid:26)(cid:1)D, return a digit
q such that (cid:0)a(cid:20)q (cid:20)a and jr(cid:1)p(cid:0)q(cid:1)Dj(cid:20)(cid:26)(cid:1)D.
The subtractive division algorithms, given divisor D and dividend X, are
characterized by the following recurrence equations for the quotient q and the
partial remainder p:
qi =r(cid:1)qi(cid:0)1+qdi
pi =r(cid:1)pi(cid:0)1(cid:0)qdi(cid:1)D (1)
where q0 = 0, p0 = X, and qdi is the quotient digit selected for iteration i. A
PVS speci(cid:12)cation of Equation1 is:
divide(X,D)(n) : RECURSIVE [# p : p_type(D), q: integer #] =
IF n=0 THEN (# p := X, q := 0 #)
ELSE (# p := r*p(divide(X,D)(n-1)) - qd_n*D,
q := r*q(divide(X,D)(n-1)) + qd_n #)
WHERE qd_n = qs(D,p(divide(X,D)(n-1))) ENDIF
MEASURE n
By using record types for the range of the function, the de(cid:12)nition is a direct
transliteration of the recurrence equations. In addition,by declaring the partial
remainderto be oftype p type(D),PVS automaticallygenerates aproofobliga-
tion to ensure that the invariantis satis(cid:12)ed. This obligationis proven using the
type constraints on the quotient selection function.
3.2 Veri(cid:12)cation of the general algorithm
To simplifylater de(cid:12)nitions we de(cid:12)ne the abbreviations:
De(cid:12)nition1.
p(X;D)(n)=^ p(divide(X;D)(n))
q(X;D)(n)=^ q(divide(X;D)(n))
PVS strategy (induct-and-simplify)proves:
(cid:0)n (cid:0)n
Lemma2. p(X;D)(n)(cid:1)r =X (cid:0)q(X;D)(n)(cid:1)r (cid:1)D
The invariantproperty of the partial remainder guarantees
(cid:0)n
p(X;D)(n)(cid:1)r (cid:0)n
Lemma3. (cid:20)r
(cid:12) D (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Tostrengthe(cid:12)n theconvergence(cid:12)property,thealgorithmincludesacorrective step
after the (cid:12)nal iteration.
De(cid:12)nition4.
(cid:0)n
(p(X;D)(n)+D)(cid:1)r if p(X;D)(n)<0
(cid:0)n
P(X;D)(n) =^ 8(p(X;D)(n)(cid:0)D)(cid:1)r if p(X;D)(n)=D
<p(X;D)(n)(cid:1)r(cid:0)n otherwise
(cid:0)n
: (q(X;D)(n)(cid:0)1)(cid:1)r if p(X;D)(n)<0
(cid:0)n
Q(X;D)(n) =^ 8(q(X;D)(n)+1)(cid:1)r if p(X;D)(n)=D
<q(X;D)(n)(cid:1)r(cid:0)n otherwise
:
Expanding the de(cid:12)nitions of P and Q, and using lemma2, we prove:
X P(X;D)(n)
Theorem5 (correctness). =Q(X;D)(n)+
D D
Similarlyexpanding the de(cid:12)nition of P and using lemma3, we prove:
P(X;D)(n) (cid:0)n
Theorem6 (convergence). <r
D
X
Thus, Q(X;D)(n) contains n radix-r digits of the quotient .
D
3.3 Example instantiations
A number of quotient selection functions have been developed for use with the
general subtractive division algorithm. These include three radix-2 algorithms:
restoring,nonrestoring,andSRT;and(cid:12)veradix-4algorithms:twousingselection
constants for comparison with p (using a = 3 and a = 2), and three distinct
radix-4 SRT lookup tables. The tables developed are for
1)Partial remainder prediction or use of a carry-save adder:
approximationof p for error fromeither computingan estimate of the
next remainder as in Taylor’scircuit [19] (previously veri(cid:12)ed in [17]),
or error froma carry-save adder [5]
2)Use of a signed-digit adder: approximationof p always overestimates,
as would be the case when using signed-digit adders to accumulate the
remainder [5]
3)Folded lookup table: lookup table folded for reduced real estate
The process of veri(cid:12)cation amounts to proving that each quotient digit select
function satis(cid:12)es the type constraints ofqs type. The radix-2instantiationsand
theradix-4usingselectionconstantswerestraightforwardveri(cid:12)cations.Thepro-
cess of verifying lookup tables in PVS was (cid:12)rst illustrated by Rue(cid:25), Srivas,and
Shankar[17] and repeated here with respect to the general division algorithm.
However, the developmentoftwonewlookuptablesproved insightfulindemon-
strating the usefulness of PVS in the design process. Not only were omissions
fromthe table detected, but PVS allowedthe faster developmentof new tables.
Each table wasdeveloped startingfromthe base table in[17]andthen modi(cid:12)ed
tomeetthenewrequirements.PVSwasusedtoindicatewhichtableentrieswere
incorrect underthenewspeci(cid:12)cations,allowingthedesignertoeasilymodifythe
table. Each new table was fully designed and veri(cid:12)ed in less than three hours
elapsed time.
4 Veri(cid:12)cation with respect to the IEEE standard
This section illustrates how to extend the above general algorithmto provide a
veri(cid:12)ed IEEE compliantimplementationof division. This does not constitute a
full proof ofcompliance,it just illustrates how non-exceptional cases of division
can be realized. This veri(cid:12)cation step is performed with respect to a formal
speci(cid:12)cation of IEEE 854 de(cid:12)ned using PVS [14]. The veri(cid:12)cation is performed
in two stages. First, a generalization of the basic guard, round, and sticky bit
rounding algorithmis shown to satisfy the requirements of the standard. Then,
the general subtractive algorithm is shown to provide su(cid:14)cient information to
utilize this rounding scheme. The veri(cid:12)ed rounding scheme is applicable for all
(cid:13)oating-point algorithms.In addition, the theory mapping the standard to the
general subtractive algorithm includes a number of intermediate results that
apply to all (cid:13)oating point division algorithms.
4.1 Rounding scheme
The IEEE Standards for (cid:13)oating-pointarithmetic[6,7] require support for four
rounding modes. The default mode is round to nearest even, and requires that
the return value be the (cid:13)oating point number nearest to the exact result. If
the exact result is halfway between two (cid:13)oating point numbers, the standards
require that it be rounded to the one with an even least signi(cid:12)cant digit. The
otherthree modesroundthe result towardspositivein(cid:12)nity,negativein(cid:12)nity,or
zero. The discussion in this section uses the followingfact about real numbers.
Lemma7. For any integer b > 1, a nonzero real number z can be uniquely
decomposed into three parts: a sign,sgn(z) 2 f(cid:0)1;1g; an integer exponent e(z);
and a signi(cid:12)cand 1(cid:20)sig(z) <b such that
e(z)
z = sgn(z)(cid:2)sig(z)(cid:2)b
The followingde(cid:12)nition is from the PVS prelude:
De(cid:12)nition8. The fractional part of a real number z is de(cid:12)ned by
fzg =^ z(cid:0)bzc
The next two de(cid:12)nitions are fromMiner’s PVS formalizationof IEEE 854[14].
The function,round, converting real numberz to aninteger foreach ofthe four
rounding modes required by the IEEE standard is de(cid:12)ned by:
De(cid:12)nition9.
round(z;to pos) =^ dze
round(z;to neg) =^ bzc
round(z;to zero) =^ sgn(z)(cid:2)bjzjc
bzc if fzg<1(cid:0)fzg
round(z;to near) =^ 8dze if 1(cid:0)fzg<fzg
><2(cid:2) dz2e otherwise
j k
>:
This de(cid:12)nition is extended to round a real number z to a real number with at
mostp base-b digits of precision using the followingfunction:
De(cid:12)nition10.
p(cid:0)1(cid:0)e(z) e(z)(cid:0)(p(cid:0)1)
round scaled(z;mode) =^ round(z(cid:1)b ;mode)(cid:1)b
A commonalgorithmforimplementingIEEE compliantroundingto pdigits
of precision uses two extra digits and a sticky-(cid:13)ag[11]. The (cid:12)rst extra digit is
called the guard digit and ensures that the computed result can be normalized
whilepreservingpdigitsofprecision.Thisisnecessary formultiplicationanddi-
visionalgorithms.Thesecond extradigitiscalledtherounddigit,andisused to
controlrounding forevery modeexcept to zero. Finally,a sticky (cid:13)agis required
to distinguish the case when the in(cid:12)nitely precise result lies halfway between
two representable values for mode to near. The PVS theory implementing the
Guard,Round,Sticky (GRS)scheme has been generalized to allowanarbitrary
even radix. Thus it works for both base-2 and base-10 instances of IEEE 854.
The principle function realizing the GRS rounding scheme is:
De(cid:12)nition11. For s2f(cid:0)1;1g,n2N, base-b digit d, and boolean sticky
n+1if s(cid:21)0^((d>0)_sticky)
grs(s;n;d;sticky;to pos) =^
(cid:26)n otherwise
n+1if s<0^((d>0)_sticky)
grs(s;n;d;sticky;to neg) =^
(cid:26)n otherwise
grs(s;n;d;sticky;to zero) =^ n
b b
grs(s;n;d;sticky;to near) =^ n+1if (d> 2)_(d= 2 ^(sticky_odd?(n)))
(cid:26)n otherwise
To complete the de(cid:12)nition of rounding using the GRS scheme, we use the fol-
lowing functions to scale the result and extract the relevant (cid:12)elds.
De(cid:12)nition12.
p(cid:0)1(cid:0)e(z)
scaled(z) =^ bjzj(cid:2)b c
p(cid:0)e(z)
round digit(z) =^ bjzj(cid:2)b cmodb
p(cid:0)e(z)
sticky (cid:13)ag(z) =^ fjzj(cid:2)b g6=0
Function scaled extracts the p most signi(cid:12)cant base b digits of z scaled to a
naturalnumber.Functionround digitextractsthe(p+1)thdigit,andsticky (cid:13)ag
istrue ifandonlyifthere aresigni(cid:12)cantdigitsbeyondthe(p+1)th.Foraresult
in a sign and magnitude representation, functions scaled and round digit can
be realized in hardware by extracting the appropriate digits from the given
result. Implementing the sticky (cid:13)ag requires some additional logic; the actual
realization will vary depending upon the algorithm employed to compute the
result. For comparisonto round scaled, we de(cid:12)ne
De(cid:12)nition13.
round grs(z;mode)
e(z)(cid:0)(p(cid:0)1)
=^ sgn(z)(cid:2)b (cid:2)
grs(sgn(z);scaled(z);round digit(z);sticky (cid:13)ag(z);mode)
The principle result of this section is:
Theorem14.
round grs(z;mode)=round scaled(z;mode)
The PVS proof of this result consists of a fairlysimplecase analysis, except for
modeto near.
TheinitialPVSproofformodeto nearincludedacomplicatedcase analysis,
where it was di(cid:14)cult to exploit symmetry. IEEE (cid:13)oating point numbers are
de(cid:12)ned using a sign and magnitude representation. However, the de(cid:12)nition of
round scaleddoesnottakeadvantageofthisrepresentation.Thus,the(cid:12)rstproof
for mode to near included an unnatural case split on the sign of the argument.
Since the GRS rounding scheme is de(cid:12)ned in terms of a sign and magnitude
representation,thecasesfornegativeargumentsdonotaligninthesamemanner
as forthe positivearguments.Thus, there was littleopportunityto reuse proofs
fromthe corresponding positivecases. The PVS proof has been simpli(cid:12)edusing
the followinglemma(which was proven using PVS strategy (grind))
Lemma15.
round(z;to near)=sgn(z)(cid:2)round(jzj;to near)
Even without the case split due to the sign, the proof for mode to near still
involvesadi(cid:14)cultcaseanalysis.Thiscaseanalysisconsistsofrelatingthevalues
oftherounddigitandsticky(cid:13)agtothecorrespondingcasesfromthespeci(cid:12)cation
for rounding modeto near.
4.2 Relating the general algorithm to the standard
The veri(cid:12)ed rounding scheme asserts that in order to achieve IEEE compliant
rounding to p signi(cid:12)cant digits, it is su(cid:14)cient to compute (p + 1) truncated
signi(cid:12)cant digits and determine if there are any remainingnon-zero digits. The
general subtractive division algorithm ensures at least n(cid:0)1 digits of precision
afterniterations.Furthermore,ifthecomputedremainderisnonzero,thenthere
are additional digits in the in(cid:12)nitely precise result. Since it is possible for the
radix of the division algorithm to be di(cid:11)erent from that for the representation
of (cid:13)oating point numbers, the PVS theory has to relate the potential division
radices to those allowedby the standards.
There are some simple results that describe the range of possible values
for (cid:13)oating point division. Let x denote a base-b (cid:13)oating point number with
p signi(cid:12)cant digits. A (cid:12)nite (cid:13)oating point number x is represented using three
(cid:12)elds:asign(cid:27)x 2f0;1g,aninteger exponent Ex,and asigni(cid:12)canddx,where dx
is a functionfromf0;:::;(p(cid:0)1)g to f0;:::;(b(cid:0)1)g.The numericalvalue ofdx
is given by
p(cid:0)1
(cid:0)i
vd(x)= dx(i)(cid:1)b
Xi=0
The numericalvalue of (cid:13)oating point number x is given by
(cid:27)x Ex
v(x)=((cid:0)1) (cid:1)b (cid:1)vd(x)
A (cid:13)oating point number x such that 1(cid:20)vd(x)<b is normalized.Furthermore,
ifitsexponent alsofallswithintherange allowedbythe standard,itisanormal
(cid:13)oating point number.
TheIEEE standard requires thatalloperationsbe performedasiftoin(cid:12)nite
precision and then rounded. The in(cid:12)nitelyprecise quotient oftwo (cid:13)oatingpoint
numbers x and y is v(x)=v(y). A number of general facts about (cid:13)oating point
division have been proven using PVS. These include the following:
Lemma16.
v(x) (cid:27)x(cid:0)(cid:27)y Ex(cid:0)Ey vd(x)
=((cid:0)1) (cid:1)b (cid:1)
v(y) vd(y)
This lemmastates that a (cid:13)oating point division algorithm can be decomposed
into simple operations on the sign and exponent (cid:12)elds combined with an algo-
rithm to compute the quotient of the signi(cid:12)cands.
Lemma17. For normalized (cid:13)oating point numbers x and y,
(cid:0)1 (cid:0)(p+1) vd(x) (1(cid:0)p)
b +b (cid:20) (cid:20) b(cid:0)b
vd(y)
Lemma18. For normalized (cid:13)oating point numbers x and y, if vd(x) < vd(y)
then
vd(x) (1(cid:0)p)
(cid:20) 1(cid:0)b
vd(y)
Lemmas 17 and 18 assert that the result of a (cid:13)oating point division algorithm
requiresatmostoneleftshifttonormalizetheresult,andfurthermore,theneces-
sityofpost-dividenormalizationcanbedeterminedbycomparingthemagnitude
ofthe signi(cid:12)cands.Therefore, roundingcannota(cid:11)ect the exponent (cid:12)eldforadi-
visionoperation.These results aretrueforany(cid:13)oatingpointdivisionalgorithm;
they do not depend on a particular instantiation.
These results allowus to de(cid:12)ne the followingexponent adjustment function
for the quotient of normalized (cid:13)oating point numbers x and y:
1if vd(x)<vd(y)
Adj(x;y)=
(cid:26)0otherwise
The above results allow us to de(cid:12)ne a template for a (cid:13)oating point division
algorithm
De(cid:12)nition19. For normal (cid:13)oating point numbers x and y such that Emin (cid:20)
vd(x)
Ex(cid:0)Ey(cid:0)Adj(x;y)(cid:20)Emax, let z = vd(y) in
div algorithm(x;y;mode)=^(cid:12)nite(XOR((cid:27)x;(cid:27)y);Ex(cid:0)Ey(cid:0)Adj(x;y);digit div(x;y)
where
(cid:27)x(cid:0)(cid:27)y
digit div(x;y)=^nat2d(grs(((cid:0)1) ;scaled(z);round digit(z);sticky (cid:13)ag(z);mode)