reduce-historical/r38/packages/scope/scope.tex

\documentstyle[11pt,reduce,makeidx]{article}
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\title{{\bf SCOPE 1.5\\ A Source-Code Optimization PackagE\\ for\\ REDUCE 3.6\\
\vspace{0.5cm}
=====\\
\vspace{0.5cm} User's Manual}}
\date{}
\author {\large
 \vspace{1cm} \\
J.A. van Hulzen \\ University of Twente, Department of Computer Science\\
P.O. Box 217, 7500 AE Enschede, The Netherlands \\
Email: infhvh@cs.utwente.nl}
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\begin{document}

\maketitle
\vspace{3cm}
\begin{center}
{\bf Abstract}\\
\end{center}

The facilities, offered by SCOPE 1.5, a Source-Code Optimization PackagE 
for {\REDUCE} 3.6, are presented. We discuss the user aspects of the package.
The algorithmic backgrounds are shortly summarized.
Examples of straightforward and more advanced usage are shown,  
both in algebraic and symbolic mode.
Possibilities for a combined use of GENTRAN and SCOPE are presented as well.

\vspace{1.5cm}
\copyright {\em \ \ } J.A. van Hulzen, University of Twente. All rights reserved.

\newpage
\tableofcontents
\newpage
\pagestyle{headings}
\section{Introduction}\label{SCOPE:intro}
\pagenumbering{arabic}

An important application of computer algebra systems is the generation
of code for numerical purposes via automatic or semi-automatic program
generation or synthesis. 
GENTRAN~\cite{Gates:84,Gates:85,Gates:86,Gates:91}, a flexible general-purpose
package, was especially developed to assist in such a task, when using
MACSYMA or {\REDUCE}.
\index{optimization}
Attendant to {\bf automatic program generation} is the problem 
of {\bf automatic source-code optimization}. 
This is a crucial aspect because code
generated from symbolic computations often tends to be made up of
lengthy arithmetic expressions. 
Such lengthy codes are grouped together in
blocks of straight-line code in a program for numerical purposes. The
main objective of SCOPE, our source-code optimization package, has been
minimization of the number of (elementary) arithmetic operations in such 
blocks. This can be accomplished by replacing repeatedly occuring
subexpressions, called common subexpressions or cse's for short,
\index{cse (common subexpression)}
by placeholders. We further assume that new statements of the form
"placeholder := cse" are inserted correctly in the code. This 
form of optimization is often helpful in reducing redundancy in (sets of) 
expressions. A recent application, code generation for an 
incompressible Navier-Stokes problem~\cite{Goldman:95}, showed reduction 
from 45.000 lines of FORTRAN code to 13.000 lines.

\index{optimizing compilers}
Optimizing compilers ought to deal effectively and efficiently with
the average, hand coded program. The enormous, arithmetic intensive
expressions, easily producable by a computer algebra system, fall
outside the range of the FORTRAN programs, once analyzed and discussed
by Knuth~\cite{Knuth:71}. He suggested that optimization of the arithmetic in
such a program is slightly overdone. The usual compiler optimization strategy
is based on easy detection of redundancy, without assuming the validity of
(some) algebric laws (see for instance ~\cite{Gonzales}) 
Our optimization strategy however, requires the validity of
some elementary algebraic laws. We employ heuristic techniques to
reduce the arithmetic complexity of the given representation of a set
${\rm E}_{in}$ of input statements, thus producing a set 
${\rm E}_{out}$ of output assignment statements.
${\rm E}_{in}$ and ${\rm E}_{out}$ define blocks of
code, which would compute the same exact values for the same exact
inputs, thus implicitly proving the correctness of the underlying
software.  Obviously the use of ${\rm E}_{out}$ ought to save a considerable
amount of execution time in comparison with the application of ${\rm
E}_{in}$.  Johnson et al~\cite{Johnson:79} suggest that such
transformations do not destabilize the computations.  However this is only
apparent after a careful error analysis of both ${\rm E}_{in}$ and ${\rm
E}_{out}$.  In view of the size of both ${\rm E}_{in}$ and ${\rm E}_{out}$
such an analysis has to be automatized as well.  Work in this direction is
in progress ~\cite{Hulshof,Molenkamp:91,Molenkamp:94}.
\index{error analysis}
\index{arithmetic complexity}

Although the use of SCOPE can considaribly reduce the arithmetic complexity
of a given piece of code,
we have to be aware of possible numerical side effects. In addition we have
to realize that a mapping is performed from one source language to another 
source language, seemingly without taking into account the platform the resulting 
numerical code has to be executed on. Seemingly, because we implemented some
facilities for regulating minimal expression length  and for producing
vector code.
\index{vector code}

This manual is organized as follows. We begin with some preliminary remarks
in section~\ref{SCOPE:prel}. The history and the present status, 
the optimization
strategy and the interplay between GENTRAN and SCOPE are shortly summarized.
The basic algebraic mode facilities are presented in
section~\ref{SCOPE:basic}. Special SCOPE features are discussed in 
section~\ref{SCOPE:soph}. Besides facilities for Horner-rules and an extended
version of the REDUCE function {\tt structr}, we introduce some tools for
extending  SCOPE with user defined specialties. File management follows in
section~\ref{SCOPE:files}. In section~\ref{SCOPE:decl} declaration handling
and related issues are discussed, before illustrating our strategy concerning
data dependencies and generation of vector code in section~\ref{SCOPE:dda}. 
In section~\ref{SCOPE:gopt} is shown how a combined used of GENTRAN and SCOPE 
can be profitable for pro\-gram-ge\-ne\-ra\-ti\-on. The use of SCOPE in 
symbolic mode is presented in section~\ref{SCOPE:symb}. A SCOPE syntax
summary is given in section~\ref{SCOPE:syntax}.
For completeness we present guidelines for installing the 
package in the last section.\\

{\bf Requests}
\begin{itemize}
\item Comment and complaints about possible bugs can be send to the author 
using the e-mail address infhvh@cs.utwente.nl. A compact piece with 
REDUCE code, illustrating the bug, is prefered. 
\item When SCOPE 1.5 is used in prepairing results, presented in some 
publication, reference to its use is highly appreciated. A copy of the 
published document as well.
\end{itemize}
\newpage

\section{Preliminaries}\label{SCOPE:prel}

For completeness we present a historical survey, a birds-eye view of the
overall optimization strategy and the interplay between GENTRAN and SCOPE.

\subsection{History and Present Status}\label{SCOPE:hito}

The first version of the package was designed to optimize the
description of {\REDUCE}-statements, generated by
NETFORM~\cite{Smit:81,Smit:82}. This
version was tailored to a restrictive class of problems, mainly occurring
in electrical network theory, thus implying that the right-hand
sides (rhs's) in the input were limited to elements of ${\rm {\bf Z_2}}$[V],
where V is a set of identifiers.  The second version~\cite{vanHulzen:83}
allowed rhs's from {\bf Z}[V]. For both versions the validity of the
commutative and the associative law was assumed.  A third version
evolved from the latter package by allowing to apply the distributive
law, i.e. by replacing (sub)expressions like $a.b  +  a.c$ by $a.(b 
+  c)$ whenever possible.  But the range of possible applications of
this version was really enlarged by redefining V as a set of kernels,
implying that almost any proper {\REDUCE}
expression could function as a rhs.  The mathematical capabilities of
this version are shortly summarized in~\cite{Wang:84}, in the context of code
generation and optimization for finite-element analysis. This version
was further extended~\cite{vanHulzen:89} with a declaration-module, 
in accordance with the strategy outlined in~\cite{Aho:86}, chapter 6. 
It is used \index{GENTRAN} 
in combination with GENTRAN, for 
the construction of Jacobians and Hessians~\cite{Heuvel:89,Berger:92} and 
also in experiments with strategies for code vectorization~\cite{Goldman:89}. 
In the meantime the Jacobian-Hessian production package, at present 
called GENJAC, is further extended with possibilities for global optimization
and with some form of loop-differentiation. So in stead of optimizing 
separate blocks of arithmetic we are now able to optimize complete programs,
albeit of a rather specific syntactical structure~\cite{Berger:92}.
The present 1.5 version of SCOPE, is an intermediate between the distributed first
version and the future, second version.  Version 2 is currently in 
development and will contain, besides the already existing common sub 
expression (cse) searches, facilities for structure and pattern recognition.
The 1.5 version permits -using the present REDUCE terminology- rounded
coefficients, based on the domain features, described in~\cite{Bradford:86},
discovery and adequate treatement of a variety of data dependencies, and
quotient-optimization, besides a collection of other improvements and 
refinements for using the facilities in the algebraic mode. 
Furthermore, an increased flexibility in the interplay between 
GENTRAN and SCOPE is accomplished.  It is used for experiments concerning 
automatic differentiation ~\cite{Goldman:91}, symbolic-numeric approaches to
stability analysis ~\cite{Ganzha:92,Ganzha:94} and for code generation for
numerically solving the Navier-Stokes equations for incompressible 
flows ~\cite{Goldman:95}. An interesting example of its use elsewhere can be found
in ~\cite{Dyer:94}.

\subsection{Acknowledgements}\label{SCOPE:ackn}

Many discussions with Victor V. Goldman, Jaap Smit and Paul S. Wang
have contributed to the present status of SCOPE. I express my
gratitude to the many students, who have also 
contributed to SCOPE, either by assisting in designing and implementing new 
facilities, or by applying the package in automated program generation projects
in and outside university, thus showing shortcomings and suggesting 
improvements and extensions.  I mention explicitly
Frits Berger, Johan de Boer, John Boers, Pim Borst, Barbara Gates, 
Marcel van Heerwaarden, Pim van den Heuvel, Ben Hulshof, Emiel Jongerius, 
Steffen Posthuma, Anco Smit, Bernard van Veelen and Jan Verheul.

\subsection{The Optimization Strategy in a Birds-eye View}\label{SCOPE:bird}

In~\cite{vanHulzen:81,vanHulzen:83} we described the overall
optimization strategy used for
\index{optimization strategy}
SCOPE as a composite function ${{\rm R}^{-1}} \circ {{\rm T}} 
\circ {{\rm R}}$.  The function R defines how to store the input
${{\rm E}}_{0}$ in an expression database ${{\rm D}}_{0}$.
The inverse function ${{\rm R}}^{{-1}}$ defines the output production.  
The function T defines the
optimization process itself. It essentially consists of a heuristic
remodeling of the (extendable and modifiable) expression database  
in combination with storing information required for a fast 
retrieval and correct insertion of the detected cse's in the output.  
This is accomplished by an iteratively
applied search, resulting in a stepwise reduction of the arithmetic
complexity of the input set, using an extended version of Breuer's
\index{Breuer's Algorithm}
grow factor algorithm~\cite{Breuer:69,vanHulzen:81,vanHulzen:83}.
It is applied until no further profit
is gained, i.e. until the reduction in arithmetic complexity stops.
Before producing output, a finishing touch can be performed to further
\index{finishing touch}
reduce the arithmetic complexity with some locally applied techniques.
Hence T is also a composite function.
The overall process can be summarized as follows: 
%\begin{eqnarray*}
\[ \begin{array}{rcrcl}
{\rm R} & : & {\rm E_{in}}~=~{\rm E_0} & \rightarrow & ({\rm D_0},{\rm profit_0}) \\
{\rm T_{\beta}} & : & ({\rm D_i},{\rm profit_i}) & \rightarrow & ({\rm D_{i+1}},
{\rm profit_{i+1}})~,~{\rm i}~=~0,...,~\lambda - 1.  \\
{\rm F} & : & ({\rm D_{\lambda}},{\rm profit_{\lambda}}) & \rightarrow & {D_{\lambda}}\\
{\rm R^{-1}} & : & {D_{\lambda}} & \rightarrow & {\rm E_{\lambda}}~=~{\rm E_{out}}
%\end{eqnarray*}
\end{array} \]
 ${\rm D_0}$ is created as a result of an R-application performed on 
input ${\rm E_0}$. The termination condition depends on some profit criterion
related to the arithmetic complexity of the latest version of the
input, ${{{\rm D}}_i}$. Hence we assume ${{{\rm profit}}_i} = true$
for $i =0,~\cdots , \lambda -1$ and ${{{\rm profit}}_\lambda} =
false$.  The function T is defined 
by ${\rm T} = {\rm F} \circ {\rm T_{\beta}^{\lambda}}$, where 
${{\rm T}}_{\beta}$ defines one iteration step, i.e. one application of the
extended version of Breuer's algorithm, and where F defines a
\index{extended Breuer algorithm}
finishing touch, resulting in the final version $D_{\lambda}$ of
${{\rm D}}_{0}$, used to produce the output ${{\rm E}}_{\lambda}$. 
It is stated in ~\cite{vanHulzen:83} that the computing time for
${\rm T_{\beta}^{\lambda}}$ is ${\rm O(n.m)}$, where n is the size 
of ${\rm E_{in}}$ and m the number of cse's found during this process.
Practical experience showed that the finishing touch can take about 
10 \% of the actual cpu-time and that its real profit is limited. 
Therefore its use is made optional. 

The wish to optimize source code, defining arithmetic, usually leads an attempt
to minimize the arithmetic complexity. This can be accomplished by replacing 
cse's by placeholders, assuming a new assignment 
statement "placeholder $:=$ cse" is correctly inserted in the code. 
So most of the cse-searches are done in right hand sides of 
arithmetic assignment statements.

The search strategy depends on the permissible structure of the arithmetic
expressions. We assume these expressions to be multivariate polynomials
or rational functions in a finite set of kernels, and  presented in some 
normal form. Let us further assume that scalar placeholders are substituted
for the non-scalar kernels, such that back-substitution remains possible,
using an adequate information storage mechanism. Then we are left with 
the interesting question how to define a minimal set of constituents of
multivariate polynomials in some normal form norm.  Let us take as an 
example of such a polynomial or rational function $p = 3a + 2b +  3 {b^2} c 
(3a + 2b){(c + d)^2}$.  We easily recognize linear forms, i.e. $3a + 2b$
(twice) and  $c + d$, possibly raised to some power (${(c + d)}^2$), power
products, such as ${b^2} c$, or monomial parts of products, i.e. $3 {b^2} c$.
Hence with some imagination, one realizes that every polynomial can be
decomposed in a set of linear forms and a set of power products. When 
assuming the validity of the commutative and the associative law, one can also
realize that we can associate a coefficient matrix with the linear forms and 
an exponent matrix with the power products. The rows can correspondent
with (sub)\-ex\-pressions and the columns with scalar identifiers.
The entries are either coefficients or exponents.
It is therefore conceivable to make a parser, mapping a set of REDUCE 
expressions in a database, consisting of two 
incidence matrices and a function table, such that the original expressions 
can be retrieved. Taking a group of assignmemnt statements or a list of 
equations, where in both cases the lhs's function as right hand side 
recognizers, does not confuse this idea. This rather informal indication
merely serves as a suggestion how ${\rm R}$ and its inverse operation are
designed.

So we suggest that we can consider any set of rhs's as being built with 
linear forms and power products only.  An additional remark is worth being 
made: Non-scalar kernels will in general have non-commutable arguments. 
These arguments  can in turn be arbitrary {\REDUCE}-expressions, which 
also have to be incorporated in the database.  Hence the function table 
is created recursively.

\index{cse (common subexpression)}
What is a cse and how do we locate its occurrences?  A (sub)expression
is common when it occurs repeatedly in the input. The occurrences are,
as part of the input, distributed over the matrices, and shown as
equivalent (sub)patterns.  In fact, we repeatedly search for
completely dense (sub)matrices of rank 1.  The expression $2a + 3c$
is a cse of ${e_1}  = 2a + 4b + 3c$ and ${e_2} = 4a + 6c + 5d$, 
representable by (2,4,3,0) and (4,0,6,5), respectively.  We
indeed have to assume commutativity, so as to be able to produce new
patterns (2,0,3,0,0), (0,4,0,0,1) and (0,0,0,5,2), representing $s = 2a + 3c$,
 ${e_1} = 4b + s$ and ${e_2} = 5d + 2s$,
respectivily, and thus saving one addition and one multiplication.
Such an additive cse can be a factor in a (sub)product,
which in turn can extend its monomial part, when replacing the cse 
by a new symbol. Therefore an essential part of an
optimization step is regrouping of information. This migration of
information between the matrices is performed if the Breuer-searches
are temporarily completed.  After this regrouping the distributive law
is applied, possibly also leading to a further regrouping. If at
least one of these actions leads to a rearrangement of information the
function ${\rm T} _{\beta}$ is again applied.  In view of the
iterative character of the optimization process we always accept
minimal profits.

A similar search is performed to detect multiplicative cse's, for
instance occuring in ${e_1} = {a^2} {b^4} {c^3}$ and ${e_2} =
{a^4} {c^6} {d^5}$.  However, given a power product $\prod_{i=1}^m
{x_i}^{{a}_i}$, any product $\prod_{i=1}^m {x_i}^{{b}_i}$, such that
some ${b_i} \leq  {a_i}$, for i = 1(1)m, can function as a cse.  We
therefore extend the search for multiplicative cse's by employing this
property, and as indicated in~\cite{vanHulzen:83}.

The finishing touch F is made to perform one-row and/or
one-column searches. Once the extended Breuer-searches do not lead to
further reduction in the arithmetic complexity we try -applying it- to
improve what is left.  The coefficients in (sub)sums can have,
possibly locally, a gcd, which can be factored out.  One-column
operations serve to discover and to replace properly constant multiples of
identifiers.  As part of the output-process we subject all
exponentiations left - at most one for each identifier - to an
addition chain algorithm.  

\subsection{The Interplay between GENTRAN and SCOPE 1.5}\label{SCOPE:inter}

The current version of SCOPE is written in RLISP. 
Like GENTRAN, it can be used as an extension of {\REDUCE}.
When SCOPE is loaded GENTRAN is also activated.

If we start a REDUCE session, we create an initial algebraic mode 
programming environment. All switches get their initial value, such as 
{\tt ON EXP,PERIOD} and {\tt OFF FORT}. 
Certain REDUCE commands serve to modify or to enrich the current environment.
Others are used to perform calculations, producing formulae. Such a
calculation follows a standard pattern, although parts of this repertoire 
can be influenced by the user, for instance by changing the value of 
certain switches. Usually execution is a three-step process.
First the infix text is parsed into a prefix form.  Then the internal algebra 
is applied on this form, leading to a so-called standard quotient.
This quotient is stored on the property list of the identifier 
functioning as assigned variable for this value. The last step
defines the inverse route from internal existence to external presentation 
in infix form. Occurrences of identifiers are recursively
replaced by their standard quotient representation when the internal algebra
is applied. Hence the REDUCE simplification strategy follows the imperative
programming paradigm.

When loading SCOPE, and thus GENTRAN, the environment is enriched with 
features for program generation and program optimization. 
Evaluation of GENTRAN and SCOPE commands differs from the standard REDUCE
approach to evaluation. Both packages employ their own storage mechanism.
The output is normally produced as a side-effect of the command evaluation.
The output is directed to some medium, a file or a screen. Command evaluation 
is similar in GENTRAN and in SCOPE.

The code generation process of GENTRAN can be viewed as the application
\index{GENTRAN ! code generation process}
of a composite function to an argument, which is almost equivalent with a 
piece of REDUCE code. Almost, because some GENTRAN specific facilities 
can be used. We can distinguish between the preprocessing phase, 
the translation phase and the postprocessing phase.
During preprocessing relevant parts of the input are evaluated prior to 
translation into prefix form. Such a locally performed evaluation can be 
accomplished through recognition of certain "evaluation markers", i.e. 
modifications of the traditional assignment symbol {\tt :=}, 
such as {\tt ::=}, {\tt :=:} and {\tt ::=:}. The {\tt :=} operator
"orders" GENTRAN to translate the statement literally. Addition of an extra 
colon to the left hand side orders subscript expression evaluation before
translation. An extra colon to the right hand side leads to right hand side 
evaluation, but without application of the storage mechanism of REDUCE. 
Hence evaluations
remain anonymous and are only incorporated in the translatable "text".
Another aspect of preprocessing is initialization of the symbol table, 
using information provided by a {\tt DECLARE} statement.
\index{GENTRAN ! {\tt DECLARE} statement}
GENTRAN also allows to further rewrite (sets of) arithmetic assignment
statements, using the switches {\tt GENTRANOPT} and {\tt GENTRANSEG}, 
\index{{\tt GENTRANOPT} switch}
\index{{\tt GENTRANSEG} switch} 
introduced for code optimization (using SCOPE) and segmentation, respectively. 
\index{GENTRAN ! code segmentation}
It possibly leads to storage of additional information in the symbol table.
During the translation phase the final internal form
of the code is produced, in combination with formatting specifications and
instructions to produce declarations.
Postprocessing finally does produce formatted code strings. So essentially,
each GENTRAN command has its own seperate translation process, implying that
the symbol table, required for the production of declarations, is fresh
at the beginning of a GENTRAN command evaluation.

As stated before, a SCOPE command evaluation is also a composite operation. 
The role of the assignment operators in both GENTRAN and SCOPE is similar.
In SCOPE, the locally performed evaluation provides
information to be entered in the database ${\rm D_0}$.
If the declaration feature is activated, the symbol table generation and 
maintenance mechanism is borrowed from GENTRAN. 
For output production, we can make a choice from 
GENTRAN's target language repertoire. When declarations are required, we simply 
obey the GENTRAN regime as well. ${D_{\lambda}}$ is used to update the symbol 
table. All cse-names, generated during the optimization process, are typed in 
accordance with the strategy for dynamic typing, which is discussed 
in~\cite{Aho:86}, chapter 6. We assume all relevant identifiers 
of ${\rm E_{in}}$ to be adequately typed, using SCOPE's {\tt DECLARE} facility,
an equivalent of GENTRAN's {\tt DECLARE} statement. The
\index{SCOPE ! {\tt DECLARE} facility}
\index{GENTRAN ! {\tt DECLARE} statement}
production of ${D_{\lambda}}$ is completely decoupled from the normal
REDUCE simplification strategy, because we employ our own expression database.

In principle, the status of REDUCE before and after a GENTRAN or SCOPE
command execution is unaltered. In principle, because some minor modifications,
although user controlable, may be necessary. The special assignment symbols 
 -also usable in SCOPE- were only introduced as a syntactical instrument to 
allow internal algebraic actions, decoupled from the standard REDUCE 
expression processing.

This short excursion into the different evaluation strategies is added to
assist in understanding the functioning of the different SCOPE commands and 
facilities, to be introduced in the next sections.
\newpage

\section{The Basic SCOPE 1.5 Facilities in the Algebraic Mode}\label{SCOPE:basic}

{\REDUCE} allows, roughly speaking, two modes of operation in algebraic
mode: {\tt ON EXP} or {\tt OFF EXP}.  
The first is the default setting, leading to
expanded forms.  The latter gives unexpanded forms, as discussed by Hearn
in some detail~\cite{Hearn:85,Hearn:86}.  It is obvious that the {\tt OFF
EXP} setting is in general preferable over the {\tt ON
EXP} setting when attempting to optimize the description of a set of
assignment statements.

\index{{\tt ACINFO} switch} 
\index{{\tt ROUNDED} switch}
\index{{\tt DOUBLE} switch}
\index{{\tt INPUTC} switch}
\index{{\tt PRIMAT} switch}
\index{{\tt PRIALL} switch}
\index{{\tt EXP} switch} 
\index{{\tt FORT} switch}
\index{{\tt FTCH} switch}
\index{{\tt AGAIN} switch}
\index{{\tt SIDREL} switch}
\index{{\tt VECTORC} switch}
\index{{\tt NAT} switch}
\index{{\tt PERIOD} switch} 
\index{{\tt GENTRANOPT} switch}
\index{{\tt PREFIX} switch}
\index{{\tt INTERN} switch}
\index{{\tt EVALLHSEQP} switch}
\index{{\tt ROUNDBF} switch} 

The result of an application of SCOPE can be influenced by the use of
certain {\REDUCE}- or SCOPE-switches. The influence of {\tt EXP} is obvious:
unexpanded input is more compact than expanded.
{\tt ON ACINFO} serves to produce tables with the numbers of arithmetic 
operations, oc\-cu\-ring in ${{{\rm E}}_0}$ and ${{\rm E}}_{\lambda}$, 
respectively.
{\tt ON INPUTC} serves to echo the input, processed by SCOPE. The actual
form of the input can be the consequence of locally performed evaluations, 
before the actual parsing into the database takes place.
{\tt ON  PRIMAT} can be used to visualize 
both ${{\rm D}}_{0}$ and $D_{\lambda}$.  
{\tt ON  PRIALL} finally, can be used instead of
{\tt ON ACINFO,INPUTC,PRIMAT}. These SCOPE-switches are initially all turned
{\tt OFF}. SCOPE has a facility to visualize the status of all
SCOPE-switches and some relevant {\REDUCE}-switches. The current status
of all relevant switches can be presented with the command

\hspace*{1cm} {\tt SCOPE\_SWITCHES}\verb+$+
\index{SCOPE function ! {\tt SCOPE\_SWITCHES}}


\example\label{ex:3.1.1}
\index{SCOPE ! example}

The start of a {\REDUCE} session shows the initial state
of {\REDUCE}, directly after loading the SCOPE package. 
The set of relevant switches is made visible. 
Besides the {\REDUCE} switches {\tt EVALLHSEQP}, {\tt EXP}, {\tt FORT}, 
{\tt NAT}, {\tt PERIOD}, {\t ROUNDBF}
and {\tt ROUNDED} six additional SCOPE switches, i.e. {\tt AGAIN}, {\tt FTCH}, 
{\tt INTERN}, {\tt PREFIX}, {\tt SIDREL} and {\tt VECTORC}, and the GENTRAN
switches {\tt DOUBLE} and {\tt GENTRANOPT} are thus presented. 
They all wil be discussed in more 
detail below.

{\small
\begin{verbatim}
REDUCE 3.6, 15-Jul-95 ...

1: load_package nscope$

2: SCOPE_SWITCHES$

 ON  :  exp  ftch  nat  period  
 OFF :  acinfo  again  double  evallhseqp  fort  gentranopt  inputc  
        intern  prefix  priall  primat  roundbf  rounded  sidrel  
        vectorc  

3: % etc. ...
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}

Output is by default given in {\REDUCE} syntax, but FORTRAN syntax is
possible in the usual way, e.g. {\tt ON FORT} and {\tt OFF PERIOD}, for 
instance.  The use of other target languages
from the GENTRAN repertoire is discussed in section~\ref{SCOPE:decl}.

\subsection{The {\tt OPTIMIZE} command: Straightforward use}\label{SCOPE:optim}

\index{{\tt OPTIMIZE} command}
\index{{\tt INAME} option}
\index{REDUCE function ! {\tt gensym}}
\index{SCOPE function ! {\tt SETLENGTH}}
A SCOPE application is easily performed and based on the use of 
the following syntax:

\begin{center}
\begin{tabular}{lcl}
 $<$SCOPE\_application$>$ & $::=$ & {\tt OPTIMIZE} $<$object\_seq$>$
 [{\tt INAME} $<$cse\_prefix$>$]\\
 $<$object\_seq$>$ & $::=$ & $<$object$>$[,$<$object\_seq$>$]\\
 $<$object$>$ & $::=$ & $<$stat$>~\mid~<$alglist$>~\mid~<$alglist\_production$>$ \\
 $<$stat$>$ & $::=$ & $<$name$>~<$assignment operator$>~<$expression$>$\\
 $<$assignment operator$>$ & $::=$ & $:=~\mid~::=~\mid~::=:~\mid~:=:$\\
 $<$alglist$>$ & $::=$ & \{$<$eq\_seq$>$\}\\
 $<$eq\_seq$>$ & $::=$ & $<$name$>~=~<$expression$>$[,$<$eq\_seq$>$]\\
 $<$alglist\_production$>$ & $::=$ & $<$name$>~\mid~<$function\_application$>$\\
 $<$name$>$ & $::=$ & $<$id$>~\mid~<$id$>(<$a\_subscript\_seq$>)$\\
 $<$a\_subscript\_seq$>$ & $::=$ & $<$a\_subscript$>$[,$<$a\_subscript\_seq$>$]\\
 $<$a\_subscript$>$ & $::=$ & $<$integer$>~\mid~<$integer infix\_expression$>$\\
 $<$cse\_prefix$>$ & $::=$ & $<$id$>$
\end{tabular}
\end{center}

A SCOPE action can be applied on one assignment statement. 
The assigned variable
is either a scalar identifier, like {\tt z} in example~\ref{ex:3.1.2}, or a 
name with subscripts, such as {\tt a(1,1)} in example~\ref{ex:3.1.3}. 
In stead of one
statement a sequence of such statements, separated by comma's, is possible.
An alternative is provided by the use of an algebraic mode list, consisting
of {\REDUCE} equations. An assigned variable, identifying such a list, is 
allowed as well. Examples are presented in section~\ref{SCOPE:algo}.
The function\_application is introduced in section~\ref{SCOPE:soph}. Such an 
application ought to produce an alglist.

The expressions, i.e. rhs's in assignments or equations are legal {\REDUCE}
expressions or ought to evaluate to such expressions. Statements inside 
expressions are allowed, but only useful if these expressions are evaluated,
before being optimized. Only integer or rounded coefficients are supported
by SCOPE. So we either suppose the default integer setting or allow the switch
{\tt ROUNDED} to be turned {\tt ON}.

The optional use of the {\tt INAME} extension in an {\tt OPTIMIZE} command
is introduced to allow the user to influence the generation of cse-names.
The cse\_prefix is an identifier, used to generate cse-names, by
extending it with an integer part. If the cse\_prefix consists of letters
only, the initially selected integer part is 0. All following integer parts are
obtained by incrementing the previous integer part by 1. If the user-supplied
cse\_prefix ends with an integer its value functions as initial integer part.
The {\tt gensym}-function is applied when the {\tt INAME}-extension is omitted.
The three alternatives are illustrated in example~\ref{ex:3.1.2}.

As stated before minimal profits are accepted during all stages of the
optimization process: many small steps may lead to impressive final results.
But it can also lead to unwanted details. Therefore, it can be desirable
to rerun an optimization request with a restriction on the minimal size of 
the rhs's. The command 

\hspace*{1cm} {\tt SETLENGTH} $<$integer$>$\$  

can be used to produce rhs's with a minimal arithmetic complexity, 
dictated by the value of 
its integer argument. Statements, used to rename function applications, are 
not affected by the {\tt SETLENGTH} command. The default setting is restored 
with the command 

\hspace*{1cm} {\tt RESETLENGTH}\$
\index{SCOPE function ! {\tt RESETLENGTH}}

We now illustrate the use of the {\tt OPTIMIZE} command through a number of
small examples, being parts of {\REDUCE} sessions. We show in
example~\ref{ex:3.1.2} the effect of the different visualization switches,
the use of {\tt SETLENGTH} and {\tt RESETLENGTH} and of the three {\tt INAME}
alternatives. In example~\ref{ex:3.1.3} the effect of some of the GENTRAN and 
SCOPE input processing features is presented. Some finishing touch activities 
are illustrated in the examples  ~\ref{ex:3.1.4} and ~\ref{ex:3.1.5}. 
The approach towards rational exponents is presented in 
example~\ref{ex:3.1.6}, while some form of quotient optimization is 
illustrated in example~\ref{ex:3.1.7}. Finally, we present the differences 
in {\tt ON/OFF EXP} processing in example~\ref{ex:3.1.8}.

\example\label{ex:3.1.2}
\index{SCOPE ! example}

The multivariate polynomial {\tt z} is a sum of 6 terms. These
terms, monomials, are constant multiples of power products.  A
picture of ${{\rm D}}_{0}$ is shown after the input echo. The
sums-matrix consists of only one row, identifiable by its Fa(the)r {\tt z},
the lhs. Its exponent is given in the EC (Exponent or Coefficient)
field. The 6 monomials are stored in the products-matrix. The
coefficients are stored in the EC-fields and the predecessor row
index, 0, is given in the Far-field. Before the $D_{\lambda}$ picture
is given the effect of the optimization process, the output and the
operator counts are shown. The optimized form of {\tt z} is obtained by
applying the distributive law. The output also shows applications of
an addition chain algorithm (\cite{Knuth:80} 441-466) as part of ${{\rm
R}}^{{-1}}$, although its use in example~\ref{ex:3.1.4} is more apparent.
\index{addition chain algorithm}

Observe that the output illustrates the heuristic character of the
optimization process: In this particular case the rhs can be written
as a polynomial in {\tt g4}, thus saving one extra multiplication.

The {\tt SETLENGTH} command is illustrated too. See also example~\ref{ex:3.2.6}.
Application of a Horner-rule may be profitable as well. See, for instance
example~\ref{ex:4.1.3}.
\index{{\tt PRIALL} switch}
{\small
\begin{verbatim}
ON PRIALL$

z:=a^2*b^2+10*a^2*m^6+a^2*m^2+2*a*b*m^4+2*b^2*m^6+b^2*m^2;

      2  2       2  6    2  2          4      2  6    2  2
z := a *b  + 10*a *m  + a *m  + 2*a*b*m  + 2*b *m  + b *m

OPTIMIZE z:=:z$

      2  2       2  6    2  2          4      2  6    2  2
z := a *b  + 10*a *m  + a *m  + 2*a*b*m  + 2*b *m  + b *m

Sumscheme :

   || EC|Far
------------
  0||  1| z
------------
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
Productscheme :

   |  0  1  2| EC|Far
---------------------
  1|     2  2|  1| 0
  2|  6     2| 10| 0
  3|  2     2|  1| 0
  4|  4  1  1|  2| 0
  5|  6  2   |  2| 0
  6|  2  2   |  1| 0
---------------------
0  : m
1  : b
2  : a

Number of operations in the input is: 

Number of (+/-) operations      : 5
Number of unary - operations    : 0
Number of * operations          : 10
Number of integer ^ operations  : 11
Number of / operations          : 0
Number of function applications : 0

g1 := b*a
g5 := m*m
g2 := g5*b*b
g3 := g5*a*a
g4 := g5*g5
z := g2 + g3 + g1*(2*g4 + g1) + g4*(2*g2 + 10*g3)

Number of operations after optimization is:

Number of (+/-) operations      : 5
Number of unary - operations    : 0
Number of * operations          : 12
Number of integer ^ operations  : 0
Number of / operations          : 0
Number of function applications : 0

Sumscheme :

   |  0  3  4  5| EC|Far
------------------------
  0|        1  1|  1| z
 15|        2 10|  1| 14
 17|  2  1      |  1| 16
------------------------
0  : g4
3  : g1
4  : g2
5  : g3
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
Productscheme :

   |  8  9 10 11 17 18 19 20| EC|Far
------------------------------------
  7|                    1  1|  1| g1
  8|  1                 2   |  1| g2
  9|  1                    2|  1| g3
 10|  2                     |  1| g4
 11|                 2      |  1| g5
 14|     1                  |  1| 0
 16|              1         |  1| 0
------------------------------------
8  : g5
9  : g4
10 : g3
11 : g2
17 : g1
18 : m
19 : b
20 : a

OFF PRIALL$
SETLENGTH 2$

OPTIMIZE z:=:z INAME s$

       2  2
s1 := b *m
       2  2
s2 := a *m
               4                       4
z := (a*b + 2*m )*a*b + 2*(s1 + 5*s2)*m  + s1 + s2

RESETLENGTH$

OPTIMIZE z:=:z INAME s1$

s1 := b*a
s5 := m*m
s2 := s5*b*b
s3 := s5*a*a
s4 := s5*s5
z := s2 + s3 + s1*(2*s4 + s1) + s4*(2*s2 + 10*s3)
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\index{SCOPE function ! {\tt SETLENGTH}}
\index{SCOPE function ! {\tt RESETLENGTH}}
\example\label{ex:3.1.3} 
\index{SCOPE ! example}

The input echo below shows the literal copy of the first assignment.
This is in accordance with role the GENTRAN assignment operator {\tt :=}
ought to play.  The second assignment, this time using the operator {\tt ::=:},
leads to rhs evaluation (expansion) and lhs subscript-value substitution.
Application of the distributive law is refected by the rhs of {\tt a(1,1)} 
in the presented result.
\index{{\tt INPUTC} switch}
{\small
\begin{verbatim}
OPERATOR a$ k:=j:=1$ u:=c*x+d$ v:=sin(u)$
ON INPUTC$

OPTIMIZE a(k,j):=v*(v^2*cos(u)^2+u), a(k,j)::=:v*(v^2*cos(u)^2+u) INAME s;

              2       2
a(k,j) := v*(v *cos(u)  + u)

                      2             3
a(1,1) := cos(c*x + d) *sin(c*x + d)  + sin(c*x + d)*c*x + sin(c*x + d)*d

s9 := cos(u)*v
a(k,j) := v*(u + s9*s9)
s6 := x*c + d
s5 := sin(s6)
s10 := s5*cos(s6)
a(1,1) := s5*(s6 + s10*s10)
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\example\label{ex:3.1.4}
\index{SCOPE ! example}

The effect is shown of a finishing touch application, in combination
with FORTRAN output. The value of {\tt S0} is rewritten during output 
preparation, and the earlier mentioned addition chain algorithm is applied.
When turning {\tt OFF} the switch {\tt FTCH} the latter activity is skipped.
\index{{\tt FTCH} switch}
\index{{\tt FORT} switch}
\index{{\tt PERIOD} switch}
{\small
\begin{verbatim}
ON FORT$ OFF PERIOD$

OPTIMIZE z:=(6*a+18*b+9*c+3*d+6*f+18*g+6*h+5*j+5*k+3)^13 INAME s;

      S0=5*(J+K)+3*(3*C+D+1+6*(B+G)+2*(A+F+H))
      S3=S0*S0*S0
      S2=S3*S3
      Z=S0*S2*S2

OFF FTCH$

OPTIMIZE z:=(6*a+18*b+9*c+3*d+6*f+18*g+6*h+5*j+5*k+3)^13 INAME s;

      Z=(5*(J+K)+3*(3*C+D+1+6*(B+G)+2*(A+F+H)))**13
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\example\label{ex:3.1.5}
\index{SCOPE ! example}

Recovery of repeatedly occurring integer multiples of identifiers,
as part of the finishing touch, is illustrated. This facility is
part of the finishing touch and will seemingly be neglected when
using {\tt SETLENGTH} 2\verb+$+ instruction in stead of {\tt OFF FTCH}.
\index{{\tt FTCH} switch}
{\small
\begin{verbatim}
OPTIMIZE x:=3*a*p, y:=3*a*q, z:=6*a*r+2*b*p,
         u:=6*a*d+2*b*q, v:=9*a*c+4*b*d, w:=4*b INAME s;

s2 := 3*a
x := s2*p
y := s2*q
s0 := 2*b
s3 := 6*a
z := s0*p + s3*r
u := s0*q + s3*d
w := 4*b
v := w*d + 9*c*a

OFF FTCH$

OPTIMIZE x:=3*a*p, y:=3*a*q, z:=6*a*r+2*b*p,
         u:=6*a*d+2*b*q, v:=9*a*c+4*b*d, w:=4*b INAME t;

x := 3*p*a
y := 3*q*a
z := 2*b*p + 6*r*a
u := 2*b*q + 6*d*a
v := 4*d*b + 9*c*a
w := 4*b

ON FTCH$
SETLENGTH 2$

OPTIMIZE x:=3*a*p, y:=3*a*q, z:=6*a*r+2*b*p,
         u:=6*a*d+2*b*q, v:=9*a*c+4*b*d, w:=4*b INAME t;

x := 3*p*a
y := 3*q*a
z := 2*b*p + 6*r*a
u := 2*b*q + 6*d*a
v := 4*d*b + 9*c*a
w := 4*b
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\index{{\tt FTCH} switch}
\index{SCOPE function ! {\tt SETLENGTH}}
\example\label{ex:3.1.6}
\index{SCOPE ! example}

This example serves to show how SCOPE deals with rational 
exponents.  All rational exponents of an identifier are collected. 
\index{rational exponents}
The least common
multiple lcm of the denominators of these rationals is
computed and the variable is replaced by a possibly newly selected
variable name, denoting the variable raised to the power 1/lcm. This
facility is only efficient for what we believe to be problems
occurring in computational practice. That is easily verified by
extending the sum we are elaborating here with some extra terms.
\newpage
{\small
\begin{verbatim}
ON INPUTC,FORT$

OPTIMIZE z:=:FOR j:=2:6 SUM q^(1/j) INAME s;

      1/6    1/5    1/4    1/3
z := q    + q    + q    + q    + sqrt(q)

      S0=Q**(1.0/60.0)
      S8=S0*S0
      S7=S8*S0
      S5=S8*S7
      S3=S5*S5
      S2=S8*S3
      S1=S7*S2
      S4=S5*s1
      Z=S4+S1+S2+S3+S4*S3
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\example\label{ex:3.1.7}
\index{SCOPE ! example}

\index{{\tt INPUTC} switch}
\index{{\tt ROUNDED} switch}
\index{{\tt FORT} switch}
The special attention, given to rational exponents, is not extended to 
rational coefficients. The script in this example shows four different 
approaches for dealing with such coefficients using the expressions assigned to
{\tt f} and {\tt g}. We start with a literal parsing of the two assignments,
leading to a form of ${\rm D}_0$, which is  based on the present REDUCE 
strategy for dealing with fixed float numbers in the default integer 
coefficient domain setting.
The four rational numbers $\frac{31}{5} ,~ \frac{31}{10} ,~\frac{93}{5}~
{\rm and}~\frac{93}{10}$ are just like ${\rm b}^{\frac{1}{5}}$, $\sqrt {\rm 
sin(\cdots)}$ and ${\rm sin(\cdots)}^{\frac{5}{3}}$ considered as kernels.

The second approach illustrates the effect of simplification in 
an {\tt OFF ROUNDED} mode prior to parsing. The input expressions are 
remodeled into rational expressions, the usual internal standard quotient form.

After turning {\tt ON} the switch  {\tt ROUNDED}  we repeat the previous 
commands. Again some differences in evaluation can be observed. Literally
taken input, the third approach,  shows rational exponent optimizations
prior to the production of rounded exponents in the output.
The last approach, simplification before parsing, leads to a float 
representation for the rational exponents. SCOPE's exponent optimization
features are designed for integer and rational exponents only. Floating
point exponentiation is therefore assumed to be a function application.

Further illustrations of operations on quotients are shown in
example~\ref{ex:8.2}.
{\small
\begin{verbatim}
ON INPUTC$

OPTIMIZE
f:= cos(6.2*a+18.6*(b)^(1/5))/sqrt(sin(3.1*a+9.3*(b)^(1/5))),
g:= sin(6.2*a+18.6*(b)^(1/5))/sin(3.1*a+9.3*(b)^(1/5))^(5/3)
INAME s$

\end{verbatim}}
\newpage
{\small
\begin{verbatim}
              31       93   1/5
         cos(----*a + ----*b   )
              5        5
f := -------------------------------
                31       93   1/5
      sqrt(sin(----*a + ----*b   ))
                10       10

            31       93   1/5
       sin(----*a + ----*b   )
            5        5
g := ----------------------------
           31       93   1/5 5/3
      sin(----*a + ----*b   )
           10       10


        1/5
s15 := b
            93       31
s12 := s15*---- + a*----
            5        5
           93         31
s6 := sin(----*s15 + ----*a)
           10         10
         1/6
s14 := s6
s5 := s14*s14*s14
      cos(s12)
f := ----------
         s5
      sin(s12)
g := -----------
      s5*s14*s6


OPTIMIZE
f:=: cos(6.2*a+18.6*(b)^(1/5))/sqrt(sin(3.1*a+9.3*(b)^(1/5))),
g:=: sin(6.2*a+18.6*(b)^(1/5))/sin(3.1*a+9.3*(b)^(1/5))^(5/3)
INAME t$


                  1/5
              93*b    + 31*a
         cos(----------------)
                    5
f := -----------------------------
                    1/5
                93*b    + 31*a
      sqrt(sin(----------------))
                      10
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
                           1/5
                       93*b    + 31*a
                  sin(----------------)
                             5
g := ------------------------------------------------
               1/5                      1/5
           93*b    + 31*a  2/3      93*b    + 31*a
      sin(----------------)   *sin(----------------)
                 10                       10


          1/5
t7 := 93*b    + 31*a
       t7
t2 := ----
       5
           t7
t5 := sin(----)
           10
         1/6
t11 := t5
t4 := t11*t11*t11
      cos(t2)
f := ---------
        t4
       sin(t2)
g := -----------
      t4*t11*t5


ON ROUNDED$

OPTIMIZE
f:= cos(6.2*a+18.6*(b)^(1/5))/sqrt(sin(3.1*a+9.3*(b)^(1/5))),
g:= sin(6.2*a+18.6*(b)^(1/5))/sin(3.1*a+9.3*(b)^(1/5))^(5/3)
INAME s$


                          1/5
        cos(6.2*a + 18.6*b   )
f := -----------------------------
                            1/5
      sqrt(sin(3.1*a + 9.3*b   ))

                         1/5
       sin(6.2*a + 18.6*b   )
g := --------------------------
                       1/5 5/3
      sin(3.1*a + 9.3*b   )
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
           0.2
s5 := 9.3*b    + 3.1*a
s8 := 2*s5
s4 := sin(s5)
         0.166666666667
s10 := s4
s3 := s10*s10*s10
      cos(s8)
f := ---------
        s3
       sin(s8)
g := -----------
      s3*s10*s4


OPTIMIZE
f:=: cos(6.2*a+18.6*(b)^(1/5))/sqrt(sin(3.1*a+9.3*(b)^(1/5))),
g:=: sin(6.2*a+18.6*(b)^(1/5))/sin(3.1*a+9.3*(b)^(1/5))^(5/3)
INAME t$


                 0.2
       cos(18.6*b    + 6.2*a)
f := --------------------------
               0.2         0.5
      sin(9.3*b    + 3.1*a)

                      0.2
            sin(18.6*b    + 6.2*a)
g := ------------------------------------
               0.2         1.66666666667
      sin(9.3*b    + 3.1*a)


           0.2
t6 := 9.3*b    + 3.1*a
t9 := 2*t6
t5 := sin(t6)
      cos(t9)
f := ---------
         0.5
       t5
          sin(t9)
g := -----------------
        1.66666666667
      t5
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\newpage
\example\label{ex:3.1.8}
\index{SCOPE ! example}
\index{{\tt ACINFO} switch}
\index{{\tt FORT} switch}
\index{{\tt PERIOD} switch}
\index{{\tt EXP} switch}

The effect of {\tt ON EXP} or {\tt OFF EXP} on the result of a
SCOPE-application is illustrated by optimi\-zing the representation of the
determinant of a symmetric (3,3) matrix {\tt m}.  Besides differences in
computing time we also observe that the arithmetic complexity of the
optimized version of the expanded representation of the determinant is
about the same as the not optimized form of the unexpanded representation.

{\small
\begin{verbatim}
MATRIX m(3,3)$

m(1,1):=18*cos(q3)*cos(q2)*m30*p^2-sin(q3)^2*j30y+sin(q3)^2*j30z-
        9*sin(q3)^2*m30*p^2+j1oy+j30y+m10*p^2+18*m30*p^2$

m(2,1):=
m(1,2):=9*cos(q3)*cos(q2)*m30*p^2-sin(q3)^2*j30y+sin(q3)^2*j30z-
        9*sin(q3)^2*m30*p^2+j30y+9*m30*p^2$

m(3,1):=
m(1,3):=-9*sin(q3)*sin(q2)*m30*p^2$

m(2,2):=-sin(q3)^2*j30y+sin(q3)^2*j30z-9*sin(q3)^2*m30*p^2+j30y+
        9*m30*p^2$

m(3,2):=
m(2,3):=0$

m(3,3):=9*m30*p^2+j30x$

ON ACINFO,FORT$ OFF PERIOD$

OPTIMIZE detm:=:det(m) INAME s;

Number of operations in the input is: 

Number of (+/-) operations      : 36
Number of unary - operations    : 0
Number of * operations          : 148
Number of integer ^ operations  : 84
Number of / operations          : 0
Number of function applications : 32

      S2=SIN(REAL(Q2))
      S30=S2*S2
      S3=SIN(REAL(Q3))
      S29=S3*S3
      S31=P*P
      S8=S31*M30
      S32=S8*S8
      S4=S32*J30Y
      S28=S32*S8
      S9=S29*M10
      S10=S30*S29*S29
      S44=COS(REAL(Q3))*COS(REAL(Q2))
      S11=S44*S44
      S20=S31*S8
      S23=S31*J30X
      S22=S29*J30X
      S24=S8*J1OY
      S19=M10*J30Y
      S43=81*S32*J30X
      S35=-S43-(81*S32*J1OY)
      S36=-(729*S29*S28)-(81*S29*S4)
      S37=J30Z-J30Y
      S39=9*S37
      S40=9*J30X
      S41=81*S32*J30Z
      S42=81*S4
      DETM=S42+S36-S35+729*S28+S37*(S22*J1OY+9*S29*S24+S23*S9)+S10*(S42-
     . S41)+S20*S8*81*(M10-S9)+S20*S9*(S39-S40)+S22*S8*(S39-(9*J1OY))+
     . S20*(9*S19+S40*M10)+S24*(S40+9*J30Y)+J30Y*J30X*(J1OY+9*S8)+S28*
     . 729*(S10-S11)+S29*(S41+S35)+S36*S30+S23*S19-(S43*S11)

Number of operations after optimization is:

Number of (+/-) operations      : 30
Number of unary - operations    : 0
Number of * operations          : 59
Number of integer ^ operations  : 0
Number of / operations          : 0
Number of function applications : 4

OFF EXP$

OPTIMIZE detm:=:det(m) INAME t;

Number of operations in the input is: 

Number of (+/-) operations      : 23
Number of unary - operations    : 1
Number of * operations          : 38
Number of integer ^ operations  : 21
Number of / operations          : 0
Number of function applications : 10

      T1=SIN(REAL(Q3))
      T9=T1*T1
      T8=P*P
      T5=T8*M30
      T16=9*T5
      T10=-T16-(9*T5*COS(REAL(Q3))*COS(REAL(Q2)))
      T13=(T16+J30Y-J30Z)*T9
      T15=T13-J30Y
      T0=T15+T10
      T14=T13-T16-J30Y
      T17=T5*SIN(REAL(Q2))
      DETM=81*T17*T17*T14*T9-((T16+J30X)*(T0*T0-(T14*(T15+2*T10-J1OY-(T8
     . *M10)))))

Number of operations after optimization is:

Number of (+/-) operations      : 13
Number of unary - operations    : 0
Number of * operations          : 18
Number of integer ^ operations  : 0
Number of / operations          : 0
Number of function applications : 4
\end{verbatim}}

We can also use this example to show that correctness of results is
easily verified. When storing
the result of a SCOPE application in a file, it is of course possible
to read the result in again. Then we apply a normal {\REDUCE} evaluation
strategy. This implies that all references to cse-names are automatically
replaced by their values.  We show the ``correctness'' of SCOPE by
storing the optimized version of the expanded form of the determinant
of {\tt M}, called {\tt detm1} in file {\tt out.1} and the result of a 
SCOPE-application on the unexpanded form, {\tt detm2}, in file {\tt out.2}, 
followed by reading in both files and by subtracting {\tt detm2} 
from {\tt detm1}, resulting in the value 0.
This is of course an ad hoc correctness-proof for one specific
example. It is in fact another way of testing the code of the package.
We show it as a direct continuation of the previous determinant calculations.

\index{{\tt NAT} switch}
This example also serves to show that the {\tt OPTIMIZE} command can be 
extended with the {\tt OUT} option. The keyword {\tt OUT} has to be followed
\index{{\tt OUT} option}
by a file-name. This file is properly closed and left in a readable form,
assuming printing is produced in a {\tt OFF NAT} fashion.
SCOPE's file management features are discussed in more detail in 
section~\ref{SCOPE:files}.
{\small
\begin{verbatim}
OFF ACINFO,FORT,NAT$ ON EXP$

OPTIMIZE detm1:=:det(M) OUT "out.1" INAME s;

OFF EXP$

OPTIMIZE detm2:=:det(M) OUT "out.2" INAME t;

ON NAT$
IN "out.1","out.2"$

detm1-detm2;

0
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}

So far we presented via some examples straightforward
algebraic mode use. The output is produced as a side-effect. However,
optimization results can easily be made operational in algebraic mode.
The parameterless function 

\hspace*{1cm} {\tt ARESULTS} 
\index{SCOPE function ! {\tt ARESULTS}}

delivers the result of the directly
preceding {\tt OPTIMIZE} command in the form of a list of equations,
corresponding with the sequence of assignment statements, shown either in
{\REDUCE} syntax or in the syntax of one of GENTRAN's target languages.
But, we need to operate carefully. Application of a variety of assignment 
operators can easily bring in identifiers, representing earlier produced
algebraic values. They will be substituted automatically, when referenced in 
rhs's in the list, produced with an {\tt ARESULTS} application. 
Therefore, we implemented a
protection mechanism. Before delivering output produced by {\tt ARESULTS},
we make the system temporarily deaf for such references. The
possibly game-spoiling algebraic values are stored at a seemingly 
anonymous place. All
identifiers, subjected to this special treatement, can be made visible with
the command 

\hspace*{1cm} {\tt RESTORABLES};
\index{SCOPE function ! {\tt RESTORABLES}}

Their original status can be restored, either globally with the command 

\hspace*{1cm} {\tt RESTOREALL}\verb+$+ 
\index{SCOPE function ! {\tt RESTOREALL}}

or selectively with the instruction 

\hspace*{1cm} {\tt ARESTORE} $<$subsequence$>$\verb+$+ 
\index{SCOPE function ! {\tt ARESTORE}}

This subsequence is built with names, selected  from the list 
of {\tt RESTORABLES}, and separated by comma's. Information restoration 
is only possible before the next {\tt OPTIMIZE} command.

\example\label{ex:3.1.9}
\index{SCOPE ! example}

The use of these commands is now illustrated.
A further explanation is given in the form of comment in the script. 

{\small
\begin{verbatim}
u:=a*x+2*b$ v:=sin(u)$ w:=cos(u)$ f:=v^2*w;

                                  2
f := cos(a*x + 2*b)*sin(a*x + 2*b)

OFF EXP$

OPTIMIZE f:=:f,g:=:f^2+f INAME s;

s3 := x*a + 2*b
s2 := sin(s3)
f := s2*s2*cos(s3)
g := f*(f + 1)

alst:=ARESULTS;

alst := {s3=a*x + 2*b,

         s2=sin(s3),

                     2
         f=cos(s3)*s2 ,

         g=(f + 1)*f}

% ---
% SCOPE is made deaf for the standard reference mechanism for algebraic 
% variables. However the rhs's in the list alst are simplified before 
% being shown. It explains the differences between the layout in the 
% alst items and the results, presented by the OPTIMIZE-command itself.
% ---

RESTORABLES;

{f}

f;

f

ARESTORE f$

f;
                             2
cos(a*x + 2*b)*sin(a*x + 2*b)

% ---
% f is re-associated with its original value. This can lead to a modified 
% presentation of some of the rhs's of alst.
% ---

alst;

{s3=a*x + 2*b,

 s2=sin(s3),

             2
 f=cos(s3)*s2 ,

                                2                                   2
 g=(cos(a*x + 2*b)*sin(a*x + 2*b)  + 1)*cos(a*x + 2*b)*sin(a*x + 2*b) }

OPTIMIZE f:=:f,g:=:f^2+f INAME s;

s3 := x*a + 2*b
s2 := sin(s3)
f := s2*s2*cos(s3)
g := f*(f + 1)

alst2:=ARESULTS$

OPTIMIZE f:=:f,g:=:f^2+f INAME s;

g := f*(f + 1)

% ---
% The algebraic value, which was associated with f, is permanently
% lost. It ought to be restored before a new OPTIMIZE command is given. 
% Therefore f:=:f produced an identity, which is redundant in terms of 
% code production. More details about removal of redundant code are
% given in section 7, when discussing data dependencies and related topics.
% ---

RESTOREALL$

f;

f
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}

\subsection{The function {\tt ALGOPT}: Straightforward use}\label{SCOPE:algo}

\index{SCOPE function ! {\tt ALGOPT}}
\index{SCOPE function ! {\tt RESTOREALL}}
\index{SCOPE function ! {\tt RESTORABLES}}
\index{SCOPE function ! {\tt ARESULTS}}
\index{SCOPE function ! {\tt ARESTORE}}
The function {\tt ALGOPT} accepts up to three arguments. It can be used in 
stead of the {\tt OPTIMIZE} command. It returns the optimization result,
like {\tt ARESULTS}, in the form of a list of equations. 
Since the {\tt ARESULTS} mechanism
is applied as well, the pre-{\tt ALGOPT}-application situation can be restored
with {\tt RESTOREALL} or partly and selective with {\tt ARESTORE}, using
information, providable by an application of the function {\tt RESTORABLES}.

The first argument of {\tt ALGOPT}, like the other two optional, is the 
equivalent of the alglist or alglist\_production in the earlier introduced
syntax of a SCOPE\_application.
The second argument can be used to inform SCOPE
that input from file(s) have to be processed. We survey SCOPE's file 
management features in section~\ref{SCOPE:files}. So we omit a further 
discussion now. The last argument correspondents with the cse\_prefix 
of the {\tt INAME } option of the {\tt OPTIMIZE} command. The extension
of the SCOPE\_application syntax, needed to include possible {\tt ALGOPT} 
activities, is:

\begin{tabular}{lcl}
$<$SCOPE\_application$>$ & $::=$ & $\cdots~\mid~$\\
 & & {\tt ALGOPT}($<$a\_object\_list$>$[,$<$string\_id\_list$>$][,$<$cse\_prefix$>$]) $\mid$\\
 & & {\tt ALGOPT}([$<$a\_object\_list$>$,]$<$string\_id\_list$>$[,$<$cse\_prefix$>$])\\
\end{tabular}
\begin{tabular}{lcl}
$<$a\_object\_list$>$ & $::=$ & $<$a\_object$>~\mid$ \{$<$a\_object$>$[,$<$a\_object\_seq$>$]\}\\
$<$a\_object\_seq$>$ & $::=$ & $<$a\_object$>$[,$<$a\_object\_seq$>$]\\
$<$a\_object$>$ & $::=$ & $<$id$>~\mid~<$alglist$>~\mid~<$alglist\_production$>~\mid$
 \{$<$a\_object$>$\}
\end{tabular}

We require at least one actual parameter, here the a\_object\_list. Its
syntactical structure allows to apply a GENTRAN-like repertoire in an 
algebraic mode setting. The a\_object's can
either be an alglist identifier, an alglist producing function application, 
or an alglist itself. An alglist identifier can be
either a scalar or a matrix or array entry. The alglist producing
functions will be discussed in section~\ref{SCOPE:soph}. An alglist has the
structure of an algebraic mode list; its elements are either a\_object's
or equations of the form lhs = rhs. Such equations correspondent
with the "take literal" GENTRAN operator {\tt :=} facility in the setting
of an {\tt OPTIMIZE} command (see also section~\ref{SCOPE:soph} for a further
discussion). The alternatives, i.e. uses of {\tt ::=}, {\tt :=:} or {\tt ::=:},
are also covered by the a\_object syntax.
The examples, given in this subsection, show that simplification of an algebraic
list of equations leads to right hand side simplification, corresponding with
the effect of the colon-added-to-the-right-extension of the assignment operator.
However, as illustrated by example~\ref{ex:3.2.7}, some care has to be taken
when operating in {\tt OFF EXP} mode.
Turning {\tt ON} the switch {\tt EVALLHSEXP}, can lead to lhs evaluations,
corresponding with the extra-colon-to-the-left strategy. But we have to
be aware of the instanteneous evaluation mechanism for matrix and 
array entries, when referenced. 

We present some examples of possible use of the {\tt ALGOPT} function.
In example~\ref{ex:3.2.4} a straightforward application is given.
In example~\ref{ex:3.2.5} follows an ilustration of a possible strategy 
concerning optimizing sets of array- and/or matrix-entries. Then, 
in example~\ref{ex:3.2.6}, possible SCOPE assistance in problem analysis is 
shown. Finally in example~\ref{ex:3.2.7}
some differences in simplification and their influence on optimization are
discussed. We also introduce and explain the role of the SCOPE 
switch {\tt SIDREL}.
\index{{\tt SIDREL} switch}
\index{{\tt EVALLHSEQP} switch}

\example\label{ex:3.2.4}
\index{SCOPE ! example}

A number of possible alglist elements is presented in the script.
The first three elements of the actual parameter define values, obtained
via the usual algebraic mode list evaluation mechanism. The last two will be
processed literally. So, the actual parameter for {\tt ALGOPT} is composed 
of the scalar {\tt alist}, a list consisting of the matrix element {\tt m(1,1)},
the array element {\tt ar(2,2)}, nested even deeper, and two equations. Before 
an {\tt ALGOPT} argument is optimized it is flattened by the SCOPE parser 
into a list of equations, the algebraic mode equivalent of the sequence of
assignments in the {\tt OPTIMIZE} context. Evaluation of an {\tt ALGOPT}
application leads to an algebraic mode list of equations, with optimized rhs's.
The cse\_prefix was seemingly superfluous, because all its references
disappeared by back-substitution before output-processing started. See also
example~\ref{ex:3.2.7}.

\index{{\tt ALGOPT} application}
Since an {\tt ALGOPT} application always results in an algebraic mode list,
one can not use this feature for production of code in one of GENTRAN's target
languages. To facilitate the translation of the result of an {\tt ALGOPT}
application, we extended the syntax of the {\tt OPTIMIZE} input repertoire, 
such that alglst\_production's are processable by {\tt OPTIMIZE} 
as well, as illustrated in the script of this example and 
in example~\ref{ex:3.2.7}
{\small
\begin{verbatim}
OFF EXP$
ARRAY ar(2,2)$ MATRIX m(2,2)$

alst:={p1=a+b,p2=(a+b)^2}$

m(1,1):={q1=c+d,q2=(c+d)^2}$

ar(2,2):={r1=(a+b)*(c+d),r2=(a+b)^2*(c+d)^2}$

optlst:=ALGOPT({alst,{m(1,1)},{{ar(2,2)}},
                t1=(a+b)*(c+d)^2,t2=(c+d)*(a+b)^2},s);

optlst := {p1=a + b,
                2
           p2=p1 ,
           q1=c + d,
                2
           q2=q1 ,
           r1=p1*q1,
                2
           r2=r1 ,
           t1=q1*r1,
           t2=p1*r1}

OPTIMIZE optlst$

p1 := a + b
p2 := p1*p1
q1 := c + d
q2 := q1*q1
r1 := q1*p1
r2 := r1*r1
t1 := r1*q1
t2 := r1*p1
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\example\label{ex:3.2.5}
\index{SCOPE ! example}

In example~\ref{ex:3.1.8} we introduced a symmetric (3,3)-matrix {\tt m}. We
present an alternative computation of its determinant. We start with building 
a list of equations, with rhs's, being the non-zero entries of {\tt m}, 
relevant for the computation. The lhs's are produced with 
the {\tt mkid} function. These
newly generated names are assigned to the matrix-entries as well. Finally
we add the definition of the computation of the determinant of {\tt m}, 
in terms of the redefined entries, to this list. 
For the construction of the value of {\tt mlst} we applied both the lhs and
rhs evaluation mechanism. Observe also that, due to the redefinition process,
the original values of the entries of the matrix {\tt m} are lost.
We can optimize {\tt mlst}, using
either an {\tt OPTIMIZE} command or an {\tt ALGOPT} application. The reduction
in arithmetic is not yet impressive here, certainly comparing it with the 
non-expanded, optimized form in the earlier example~\ref{ex:3.1.8}. 
(See also example~\ref{ex:3.2.7} for additional comment).
However, working with larger and non-symmetric matrices will certainly 
improve results, when applying a comparable strategy. 

Observe that the syntax of permissible {\tt ALGOPT} a\_object's does not 
allow to use matrix or array names  to compactly identify the complete set
of their entries. The script in this example shows that such a facility is
easily made. This possibility exists already for matrices in a GENTRAN 
setting (see also example~\ref{ex:8.2} in section~\ref{SCOPE:gopt}).
\index{{\tt EVALLHSEQP} switch}
{\small
\begin{verbatim}
% ---
% We assume the matrix m to be known already.
% ---

mlst:={}$ l:=-1$

OFF EXP$ ON EVALLHSEQP$

FOR j:=1:3 DO FOR k:=j:3 DO
 IF m(j,k) neq 0 THEN
  << s:=mkid(t,l:=l+1);
     mlst:=append(mlst,{s=m(j,k)});
     m(j,k):=m(k,j):=s
  >>$

OFF EVALLHSEQP$

m;

[t0  t1  t2]
[          ]
[t1  t3  0 ]
[          ]
[t2  0   t4]


mlst:=append(mlst,{detm=det(m)});

                                     2         2
mlst := {t0= - (j30y - j30z + 9*m30*p )*sin(q3)

                                    2                      2
          + 18*cos(q2)*cos(q3)*m30*p  + j10y + j30y + m10*p

                    2
          + 18*m30*p ,

                                     2         2
         t1= - (j30y - j30z + 9*m30*p )*sin(q3)

                                   2                 2
          + 9*cos(q2)*cos(q3)*m30*p  + j30y + 9*m30*p ,

                                      2
         t2= - 9*sin(q2)*sin(q3)*m30*p ,

                                     2         2                 2
         t3= - (j30y - j30z + 9*m30*p )*sin(q3)  + j30y + 9*m30*p ,

                          2
         t4=j30x + 9*m30*p ,

                         2         2
         detm=(t0*t3 - t1 )*t4 - t2 *t3}

ON ACINFO,FORT$ OFF PERIOD$
\end{verbatim}}
\index{{\tt ACINFO} switch}
\index{{\tt FORT} switch}
\index{{\tt PERIOD} switch}
{\small
\begin{verbatim}
OPTIMIZE mlst INAME s;

Number of operations in the input is: 

Number of (+/-) operations      : 19
Number of unary - operations    : 1
Number of * operations          : 33
Number of integer ^ operations  : 16
Number of / operations          : 0
Number of function applications : 9

      S0=SIN(REAL(Q3))
      S7=P*P
      S5=S7*M30
      S4=S5*COS(REAL(Q3))*COS(REAL(Q2))
      S13=9*S5
      S11=(S13+J30Y-J30Z)*S0*S0
      T0=J30Y+J10Y+18*(S4+S5)+S7*M10-S11
      T3=S13+J30Y-S11
      T1=T3+9*S4
      T2=-(S13*SIN(REAL(Q2))*S0)
      T4=S13+J30X
      DETM=T4*(T3*T0-(T1*T1))-(T2*T2*T3)

Number of operations after optimization is:

Number of (+/-) operations      : 13
Number of unary - operations    : 1
Number of * operations          : 17
Number of integer ^ operations  : 0
Number of / operations          : 0
Number of function applications : 4
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\example\label{ex:3.2.6}
\index{SCOPE ! example}

We now illustrate that information, produced by SCOPE, can possibly also play 
a role in computations in algebraic mode.  Let ${\rm A.\vec{x}~=~\vec{b}}$ 
be given by

 \[ \left[ \begin{array}{rrrrrr}
                             -1 & 2  & -2 & 1 & 3 & 2 \\
                             -2 & 4  & -4 & 2 & -2 & 3 \\
                              1 & 1  &  1 & 1 & 2 & 4 \\
                              2 & -2 & -1 & 1 & -1 & -2 \\
                              3 & 1  & -4 & 1 & 1 & 2 \\
                             -1 & -5 &  1 & 1 & 3 & 6
      \end{array}\right]~\cdot~\left[ \begin{array}{c}
                                 x1 \\ x2 \\ x3 \\ x4 \\ x5 \\ x6
                               \end{array} \right]~=~
                               \left[ \begin{array}{r}
                                 5 \\ 1 \\ 10 \\ -3 \\ 4 \\ 5
                               \end{array} \right] \cdot \]

This artificial system is constructed for illustrative purposes. 
Its solution is simply ${\rm x_i~=~1}$, ${\rm i~=~1,..,6}$. 
But straightforward inspection shows that

 \[ {\rm A}~=~\left[ \begin{array}{cc}
                          {\rm A}_1 & {\rm A}_2 \\
                          {\rm A}_3 & {\rm A}_4 \end{array}\right]~
            {\rm where}~{\rm A}_1~=~\left[ \begin{array}{cccc}
                                  -1 & 2 & -2 & 1\\ 
                                  -2 & 4 & -4 & 2  \end{array}\right]~
            {\rm and}~ {\rm A}_4~=~\left[ \begin{array}{rr}
                          2 & 4 \\
                         -1 & -2 \\
                          1 & 2 \\
                          3 & 6 \end{array} \right] \cdot \]

We can use {\tt ALGOPT} to "discover" and thus to employ this information.
The system is introduced in the form of assignment statements
  $e_i~=~(\sum_{j=1}^{6}~a_{ij}~\cdot~x_{j})~-~b_{i},~i~=~1, \cdots ,6$.
We use {\tt alst}, identifying the set of equations (see command 9), 
as actual parameter for {\tt ALGOPT}, leading to an algebraic list, 
identified by {\tt reslst} (see command 10). We recognize {\tt g2 = g6 + x5} 
{\tt (= x5 + 2x6)} and {\tt g1 = g3 + g5 + -2x3} {\tt (= -x1 + 2x2 - 2x3 + x4)}. 
Through command 12 we require
cse's to have an arithmetic complexity of a least 4. We then find {\tt g1}
directly, now called {\tt g8}, because we continue applying the
function {\tt gensym}; the cse\_prefix was left out as actual parameter. 
The {\tt solve} function is applied (command 14) to obtain {\tt rootset1}, a list
of values for {\tt x5} and {\tt x6}, expressed in the parameter {\tt g8}.
After assigning {\tt g8} its value in algebraic mode and 
resetting the algebraic values of {\tt ei}, $i~=~ 1, \cdots ,6$ 
with {\tt RESTOREALL} instructions (the commands 11 and 16), 
we can obtain the solution 
of the subsets, denoted by {\tt rootset1} and {\tt rootset2}.
\index{SCOPE function ! {\tt RESTOREALL}}
{\small
\begin{verbatim}
1: LOAD_PACKAGE nscope$

2: e1:=2*x6+3*x5+x4-2*x3+2*x2-x1-5$

3: e2:=3*x6-2*x5+2*x4-4*x3+4*x2-2*x1-1$

4: e3:=2*x5+4*x6+x1+x2+x3+x4-10$

5: e4:=-x5-2*x6+2*x1-2*x2-x3+x4+3$

6: e5:=x5+2*x6+3*x1+x2-4*x3+x4-4$

7: e6:=3*x5+6*x6-x1-5*x2+x3+x4-5$

8: solve({e1,e2,e3,e4,e5,e6},{x1,x2,x3,x4,x5,x6});

{{x1=1,x2=1,x3=1,x4=1,x5=1,x6=1}}

9: alst:={e1=e1,e2=e2,e3=e3,e4=e4,e5=e5,e6=e6}$

10: reslst:=ALGOPT alst;

reslst := {g3= - x1 + x4,

           g5=2*x2,

           g1=g3 + g5 - 2*x3,

           g6=2*x6,

           e1=g1 + g6 + 3*x5 - 5,

           e2=2*g1 - 2*x5 + 3*x6 - 1,

           g2=g6 + x5,

           g4=x2 + x4,

           e3=2*g2 + g4 + x1 + x3 - 10,

           e4= - g2 - g5 + 2*x1 - x3 + x4 + 3,

           e5=g2 + g4 + 3*x1 - 4*x3 - 4,

           e6=3*g2 + g3 - 5*x2 + x3 - 5}

11: RESTOREALL$

12: SETLENGTH 4$

13: reslst:=ALGOPT alst;

reslst := {g8= - x1 + 2*x2 - 2*x3 + x4,

           e1=g8 + 3*x5 + 2*x6 - 5,

           e2=2*g8 - 2*x5 + 3*x6 - 1,

           e3=x1 + x2 + x3 + x4 + 2*x5 + 4*x6 - 10,

           e4=2*x1 - 2*x2 - x3 + x4 - x5 - 2*x6 + 3,

           e5=3*x1 + x2 - 4*x3 + x4 + x5 + 2*x6 - 4,

           e6= - x1 - 5*x2 + x3 + x4 + 3*x5 + 6*x6 - 5}

14: rootset1:=solve({part(reslst,2,2),part(reslst,3,2)},{x5,x6});

                  g8 + 13       - 8*g8 + 13
rootset1 := {{x5=---------,x6=--------------}}
                    13              13
15: g8:=part(reslst,1,2);

g8 :=  - x1 + 2*x2 - 2*x3 + x4

16: RESTOREALL$

17: rootset2:=solve(sub(rootset1,{e3,e4,e5,e6}),{x1,x2,x3,x4});

rootset2 := {{x1=1,x2=1,x3=1,x4=1}}

18: rootset1:=sub(rootset2,rootset1);

rootset1 := {{x5=1,x6=1}}
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\index{SCOPE function ! {\tt SETLENGTH}}
\index{{\tt EXP} switch}
\example\label{ex:3.2.7}
\index{SCOPE ! example}

The script in example~\ref{ex:3.2.5} suggests that we can easily copy GENTRAN's
assignment features by some listprocessing in algebraic mode. However, we 
have to operate carefully. In the script of the present example we introduce an
expression denoted by {\tt f}. Production of a number of its partial (higher)
derivatives is a straightforward mechanism to assist in constructing a set
of assignment statements, containing lots of cse's. Inspection of the values,
in {\tt OFF EXP} mode assigned to {\tt faa}, {\tt tst1} and {\tt tst2}, 
respectively, learns that the value of {\tt mlst} in example~\ref{ex:3.2.5}
may be improvable.
{\small
\begin{verbatim}
u:=a*x+2*b$ v:=sin(u)$ w:=cos(u)$ f:=v^2*w$

OFF EXP$

faa:=df(f,a,2);

                        2                   2                  2
faa := (2*cos(a*x + 2*b)  - 7*sin(a*x + 2*b) )*cos(a*x + 2*b)*x

tst1:={faa=df(f,a,2)};

                             3  2                                  2  2
tst1 := {faa=2*cos(a*x + 2*b) *x  - 7*cos(a*x + 2*b)*sin(a*x + 2*b) *x }

tst2:={faa=(faa:=df(f,a,2))};

                              2                   2                  2
tst2 := {faa=(2*cos(a*x + 2*b)  - 7*sin(a*x + 2*b) )*cos(a*x + 2*b)*x }
\end{verbatim}}
We produce an optimized version of the value of {\tt tlst}, using {\tt ALGOPT}.
Switching {\tt ON INPUTC} and {\tt PRIMAT} results in an input echo, indeed
showing expanded rhs's and a vizualized picture of the \verb+Sumscheme+ of
$D_{\lambda}$. We skipped the ${\rm D_0}$-picture and the rest of the 
$D_{\lambda}$-picture from the script. The value of {\tt reslst} shows the
patterns {\tt s14 = -7.f.x + 2.s9.x} and {\tt fbb = -28.f + 8.s9}. 
The presented \verb+Sumscheme+ of $D_{\lambda}$ suggests that {\tt fbb} and 
{\tt s13} (see the Fa(the)r entries) seemingly have
nothing in common. But {\tt s14} stands for {\tt 2.s6 - 7.s7}, because,
column 8 has to be identified with {\tt s7}, etc. Since both {\tt S6} and
{\tt S7} occur only once in $D_{\lambda}$, their value replaces them in the 
output.
It is an illustration of the heuristic character of the optimization process.
Optimization of the value of {\tt reslst} shows that the repeated pattern 
is now recognized.
\index{{\tt INPUT} switch}
\index{{\tt PRIMAT} switch}
{\small
\begin{verbatim}
tlst:={f=f,fa=df(f,a),fb=df(f,b),faa=df(f,a,2),
       fab=df(f,a,b),fba=df(f,b,a),fbb=df(f,b,2)}$

ON INPUTC,PRIMAT$

reslst:=ALGOPT(tlst,s);

                                  2
f := cos(a*x + 2*b)*sin(a*x + 2*b)

                      2                                  3
fa := 2*cos(a*x + 2*b) *sin(a*x + 2*b)*x - sin(a*x + 2*b) *x

                      2                                  3
fb := 4*cos(a*x + 2*b) *sin(a*x + 2*b) - 2*sin(a*x + 2*b)

                       3  2                                  2  2
faa := 2*cos(a*x + 2*b) *x  - 7*cos(a*x + 2*b)*sin(a*x + 2*b) *x

                       3                                     2
fab := 4*cos(a*x + 2*b) *x - 14*cos(a*x + 2*b)*sin(a*x + 2*b) *x

                       3                                     2
fba := 4*cos(a*x + 2*b) *x - 14*cos(a*x + 2*b)*sin(a*x + 2*b) *x

                       3                                   2
fbb := 8*cos(a*x + 2*b)  - 28*cos(a*x + 2*b)*sin(a*x + 2*b)

Sumscheme :

   |  3  4  5  6  7  8  9 10 11 12 32| EC|Far
---------------------------------------------
  3|                             1  2|  1| s3
 20|         -28  8                  |  1| fbb
 37|                -7  2            |  1| s14
 38|                      -1  2      |  1| s15
---------------------------------------------
3  : s3
4  : s15
5  : s14
6  : s4
7  : s9
8  : s7
9  : s6
10 : s10
11 : s5
12 : s8
32 : b

reslst := {s3=a*x + 2*b,

           s0=cos(s3),

                3
           s9=s0 ,

           s2=sin(s3),

           s12=s0*s2,

           f=s12*s2,

                            3
           s15=2*s0*s12 - s2 ,

           fa=s15*x,

           fb=2*s15,

           s14= - 7*f*x + 2*s9*x,

           faa=s14*x,

           fab=2*s14,

           fba=fab,

           fbb= - 28*f + 8*s9}

OFF INPUTC,PRIMAT$

OPTIMIZE reslst$

s3 := 2*b + x*a
s0 := cos(s3)
s9 := s0*s0*s0
s2 := sin(s3)
s12 := s2*s0
f := s12*s2
s15 := 2*s12*s0 - s2*s2*s2
fa := s15*x
fb := 2*s15
g3 := 2*s9 - 7*f
s14 := g3*x
faa := s14*x
fab := 2*s14
fba := fab
fbb := 4*g3
\end{verbatim}}
Repeating this process, this time with an {\tt OPTIMIZE} command to begin with,
learns that the {\tt OFF EXP} mode is now effective. But this time, and for 
similar
reasons, the assignments {\tt fbb = 4.s9.s0} and {\tt s12 = s9.s0.x} still
have a subexpression in common. Now the \verb+Productscheme+ of $D_{\lambda}$ 
helps understanding the phenomenon; again we skipped for shortness the rest of
the information, provided by the {\tt ON PRIMAT} status of SCOPE.
Internally {\tt s12} denotes the product {\tt s4.s9},
where {\tt s4 = x.s0}. The cse {\tt s4} disappeared from the output. 
An {\tt ALGOPT} application leads to the "discovery" of the 
cse {\tt g10 = s0.s9}.
{\small
\begin{verbatim}
f:=v^2*w;
                                  2
f := cos(a*x + 2*b)*sin(a*x + 2*b)

ON INPUTC,PRIMAT$

OPTIMIZE f:=:f,fa:=:df(f,a),fb:=:df(f,b),faa:=:df(f,a,2),
         fab:=:df(f,a,b),fba:=:df(f,b,a),fbb:=:df(f,b,2) INAME s$

                                  2
f := cos(a*x + 2*b)*sin(a*x + 2*b)

                       2                 2
fa := (2*cos(a*x + 2*b)  - sin(a*x + 2*b) )*sin(a*x + 2*b)*x

                         2                 2
fb := 2*(2*cos(a*x + 2*b)  - sin(a*x + 2*b) )*sin(a*x + 2*b)

                        2                   2                  2
faa := (2*cos(a*x + 2*b)  - 7*sin(a*x + 2*b) )*cos(a*x + 2*b)*x

                          2                   2
fab := 2*(2*cos(a*x + 2*b)  - 7*sin(a*x + 2*b) )*cos(a*x + 2*b)*x

                          2                   2
fba := 2*(2*cos(a*x + 2*b)  - 7*sin(a*x + 2*b) )*cos(a*x + 2*b)*x

                          2                   2
fbb := 4*(2*cos(a*x + 2*b)  - 7*sin(a*x + 2*b) )*cos(a*x + 2*b)

s3 := x*a + 2*b
s0 := cos(s3)
s2 := sin(s3)
s6 := s2*s2
f := s6*s0
s14 := 2*s0*s0
s13 := (s14 - s6)*s2
fa := s13*x
fb := 2*s13
s9 := s14 - 7*s6
s12 := s9*s0*x
faa := s12*x
fab := 2*s12
fba := fab
fbb := 4*s9*s0

Productscheme :

   |  0  2  3  4  5 12 14 18 19 20 21 22 23| EC|Far
---------------------------------------------------
  0|                    1           1      |  1| f
  5|     1                             1   |  1| fa
  9|     1                                 |  2| fb
 13|        1                          1   |  1| faa
 17|  1                                    |  1| fab
 21|  1                                    |  1| fba
 25|           1                    1      |  4| fbb
 29|                                1  1   |  1| s4
 30|                                   1  1|  1| s5
 31|                             2         |  1| s6
 33|                                2      |  1| s8
 37|           1              1            |  1| s12
 38|              1              1         |  1| s13
 39|                 1                     |  2| s14
 40|        1                              |  2| s15
---------------------------------------------------
0  : s15
2  : s13
3  : s12
4  : s9
5  : s10
12 : s8
14 : s6
18 : s5
19 : s4
20 : s2=sin(s3)
21 : s0=cos(s3)
22 : x
23 : a

ALGOPT ARESULTS;

{s3=a*x + 2*b,

 s0=cos(s3),

 s2=sin(s3),

      2
 s6=s2 ,

 f=s0*s6,

         2
 s14=2*s0 ,

 s13=(s14 - s6)*s2,

 fa=s13*x,

 fb=2*s13,

 s9=s14 - 7*s6,

 g10=s0*s9,

 s12=g10*x,

 faa=s12*x,

 fab=2*s12,

 fba=fab,

 fbb=4*g10}     
\end{verbatim}}
This script is shown for different reasons. It illustrates the heuristic 
character of the optimization process. We optimize, but do not guarantee 
the optimal solution. It also shows how easily repeated SCOPE applications
can be accomplished. Hence commands like "{\tt ALGOPT} {\tt ARESULTS};", 
"{\tt ALGOPT} {\tt ALGOPT} $\cdots$ ;" 
or "{\tt OPTIMIZE} {\tt ALGOPT} $\cdots$ ;" are all possible.
However, it is sometimes better to avoid such a combination when a 
{\tt RESTOREALL} instruction has to follow the first application.
A more detailed discussion about these possibilities is given in 
section~\ref{SCOPE:soph}, and especially in section~\ref{SSF:Sl}.
An additional reason was, to stipulate that SCOPE's actual parameters
have to be built carefully.
\index{{\tt SIDREL} switch}
\index{SCOPE function ! {\tt SETLENGTH}}

This example is also used to illustrate the role, which the switch {\tt SIDREL}
can possibly play. When turned it {\tt ON} the finishing touch F (see 
subsection~\ref{SCOPE:bird}) is omitted and all non-additive cse's
are substituted back, thus producing a possibly still rewritten input set,
which consists of toplevel input and additive cse's only. A simple
straightforward backsubstitution mechanism is applied on the optimization
result before it is presented to the user. Seemingly, it can lead to surprises
as shown below by the differences between the presentations of {\tt s15}, 
{\tt s14} and {\tt fbb} when again optimizing the contents of {\tt tlst}.
This effect disappeares when using {\tt SETLENGTH}.

The switch {\tt SIDREL} was introduced in SCOPE quite long ago. By that
time Hearn was wondering ~\cite{Hearn:85,Hearn:86} if (parts of) SCOPE output, 
presented in algebraic mode, can be used as input for a Gr\"{o}bner-base
algorithm application, thus attempting to assist in expression restructuring 
leading to improved expression representations. 
{\small
\begin{verbatim}
ON SIDREL$

ALGOPT(tlst,s);

{s3=a*x + 2*b,

                  2
 f=cos(s3)*sin(s3) ,

              2                  3
 s15=2*cos(s3) *sin(s3) - sin(s3) ,

 fa=s15*x,

 fb=2*s15,

                                       2              3
 s14= - 7*cos(a*x + 2*b)*sin(a*x + 2*b) *x + 2*cos(s3) *x,

 faa=s14*x,

 fab=2*s14,

 fba=2*s14,

                                        2            3
 fbb= - 28*cos(a*x + 2*b)*sin(a*x + 2*b)  + 8*cos(s3) }

SETLENGTH 4$

ALGOPT(tlst,s);

                                2
{f=cos(a*x + 2*b)*sin(a*x + 2*b) ,

                     2                                3
 s15=2*cos(a*x + 2*b) *sin(a*x + 2*b) - sin(a*x + 2*b) ,

 fa=s15*x,

 fb=2*s15,

                     3                                    2
 s14=2*cos(a*x + 2*b) *x - 7*cos(a*x + 2*b)*sin(a*x + 2*b) *x,

 faa=s14*x,

 fab=2*s14,

 fba=2*s14,

                     3                                   2
 fbb=8*cos(a*x + 2*b)  - 28*cos(a*x + 2*b)*sin(a*x + 2*b) }
\end{verbatim}}
{\small
\begin{flushright}
$\Box$
\end{flushright}}
\newpage
\section{Special SCOPE 1.5 Features}\label{SCOPE:soph}

Part of the input syntax for the function {\tt ALGOPT} was left undiscussed
in section~\ref{SCOPE:algo}. It was the permissable form for (parts of) 
the actual parameter, defining function applications, producing 
an alglist. The alglist is an algebraic mode list, consisting 
either of equations of the form lhs = rhs or of constructs 
evaluating into an alglist or referencing an alglist.
In section~\ref{SSF:Sl} is explained which type of user defined
functions lead to permissable function applications as (part of an) actual 
parameter for an {\tt OPTIMIZE} command or an {\tt ALGOPT} application.
Tools are provided for building a SCOPE library.
Already available facilities, designed along these lines, cover
{\bf structure recognition}, presented in section~\ref{SSF:sr} and
{\bf Horner-rule} based expression rewriting, surveyed in section~\ref{SSF:Hr}.

\subsection{Towards a SCOPE 1.5 Library}\label{SSF:Sl}

\index{REDUCE function ! {\tt ENDSTAT}-type}
\index{REDUCE function ! {\tt NORMAL}-type}
\index{REDUCE function ! {\tt PSOPFN}-type}
Design and implementation of an algebraic or symbolic procedure, returning
a list of equations in algebraic mode, is straightforward as long as the 
number of formal parameters is exactly known. Let us call such procedures
functions of {\tt NORMAL}-type. When formal parameters are not required, 
the so-called {\tt ENDSTAT}-variant can be used.
One simply associates an indicator {\tt stat}, with
value {\tt endstat}, with the function name {\tt f-name},
using \verb+lisp(put('f-name,'stat,'endstat))$+ as instruction. 
Such a function will be said to be of {\tt ENDSTAT}-type.

The so-called {\tt PSOPFN}-type function is similar to the {\tt FEXPR}-type 
function in symbolic mode. It may have an arbitrary number of unevaluated
parameters.
Special attention is made possible by modifying the function {\tt reval1}, used
in both {\tt reval} and {\tt aeval}. The relevant section of the evaluator is:
{\small
\begin{verbatim}
symbolic procedure aeval u; reval1(u,nil);
symbolic procedure reval u; reval1(u,t);

symbolic procedure reval1(u,v);
.......................
else if x:=get(car u,'psopfn)
 then << u:=apply(x,list cdr u);
         if x:=get(x,'cleanupfn) then u:=apply(x,list(u,v));
         return u
      >>
.......................
\end{verbatim}}
The actual parameter {\tt u} of {\tt reval1} is a function application in
prefixform: ({\tt function-name} {\tt arg1} ... {\tt argn}). 
The {\tt function-name} is replaced by the value of the indicator {\tt psopfn}.
The thus modified S-expression is evaluated. This mechanism leaves control
over evaluation of (part of) the arguments, collected in {\tt cdr u}, to the
designer of the function hidden behind the {\tt psopfn} value. We employed
this simple mechanism to implement {\tt ALGOPT} and some of the features,
to be discussed in section~\ref{SSF:sr} and section~\ref{SSF:Hr}.
A possible re-evaluation step, based on the use of the
value of the indicator {\tt cleanupfn} was not necessary; it is not yet 
allowed in the SCOPE context.

The different combinations, suggested in example~\ref{ex:3.2.7}, such as
{\tt ALGOPT} {\tt ALGOPT} or {\tt ALGOPT} {\tt ARESULTS}, are merely examples 
of a general rule:

{\em {\tt ENDSTAT}-, {\tt NORMAL}- and {\tt PSOPFN}-type functions, delivering
an alglist when applied, are all applicable as actual parameter or
as element of an alglist, functioning as actual parameter in an
{\tt OPTIMIZE} command or an {\tt ALGOPT} application. }

\index{REDUCE function ! {\tt ENDSTAT}-type}
\index{REDUCE function ! {\tt NORMAL}-type}
\index{REDUCE function ! {\tt PSOPFN}-type}
So, in principle, special features, providing a form of preprocessing, can be
designed and implemented as extension of the default optimization repertoire.
Of course additional function types are conceivable.
We illustrate the potential of this facility with a simple example. 
Further examples follow
in section~\ref{SSF:sr} and section~\ref{SSF:Hr}. 

\example\label{ex:4.1.1}
\index{SCOPE ! example}

The procedures {\tt asquares} and {\tt repeated\_squaring}  
define the production of
lists of equations. The lhs's function as name and the rhs's as
the computational rules.
Application of these functions shows how easy a user can provide new features,
usable in a SCOPE context.
The procedures {\tt asquares} and {\tt repeated\_squaring} are essentially 
different The first has one parameter, a list of equations, while the latter
accepts an arbitrary number of such lists as actual parameters. 
The {\tt psopfn} indicator value is
{\tt repeated\_squaringeval}, the name of the function, which is actually
introduced. {\tt asquares} is of {\tt NORMAL}-type and applicable in both 
algebraic and symbolic mode.
{\small
\begin{verbatim}
OPERATOR square$

sq_rule:={square(~u) => u^2}$

ALGEBRAIC PROCEDURE asquare u;
 square(u) WHERE sq_rule$

SYMBOLIC PROCEDURE rsquare u; 
 reval asquare aeval u$

SYMBOLIC PROCEDURE asquares u;
 append(list('list),
        FOREACH el IN cdr u COLLECT list('equal,cadr el,rsquare caddr el))$

SYMBOLIC OPERATOR asquares$

SYMBOLIC PROCEDURE repeated_squaringeval u;
BEGIN SCALAR res; INTEGER j;
 j=0;
 FOREACH el IN u DO
  << j:=j+1; el:=asquares el;
     FOR k:=2:j DO el:=asquares el;
     res:= IF j=1 THEN el ELSE append(res,cdr el)
  >>;
 RETURN res
END$

LISP( put('repeated_squaring,'psopfn,'repeated_squaringeval))$

% ---
% Examples of the use of asquares and repeated_squaring.
% Although the psopfn-mechanism can be easily avoided, 
% it is used for illustrative purposes here.
% ---

OFF EXP$

asquare sin(x);

      2
sin(x)

LISP(rsquare('(sin x)));

(expt (sin x) 2)

asq:=asquares {s1=a+b,s2=(a+b)^2,s3=(a+b)^3};

                  2           4           6
asq := {s1=(a + b) ,s2=(a + b) ,s3=(a + b) }

repeated_squaring({s1=a+b,s2=(a+b)^2},{s3=(a+b)^3,s4=(a+b)^4},
                           {s5=(a+b)^5,s6=(a+b)^6});

           2
{s1=(a + b) ,

           4
 s2=(a + b) ,

           12
 s3=(a + b)  ,

           16
 s4=(a + b)  ,

           40
 s5=(a + b)  ,

           48
 s6=(a + b)  }

% ---
% The "ALGOPT asquares ...;" application is similar to the "ALGOPT asq;"
% instruction.
% ---

ALGOPT(asquares {s1=a+b,s2=(a+b)^2,s3=(a+b)^3},t);

               2      2
{t2=a + b,s1=t2 ,s2=s1 ,s3=s1*s2}

ALGOPT asq;
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
               2      2
{g6=a + b,s1=g6 ,s2=s1 ,s3=s1*s2}

% ---
% The OPTIMIZE variant is now applied on a repeated_squaring application.
% ---

OPTIMIZE repeated_squaring({s1=a+b,s2=(a+b)^2},{s3=(a+b)^3,s4=(a+b)^4},
                           {s5=(a+b)^5,s6=(a+b)^6}) INAME t;
t5 := a + b
s1 := t5*t5
s2 := s1*s1
t12 := s2*s2
s3 := s2*t12
s4 := s2*s3
s5 := t12*t12*t12*s4
s6 := t12*s5
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}

\subsection{Structure Recognition: 
{\tt GSTRUCTR} and {\tt ALGSTRUCTR}} \label{SSF:sr}

\index{REDUCE function ! {\tt GSTRUCTR}}
\index{REDUCE function ! {\tt structr}}
\index{SCOPE function ! {\tt ALGSTRUCTR}}
The {\tt structr} command in REDUCE 3.6 (see the manual, section 8.3.8)
can be used
to display the skeletal structure of its evaluated argument, a single 
expression. After setting {\tt ON SAVESTRUCTR} a {\tt structr} command 
will return a list, whose first element is a presentation for the expression 
and subsequent elements are the subexpression relations.\\
A special SCOPE feature provides an extended display facility, called
{\tt GSTRUCTR}. The syntax of this generalized command is:

\begin{center}
\begin{tabular}{lcl}
$<${\tt REDUCE}\_command$>$ & $::=$ & $\cdots~\mid$\\
& & {\tt GSTRUCTR} $<$stat\_group$>$ [{\tt NAME} $<$cse\_prefix$>$]\\ 
$<$stat\_group$>$ & $::=$ & $\ll~<$stat\_list$>~\gg$\\
$<$stat\_list$>$ & $::=$ & $<$gstat$>$ [; $<$stat\_list$>$]\\
$<$gstat$>$ & $::=$ & $<$name$>~:=~<$ expression$>~\mid~<$matrix\_id$>$
\end{tabular}
\end{center}

The stat\_group consists
of one assignment statement or a group of such statements. Application of 
a {\tt GSTRUCTR} command provides a display of the structure of the whole 
set of assignments. Such an assignment can be replaced by a matrix reference.  
That leads to the display of all the non-zero entries of the referenced
matrix as well. The {\tt NAME} part is optional. The cse-name mechanism 
is applied in the usual way.

The equivalent of a possible {\tt ON SAVESTRUCTR} setting is provided in the
form of a {\tt PSOPFN}-type function, called {\tt ALGSTRUCTR}. 
Its syntax is:

\begin{center}
\begin{tabular}{lcl}
$<$function\_application$>$ & $::=$ & 
{\tt ALGSTRUCTR} ($<$arg\_list$>$ [, $<$cse\_prefix$>$ ]) \\
$<$arg\_list$>$ & $::=$ & $<$arg\_list\_name$>~\mid~$\{$<$arg\_seq$>$\}\\
$<$arg\_seq$>$ & $::=$ & $<$arg$>$[,$<$arg\_seq$>$]\\
$<$arg$>$ & $::=$ & $<$matrix\_id$>~\mid~<$name$>$=$<$expression$>$\\
$<$arg\_list\_name$>$ & $::=$ & $<$id$>$
\end{tabular}
\end{center}

The result is presented in the form of an algebraic mode list.

Earlier SCOPE-versions allowed to use a {\tt GSTRUCTR} command as (part of an)
actual parameter for an {\tt OPTIMIZE} command. This facility is not longer
supported. In stead, an {\tt ALGSTRUCTR} application can now be used as (part
of an) actual parameter in both an {\tt OPTIMIZE} command or an {\tt ALGOPT} 
application. We now illustrate these features in:
\index{REDUCE function ! {\tt GSTRUCTR}}
\index{SCOPE function ! {\tt ALGSTRUCTR}}

\example\label{ex:4.1.2}
\index{SCOPE ! example}

The script hardly requires explanation. However, observe 
that {\tt v1}, {\tt v3}, {\tt v4}, {\tt v6} and {\tt v7} occur only once 
in the result of the {\tt GSTRUCTR} application. When this application
is used as actual parameter for an {\tt OPTIMIZE} command
these redundancies are removed before the actual optimization process starts.
Likewise,
an {\tt ALGSTRUCTR} application only leads to identification of repeatedly
occuring sub-structures in its input. 
{\tt ALGSTRUCTR}, {\tt ALGHORNER}, see the next subsection, and {\tt ALGOPT}
all apply the same output production strategy, i.e. it might be necessary to
restore the previous algebraic mode status by applying the 
function {\tt RESTOREALL}.
{\small
\begin{verbatim}
OFF EXP,PERIOD$

MATRIX a(2,2);

a:=mat((x+y+z,x*y),((x+y)*x*y,(x+2*y+3)^3-x));

     [ x + y + z          x*y        ]
     [                               ]
a := [                          3    ]
     [(x + y)*x*y  (x + 2*y + 3)  - x]

GSTRUCTR <<a;b:=(x+y)^2;c:=(x+y)*(y+z);d:=(x+2*y)*(y+z)*(z+x)^2>> NAME v$

a(1,1) := v1
a(1,2) := x*y
a(2,1) := v2*x*y
a(2,2) := v4
       2
b := v2
c := v2*v5
          2
d := v6*v7 *v5

   where

      v7 := x + z
      v6 := x + 2*y
      v5 := y + z
              3
      v4 := v3  - x
      v3 := x + 2*y + 3
      v2 := x + y
      v1 := x + y + z

ALGSTRUCTR({a,b=(x+y)^2,c=(x+y)*(y+z),d=(x+2*y)*(y+z)*(z+x)^2},v);

{a(1,1)=x + y + z,

 a(1,2)=x*y,

 v2=x + y,

 a(2,1)=v2*x*y,

                     3
 a(2,2)=(x + 2*y + 3)  - x,

     2
 b=v2 ,

 v5=y + z,

 c=v2*v5,

                    2
 d=(x + 2*y)*(x + z) *v5}
\end{verbatim}}
\index{SCOPE function ! {\tt RESTORABLES}}
\index{SCOPE function ! {\tt ARESTORE}}
{\small
\begin{verbatim}
RESTORABLES;

{a}

ARESTORE a$

alst:=
 ALGOPT(ALGSTRUCTR({a,b=(x+y)^2,c=(x+y)*(y+z),d=(x+2*y)*(y+z)*(z+x)^2},v),s);

*** a declared operator 

alst := {s5=x + z,

         a(1,1)=s5 + y,

         a(1,2)=x*y,

         v2=x + y,

         a(2,1)=a(1,2)*v2,

         s6=x + 2*y,

         s4=s6 + 3,

                  3
         a(2,2)=s4  - x,

             2
         b=v2 ,

         v5=y + z,

         c=v2*v5,

             2
         d=s5 *s6*v5}

% ---
% The above delivered warning is caused by the decloupling of a and its
% status as matrix. Therefore a(1,2) can function in the rhs of a(2,1).
% After an ARESTORE instruction a can restart its life as matrix_id.
% ---

a;

a

ARESTORE a$

a;

[ x + y + z          x*y        ]
[                               ]
[                          3    ]
[(x + y)*x*y  (x + 2*y + 3)  - x]
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}

\subsection{Horner-rules: {\tt GHORNER} and {\tt ALGHORNER}} \label{SSF:Hr}

\index{Horner-rules} 
\index{REDUCE function ! {\tt GHORNER}} 
\index{SCOPE function ! {\tt ALGHORNER}}
Horner-rule based expression modification is a SCOPE facility, 
called {\tt GHORNER}. The syntax of the command is similar to the
{\tt GSTRUCTR} syntax: 

\begin{center}
\begin{tabular}{lcl}
$<${\tt REDUCE}\_command$>$ & $::=$ & $\cdots~\mid$\\
& & {\tt GHORNER} $<$stat\_group$>$ [{\tt VORDER} $<$id\_seq$>$];
\end{tabular}
\end{center}

The {\tt VORDER} part is optional. Application of 
a (generalized) Horner-rule assumes an identifier ordering. The syntax
of the identifier sequence is: 

\hspace{1cm} $<$id\_seq$>~::=~<$id$>$[,$<$id\_seq$>$].

We assume the rhs's in the stat\_group to be polynomials in
the identifiers, partly or completely given in the id\_seq. The 
left-to-right ordering of this sequence replaces the existing system 
identifier ordering. Identifiers, omitted from the {\tt vorder} sequence
have a lower preference and follow the existing system ordering. The
rewritten rhs's are presented as a side-effect. FORTRAN notation is of
course permitted. It is simply an extended print facility.

The {\tt PSOPFN}-type variant of the {\tt GHORNER} command is 
called {\tt ALGHORNER}. Its syntax is:

\begin{center}
\begin{tabular}{lcl}
$<$function\_application$>$ & $::=$ & $\cdots~\mid$\\
 & & {\tt ALGHORNER} ($<$arg\_list$>$ [,\{$<$id\_seq$>$\}]) \\
\end{tabular}
\end{center}

The syntax for the arg\_list can be found in subsection~\ref{SSF:sr}.
The result is presented in the form of an algebraic mode list.
An {\tt ALGHORNER} application can be used as (part of an) actual parameter 
of either an {\tt OPTIMIZE} command or an {\tt ALGOPT} application.

\example\label{ex:4.1.3}
\index{SCOPE ! example}

We illustrate the Horner-facilities by rewriting the expression of 
example~\ref{ex:3.1.2}, before optimizing it. Observe that 
application of {\tt ALGHORNER} in the default algebraic mode setting 
is useless. Due to  the algebraic mode regime the rewritten expression 
is expanded again. We also show some Taylor-series remodelling.
{\small
\begin{verbatim}
ON EXP$

z:=a^2*b^2+10*a^2*m^6+a^2*m^2+2*a*b*m^4+2*b^2*m^6+b^2*m^2;

      2  2       2  6    2  2          4      2  6    2  2
z := a *b  + 10*a *m  + a *m  + 2*a*b*m  + 2*b *m  + b *m

GHORNER z:=z VORDER a;

         2  6    2  2            4       2       6    2
z := (2*b *m  + b *m ) + a*(2*b*m  + a*(b  + 10*m  + m ))

GHORNER z:=z VORDER b;

          2  6    2  2            4       2      6    2
z := (10*a *m  + a *m ) + b*(2*a*m  + b*(a  + 2*m  + m ))

hlst:={z=z}$

ALGHORNER(hlst,{a,b,m});

    2  2       2  6    2  2          4      2  6    2  2
{z=a *b  + 10*a *m  + a *m  + 2*a*b*m  + 2*b *m  + b *m }

OPTIMIZE ALGHORNER(hlst,{a,b,m}) INAME s;

s1 := m*m
s0 := s1*s1
s2 := b*b
s4 := 2*s0
z := a*(a*(s2 + s1*(10*s0 + 1)) + s4*b) + s2*s1*(s4 + 1)

OPTIMIZE ALGHORNER(hlst,{b,m}) INAME s;

s2 := m*m
s0 := s2*s2
s1 := a*a
s4 := 2*s0
z := b*(b*(s1 + s2*(s4 + 1)) + s4*a) + s2*(s1 + 10*s1*s0)
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
% Hornering Taylor-series:

PROCEDURE taylor(fx,x,x0,n);
sub(x=x0,fx)+(FOR k:=1:n SUM(sub(x=x0,df(fx,x,k))*(x-x0)^k/factorial(k)))$

hlst2:={f1=taylor(e^x,x,0,4),f2=taylor(cos x,x,0,6)};

               4      3       2
              x  + 4*x  + 12*x  + 24*x + 24
hlst2 := {f1=-------------------------------,
                           24

                  6       4        2
               - x  + 30*x  - 360*x  + 720
          f2=------------------------------}
                          720

OPTIMIZE ALGHORNER(hlst2,{x});


       24 + x*(24 + x*(12 + x*(4 + x)))
f1 := ----------------------------------
                      24
g7 := x*x
       720 + g7*(g7*(30 - g7) - 360)
f2 := -------------------------------
                    720

ON ROUNDED$


hlst2:=hlst2;

                              4                   3        2
hlst2 := {f1=0.0416666666667*x  + 0.166666666667*x  + 0.5*x  + x + 1,

                                  6                    4        2
          f2= - 0.00138888888889*x  + 0.0416666666667*x  - 0.5*x  + 1

          }

OPTIMIZE ALGHORNER(hlst2,{x});


f1 := 1 + x*(1 + x*(0.5 + x*(0.0416666666667*x + 0.166666666667)))
g9 := x*x
f2 := 1 + g9*(g9*(0.0416666666667 - 0.00138888888889*g9) - 0.5)
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\index{{\tt ROUNDED} switch}

\newpage
\section{File Management and Optimization Strategies}\label{SCOPE:files}

Both the {\tt OPTIMIZE} command and the {\tt ALGOPT} function accept
input from file(s). Obviously, this input ought to obey the
usual syntactical rules, as introduced in the previous (sub)sections.

The {\tt OPTIMIZE} command is designed as a syntactical extension of {\REDUCE}
itself, i.e. the meaning of its actual parameters is understood from the
token-context in the command. However, an {\tt ALGOPT}
application requires one, two or three actual parameters without additional
provisions or conditions.
The {\tt ALGOPT} facility is added to provide a simple, user friendly, 
algebraic mode tool. Therefore -in contrast with the {\tt OPTIMIZE} command- 
it does not allow to direct output to a file; the default {\REDUCE} features
for dealing with output files can be applied.
\index{file management}
\index{{\tt IN} option}
\index{{\tt OUT} option}
\index{{\tt INAME} option}
\index{{\tt AGAIN} switch}
The previously given syntax requires some extensions:

$<$SCOPE\_application$>~::=$ $<${\tt OPTIMIZE} command$>~\mid~<${\tt ALGOPT} application$>$\\
$<${\tt OPTIMIZE} command$>~::=$\\
\hspace*{1cm}  {\tt OPTIMIZE} $<$object\_seq$>$ [{\tt IN} $<$file\_id\_seq$>$] [{\tt OUT} $<$file\_id$>$] [{\tt INAME} $<$cse\_prefix$>$] $\mid$\\
\hspace*{1cm}  {\tt OPTIMIZE} [$<$object\_seq$>$] {\tt IN} $<$file\_id\_seq$>$ [{\tt OUT} $<$file\_id$>$] [{\tt INAME} $<$cse\_prefix$>$]\\
$<${\tt ALGOPT} application$>~::=$\\
\hspace*{1cm}  {\tt ALGOPT}($<$a\_object\_list$>$[,$<$string\_id\_list$>$][,$<$cse\_prefix$>$]) $\mid$\\
\hspace*{1cm}  {\tt ALGOPT}([$<$a\_object\_list$>$,]$<$string\_id\_list$>$[,$<$cse\_prefix$>$])

The different variations for the object\_seq and
the a\_object\_list and the meaning of cse\_prefix are introduced
in the subsections ~\ref{SCOPE:optim} and ~\ref{SCOPE:algo}. 
The syntax of the file handling features is: 
\begin{center}
\begin{tabular}{lcl}
$<$file\_id\_seq$>$ & $::=$ & $<$file\_id$>$ [,$<$file\_id\_seq$>$]\\
$<$file\_id$>$ & $::=$ & $<$id$>$ $\mid$ $<$string\_id$>$\\
$<$string\_id\_list$>$ & $::=$ & $<$string\_id$>$ $\mid$ 
 \{$<$string\_id\_seq$>$\}\\
$<$string\_id\_seq$>$ & $::=$ & $<$string\_id$>$ [,$<$string\_id\_seq$>$]\\
$<$string\_id$>$ & $::=$ & {\tt "}$<$id$>${\tt "} $\mid$ 
{\tt "}$<$id$>.<$f\_extension$>${\tt "}
\end{tabular}
\end{center}

The differences in input-file management are introduced for practical reasons.
As stated above, the {\tt ALGOPT} function can have up to three arguments.
To be able to distinguish the optional second argument from the first and the
last requires file-names to be  given in the form of strings. The
{\tt OPTIMIZE} command follows the ordinary {\REDUCE} rules for file names.

File management can be used as a tool for input partioning.  If $m>1$ then
$N^m>\sum_{i=1}^k {n^k}_i$ for positive integers $N$ and $n_i$ , such that
$N=\sum_{i=1}^k n_i$. In view of the time-complexity of the
optimization algorithm, it may be worth the effort to partition
SCOPE input of size $N$ in $k$ partitions, of sizes $n_i,~i=1,...,k$. We can
start optimizing the contents of file fi.1, containing the initial $n_1$-sized
piece of code, and store the result of this operation in file fo.1. 
Consecutive steps provide an optimization of the combined contents of the 
files fo.i and fi.(i+1), i=1,..., k-1. 
During this iterative process, or during variations 
of this strategy, it is better not to perform a finishing touch. The switch
{\tt AGAIN}, which is normally {\tt OFF}, can be used, when set {\tt ON}, to 
avoid this. The switch serves an additional purpose. When switched {\tt ON} 
storage of partly optimized code in a file will include all relevant 
information, needed to restore the required status of system generated 
sub-expression names.

We illustrate SCOPE's file management facilities with example.

\example\label{ex:5.1}
\index{SCOPE ! example}

We assume to have three files, called f1, f2 and f3. Each file contains only
one assignment. We simply show different variations of the use of these files.
\index{{\tt INPUTC} switch}
\index{{\tt IN} option}
With {\tt ON INPUTC} the contents of the files is made visible.
{\small
\begin{verbatim}
ON INPUTC$

OPTIMIZE IN f1,f2,f3 INAME s;
 

                                 2
                 2        (x + y)              8        2      2
        2*(sin(x)  - cos(e        ) + 3*cos(x)) *(x + y)  + 4*y  + 4*y
e1 := ----------------------------------------------------------------
                                  3*x + 2*y

                                2
                2        (x + y)              2        3
e2 := (4*(sin(x)  - cos(e        ) + 2*cos(x)) *(x + y)

              2                   2
        + (4*x  - 4*y) - 6*x)/(8*x  + 3*y - 2*x)

                         2
                  (x + y)                       2           2
       4*sin(cos(e        )) + sin(x + y) + (4*x  - x + 2*y)
e3 := --------------------------------------------------------
                      3*y + f(x,g( - cos(x)))

s3 := sin(x)
s20 := x + y
s6 := s20*s20
           s6
s4 := cos(e  )
s8 := cos(x)
s31 := s3*s3 - s4
s2 := s31 + 3*s8
s44 := s2*s2
s43 := s44*s44
s36 := 4*y
s34 := 2*y
s10 := s34 + 3*x
       s36 + s36*y + 2*s6*s43*s43
e1 := ----------------------------
                  s10
s13 := s31 + 2*s8
s33 := 4*x*x
s30 := s33 - x
s35 := 3*y
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
       s33 - 2*s10 + 4*s6*s20*s13*s13
e2 := --------------------------------
                s35 + 2*s30
s21 := s34 + s30
       4*sin(s4) + sin(s20) + s21*s21
e3 := --------------------------------
            s35 + f(x,g( - s8))

\end{verbatim}}

We repeat the same process. However, this time we apply input partitioning.
The switch {\tt AGAIN} is turned {\tt ON}. Output is redirected to the
output file {\tt fo.1} in an {\tt OFF NAT} fashion and ended with the required
{\tt ;end;} closure, thus made ready for re-use during a next step. 
The default mode of operation is {\tt OFF AGAIN} and {\tt ON NAT}. If the
switch {\tt NAT} is turned {\tt OFF} file output is automatically ended
by {\tt ;end;}.
\index{{\tt AGAIN} switch}
\index{{\tt NAT} switch}

Due to the {\tt ON INPUTC} effect we can also observe that the identifiers
{\tt gsym} and {\tt cses} are apparently used to store relevant information
about cse names.

{\small
\begin{verbatim}
ON AGAIN,INPUTC$

OPTIMIZE IN f1 OUT "fo.1" INAME s$


                                2
                2        (x + y)              8        2      2
       2*(sin(x)  - cos(e        ) + 3*cos(x)) *(x + y)  + 4*y  + 4*y
e1 := ----------------------------------------------------------------
                                 3*x + 2*y


OPTIMIZE IN "fo.1",f2 OUT "fo.2" INAME t$

gsym := g0001
cses := s6
             2
s6 := (x + y)
                2                          2        s6  8
       4*y + 4*y  + 2*s6*(3*cos(x) + sin(x)  - cos(e  ))
e1 := ----------------------------------------------------
                           3*x + 2*y
                                2
                2        (x + y)              2        3
e2 := (4*(sin(x)  - cos(e        ) + 2*cos(x)) *(x + y)

              2                   2
        + (4*x  - 4*y) - 6*x)/(8*x  + 3*y - 2*x)

OFF AGAIN$

ALGOPT({"fo.2","f3"},u);

gsym := g0002

cses := t23 + t11 + t26 + t7 + t17 + t19

t19 := x + y

          2
t17 := t19

t7 := cos(x)

             2        t17
t26 := sin(x)  - cos(e   )

t11 := 3*x + 2*y

                2                 8
       4*y + 4*y  + 2*(t26 + 3*t7) *t17
e1 := ----------------------------------
                     t11

        2
t23 := x

                                         2
       4*t23 - 2*t11 + 4*t19*(t26 + 2*t7) *t17
e2 := -----------------------------------------
                  8*t23 - 2*x + 3*y

                         2
                  (x + y)                       2           2
       4*sin(cos(e        )) + sin(x + y) + (4*x  - x + 2*y)
e3 := --------------------------------------------------------
                      3*y + f(x,g( - cos(x)))


*** f declared operator 

*** g declared operator 

{u23=x + y,

        2
 u20=u23 ,

 t7=cos(x),

 u5=sin(x),

         u20
 u6=cos(e   ),

\end{verbatim}}
\newpage
{\small
\begin{verbatim}
       2
 t26=u5  - u6,

 u33=2*y,

 t11=u33 + 3*x,

 u10=t26 + 3*t7,

        2
 u46=u10 ,

        2
 u45=u46 ,

 u35=4*y,

              2
     2*u20*u45  + u35*y + u35
 e1=--------------------------,
               t11

      2
 t23=x ,

 u13=t26 + 2*t7,

 u36=4*t23,

 u31=u36 - x,

 u34=3*y,

                     2
      - 2*t11 + 4*u13 *u20*u23 + u36
 e2=---------------------------------,
               2*u31 + u34

 u24=u31 + u33,

                               2
     sin(u23) + 4*sin(u6) + u24
 e3=-----------------------------}
         f(x,g( - t7)) + u34
\end{verbatim}}
\noindent
Observe that the initial characters of the sub-expression names indicate
their moment of generation. We used {\tt f} and {\tt g} as operators. 
Therefore, a warning was produced ahead of the {\tt ALGOPT} output.
Since an {\tt OPTIMIZE} command produces output as a side-effect these warnings
were not given earlier.
{\small
\begin{flushright}
$\Box$
\end{flushright}}
\newpage
\section{Generation of Declarations}\label{SCOPE:decl}

GENTRAN's {\tt DECLARE} statement can be used as an optional extension of the
{\tt OPTIMIZE} command, and as ilustrated in example~\ref{ex:6.1}. The syntax 
of such an extension is in accordance with the GENTRAN rules:
\index{GENTRAN ! {\tt DECLARE} statement}
\index{SCOPE ! {\tt DECLARE} facility}
\index{{\tt DECLARE} option}
\index{{\tt IN} option}
\index{{\tt OUT} option}
\index{{\tt INAME} option}
\index{{\tt IMPLICIT} type}
\index{{\tt integer} type}
\index{{\tt real} type}
\index{{\tt real*8} type}
\index{{\tt complex} type}
\index{{\tt complex*16} type}
\index{SCOPE function ! {\tt OPTLANG}}
\index{SCOPE target language ! {\tt fortran77}}
\index{SCOPE target language ! {\tt fortran90}}
\index{SCOPE target language ! {\tt f90}}
\index{SCOPE target language ! {\tt c}}
\index{SCOPE target language ! {\tt pascal}}
\index{SCOPE target language ! {\tt ratfor}}
\index{SCOPE target language ! {\tt nil}}

$<${\tt OPTIMIZE} command$>~::=$\\
 \hspace*{1cm} {\tt OPTIMIZE} $<$object\_seq$>$ [{\tt IN} $<$file\_id\_seq$>$] 
 [{\tt OUT} $<$file\_id$>$]\\ 
 \hspace*{3cm} [{\tt INAME} $<$cse\_prefix$>$] 
  [{\tt DECLARE} $<$declaration\_group$>$] $\mid$\\
 \hspace*{1cm} {\tt OPTIMIZE} [$<$object\_seq$>$] {\tt IN} $<$file\_id\_seq$>$ 
 [{\tt OUT} $<$file\_id$>$]\\
 \hspace*{3cm} [{\tt INAME} $<$cse\_prefix$>$] 
 [{\tt DECLARE} $<$declaration\_group$>$]\\

The syntax of the declaration\_group is:
\begin{center}
\begin{tabular}{lcl}
$<$declaration\_group$>$ & $::=$ & $<$declaration$>~\mid~\ll~<$declaration\_list$>~\gg$\\
$<$declaration\_list$>$ & $::=$ & $<$declaration$>$[$;<$declaration\_list$>$]\\
$<$declaration$>$ & $::=$ & $<$range\_list$>:$ {\tt IMPLICIT} $<$type$>~\mid$
 $<$id\_list$>:<$type$>$\\
$<$range\_list$>$& $::=$ & $<$range$>$[,$<$range\_list$>$]\\
$<$range$>$ & $::=$ & $<$id$>~\mid~<$id$>-<$id$>$\\
$<$id\_list$>$ & $::=$ & $<$id$>$[,$<$id\_list$>$]\\
$<$type$>$ & $::=$ & {\tt integer} $\mid$ {\tt real} $\mid$ {\tt complex} $\mid$ {\tt real*8} $\mid$ {\tt complex*16}
\end{tabular}
\end{center}

The symbol table features of GENTRAN are used. During the subtask
R (see subsection ~\ref{SCOPE:bird}) of an {\tt OPTIMIZE} command evaluation, 
all typing information is installed in the symbol table. Once optimization
is ready all relevant information for completing the declarations ought to
be known, i.e. the contents of the symbol table and the result of the
optimization operations, collected in prefix form in a list, 
called {\tt prefixlist}. This {\tt prefixlist} is employed do decide which not yet typed identifiers and system selected cse names have to be entered in the
symbol table. We make use of earlier provided information, delivered via the
{\tt DECLARE} option, (sub)expression structure and the normal hierarchy in data
types. The strategy to achieve this form of dynamic typing is based on 
chapter 6 of ~\cite{Aho:86}.
Once the table is completed a list of declarations is produced and precedes the
other SCOPE output. SCOPE output is by default given in {\REDUCE} notation.
Therefore such lists of declarations are also given in {\REDUCE} text.
Incomplete initial typing information can lead to overtyping after 
optimization, such as {\tt complex} in stead of {\tt real}, for instance. 
It can therefore lead to erroneous results and even to an error message. 
A safe procedure is to
use the {\tt DECLARE} option of the {\tt OPTIMIZE} command for typing
all identifiers, occuring in the input set ${\rm E}_0$
\index{dynamic typing}

Alternative output can be obtained via an application of the function 
{\tt OPTLANG}. This function accepts one argument from the set \{{\tt fortran},
{\tt c}, {\tt ratfor}, {\tt pascal}\footnote{The {\tt pascal} module of
GENTRAN is not error free. Especially the template file features do not
function correctly.}, {\tt f90}, {\tt nil}\}.
The {\tt fortran}(77) choice can also be made by turning {\tt ON} the 
switch {\tt FORT}. The {\tt nil} option is necessary if one wants to
switch back to the usual {\REDUCE} output. 
not yet generally available. The output modules of GENTRAN are used for 
producing formatted code in the user selected target language. The
{\tt f90} option, for the production of {\tt fortran90} code, 
is not yet provided by the standard GENTRAN version ~\cite{Borst:94}.

Especially the above given syntax rules for typing require some additional 
explanation:
\begin{itemize}
\item The corresponding types in Fortran are {\tt integer}, {\tt real},
{\tt complex}, {\tt double precision} and {\tt complex*16}.
\item The GENTRAN switch {\tt DOUBLE} is automatically turned {\tt ON},
when a type {\tt real*8} or  type {\tt complex*16}  is introduced in
a {\tt DECLARE} option. The same mode of operation is introduced when 
floating point numbers appear in SCOPE input. Fixed floats do not produce 
this side effect.
\item When generating {\tt fortran} code we have to be aware of a possibly 
existing statement length limitation. If one is afraid that a declaration statement 
will become 
too long, for instance due to a huge number, dynamically added cse-names, 
it may be better to use {\tt IMPLICIT} typing.
\item C neither supports {\tt IMPLICIT} types nor has the 
types {\tt complex} and {\tt complex*16}. The remaining types are denoted
by {\tt int}, {\tt float} and {\tt double}, respectively.
\item Array and/or matrix definitions are also considered to be id's in
 id\_list's in declarations. 
However, we have to be aware of the instantaneous replacement of array-
and/or matrix entries, when  expressions are simplified. Therefore, we have to
use operators, functioning as array and/or matrix names in code we want to
optimize. We return to this question in the sections ~\ref{SCOPE:dda} and
 ~\ref{SCOPE:gopt}.
\end{itemize}
\index{{\tt ROUNDED} switch}
\index{{\tt DOUBLE} switch} 
When the {\tt ON/OFF AGAIN} strategy is applied we have to be aware of the
above outlined declaration strategy. The last {\tt OPTIMIZE} command,
executed directly after choosing {\tt OFF AGAIN}, has to be extended 
with the {\tt DECLARE} option. 

Array and/or matrix names only occur in literally parsed information. In all
other situations we have to make use of {\REDUCE} {\tt operators}. Normally,
function applications inside SCOPE input are instantaneously replaced by
newly selected cse names after putting them in the function table.
Usually array and/or matrix entries are considered to be function applications.
However, when due to a {\tt DECLARE} option array and/or matrix names
are known via the contents of the symbol table, such entries are 
substituted back before SCOPE produces output.

\index{{\tt IMPLICIT} type}
\index{{\tt integer} type}
\index{{\tt real} type}
\index{{\tt real*8} type}
\index{{\tt complex} type}
\index{{\tt double precision} type}
\index{{\tt int} type}
\index{{\tt float} type}
\index{{\tt double} type}
\index{{\tt DOUBLE} switch}
\index{{\tt AGAIN} switch}
\index{SCOPE function ! {\tt OPTLANG}}
\example\label{ex:6.1}
\index{SCOPE ! example}

A simple {\tt OPTIMIZE} command, extended with a {\tt DECLARE} option, is
executed for the various output options of GENTRAN, including the {\tt f90}
alternative.
{\small
\begin{verbatim}
OPTLANG fortran$
OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << a(4,4),x(4),y(5):real; b(5):integer>>$

      INTEGER B(5),I,S10,S9
      REAL A(4,4),X(4),Y(5)
      S10=I+1
      S9=I-1
      X(S10)=A(S10,S9)+B(I)
      Y(S9)=A(S9,S10)-B(i)

OPTLANG ratfor$
OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << a(4,4),x(4),y(5):real; b(5):integer>>$

integer b(5),i,s10,s9
real a(4,4),x(4),y(5)
{
    s10=i+1
    s9=i-1
    x(s10)=a(s10,s9)+b(i)
    y(s9)=a(s9,s10)-b(i)
}

OPTLANG c$
OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << a(4,4),x(4),y(5):real; b(5):integer>>$

int b[6],i,s10,s9;
float a[5][5],x[5],y[6];
{
    s10=i+1;
    s9=i-1;
    x[s10]=a[s10][s9]+b[i];
    y[s9]=a[s9][s10]-b[i];
}

OPTLANG pascal$
OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << a(4,4),x(4),y(5):real; b(5):integer>>$

var
    s9,s10,i: integer;
    b: array[0..5] of integer;
    y: array[0..5] of real;
    x: array[0..4] of real;
    a: array[0..4,0..4] of real;
begin
    s10:=i+1;
    s9:=i-1;
    x[s10]:=a[s10,s9]+b[i];
    y[s9]:=a[s9,s10]-b[i]
end;

OPTLANG nil$
OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << a(4,4),x(4),y(5):real; b(5):integer>>$

integer b(5),i,s10,s9
real a(4,4),x(4),y(5)

s10 := i + 1
s9 := i - 1
x(s10) := a(s10,s9) + b(i)
y(s9) := a(s9,s10) - b(i)

OPTLANG fortran$
OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << a(4,4),x(4),y(5):real*8; b(5):integer>>$

      INTEGER B(5),I,S10,S9
      DOUBLE PRECISION A(4,4),X(4),Y(5)
      S10=I+1
      S9=I-1
      X(S10)=A(S10,S9)+B(I)
      Y(S9)=A(S9,S10)-B(I)

OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << x(4),y(5):real; b(5):complex>>$

***** Type error:
real x(4),y(5)
complex b(5)
(integer all) s9
integer s5,i
real := complex(all)
***** Wrong typing 
Cont? (Y or N)
\end{verbatim}}

We can restart \REDUCE and rerun the example with
the {\tt Fortran90} version of SCOPE. It results in:
\index{{\tt scope90}}
{\small
\begin{verbatim}
LOAD_PACKAGE scope90$
OPTLANG f90$
OPTIMIZE x(i+1):=a(i+1,i-1)+b(i),y(i-1):=a(i-1,i+1)-b(i)
INAME s
DECLARE << a(4,4),x(4),y(5):real; b(5):integer>>$

      REAL,DIMENSION(4,4)::A
      INTEGER,DIMENSION(5)::B
      INTEGER::I,S10,S9
      REAL,DIMENSION(4)::x
      REAL,DIMENSION(5)::y
      S10=I+1
      S9=I-1
      X(S10)=A(S10,S9)+B(I)
      Y(S9)=A(S9,S10)-B(I)
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}

\subsection{Coefficient Arithmetic and Precision Handling}\label{SCOPE:caph}

{\REDUCE} knows a variety of coefficient domains, as presented in 
subsection 9.11 of the {REDUCE} 3.6 manual ~\cite{Hearn:95}, entitled
 {\em Polynomial Coefficient Arithmetic}. As stated in subsection~\ref{SCOPE:optim} 
SCOPE supports integer and real coefficients.  By turning {\tt ON} the switch 
 {\tt ROUNDED} we introduce float arithmetic for coefficients.
The operator {\tt PRECISION} can be applied to change the default,
machine dependent {\em precision}. Internally, {\REDUCE} uses floating point 
numbers up to the precison supported by the underlying machine hardware, and
so-called {\em bigfloats} for higher precision. The internal precision is two decimals 
greater than the exernal precision to guard against roundoff inaccuracies.
Rounded numbers are normally printed to the specified precision. If the user
wishes to print such numbers with less precision, the printing
precision can be set by the command {\tt PRINT\_PRECISION}. If a case
arises where use of the machine arithmetic leads to problems, a user can force
{\REDUCE} to use the bigfloat representation by turning {\tt ON} the switch
{\tt ROUNDBF}, which is normally {\tt OFF}. GENTRAN, and thus SCOPE as well,
support bigfloat notation. However the precision is a responsibility
of the user. A possibility is to use the {\tt PRINT\_PRECISION} command, both
for algebraic mode output and for output in a selected target language, 
like {\tt fortran77}. SCOPE uses the given precision for selecting cse's. 
Although complex arithmetic is not supported in SCOPE, a simple alternative
is provided. When using float arithmetic in {\REDUCE} the protected name
{\tt I} can be used to denote $\sqrt -1$. If the {\tt I} is included in a
declaration list as an identifier of type {\tt complex}({\tt !*}), its assumed value is 
automatically put ahead of the resulting optimized code.
We illustrate the different possibilities in example ~\ref{ex:6.2}. Comment
is  included.
\index{{\tt ROUNDBF} switch}
\index{REDUCE function ! {\tt PRECISION}}
\index{REDUCE function ! {\tt PRINT\_PRECISION}}
\index{bigfloats}
\index{coefficient arithmetic}
\index{machine precision}

\example\label{ex:6.2}
\index{SCOPE ! example}
{\small
\begin{verbatim}
OPTLANG fortran$
ON ROUNDED, DOUBLE$

% ---
% We start with precision 6. The returned value is the internal
% precision supported by the underlying machine hardware.
% ---

PRECISION 6;

12

OPTIMIZE x1:= 2        *a + 10         * b,
         x2:= 2.00001  *a + 10         * b,
         x3:= 2        *a + 10.00001   * b,
         x4:= 6        *a + 30         * b,
         x5:= 2.0000001*a + 10.000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,X1,X2,X3,X4,X5
      S1=10*B
      X1=S1+2*A
      X2=S1+2.00001D0*A
      X3=X1
      X4=3*X1
      X5=X1

% ---
% Explanation: X1 is a cse of X3, X4 and X5, but not of X2, because
% the coefficient 2.00001 is given in 6 decimal digits. 
% Increase in precision will show this.
% ---

PRECISION 7$

OPTIMIZE x1:= 2        *a + 10         * b,
         x2:= 2.00001  *a + 10         * b,
         x3:= 2        *a + 10.00001   * b,
         x4:= 6        *a + 30         * b,
         x5:= 2.0000001*a + 10.000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.00001D0*A
      X3=S1+1.000001D1*B
      X4=3*X1
      X5=X1

PRECISION 8$

OPTIMIZE x1:= 2        *a + 10         * b,
         x2:= 2.00001  *a + 10         * b,
         x3:= 2        *a + 10.00001   * b,
         x4:= 6        *a + 30         * b,
         x5:= 2.0000001*a + 10.000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.00001D0*A
      X3=S1+1.000001D1*B
      X4=3*X1
      X5=2.0000001D0*A+1.0000001D1*B

% ---
% All rhs's were taken literally. Let us now increase precision and
% simplify the rhs's before optimization. It is in fact a repetition
% of the examples above, this time with a larger precision.
% ---

PRECISION 20$

OPTIMIZE x1:=:2                      *a + 10                       * b,
         x2:=:2.0000000000000000001  *a + 10                       * b,
         x3:=:2                      *a + 10.0000000000000000001   * b,
         x4:=:6                      *a + 30                       * b,
         x5:=:2.000000000000000000001*a + 10.000000000000000000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.0000000000000000001D0*A
      X3=S1+1.0D1*B
      X4=3*X1
      X5=1.0D0*X1

PRECISION 21$

OPTIMIZE x1:=:2                      *a + 10                       * b,
         x2:=:2.0000000000000000001  *a + 10                       * b,
         x3:=:2                      *a + 10.0000000000000000001   * b,
         x4:=:6                      *a + 30                       * b,
         x5:=:2.000000000000000000001*a + 10.000000000000000000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.0000000000000000001D0*A
      X3=S1+1.00000000000000000001D1*B
      X4=3*X1
      X5=2.0D0*A+1.0D1*B

PRECISION 22$

OPTIMIZE x1:=:2                      *a + 10                       * b,
         x2:=:2.0000000000000000001  *a + 10                       * b,
         x3:=:2                      *a + 10.0000000000000000001   * b,
         x4:=:6                      *a + 30                       * b,
         x5:=:2.000000000000000000001*a + 10.000000000000000000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.0000000000000000001D0*A
      X3=S1+1.00000000000000000001D1*B
      X4=3*X1
      X5=2.000000000000000000001D0*A+1.0D1*B

% ---
% However, we can observe some differences in both modes of operation, when
% selecting a precision around the precision supported by the undelying
% machine hardware. Then the switch ROUNDBF can better be turned ON.
% ---
\end{verbatim}}
\index{{\tt ROUNDBF} switch}
\index{{\tt complex} type}
{\small
\begin{verbatim}
OFF ROUNDBF$
PRECISION 12$

OPTIMIZE x1:= 2.00         *a + 10.00       * b,
         x2:= 2.00000000001*a + 10         * b,
         x3:= 2           *a + 10.000000001* b,
         x4:= 6           *a + 30         * b,
         x5:= 2.0000000000001*a + 10.0000000000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.00000000001D0*A
      X3=S1+1.0000000001D1*B
      X4=3*X1
      X5=X1


OPTIMIZE x1:=:2.00         *a + 10.00       * b,
         x2:=:2.00000000001*a + 10         * b,
         x3:=:2           *a + 10.000000001* b,
         x4:=:6           *a + 30         * b,
         x5:=:2.0000000000001*a + 10.0000000000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.0D0*A
      X3=S1+10.0D0*B
      X4=3*X1
      X5=X1

% ---
% Observe that simplification prior to optimization leads to internal
% roundings, which differ from the rounding used for literally taken
% coefficients. This difference disappeares with ON ROUNDBF.
% ---

ON ROUNDBF$

OPTIMIZE x1:= 2.00         *a + 10.00       * b,
         x2:= 2.00000000001*a + 10         * b,
         x3:= 2           *a + 10.000000001* b,
         x4:= 6           *a + 30         * b,
         x5:= 2.0000000000001*a + 10.0000000000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.00000000001D0*A
      X3=S1+1.0000000001D1*B
      X4=3*X1
      X5=X1

OPTIMIZE x1:=:2.00         *a + 10.00       * b,
         x2:=:2.00000000001*a + 10         * b,
         x3:=:2           *a + 10.000000001* b,
         x4:=:6           *a + 30         * b,
         x5:=:2.0000000000001*a + 10.0000000000001 * b
INAME s
DECLARE <<x1,x2,x3,x4,x5,a,b: real>>$

      DOUBLE PRECISION A,B,S1,S2,X1,X2,X3,X4,X5
      S1=2*A
      S2=10*B
      X1=S2+S1
      X2=S2+2.00000000001D0*A
      X3=S1+1.0000000001D1*B
      X4=3*X1
      X5=X1

% ---
% Complex arithmetic is not supported in SCOPE. However the Fortan equivalent
% of I, a protected name in REDUCE, is automatically created, ahead of the 
% optimized code, whenever I is included in the declaration as a type complex 
% or a type complex*16 identifier.
% ---

OPTIMIZE a:=b+c 
INAME s
DECLARE <<a,b,i,c: complex>>;

      COMPLEX*16 B,I,C,A
      I=(0.0D0, 1.0D0)
      A=B+C

OFF DOUBLE$

OPTIMIZE a:=b+c 
INAME s
DECLARE <<a,b,i,c: complex>>;

      COMPLEX B,I,C,A
      I=(0.0, 1.0)
      A=B+C
\end{verbatim}}
\begin{flushright}
$\Box$
\end{flushright}
\newpage
\section{Dealing with Data Dependencies}\label{SCOPE:dda}
\index{flow dependency}
\index{anti dependency}
\index{used identifiers}
\index{defined identifiers}
\index{dead code}
\index{data dependency analysis}

SCOPE is designed to optimize blocks $B$ of straight line code, i.e.
sequences of $n$ assignment statements $S_i$ of the 
form ${\lambda}_i := {\rho}_i$, where $i = 1, \cdots , n$. If an identifier 
occurs in ${\lambda}_i$, it is said to be {\em defined} in $S_i$. All 
identifiers occuring in ${\rho}_i$ are said to be {\em used} in $S_i$. 
The set DEF($S_i$)
is formed by the identifiers defining $S_i$, usually only one.
The set USE($S_i$) is formed by the identifiers, which are used in $S_i$.
The relation DEF($S_i$) $\in$ USE($S_j$), for $1 \leq i < j \leq n$, is 
called  a {\em flow dependency} and denoted by $S_i \rightarrow S_j$. The
relation DEF($S_i$) $\in$ USE($S_j$), for $1 \leq j \leq i \leq n$ is called
an {\em anti dependency} and denoted by $S_i$ \ad $S_j$. The {\em set of inputs}
of $B$, denoted by $I(B)$, consists of identifiers, which are used
in $B$, before being defined, if defined at all. The {\em set of outputs} of
$B$, denoted by $O(B)$, consists of the set of all last definitions of
identifiers, occuring in $B$. So a block of straight line code can be 
introduced as a triple $B = \{ S, I, O \}$, where $S$ stands for the sequence
$S_1 ; S_2 ; \cdots ; S_{n-1} ; S_n$, and where $I$ and $O$ define the
inputs and outputs, respectively. When optimizing source code defined by $B$,
i.e. the sequence $S$, the intention is to mechanically produce an equivalent,
but computationally less complex sequence, preserving the relation between
inputs and outputs. Due to anti dependencies, i.e. redefinitions of the rules
for computing identifier values or stepwise computing such values, 
 $\mid O(B) \mid < n$ is possible.  But that in turn implies that some of the
used identifiers, although being literally identical, represent different
values. Therefore, a mechanical search for cse's can only be maintained if 
these critical identifiers are adequately renamed internally before the 
optimization process itself is started. As long as the relation between
$I(B)$ and $O(B)$ is preserved it is even allowed to partly maintain these
additional names, when presenting the results of an optimization operation.
Furthermore it is worth noting that {\em dead code} can be left out, when
ever occuring. Such code can be introduced through redundant assignments.
The SCOPE features for dealing with data dependencies are illustrated, using 
the following artificial piece of code:
\begin{center}
\[ \begin{array}{lclcl}
 S_1 & : & a_{1,x+1} & := & g + h . r^f \\
 S_2 & : & b_{y+1} & := & a_{1, 2.x+1} .(g + h . r^f) \\
 S_3 & : & c1 & := &  h.r. a_{2,1+x}/g \\
 S_4 & : & c2 & := & c1 . a_{1,x+1} + sin(d) \\
 S_5 & : & a_{1,x+1} & := & c1 + 2 \\
 S_6 & : & d & := & b_{y+1} . a_{1,x+1} \\
 S_7 & : & a_{1,1+2.x} & := & a_{1,x+1} . b_{y+1} . c/(d . g^2)  \\
 S_8 & : & b_{y+1} & := & a_{1.x+1} + b_{y+1} + sin(d) \\
 S_9 & : & a_{1,x+1} & := & b_{y+1} . c + h/(g  + sin(d))\\
 S_{10} & : & d & := & k . e + d . (a_{1,1+x} + 3) \\
 S_{11} & : & e & := &  d . (a_{1,1+x} + 3) + sin(d) \\
 S_{12} & : & f & := &  d . (a_{1,1+x} + 3) + sin(d) \\
 S_{13} & : & g & := &  d . (3 + a_{1,1+x}) + f \\
\end{array}\]
\end{center}

The different flow and anti dependencies can be vizualized in the following way:
\begin{center}
\begin{tabular}{lcl}
 $x$ & : & $\{ {\lambda}_1 , {\rho}_2 , {\rho}_3 , {\rho}_4 , {\lambda}_5 ,
 {\rho}_6 , {\lambda}_7 , {\rho}_7 , {\rho}_8 ,{\lambda}_9 , {\rho}_{10} ,
 {\rho}_{11} , {\rho}_{12} , {\rho}_{13} \}$ \\
 $y$ & : & $\{ {\lambda}_2 , {\rho}_6 , {\rho}_7 , {\lambda}_8 , {\rho}_8 ,
 {\rho}_9 \}$ 
\end{tabular}
\end{center}

The identifiers, occuring in the piece of code given above, can be defined,
can be used or can play both roles. Identifiers, used in one or more of the
$\lambda_i , i = 1 \cdots n$ figure in subscript expressions. The set 
notation \{ $\cdots$ \} is used to explicitly describe USE sets. Since
all DEF sets consist of one element only, we omitted the set notation for the
DEF sets. This provides a simple tool to distinguish flow and anti dependencies:

\begin{center}
\begin{tabular}{lcl}
 $a_{1,x+1}$ & : & ${\lambda}_1 \rightarrow \{ {\rho}_3 , {\rho}_4 \}$ \ad 
 ${\lambda}_5 \rightarrow \{ {\rho}_6 , {\rho}_7 , {\rho}_8 \}$ \ad 
 ${\lambda}_9 \rightarrow \{ {\rho}_{10} , {\rho}_{11} , {\rho}_{12} , 
 {\rho}_{13} \}$ \\
 $g$ & : & $\{ {\rho}_1 , {\rho}_2 , {\rho}_3 , {\rho}_7 , {\rho}_9 \}$ \ad
 ${\lambda}_{13}$ \\
 $h$ & : & $\{ {\rho}_1 , {\rho}_2 , {\rho}_3 , {\rho}_9 \}$ \\
 $r$ & : & $\{ {\rho}_1 , {\rho}_2 , {\rho}_3 \}$ \\
 $f$ & : & $\{ {\rho}_1 , {\rho}_2 \}$ \ad 
 ${\lambda}_{12} \rightarrow {\rho}_{13}$ \\
 $b_{y+1}$ & : & ${\lambda}_2 \rightarrow \{ {\rho}_6 , {\rho}_7 , {\rho}_8 \}$  \ad ${\lambda}_8 \rightarrow \{ {\rho}_9 \}$ \\
 $a_{1,2.x+1}$ & : & $\{ {\rho}_2 \}$ \ad ${\lambda}_7$ \\
 $c1$ & : & ${\lambda}_3 \rightarrow \{ {\rho}_4 , {\rho}_5 \}$ \\
 $a_{2,1+x}$ & : & $\{ {\rho}_3 \}$ \\
 $c2$ & : & ${\lambda}_4$ \\
 $d$ & : & $\{ {\rho}_4 \}$ \ad ${\lambda}_6 \rightarrow \{ {\rho}_7 , 
 {\rho}_8 , {\rho}_9 , {\rho}_{10} \}$ \ad ${\lambda}_{10} \rightarrow 
  \{ {\rho}_{11} , {\rho}_{12} , {\rho}_{13} \}$ \\
 $c$ & : & $\{ {\rho}_7 , {\rho}_9 \}$ \\
 $k$ & : & $\{ {\rho}_{10} \}$ \\
 $e$ & : & $\{ {\rho}_{10} \}$ \ad ${\lambda}_{12}$ 
\end{tabular}
\end{center}

We observe that
\[ I(B) = \{ x, y, g, h, r, f, a_{1,2.x+1} , a_{2,1+x} , d, c, k, e \} ,\]
\[ O(B) = \{ a_{1,x+1} , g, f, b_{y+1} , a_{1,2.x+1} , c1 , c2 , d, e\}\]
and thus, that
\[ I(B) \cap O(B) \neq \emptyset .\]
The identifiers $a_{1,1+x}$ and $d$ are redefined twice and the identifier 
$b_{y+1}$ once. Only the input occurrences and the last
output definitions need to be preserved.
\index{output definition preservation}
\index{{\tt VECTORC} switch}
\index{SCOPE function ! {\tt VECTORCODE}}

We also observe that some of the identifiers are subscripted. In our example
the subscript expressions are constructed with input identifiers only. More
general situations are conceivable. The set of subscript expressions contains 
cse's. Since our optimization strategy is based on an all level approach 
expressions, like $1+x$ and $y+1$ are certainly discovered as cse's.

An {\tt OPTIMIZE} command can be extended with a {\tt DECLARE} option, 
indicating that both $a$ and $b$ are array names. Their subscript expressions
are optimized. In addition, the $a$ and $b$ entries are considered to be
array entries, and not as function applications. The latter will happen when
the {\tt DECLARE} option is omitted. Vector architectures make often use of
machine specific optimizing compilers. Therefore it may be better not to 
optimize the subset of subscript expressions. We implemented some facilities 
to take such machine specific limitations into account.  When turning {\tt ON}
the switch {\tt VECTORC} the finishing touch is omitted and subscript 
expressions are not optimized. The function
\index{{\tt VECTORC} switch}

\hspace*{1cm} {\tt VECTORCODE} $<$a\_or\_m\_id\_seq$>$

can be used to operate more selectively. The a\_or\_m\_id\_seq consists of
one or more array and/or matrix names, separated by comma's. Only the
actual parameters of {\tt VECTORCODE} are assumed to be  names of arrays. So
only their subscript expressions are disregarded during an optimization
process. We can undo the effect of the {\tt VECTORCODE} setting with a
command of the form:

\hspace*{1cm} {\tt VCLEAR} $<$a\_or\_m\_id\_(sub)seq$>$
\index{SCOPE function ! {\tt VCLEAR}}

The actual parameters are supposed to be taken from the sequence of actual
parameters of its counterpart, the function {\tt VECTORCODE}.

The different settings are now illustrated in:

\example\label{ex:7.1}
\index{SCOPE ! example}

The above given block of code $B = \{ S, I, O \}$ is optimized in five
different situations. To start with we hand it over to SCOPE, using an
{\tt OPTIMIZE} command, without using the {\tt DECLARE} option.
We observe, looking at the results, that some renaming of non significant 
identifier
definitions have been performed. The first occurrence of $a_{1,1+x}$ is replaced
by {\tt s34}, the second by  {\tt s4}. The first occurrence of $b_{y+1}$ is
replaced by {\tt s3}, but the occurrences of $d$, a scalar identifier,
are maintained.  

Especially the role of the scalar $d$ is quite interesting. The first definition
of $d$ is given in $S_6$ In $S_7$ $d$ is used twice: explicitly in the 
denominator and in a hidden way in the numerator as well. The optimized 
version of $S_7$ shows a factor $d/d$. The reason is that $d$ locally replaces
an internally created cse name, substituted for ${\rho}_6$ and for part of the 
numerator of ${\rho}_7$. Like illustrated in example~\ref{ex:3.2.7} a second
SCOPE application can further simplify ${\rho}_7$. The scalar $d$ is also used
as argument of the $sin$-function in $S_4 , S_8 , S_9 , S_{11}$ and $S_{12}$. 
The $d$-values in $S_4$, in $\{ S_8 , S_9 \}$ and in $\{ S_{11} , S_{12} \}$ 
are all different, due to the redefinitions in $S_6$ and in $S_{10}$. Therefore 
we recognize two different cse's, containing $sin(d)$: {\tt s24} and {\tt e}.
The role of $a_{1,x+1}$ is of course very similar, albeit less transparant,
due to the renamings.

The second run showes that adding a {\tt DECLARE}
option does not influence the form of the output in this particular case. 
Besides the production of declarations, the result of both runs is identical.
All relevant input and output names are preserved in both runs.
{\small
\begin{verbatim}
OPTIMIZE a(1,x+1)  := g+h*r^f,
         b(y+1)    := a(1,2*x+1)*(g+h*r^f),
         c1        := (h*r)/g*a(2,1+x),
         c2        := c1*a(1,x+1) + sin(d),
         a(1,x+1)  := c1^(5/2),
         d         := b(y+1)*a(1,x+1),
         a(1,1+2*x):= (a(1,x+1)*b(y+1)*c)/(d*g^2),
         b(y+1)    := a(1,1+x)+b(y+1) + sin(d),
         a(1,x+1)  := b(y+1)*c+h/(g + sin(d)),
         d         := k*e+d*(a(1,1+x)+3),
         e         := d*(a(1,1+x)+3) + sin(d),
         f         := d*(3+a(1,x+1)) + sin(d),
         g         := d*(3+a(1,x+1))+f
INAME s$


s0 := x + 1
        f
s34 := r *h + g
s2 := 1 + y
s6 := 2*x + 1
s3 := s34*a(1,s6)
               r*h
c1 := a(2,s0)*-----
                g
c2 := sin(d) + s34*c1
s4 := sqrt(c1)*c1*c1
d := s4*s3
             d*c
a(1,s6) := -------
            g*g*d
s24 := sin(d)
b(s2) := s4 + s3 + s24
               h
a(1,s0) := --------- + b(s2)*c
            g + s24
s29 := 3 + a(1,s0)
d := s29*d + e*k
s33 := s29*d
e := s33 + sin(d)
f := e
g := s33 + f

ON FORT$

OPTIMIZE a(1,x+1)  := g+h*r^f,
         b(y+1)    := a(1,2*x+1)*(g+h*r^f),
         c1        := (h*r)/g*a(2,1+x),
         c2        := c1*a(1,x+1) + sin(d),
         a(1,x+1)  := c1^(5/2),
         d         := b(y+1)*a(1,x+1),
         a(1,1+2*x):= (a(1,x+1)*b(y+1)*c)/(d*g^2),
         b(y+1)    := a(1,1+x)+b(y+1) + sin(d),
         a(1,x+1)  := b(y+1)*c+h/(g + sin(d)),
         d         := k*e+d*(a(1,1+x)+3),
         e         := d*(a(1,1+x)+3) + sin(d),
         f         := d*(3+a(1,x+1)) + sin(d),
         g         := d*(3+a(1,x+1))+f
INAME s
DECLARE <<a(5,5),b(7),c,c1,c2,d,f,g,h,r:real*8; x,y:integer>>$

      INTEGER X,Y,S0,S2,S6
      DOUBLE PRECISION C,H,R,S34,S3,C1,C2,S4,S24,B(7),A(5,5),S29,K,D,S33
     . ,E,F,g
      S0=X+1
      S34=R**F*H+G
      S2=1+Y
      S6=2*X+1
      S3=S34*A(1,S6)
      C1=A(2,S0)*((R*H)/G)
      C2=DSIN(D)+S34*C1
      S4=DSQRT(C1)*C1*C1
      D=S4*S3
      A(1,S6)=(D*C)/(G*G*D)
      S24=DSIN(D)
      B(S2)=S4+S3+S24
      A(1,S0)=H/(G+S24)+B(S2)*C
      S29=3+A(1,S0)
      D=S29*D+DEXP(1.0D0)*K
      S33=S29*D
      E=S33+DSIN(D)
      F=DEXP(1.0D0)
      G=S33+F

OFF FORT$
\end{verbatim}}

Observe the differences in the {\tt f} and {\tt F} assignments. When generating
{\tt fortran77} code all right hand side occurrences of {\tt e} are 
automatically considered as appearances of the base of the natural logarithm
and are translated accordingly.

\index{{\tt VECTORC} switch}
\index{SCOPE function ! {\tt VECTORCODE}}
\index{SCOPE function ! {\tt VCLEAR}}
The third run is influenced by turning {\tt ON} the switch {\tt VECTORC}.
We observe that all array references are substituted back, without having
replaced repeatedly occuring identical subscript expressions by internally 
selected cse names.

The fourth and the last run are governed by the functions {\tt VECTORCODE} and 
{\tt VCLEAR}, after having turned {\tt OFF} the switch {\tt VECTORC}.
{\small
\begin{verbatim}
ON VECTORC$

OPTIMIZE a(1,x+1)  := g+h*r^f,
         b(y+1)    := a(1,2*x+1)*(g+h*r^f),
         c1        := (h*r)/g*a(2,1+x),
         c2        := c1*a(1,x+1) + sin(d),
         a(1,x+1)  := c1^(5/2),
         d         := b(y+1)*a(1,x+1),
         a(1,1+2*x):= (a(1,x+1)*b(y+1)*c)/(d*g^2),
         b(y+1)    := a(1,1+x)+b(y+1) + sin(d),
         a(1,x+1)  := b(y+1)*c+h/(g + sin(d)),
         d         := k*e+d*(a(1,1+x)+3),
         e         := d*(a(1,1+x)+3) + sin(d),
         f         := d*(3+a(1,x+1)) + sin(d),
         g         := d*(3+a(1,x+1))+f
INAME s$

               f
a(1,x + 1) := r *h + g
b(y + 1) := a(1,x + 1)*a(1,2*x + 1)
                  r*h
c1 := a(2,x + 1)*-----
                   g
c2 := sin(d) + a(1,x + 1)*c1
                         2
a(1,x + 1) := sqrt(c1)*c1
d := a(1,x + 1)*b(y + 1)
\end{verbatim}}
\newpage
{\small
\begin{verbatim}
                 d*c
a(1,1 + 2*x) := ------
                  2
                 g *d
s20 := sin(d)
b(y + 1) := a(1,x + 1) + b(y + 1) + s20
                  h
a(1,x + 1) := --------- + b(y + 1)*c
               g + s20
s25 := a(1,x + 1) + 3
d := s25*d + e*k
s28 := s25*d
e := s28 + sin(d)
f := e
g := s28 + f

OFF VECTORC$
VECTORCODE a,b$

OPTIMIZE a(1,x+1)  := g+h*r^f,
         b(y+1)    := a(1,2*x+1)*(g+h*r^f),
         c1        := (h*r)/g*a(2,1+x),
         c2        := c1*a(1,x+1) + sin(d),
         a(1,x+1)  := c1^(5/2),
         d         := b(y+1)*a(1,x+1),
         a(1,1+2*x):= (a(1,x+1)*b(y+1)*c)/(d*g^2),
         b(y+1)    := a(1,1+x)+b(y+1) + sin(d),
         a(1,x+1)  := b(y+1)*c+h/(g + sin(d)),
         d         := k*e+d*(a(1,1+x)+3),
         e         := d*(a(1,1+x)+3) + sin(d),
         f         := d*(3+a(1,x+1)) + sin(d),
         g         := d*(3+a(1,x+1))+f
INAME s$

               f
a(1,x + 1) := r *h + g
b(y + 1) := a(1,x + 1)*a(1,2*x + 1)
                  r*h
c1 := a(2,x + 1)*-----
                   g
c2 := sin(d) + a(1,x + 1)*c1
a(1,x + 1) := sqrt(c1)*c1*c1
d := a(1,x + 1)*b(y + 1)
                  d*c
a(1,1 + 2*x) := -------
                 g*g*d
s22 := sin(d)
b(y + 1) := a(1,x + 1) + b(y + 1) + s22
                  h
a(1,x + 1) := --------- + b(y + 1)*c
               g + s22
s27 := 3 + a(1,x + 1)
d := s27*d + e*k
s30 := s27*d
e := s30 + sin(d)
f := e
g := s30 + f


VCLEAR b$

OPTIMIZE a(1,x+1)  := g+h*r^f,
         b(y+1)    := a(1,2*x+1)*(g+h*r^f),
         c1        := (h*r)/g*a(2,1+x),
         c2        := c1*a(1,x+1) + sin(d),
         a(1,x+1)  := c1^(5/2),
         d         := b(y+1)*a(1,x+1),
         a(1,1+2*x):= (a(1,x+1)*b(y+1)*c)/(d*g^2),
         b(y+1)    := a(1,1+x)+b(y+1) + sin(d),
         a(1,x+1)  := b(y+1)*c+h/(g + sin(d)),
         d         := k*e+d*(a(1,1+x)+3),
         e         := d*(a(1,1+x)+3) + sin(d),
         f         := d*(3+a(1,x+1)) + sin(d),
         g         := d*(3+a(1,x+1))+f
INAME s$

               f
a(1,x + 1) := r *h + g
s1 := y + 1
s2 := a(1,x + 1)*a(1,1 + 2*x)
                  r*h
c1 := a(2,1 + x)*-----
                   g
c2 := sin(d) + a(1,x + 1)*c1
a(1,x + 1) := sqrt(c1)*c1*c1
d := a(1,x + 1)*s2
                  d*c
a(1,1 + 2*x) := -------
                 g*g*d
s23 := sin(d)
b(s1) := a(1,x + 1) + s2 + s23
                  h
a(1,x + 1) := --------- + b(s1)*c
               g + s23
s28 := 3 + a(1,x + 1)
d := s28*d + e*k
s31 := s28*d
e := s31 + sin(d)
f := e
g := s31 + f
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\newpage
\section{A Combined Use of GENTRAN and SCOPE 1.5}\label{SCOPE:gopt}
\index{{\tt GENTRANOPT} switch}

As already stated in subsection~\ref{SCOPE:inter} each GENTRAN command is 
evaluated separately, implying that the symbol table, required 
for the production of declarations, is fresh at the beginning of a GENTRAN 
command evaluation. Turning {\tt ON} the switch {\tt GENTRANOPT} leads to
the optimization of the arithmetic code, defined in  the GENTRAN command,
obeying the new {\tt GENTRANOPT} regime. In addition, we can observe that 
each separate GENTRAN command can produce its own declarations.
\index{{\tt GENTRANOPT} switch}

To increase flexibility we introduced facilities for making declarations,
associated with a group of GENTRAN commands and for the optimization of
the arithmetic in such a group as well.
We implemented two function pairs of parameter-less functions:

\hspace{1cm} ({\tt DELAYDECS}, {\tt MAKEDECS})
\index{SCOPE function ! {\tt DELAYDECS}}
\index{SCOPE function ! {\tt MAKEDECS}}

and 

\hspace{1cm} ({\tt DELAYOPTS}, {\tt MAKEOPTS}).
\index{SCOPE function ! {\tt DELAYOPTS}}
\index{SCOPE function ! {\tt MAKEOPTS}}

Both pairs function as "brackets" around groups of statements. The {\tt OPTS}
pair can be used (repeatedly) inside a {\tt DECS} pair. Both pairs achieve
an alterned GENTRAN mode of operation. All GENTRAN productions between such
a pair are collected in an internal format, say {\tt if\_list}. 
The {\tt DELAY...} functions initialize the modified mode of operation. 
The {\tt MAKE...} functions restore the previous mode of operation in 
combination with the production either of declarations, associated with the
contents of {\tt if\_list}, or of an optimized representation  of the
contents of {\tt if\_list}.

Example~\ref{ex:8.1} serves to introduce a variety of approaches to code 
generation, based on the use of these pairs of brackets and of the 
switch {\tt GENTRANOPT}. We illustrate a more realistic use in 
example~\ref{ex:8.2}: generation of optimized fortran77 code for the 
computation of the entries of the inverse of a symmetric (3,3) matrix. 
It is a continuation of example~\ref{ex:3.1.8} in 
subsection~\ref{SCOPE:optim} and example~\ref{ex:3.2.5} in 
subsection ~\ref{SCOPE:algo}. It was also presented 
in ~\cite{Gates:85}, albeit in a slightly different form.

\example\label{ex:8.1}
\index{SCOPE ! example}

The output of combined GENTRAN and SCOPE operations is by default given 
in fortran77 notation. We illustrate the different possibilities in the
form of small pieces of code.
\begin{itemize}
\item The pair ({\tt DELAYDECS}, {\tt MAKEDECS}) encloses four GENTRAN commands.
The first is needed to initialize the symbol table. The literal translation in
internal format of the last three commands is stored in the {\tt if\_list}.
The application of {\tt MAKEDECS} leads to the restoration of the default
GENTRAN regime, applied to the {\tt if\_list} and leading to the result,
presented in the form of fortran77 code.
{\small
\begin{verbatim}
DELAYDECS$

  GENTRAN DECLARE <<a,b,c,d,q,w:real>>$
  GENTRAN a:=b+c$
  GENTRAN d:=b+c$
  GENTRAN <<q:=b+c;w:=b+c>>$

MAKEDECS$

      REAL A,B,C,D,Q,W
      A=B+C
      D=B+C
      Q=B+C
      W=B+C
\end{verbatim}}
\item We repeat the previous commands, but execute them in a slightly different
setting by turning {\tt ON} the switch {\tt GENTRANOPT}. This time the
arithmetical rules in each of the three last GENTRAN commands are optimized 
separately.  This is illustrated by the form of the output. The last piece of 
code contains the cse {\tt B+C}, which is presented under the name {\tt Q} 
in the fortran77 output.
\index{{\tt GENTRANOPT} switch}
\index{SCOPE function ! {\tt DELAYDECS}}
\index{SCOPE function ! {\tt MAKEDECS}}
\index{SCOPE function ! {\tt DELAYOPTS}}
\index{SCOPE function ! {\tt MAKEOPTS}}
{\small
\begin{verbatim}
ON GENTRANOPT$

DELAYDECS$

  GENTRAN DECLARE <<a,b,c,d,q,w:real>>$
  GENTRAN a:=b+c$
  GENTRAN d:=b+c$
  GENTRAN <<q:=b+c;w:=b+c>>$

MAKEDECS$

      REAL B,C,A,D,Q,W
      A=B+C
      D=B+C
      Q=B+C
      W=Q

OFF GENTRANOPT$
\end{verbatim}}
\item We can improve the optimization strategy by using the function pair
({\tt DELAYOPTS}, {\tt MAKEOPTS}) in stead of the
pair ({\tt DELAYDECS}, {\tt MAKEDECS}). All blockwise combinable arithmetic
is collected. These blocks of straight line code are optimized as separate
input sets ${\rm E_{in}}$, when activating {\tt MAKEOPTS}.
{\small
\begin{verbatim}
DELAYOPTS$

  GENTRAN a:=b+c$
  GENTRAN d:=b+c$
  GENTRAN <<q:=b+c;w:=b+c>>$

MAKEOPTS$

      A=B+C
      D=A
      Q=A
      W=A
\end{verbatim}}
\item We can furhter improve the optimization strategy by using the 
function pair ({\tt DELAYOPTS}, {\tt MAKEOPTS}) inside the 
pair ({\tt DELAYDECS}, {\tt MAKEDECS}). All blockwise combinable arithmetic
is collected. These blocks of straight line code are optimized as separate
input sets ${\rm E_{in}}$, when activating {\tt MAKEOPTS}. But this time the
results are added in internal format to the {\tt if\_list} version, 
being maintained, as to obey the {\tt DELAYDECS} regime.
{\small
\begin{verbatim}
DELAYDECS$

  GENTRAN DECLARE <<a,b,c,d,q,w:real>>$

  DELAYOPTS$

    GENTRAN a:=b+c$
    GENTRAN d:=b+c$
    GENTRAN <<q:=b+c;w:=b+c>>$

  MAKEOPTS$

MAKEDECS$

      REAL B,C,A,D,Q,W
      A=B+C
      D=A
      Q=A
      W=A
\end{verbatim}}
\item A slightly more realistic example suggests how easily optimized code
for sets of matrix entries can be obtained. We use two identical matrices
{\tt a} and {\tt d}. The latter is not introduced at the {\REDUCE} level,
but simply used inside a GENTRAN command. The entries of {\tt a} are
initialized inside a double for-loop. After each initialization a GENTRAN
command is applied, using the special assignment operator {\tt ::=:} for
correctly combining entry names and entry values. The REDUCE algebraic mode 
assignments are again used, when identifying the matrix {\tt d} with the 
matrix {\tt a}, applying the special assignment operator {\tt :=:} in a 
separate GENTRAN command. The latter command is internally expanded into
separate GENTRAN commands for each entry assignment. By using the pair
({\tt DELAYOPTS}, {\tt MAKEOPTS}) one block of straigt line code is
constructed and optimized. It consists of two sets of assignments, one 
for the entries of the matrix {\tt a} and another for the entries of the 
matrix {\tt d}. The presented output shows that both matrices are indeed 
identical.
{\small
\begin{verbatim}
MATRIX a(4,4);

DELAYDECS$

  GENTRAN DECLARE <<a(4,4),d(4,4),b,c:real>>$

  DELAYOPTS$

    FOR i:=1:4 DO FOR j:=1:4 DO << a(i,j):=(i+j)*(b+c)+i*b+j*c;
                                   GENTRAN a(i,j)::=:a(i,j)>>$
    GENTRAN d:=:a$

  MAKEOPTS$

MAKEDECS$

      REAL B,C,G56,G54,G57,G55,A(4,4),D(4,4)
      A(1,1)=3.0*(B+C)
      G56=5.0*C
      A(1,2)=G56+4.0*B
      G54=5.0*B
      G57=7.0*C
      A(1,3)=G57+G54
      A(1,4)=6.0*B+9.0*C
      A(2,1)=G54+4.0*c
      A(2,2)=2.0*A(1,1)
      G55=7.0*B
      A(2,3)=G55+8.0*C
      A(2,4)=2.0*A(1,2)
      A(3,1)=G56+G55
      A(3,2)=G57+8.0*B
      A(3,3)=3.0*A(1,1)
      A(3,4)=10.0*B+11.0*C
      A(4,1)=9.0*B+6.0*C
      A(4,2)=2.0*A(2,1)
      A(4,3)=11.0*B+10.0*C
      A(4,4)=4.0*A(1,1)
      D(1,1)=A(1,1)
      D(1,2)=A(1,2)
      D(1,3)=A(1,3)
      D(1,4)=A(1,4)
      D(2,1)=A(2,1)
      D(2,2)=A(2,2)
      D(2,3)=A(2,3)
      D(2,4)=A(2,4)
      D(3,1)=A(3,1)
      D(3,2)=A(3,2)
      D(3,3)=A(3,3)
      D(3,4)=A(3,4)
      D(4,1)=A(4,1)
      D(4,2)=A(4,2)
      D(4,3)=A(4,3)
      D(4,4)=A(4,4)
\end{verbatim}}
\item Finally, and again only for illustrative purposes the fifth piece of
code is again executed in an almost identical manner. We omit the declarations
and replace the instruction {\tt GENTRAN d:=:a}\verb+$+ by the command
{\tt GENTRAN a:=:a}\verb+$+. Hence the matrix a is simply redefined. As
stated in section~\ref{SCOPE:dda} redundant assignments --- producing dead
code, for instance by copying previous assignments --- are automatically 
removed. as part of the optimization process. Therefore the optimized
entry values of the matrix {\tt a} are presented only once.
{\small
\begin{verbatim}
DELAYOPTS$

  FOR i:=1:4 DO FOR j:=1:4 DO << a(i,j):=(i+j)*(b+c)+i*b+j*c;
                                 GENTRAN a(i,j)::=:a(i,j)>>$
  GENTRAN a:=:a$

MAKEOPTS$

      A(1,1)=3.0*(B+C)
      G111=5.0*C
      A(1,2)=G111+4.0*B
      G109=5.0*B
      G112=7.0*C
      A(1,3)=G112+G109
      A(1,4)=6.0*B+9.0*C
      A(2,1)=G109+4.0*C
      A(2,2)=2.0*A(1,1)
      G110=7.0*B
      A(2,3)=G110+8.0*C
      A(2,4)=2.0*A(1,2)
      A(3,1)=G111+G110
      A(3,2)=G112+8.0*B
      A(3,3)=3.0*A(1,1)
      A(3,4)=10.0*B+11.0*C
      A(4,1)=9.0*B+6.0*C
      A(4,2)=2.0*A(2,1)
      A(4,3)=11.0*B+10.0*C
      A(4,4)=4.0*A(1,1)
\end{verbatim}}
\end{itemize}
{\small
\begin{flushright}
$\Box$
\end{flushright}}

\example\label{ex:8.2}
\index{SCOPE ! example}

Let us now again look at the symmetric (3,3) matrix {\tt m}, already used in
the examples ~\ref{ex:3.1.8} and ~\ref{ex:3.2.5}. For completeness we begin
by showing the entry values. We generate optimized fortran77 code
for the inverse {\tt mnv} of {\tt m} and store it in a file, 
named {\tt inverse.code}, using the function pair ({\tt GENTRANOUT}, 
{\tt GENTRANSHUT}). Inside this pair we apply the pair ({\tt DELAYDECS},
{\tt MAKEDECS}). The latter pair encloses in turn the pair ({\tt DELAYOPTS},
{\tt MAKEOPTS}). Inside these innermost brackets four different sections of 
code can be distinguished. The first section consists of three {\tt LITERAL} 
commands, for inserting comment in the target code. The second is formed by
a double for-loop.  Essential are the applications of the GENTRAN functions
{\tt tempvar} and {\tt markvar} for assigning internal names to matrix entry
values. These applications are very similar to the approach, chosen in
example~\ref{ex:3.2.5}. The third section is again formed by {\tt LITERAL}
commands and the last orders the creation of the entries of the inverse
matrix {\tt mnv}. Before introducing the file {\tt inverse.code} we selected
{\tt S0} as initial cse name, using the function {\tt INAME}, and 
turned {\tt ON} the switches {\tt ACINFO} and {\tt DOUBLE}.
\index{GENTRAN function ! {\tt GENTRANOUT}}
\index{GENTRAN function ! {\tt GENTRANSHUT}}
\index{{\tt ACINFO} switch}
\index{{\tt DOUBLE} switch}
\index{SCOPE function ! {\tt INAME}}
\index{GENTRAN's {\tt DECLARE} statement}
\index{GENTRAN's {\tt LITERAL} statement}
\index{REDUCE function ! {\tt gensym}}

Observe that directly after the {\tt MAKEOPTS} instruction two sets of tables
are printed with information about the arithmetic complexity of the two blocks 
of code, which are generated in the second and the last section.  We 
activated {\tt ACINFO} to show this effect.  The tables are printed as 
a side effect. The output itself is directed to the file {\tt inverse.code}.
Looking at the contents of this file, given below, shows three different 
kinds of internally generated names. A number of {\tt S}-names is created 
during the optimization of the first block of straight line code, created with
the second section. In this piece of code we also notice {\tt T}-names,
generated by {\tt tempvar} applications. The intial {\tt T} character is the
default internal GENTRAN selection for {(temporarily needed) names.
And finally {\tt G}-names, introduced by
applications of {\tt gensym}, during the second optimization operation for
reducing the arithmetic complexity of the entries of {\tt mnv}. Because the
code splitting is internally performed, the second SCOPE application 
is missing its {\tt INAME} initialisation, thus leading to the application of
{\tt gensym}. Observe as well that {\tt INAME} can be used as a separate 
facility as well.  
{\small
\begin{verbatim}
OFF EXP$

MATRIX m(3,3),mnv(3,3)$
IN "matrix.M"$

m(1,1) :=  - ((j30y - j30z + 9*m30*p )*sin(q3)

                                         2                      2
               - 18*cos(q2)*cos(q3)*m30*p  - j10y - j30y - m10*p

                         2
               - 18*m30*p )

                                              2         2
m(2,1) := m(1,2) :=  - ((j30y - j30z + 9*m30*p )*sin(q3)

                             2                 2
    - 9*cos(q2)*cos(q3)*m30*p  - j30y - 9*m30*p )

                                              2
m(3,1) := m(1,3) :=  - 9*sin(q2)*sin(q3)*m30*p

                                    2         2                 2
m(2,2) :=  - ((j30y - j30z + 9*m30*p )*sin(q3)  - j30y - 9*m30*p )

m(3,2) := m(2,3) := 0

                        2
m(3,3) := j30x + 9*m30*p

INAME s0$
ON ACINFO,DOUBLE$

GENTRANOUT "inverse.code"$

DELAYDECS$

  GENTRAN DECLARE <<mnv(3,3),p,m30,j30y,j30z,q3,m10,q2,j10y,j30x:real>>$
  DELAYOPTS$

    GENTRAN LITERAL "C",tab!*,"  ",cr!*$
    GENTRAN LITERAL "C",tab!*," -- Computation of relevant M-entries --",cr!*$
    GENTRAN LITERAL "C",tab!*,"  ",cr!*$

    FOR j:=1:3 DO FOR k:=j:3 DO
     IF m(j,k) NEQ 0 THEN
      << s:=tempvar('real); markvar s;
         GENTRAN eval(s):=:m(j,k);
         m(j,k):=m(k,j):=s
      >>$
\end{verbatim}}
\index{GENTRAN function ! {\tt markvar}}
\index{GENTRAN function ! {\tt tempvar}}
{\small
\begin{verbatim}
    GENTRAN LITERAL "C",tab!*,"  ",cr!*$
    GENTRAN LITERAL "C",tab!*," -- Computation of the inverse of M --",cr!*$
    GENTRAN LITERAL "C",tab!*,"  ",cr!*$

    GENTRAN mnv:=:m^(-1)$

  MAKEOPTS$

Number of operations in the input is: 

Number of (+/-) operations      : 17
Number of unary - operations    : 4
Number of * operations          : 30
Number of integer ^ operations  : 14
Number of / operations          : 0
Number of function applications : 9


Number of operations after optimization is:

Number of (+/-) operations      : 11
Number of unary - operations    : 1
Number of * operations          : 12
Number of integer ^ operations  : 0
Number of / operations          : 0
Number of function applications : 4


Number of operations in the input is: 

Number of (+/-) operations      : 20
Number of unary - operations    : 4
Number of * operations          : 45
Number of integer ^ operations  : 20
Number of / operations          : 9
Number of function applications : 0


Number of operations after optimization is:

Number of (+/-) operations      : 4
Number of unary - operations    : 2
Number of * operations          : 11
Number of integer ^ operations  : 0
Number of / operations          : 4
Number of function applications : 0

MAKEDECS$

GENTRANSHUT "inverse.code"$
\end{verbatim}}

The contents of the file {\tt inverse.code} is:
{\small
\begin{verbatim}
      DOUBLE PRECISION P,M30,J30Y,J30Z,Q3,M10,Q2,J10Y,J30X,S0,S7,S5,S4,
     . S13,S11,T0,T3,T1,T2,T4,G153,G152,G151,G147,G155,G156,MNV(3,3)
C       
C      -- Computation of relevant M-entries --
C       
      S0=DSIN(Q3)
      S7=P*P
      S5=S7*M30
      S4=S5*DCOS(Q3)*DCOS(Q2)
      S13=9.0D0*S5
      S11=(S13+J30Y-J30Z)*S0*S0
      T0=J30Y+J10Y+18.0D0*(S4+S5)+S7*M10-S11
      T3=S13+J30Y-S11
      T1=T3+9.0D0*S4
      T2=-(S13*DSIN(Q2)*S0)
      T4=S13+J30X
C       
C      -- Computation of the inverse of M --
C       
      G153=T2*T2
      G152=T1*T1
      G151=T0*T4
      G147=G151*T3-(G153*T3)-(G152*T4)
      G155=T4/G147
      MNV(1,1)=G155*T3
      MNV(1,2)=-(G155*T1)
      G156=T2/G147
      MNV(1,3)=-(G156*T3)
      MNV(2,1)=MNV(1,2)
      MNV(2,2)=(G151-G153)/G147
      MNV(2,3)=G156*T1
      MNV(3,1)=MNV(1,3)
      MNV(3,2)=MNV(2,3)
      MNV(3,3)=(T0*T3-G152)/G147
\end{verbatim}}

Let us now repeat the generation process in a slightly different setting.
We leave out the comment generating instructions, thus creating only one
block of straight line code to be optimized. We choose for an {\tt S}-name
selection based on {\tt tempvar} applications and for {\tt T}-names for cse's.
This time the default use of {\tt gensym} is not necessary.

The contents' of both output files illustrate quotient optimization.
All denominators, being the determinant of the matrix {\tt m}, are identical.
The set of rational entries of {\tt MNV} contains the cse's {\tt G155 (T45)}
and {\tt G156 (T46)}.
\index{GENTRAN identifier !{\tt TEMPVARNAME*}}
\index{GENTRAN identifier !{\tt TEMPVARNUM*}}
\index{GENTRAN function ! {\tt GENTRANOUT}}
\index{SCOPE function ! {\tt INAME}}
{\small
\begin{verbatim}
TEMPVARNAME!*:='s$
TEMPVARNUM!*:=0$
INAME t0$

GENTRANOUT "inverse.code2"$

DELAYDECS$

  GENTRAN DECLARE <<mnv(3,3),p,m30,j30y,j30z,q3,m10,q2,j10y,j30x:real>>$
  DELAYOPTS$

    FOR j:=1:3 DO FOR k:=j:3 DO
     IF m(j,k) NEQ 0 THEN
      << s:=tempvar('real); markvar(s);
         GENTRAN eval(s):=:m(j,k);
         m(j,k):=m(k,j):=s
      >>$

    GENTRAN mnv:=:m^(-1)$

  MAKEOPTS$

Number of operations in the input is: 

Number of (+/-) operations      : 37
Number of unary - operations    : 8
Number of * operations          : 75
Number of integer ^ operations  : 34
Number of / operations          : 9
Number of function applications : 9


Number of operations after optimization is:

Number of (+/-) operations      : 15
Number of unary - operations    : 3
Number of * operations          : 23
Number of integer ^ operations  : 0
Number of / operations          : 4
Number of function applications : 4

MAKEDECS$

GENTRANSHUT "inverse.code2"$
\end{verbatim}}

The contents of the file {\tt inverse.code2} is:
{\small
\begin{verbatim}
      DOUBLE PRECISION P,M30,J30Y,J30Z,Q3,M10,Q2,J10Y,J30X,T9,T40,T32,
     . T31,T49,T47,S0,S3,S1,S2,S4,T39,T38,T36,T30,T45,T46,MNV(3,3)
      T9=DSIN(Q3)
      T40=P*P
      T32=T40*M30
      T31=T32*DCOS(Q3)*DCOS(Q2)
      T49=9.0D0*T32
      T47=(T49+J30Y-J30Z)*T9*T9
      S0=J30Y+J10Y+18.0D0*(T31+T32)+T40*M10-T47
      S3=T49+J30Y-T47
      S1=S3+9.0D0*T31
      S2=-(T49*DSIN(Q2)*T9)
      S4=T49+J30X
      T39=S2*S2
      T38=S1*S1
      T36=S0*S4
      T30=T36*S3-(T39*S3)-(T38*S4)
      T45=S4/T30
      MNV(1,1)=T45*S3
      MNV(1,2)=-(T45*S1)
      T46=S2/T30
      MNV(1,3)=-(T46*S3)
      MNV(2,1)=MNV(1,2)
      MNV(2,2)=(T36-T39)/T30
      MNV(2,3)=T46*S1
      MNV(3,1)=MNV(1,3)
      MNV(3,2)=MNV(2,3)
      MNV(3,3)=(S0*S3-T38)/T30
\end{verbatim}}

A comparison between the arithmetic complexities given here and in example
~\ref{ex:3.2.5} shows that computing the entries of {\tt MNV}(=${\tt M}^{-1}$) 
instead of the value of the determinant of {\tt M}, only requires 2 
extra additions, 1 extra negation, 6 extra multilications and 4 extra 
divisions. 
{\small
\begin{flushright}
$\Box$
\end{flushright}}

Other examples of this combined use of GENTRAN and SCOPE can be found 
in ~\cite{vanHulzen:95,Ganzha:94}. The symbolic-numeric strategy discussed in
~\cite{Ganzha:94} also relies on the {\tt ALGOPT} facilities, which were 
introduced earlier. 
\newpage
\section{Symbolic Mode Use of SCOPE 1.5}~\label{SCOPE:symb}

Both the {\tt OPTIMIZE} command and the {\tt ALGOPT} function are transformed
into the same symbolic mode function, called {\tt SYMOPTIMIZE}. It is this
function, which governs the optimization process as a whole, delivering
the results of an optimization run as a side effect, for instance by making
it visible on a screen or by storing it in a file. Using {\tt SYMOPTIMIZE}
is straighforward, once the syntax for its five actual parameters is known.
If we set {\tt ON INTERN} a {\tt SYMOPTIMIZE} application will deliver a
list, containing the correctly ordered results of an optimization operation 
in the form of assignment statements in prefix form in Lisp notation. The thus
provided results can function as one of the five actual parameters for
{\tt SYMOPTIMIZE} as well. This simple feature helps avoiding file traffic
when stepwise optimizing code and as illustrated earlier in example~\ref{ex:5.1}
in section ~\ref{SCOPE:files}. Before illustrating that in example~\ref{ex:9.1}
we present the syntax of the actual parameters for {\tt SYMOPTIMIZE}:
\index{{\tt INTERN} switch}
\index{SCOPE function ! {\tt SYMOPTIMIZE}}

$<$SCOPE\_application$>~::=~\cdots~\mid$\\
\hspace*{1cm} {\tt SYMOPTIMIZE}($<$ssetq\_list$>,<$infile\_list$>,<$outfile\_name$>,<$cse\_prefix$>,\\
\hspace*{6cm} <$declaration\_list$>$)

\begin{center}
\begin{tabular}{lcl}
$<$ssetq\_list$>$ & $::=$ & ($<$ssetq\_seq$>$)\\
$<$ssetq\_seq$>$ & $::=$ & $<$ssetq\_stat$>~[<$ssetq\_seq$>]$\\
$<$ssetq\_stat$>$ & $::=$ & ({\tt setq} $<$lhs\_id$>~<$rhs$>)~\mid$
({\tt rsetq} $<$lhs\_id$>~<$rhs$>)~\mid$\\
 & & ({\tt lsetq} $<$lhs\_id$>~<$rhs$>)~\mid$
({\tt lrsetq} $<$lhs\_id$>~<$rhs$>)$\\
$<$lhs\_id$>$ & $::=$ & $<$id$>~\mid~<$subscripted\_id$>$\\
$<$subscripted\_id$>$ & $::=$ & $(<$id$>~<$s\_subscript\_seq$>)$\\
$<$s\_subscript\_seq$>$ & $::=$ & $<$s\_subscript$>[~<$s\_subscript\_seq$>$]\\
$<$s\_subscript$>$ & $::=$ & $<$integer$>~\mid~<$integer prefix\_expression$>$\\
$<$rhs$>$ & $::=$ & $<$prefix\_expression$>$\\
$<$infile\_list$>$ & $::=$ & $(<$infile\_seq$>)$\\
$<$infile\_seq$>$ & $::=$ & $<$infile\_name$>[~<$infile\_seq$>]$\\
$<$infile\_name$>$ & $::=$ & $<$string\_id$>$\\
$<$outfile\_name$>$ & $::=$ & $<$string\_id$>$\\
$<$declaration\_list$>$ & $::=$ & $(<$declaration\_seq$>)$\\
$<$declaration\_seq$>$ & $::=$ & $<$declaration$>~<$declaration\_seq$>$\\
$<$declaration$>$ & $::=$ & $(<$type$>~<$lhs\_id\_seq$>)$\\
$<$lhs\_id\_seq$>$ & $::=$ & $<$lhs\_id$>~<$lhs\_id\_seq$>$
\end{tabular}
\end{center}

The above given syntax requires some explanation:
\begin{itemize}
\item The presented ssetq syntax is incomplete. The prefix equivalent of 
any object, introduced in subsection~\ref{SCOPE:optim} and of any a\_object,
defined in subsection~\ref{SCOPE:algo}, is accepted as ssetq item. Such prefix
equivalents can be obtained quite easily by using the function {\tt show}:

\hspace*{1cm} {\tt SYMBOLIC PROCEDURE show u; prettyprint u}\verb+$+ \\
\hspace*{1cm} {\tt SYMBOLIC OPERATOR show}\verb+$+

The explicit presentation of a subset of the syntax rules for ssetq
is given to suggest that local simplification in symbolic mode can be brought 
in easily by using the assignment operators {\tt lsetq}, {\tt lrsetq} 
and {\tt rsetq}. The algebraic mode equivalent of these operators is
{\tt ::=}, {\tt ::=:} and {\tt :=:}, respectively. Their effect on
simplification is discussed in subsection~\ref{SCOPE:inter} and already 
shown in a number of examples. In addition it is worth noting that 
any (sub)expression
in a lhs\_id or a rhs may contain any number of calls to {\tt eval}. 
These calls lead to simplification of their arguments, prior to optimization.
Details about the use of {\tt eval} are presented in the 
GENTRAN manual~\cite{Gates:91}.
\item Since we operate in symbolic mode the last four formal parameters
have possibly to be replaced by quoted actual parameters. This is illustrated 
in example ~\ref{ex:9.1}.
\item  The infile\_seq consists of  file names in string notation. The contents'
of such input files may contain any form of infix input, obeying the syntax
rules for objects and/or a\_objects, as introduced in the subsections
 ~\ref{SCOPE:optim} and ~\ref{SCOPE:algo}, respectively.
\item The single output file name  outfile\_name ought to be given in string
notation as well. The outfile\_name  is properly closed. The default output
is REDUCE infix in an {\tt ON NAT} fashion. Alternatives are discussed above:
{\tt ON AGAIN} or {\tt OFF NAT}, both leading to re-readable output, or an 
application of {\tt OPTLANG} for a non {\tt nil} argument.
\item The declaration list presents declarations in prefix notation. The list
is used to initialize the symbol table prior to optimization. This 
information is used for dynamically typing  the result of an optimization
process. In addition it is used to determine wether subscripted names denote
array elements or a function call. The latter is replaced by a cse name in the
presented output, whereas the former is not.
\item  The five parameters of {\tt SYMOPTIMIZE} correspondent with optional
extensions of the {\tt OPTIMIZE} command. When part of these options remains
unused, {\tt nil} has to taken as value for the corresponding actual parameters.
\end{itemize}
\index{{\tt AGAIN} switch}
\index{{\tt INPUTC} switch}
\index{{\tt INTERN} switch}
\index{{\tt NAT} switch}
\index{SCOPE function ! {\tt OPTLANG}}
\index{GENTRAN function ! {\tt lsetq}}
\index{GENTRAN function ! {\tt lrsetq}}
\index{GENTRAN function ! {\tt rsetq}}

We illustrate the symbolic mode variant of the {\tt OPTIMIZE} command by
repeating example~\ref{ex:5.1} from section~\ref{SCOPE:files}, albeit in
a modified setting.

\example\label{ex:9.1}
\index{SCOPE ! example}

The script explains itself. 
{\small
\begin{verbatim}
LISP$
ON ACINFO,INPUTC,INTERN,AGAIN$

prettyprint(prefixlist:=SYMOPTIMIZE(nil,'("f1" "f2"),nil,'c,nil))$

                                2
                2        (x + y)              8        2      2
       2*(sin(x)  - cos(e        ) + 3*cos(x)) *(x + y)  + 4*y  + 4*y
e1 := ----------------------------------------------------------------
                                 3*x + 2*y

                                2
                2        (x + y)              2        3
e2 := (4*(sin(x)  - cos(e        ) + 2*cos(x)) *(x + y)

              2                   2
        + (4*x  - 4*y) - 6*x)/(8*x  + 3*y - 2*x)

Number of operations in the input is: 

Number of (+/-) operations      : 16
Number of unary - operations    : 0
Number of * operations          : 16
Number of integer ^ operations  : 11
Number of / operations          : 2
Number of function applications : 8


Number of operations after optimization is:

Number of (+/-) operations      : 16
Number of unary - operations    : 0
Number of * operations          : 16
Number of integer ^ operations  : 6
Number of / operations          : 2
Number of function applications : 4

((setq gsym c23)
   (setq cses (plus c18 c10 c20 c8 c6 c14))
   (setq c14 (plus x y))
   (setq c6 (expt c14 2))
   (setq c8 (cos x))
   (setq c20 (plus (expt (sin x) 2) (minus (cos (expt e c6)))) )
   (setq c10 (plus (times 3 x) (times 2 y)))
   (setq e1
      (quotient
         (plus
            (times 4 y)
            (times 4 (expt y 2))
            (times 2 c6 (expt (plus c20 (times 3 c8)) 8)))
         c10))
   (setq c18 (expt x 2))
   (setq e2
      (quotient
         (plus
            (times 4 c18)
            (minus (times 2 c10))
            (times 4 c6 c14 (expt (plus c20 (times 2 c8)) 2)))
         (plus (times 8 c18) (minus (times 2 x)) (times 3 y)))) )
\end{verbatim}}
\index{{\tt DOUBLE} switch}
\index{{\tt FORT} switch}
{\small
\begin{verbatim}
OFF INTERN,AGAIN,PERIOD$
ON DOUBLE,FORT$

SYMOPTIMIZE(prefixlist,'("f3"), '"f7",'d,'((real e1 e2 e3 x y)))$

gsym := c23

cses := c18 + c10 + c20 + c8 + c6 + c14

c14 := x + y

         2
c6 := c14

c8 := cos(x)

             2        c6
c20 := sin(x)  - cos(e  )

c10 := 3*x + 2*y

                2                    8
       4*y + 4*y  + 2*c6*(c20 + 3*c8)
e1 := ---------------------------------
                     c10

        2
c18 := x

                                            2
       4*c18 - 2*c10 + 4*c6*c14*(c20 + 2*c8)
e2 := ----------------------------------------
                 8*c18 - 2*x + 3*y

                         2
                  (x + y)                       2           2
       4*sin(cos(e        )) + sin(x + y) + (4*x  - x + 2*y)
e3 := --------------------------------------------------------
                      3*y + f(x,g( - cos(x)))

Number of operations in the input is: 

Number of (+/-) operations      : 23
Number of unary - operations    : 1
Number of * operations          : 20
Number of integer ^ operations  : 9
Number of / operations          : 3
Number of function applications : 11


Number of operations after optimization is:

Number of (+/-) operations      : 15
Number of unary - operations    : 1
Number of * operations          : 24
Number of integer ^ operations  : 0
Number of / operations          : 3
Number of function applications : 8
\end{verbatim}}

The contents of the output file {\tt f7} is:
{\small
\begin{verbatim}
      DOUBLE PRECISION X,Y,D19,D16,C8,D1,D2,C20,D29,C10,D6,D38,D37,D31,
     . E1,C18,D9,D32,D27,D30,E2,D20,E3
      D19=X+Y
      D16=D19*D19
      C8=DCOS(X)
      D1=DSIN(X)
      D2=DCOS(DEXP(D16))
      C20=D1*D1-D2
      D29=2*Y
      C10=D29+3*X
      D6=C20+3*C8
      D38=D6*D6
      D37=D38*D38
      D31=4*Y
      E1=(D31+D31*Y+2.0D0*D16*D37*D37)/C10
      C18=X*X
      D9=C20+2*C8
      D32=4*C18
      D27=D32-X
      D30=3*Y
      E2=(D32-(2.0D0*C10)+4.0D0*D9*D9*D19*D16)/(D30+2.0D0*D27)
      D20=D29+D27
      E3=(4.0D0*DSIN(D2)+DSIN(D19)+D20*D20)/(D30+F(X,G(-C8)))
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}

We especially designed these symbolic mode facilities for our joint 
research with Delft Hy\-draulics concerning code generation for an 
incompressible Navier-Stokes problem ~\cite{Goldman:95}.

\index{{\tt PREFIX} switch}
\index{{\tt INTERN} switch}
A final remark: The {\tt ON PREFIX} mode of operation, in both 
algebraic and symbolic mode causes the results of a SCOPE 
application to be presented in the form of an association  list,  
called {\tt Prefixlist}.  The pairs are formed by lhs\_id's and 
rhs values in prefixform. This lisp S-expression can be used to 
create an alternative version of the optimization results, 
in whatever target language the user prefers to choose. 

\example\label{ex:9.2}
\index{SCOPE ! example}

We show the {\tt ON PREFIX} effect. When switching to symbolic mode
(command 5) we can again obtain the output, assigned as value to the global
identifer {\tt prefixlist}. The {\tt ON PREFIX} facility allows storage
in a file for later use. When working in symbolic mode it is of course
possible to apply {\tt ON INTERN} in stead and to remove the {\tt setq} 
extensions from the provided output value, if desired.
{\small
\begin{verbatim}
REDUCE 3.6, 15-Jul-95 ...

1: LOAD_PACKAGE nscope$

2: OPTIMIZE a:=b+c*sin(x), d:=c*sin(x)*cos(y);

g7 := sin(x)*c
a := g7 + b
d := g7*cos(y)

3: ON PREFIX$

4: input 2;

Prefixlist:=
((g3 times (sin x) c) (a plus g3 b) (d times g3 (cos y)))

5: LISP$

6: prettyprint prefixlist$
((g3 times (sin x) c) (a plus g3 b) (d times g3 (cos y)))

7: caar prefixlist;

g3

7: cdar prefixlist;

(times (sin x) c)

9: BYE;
\end{verbatim}
\begin{flushright}
$\Box$
\end{flushright}}
\newpage
\section{ A Syntax Summary of SCOPE 1.5}~\label{SCOPE:syntax}

{\REDUCE} is extended with some commands, designed to apply the facilities
offered by SCOPE in a flexible way. The syntactical rules, defining how to
activate SCOPE in both algebraic and symbolic mode, are given in subsection
~\ref{SCOPE:srules}. A short overview of the set of additional functions is   
given in subsection~\ref{SCOPE:arules} and the relevant switches are again
presented in subsection~\ref{SCOPE:switches}.

\subsection{SCOPE's Toplevel Commands}~\label{SCOPE:srules}

We assume the syntax of {\tt id}'s, {\tt integer}'s and the like to be already 
known. Hence we do not present an exhaustive description of the rules.\\
\index{SCOPE function ! {\tt OPTIMIZE}}
\index{SCOPE function ! {\tt ALGOPT}}
\index{SCOPE function ! {\tt SYMOPTIMIZE}}
\index{REDUCE function ! {\tt GSTRUCTR}}
\index{REDUCE function ! {\tt GHORNER}}

$<${\tt REDUCE}\_command$>~::=~ \cdots ~ \mid~<${\tt SCOPE}\_application$>$\\
\hspace*{1cm} $<${\tt GSTRUCTR}\_application$>~\mid$
$<${\tt GHORNER}\_application$>~\mid$\\
$<${\tt SCOPE}\_application$>~::=~<${\tt OPTIMIZE} command$>~\mid$\\
\hspace*{1cm} 
$<${\tt ALGOPT} application$>~\mid$
$<${\tt SYMOPTIMIZE} application$>$\\

$<${\tt OPTIMIZE} command$>~::=$\\
\hspace*{1cm}{\tt OPTIMIZE} $<$object\_seq$>$ [{\tt IN} $<$file\_id\_seq$>$]
 [{\tt OUT} $<$file\_id$>$]\\
\hspace*{3cm}[{\tt INAME} $<$cse\_prefix$>$]
[{\tt DECLARE} $<$declaration\_group$>$] $\mid$\\
\hspace*{1cm}{\tt OPTIMIZE} [$<$object\_seq$>$] {\tt IN} $<$file\_id\_seq$>$
 [{\tt OUT} $<$file\_id$>$]\\ 
\hspace*{3cm}[{\tt INAME} $<$cse\_prefix$>$]
[{\tt DECLARE} $<$declaration\_group$>$]\\

$<${\tt ALGOPT} application$>~::=$\\
\hspace*{1cm}{\tt ALGOPT}($<$a\_object\_list$>$[,$<$string\_id\_list$>$][,$<$cse\_prefix$>$]) $\mid$\\
\hspace*{1cm}{\tt ALGOPT}([$<$a\_object\_list$>$,]$<$string\_id\_list$>$[,$<$cse\_prefix$>$])\\

$<${\tt SYMOPTIMIZE} application$>~::=$\\
\hspace*{0.3cm} {\tt SYMOPTIMIZE}($<$ssetq\_list$>,<$infile\_list$>,<$outfile\_name$>,<$cse\_prefix$>$,\\
\hspace*{2.3cm} $<$declaration\_list$>$)\\

\begin{tabular}{lcl}
$<${\tt GSTRUCTR}\_application$>$ & $::=$ & 
 {\tt GSTRUCTR} $<$stat\_group$>$ [{\tt NAME} $<$cse\_prefix$>$]\\ 
$<$stat\_group$>$ & $::=$ & $\ll~<$stat\_list$>~\gg$\\
$<$stat\_list$>$ & $::=$ & $<$gstat$>$ [; $<$stat\_list$>$]\\
$<$gstat$>$ & $::=$ & $<$name$>~:=~<$ expression$>~\mid~<$matrix\_id$>$\\
$<${\tt GHORHER}\_application$>$ & $::=$ &
 {\tt GHORNER} $<$stat\_group$>$ [{\tt VORDER} $<$id\_seq$>$]\\
$<$id\_seq$>$ & $::=$ & $<$id$>$[,$<$id\_seq$>$]\\
\end{tabular}
\index{SCOPE ! {\tt DECLARE} facility}
\index{{\tt DECLARE} option}
\index{{\tt IN} option}
\index{{\tt OUT} option}
\index{{\tt INAME} option}

\newpage
\index{GENTRAN function ! {\tt lsetq}}
\index{GENTRAN function ! {\tt lrsetq}}
\index{GENTRAN function ! {\tt rsetq}}
\index{SCOPE function ! {\tt ALGSTRUCTR}}
\index{SCOPE function ! {\tt ALGHORNER}}
\index{{\tt IMPLICIT} type}
\index{{\tt integer} type}
\index{{\tt real} type}
\index{{\tt real*8} type}
\index{{\tt complex} type}
\index{{\tt complex*16} type}
\begin{center}
\begin{tabular}{lcl}
 $<$object\_seq$>$ & $::=$ & $<$object$>$[,$<$object\_seq$>$]\\
 $<$object$>$ & $::=$ & $<$stat$>~\mid~<$alglist$>~\mid~<$alglist\_production$>$ \\
 $<$stat$>$ & $::=$ & $<$name$>~<$assignment operator$>~<$expression$>$\\
 $<$assignment operator$>$ & $::=$ & $:=~\mid~::=~\mid~::=:~\mid~:=:$\\
 $<$alglist$>$ & $::=$ & \{$<$eq\_seq$>$\}\\
 $<$eq\_seq$>$ & $::=$ & $<$name$>~=~<$expression$>$[,$<$eq\_seq$>$]\\
 $<$alglist\_production$>$ & $::=$ & $<$name$>~\mid~<$function\_application$>$\\
 $<$name$>$ & $::=$ & $<$id$>~\mid~<$id$>(<$a\_subscript\_seq$>)$\\
 $<$a\_subscript\_seq$>$ & $::=$ & $<$a\_subscript$>$[,$<$a\_subscript\_seq$>$]\\
 $<$a\_subscript$>$ & $::=$ & $<$integer$>~\mid~<$integer infix\_expression$>$\\
 $<$cse\_prefix$>$ & $::=$ & $<$id$>$\\
 & & $\ \ $\\
$<$a\_object\_list$>$ & $::=$ & $<$a\_object$>~\mid$ \{$<$a\_object$>$[,$<$a\_object\_seq$>$]\}\\
$<$a\_object\_seq$>$ & $::=$ & $<$a\_object$>$[,$<$a\_object\_seq$>$]\\
$<$a\_object$>$ & $::=$ & $<$id$>~\mid~<$alglist$>~\mid~<$alglist\_production$>~\mid$\\
& & \{$<$a\_object$>$\}\\
 & & $\ \ $\\
$<$function\_application$>$ & $::=$ & 
{\tt ALGSTRUCTR} ($<$arg\_list$>$ [, $<$cse\_prefix$>$ ]) $\mid$\\
 & & {\tt ALGHORNER} ($<$arg\_list$>$ [,\{$<$id\_seq$>$\}]) $\mid$ \\
 & & $\cdots$\\
$<$arg\_list$>$ & $::=$ & $<$arg\_list\_name$>~\mid~$\{$<$arg\_seq$>$\}\\
$<$arg\_seq$>$ & $::=$ & $<$arg$>$[,$<$arg\_seq$>$]\\
$<$arg$>$ & $::=$ & $<$matrix\_id$>~\mid~<$name$>$=$<$expression$>$\\
$<$arg\_list\_name$>$ & $::=$ & $<$id$>$\\
 & & $\ \ $\\
$<$file\_id\_seq$>$ & $::=$ & $<$file\_id$>$ [,$<$file\_id\_seq$>$]\\
$<$file\_id$>$ & $::=$ & $<$id$>$ $\mid$ $<$string\_id$>$\\
$<$string\_id\_list$>$ & $::=$ & $<$string\_id$>$ $\mid$ 
 \{$<$string\_id\_seq$>$\}\\
$<$string\_id\_seq$>$ & $::=$ & $<$string\_id$>$ [,$<$string\_id\_seq$>$]\\
$<$string\_id$>$ & $::=$ & {\tt "}$<$id$>${\tt "} $\mid$ 
{\tt "}$<$id$>.<$f\_extension$>${\tt "}\\
 & & $\ \ $\\
$<$declaration\_group$>$ & $::=$ & $<$declaration$>~\mid~\ll~<$declaration\_list$>~\gg$\\
$<$declaration\_list$>$ & $::=$ & $<$declaration$>$[$;<$declaration\_list$>$]\\
$<$declaration$>$ & $::=$ & $<$range\_list$>:$ {\tt IMPLICIT} $<$type$>~\mid$
 $<$id\_list$>:<$type$>$\\
$<$range\_list$>$& $::=$ & $<$range$>$[,$<$range\_list$>$]\\
$<$range$>$ & $::=$ & $<$id$>~\mid~<$id$>-<$id$>$\\
$<$id\_list$>$ & $::=$ & $<$id$>$[,$<$id\_list$>$]\\
$<$type$>$ & $::=$ & {\tt integer} $\mid$ {\tt real} $\mid$ {\tt complex} $\mid$ 
 {\tt real*8} $\mid$ {\tt complex*16}\\
 & & $\ \ $\\
$<$ssetq\_list$>$ & $::=$ & ($<$ssetq\_seq$>$)\\
$<$ssetq\_seq$>$ & $::=$ & $<$ssetq\_stat$>~[<$ssetq\_seq$>]$\\
$<$ssetq\_stat$>$ & $::=$ & ({\tt setq} $<$lhs\_id$>~<$rhs$>)~\mid$
({\tt rsetq} $<$lhs\_id$>~<$rhs$>)~\mid$\\
 & & ({\tt lsetq} $<$lhs\_id$>~<$rhs$>)~\mid$
({\tt lrsetq} $<$lhs\_id$>~<$rhs$>)$\\
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{lcl}
$<$lhs\_id$>$ & $::=$ & $<$id$>~\mid~<$subscripted\_id$>$\\
$<$subscripted\_id$>$ & $::=$ & $(<$id$>~<$s\_subscript\_seq$>)$\\
$<$s\_subscript\_seq$>$ & $::=$ & $<$s\_subscript$>[~<$s\_subscript\_seq$>$]\\
$<$s\_subscript$>$ & $::=$ & $<$integer$>~\mid~<$integer prefix\_expression$>$\\
$<$rhs$>$ & $::=$ & $<$prefix\_expression$>$\\
$<$infile\_list$>$ & $::=$ & $(<$infile\_seq$>)$\\
$<$infile\_seq$>$ & $::=$ & $<$infile\_name$>[~<$infile\_seq$>]$\\
$<$infile\_name$>$ & $::=$ & $<$string\_id$>$\\
$<$outfile\_name$>$ & $::=$ & $<$string\_id$>$\\
$<$declaration\_list$>$ & $::=$ & $(<$declaration\_seq$>)$\\
$<$declaration\_seq$>$ & $::=$ & $<$declaration$>~<$declaration\_seq$>$\\
$<$declaration$>$ & $::=$ & $(<$type$>~<$lhs\_id\_seq$>)$\\
$<$lhs\_id\_seq$>$ & $::=$ & $<$lhs\_id$>~<$lhs\_id\_seq$>$
\end{tabular}
\end{center}

\subsection{Additional SCOPE-functions}~\label{SCOPE:arules}

Fifteen additional functions can be used. We shortly summarize their name and use: 

\begin{center}
\begin{tabular}{| l | l | l |} \hline 
Name(s) & Introduced in & See the examples:\\ \hline \hline
{\tt SCOPE\_SWITCHES} & ~\ref{SCOPE:basic} & ~\ref{ex:3.1.1}\\
{\tt SETLENGTH, RESETLENGTH} & ~\ref{SCOPE:optim} & ~\ref{ex:3.1.2}, 
 ~\ref{ex:3.1.5}, ~\ref{ex:3.2.6} and ~\ref{ex:3.2.7}\\
{\tt ARESULTS, RESTORABLES, ARESTORE, RESTOREALL} & ~\ref{SCOPE:optim} & 
 ~\ref{ex:3.1.9}, ~\ref{ex:3.2.6} and ~\ref{ex:4.1.2}\\
{\tt OPTLANG} & ~\ref{SCOPE:decl} & ~\ref{ex:6.1}\\
{\tt VECTORCODE, VCLEAR} & ~\ref{SCOPE:dda} & ~\ref{ex:7.1}\\
{\tt DELAYDECS, MAKEDECS, DELAYOPTS, MAKEOPTS} & ~\ref{SCOPE:gopt} &
 ~\ref{ex:8.1} and ~\ref{ex:8.2}\\
{\tt INAME} & ~\ref{SCOPE:gopt} & ~\ref{ex:8.2}\\ \hline
\end{tabular}
\end{center}
\index{SCOPE function ! {\tt SCOPE\_SWITCHES}}
\index{SCOPE function ! {\tt SETLENGTH}}
\index{SCOPE function ! {\tt RESETLENGTH}}
\index{SCOPE function ! {\tt ARESULTS}}
\index{SCOPE function ! {\tt RESTORABLES}}
\index{SCOPE function ! {\tt ARESTORE}}
\index{SCOPE function ! {\tt RESTOREALL}}
\index{SCOPE function ! {\tt OPTLANG}}
\index{SCOPE function ! {\tt VECTORCODE}}
\index{SCOPE function ! {\tt VCLEAR}}
\index{SCOPE function ! {\tt DELAYDECS}}
\index{SCOPE function ! {\tt MAKEDECS}}
\index{SCOPE function ! {\tt DELAYOPTS}}
\index{SCOPE function ! {\tt MAKEOPTS}}
\index{SCOPE function ! {\tt INAME}}

\subsection{The relevant REDUCE, GENTRAN and SCOPE switches}~\label{SCOPE:switches}

We also shortly summarize the use of the switches, which were introduced 
in section ~\ref{SCOPE:basic} in example~\ref{ex:3.1.1}:

\begin{center}
\begin{tabular}{| l | l | l |}\hline 
Name & Origin & Illustrated in the examples:\\ \hline \hline
{\tt ACINFO} & SCOPE & ~\ref{ex:3.1.8}, ~\ref{ex:3.2.5}, ~\ref{ex:8.2}
 and ~\ref{ex:9.1}\\
{\tt AGAIN} & SCOPE & ~\ref{ex:5.1} and ~\ref{ex:9.1}\\
{\tt DOUBLE} & GENTRAN & ~\ref{ex:9.1}\\
{\tt EVALLHSEQP} & REDUCE & ~\ref{ex:3.2.5}\\
{\tt EXP} & REDUCE & ~\ref{ex:3.1.8}, ~\ref{ex:3.2.7}, ~\ref{ex:4.1.3} and 
 ~\ref{ex:8.2}\\
{\tt FORT} & REDUCE & ~\ref{ex:3.1.4}, ~\ref{ex:3.1.6}, ~\ref{ex:3.1.8},
 ~\ref{ex:3.2.5}, ~\ref{ex:7.1} and ~\ref{ex:9.1}\\
{\tt FTCH} & SCOPE & ~\ref{ex:3.1.4} and ~\ref{ex:3.1.5}\\ \hline
\end{tabular}
\end{center}
\index{{\tt ACINFO} switch}
\index{{\tt AGAIN} switch}
\index{{\tt DOUBLE} switch}
\index{{\tt EVALLHSEQP} switch}
\index{{\tt EXP} switch}
\index{{\tt FORT} switch}
\index{{\tt FTCH} switch}
\newpage
\index{{\tt GENTRANOPT} switch}
\index{{\tt INPUTC} switch}
\index{{\tt INTERN} switch}
\index{{\tt NAT} switch}
\index{{\tt PERIOD} switch}
\index{{\tt PREFIX} switch}
\index{{\tt PRIALL} switch}
\index{{\tt PRIMAT} switch}
\index{{\tt ROUNDBF} switch}
\index{{\tt ROUNDED} switch}
\index{{\tt SIDREL} switch}
\index{{\tt VECTORC} switch}
\begin{center}
\begin{tabular}{| l | l | l |}\hline 
Name & Origin & Illustrated in the examples:\\ \hline \hline
{\tt GENTRANOPT} & GENTRAN & ~\ref{ex:8.1}\\
{\tt INPUTC} & SCOPE & ~\ref{ex:3.1.3}, ~\ref{ex:3.1.6}, ~\ref{ex:3.1.7},
 ~\ref{ex:3.2.7}, ~\ref{ex:5.1} and ~\ref{ex:9.1}\\
{\tt INTERN} & SCOPE & ~\ref{ex:9.1}\\
{\tt NAT} & REDUCE & ~\ref{ex:3.1.8} and ~\ref{ex:5.1}\\
{\tt PERIOD} & REDUCE & ~\ref{ex:3.1.4}\\
{\tt PREFIX} & SCOPE & ~\ref{ex:9.2}\\
{\tt PRIALL} & SCOPE & ~\ref{ex:3.1.2}\\
{\tt PRIMAT} & SCOPE & ~\ref{ex:3.2.7}\\
{\tt POUNDBF} & REDUCE & ~\ref{ex:6.2}\\
{\tt ROUNDED} & REDUCE & ~\ref{ex:3.1.7} and ~\ref{ex:8.2}\\
{\tt SIDREL} & SCOPE & ~\ref{ex:3.2.7}\\
{\tt VECTORC} & SCOPE & ~\ref{ex:7.1}\\ \hline
\end{tabular}
\end{center}
\newpage
\section{SCOPE 1.5 Installation Guide}~\label{SCOPE:inst}

SCOPE 1.5 is easily installed. The usual code compilation facilities of {\REDUCE}
can be applied. In view of the frequent interaction between SCOPE and GENTRAN
a compiled version of GENTRAN is required during the creation of the load module
for SCOPE 1.5.  The compilation process itself is vizualized below.

{\small
\begin{verbatim}
faslout "~infhvh/mkscope/scope_15";

lisp in "~infhvh/mkscope/cosmac.red"$
lisp in "~infhvh/mkscope/codctl.red"$
lisp in "~infhvh/mkscope/codmat.red"$
lisp in "~infhvh/mkscope/codopt.red"$
lisp in "~infhvh/mkscope/codad1.red"$
lisp in "~infhvh/mkscope/codad2.red"$
lisp in "~infhvh/mkscope/coddec.red"$
lisp in "~infhvh/mkscope/codpri.red"$
lisp in "~infhvh/mkscope/codgen.red"$
lisp in "~infhvh/mkscope/codhrn.red"$
lisp in "~infhvh/mkscope/codstr.red"$
lisp in "~infhvh/mkscope/coddom.red"$
%lisp in "~infhvh/mkscope/apatch.red"$
algebraic;

faslend ;
end;
\end{verbatim}}

The subdirectory {\tt mkscope} in the author's directory system 
contains the files with the source code of SCOPE 1.5. 
The order in which the files are read in is irrelevant except
the first and the last. The file {\tt cosmac.red} contains
one module, named {\tt cosmac}, which consists of a set of
smacro procedures, designed to simplify access to the lower
levels of the expression data base, employed during optimization.
These smacro's are used in all other code sections.
The last file in optional and usually executed to include new
patches into a recompilable version of the package.
Once it is stored in the {\tt fasl} directory of the local
REDUCE system it is available as a {\tt load\_package}.

In short, the files contain the following code sections: 

\begin{itemize}
\item {\tt cosmac.red} contains the module {\tt cosmac}, consisting of
smacro procedures, allowing access to the expression data base.
\item {\tt codctl.red} consists of the three modules {\tt codctl},
{\tt restore} and {\tt minlenght}. The first is a large module,
containing the optimization process managing facilities. The second
module is added to regulate the interplay with the REDUCE simplifier,
when entering optimizer output in algebraic mode using functions like
{\tt ARESULTS}. The last module serves to vary the minimal length of
cse's using {\tt SETLENGTH} and {\tt RESETLENGTH}.
\item {\tt codmat.red} contains the module {\tt codmat}, definig the
parsing process and the expression data base setup and access facilities.
\item {\tt codopt.red}'s content is formed by the module {\tt codopt}.
It is the kernel of the optimization process, the implementation of the
extended Breuer algorithm.
\item {\tt codad1.red} contains the module {\tt codad1}, formed by additional
facilities for improving the lay-out of the overall result, for information
migration between different sections of the expression data base  and for the
application of the distributive law to remodel and compactify (sub)expression 
structure  at any level.
\item {\tt codad2.red} contains the module {\tt codad2}. This module defines
five different possible activities during the optimization process. The first
four regulate the so called {\em finishing touch}. The last one is a new
section, defining how to optimize {\em rational forms} as part of the overall
optimization process.
\item {\tt coddec.red} covers the declaration facilities, presented in 
section ~\ref{SCOPE:decl}, collected in the module {\tt coddec} and 
based on chapter 6 of ~\cite{Aho:86}. The symbol table setup of GENTRAN is used.
\item {\tt codpri.red} is also formed by one module, called {\tt codpri}.
It covers all printing facilities. The first section is applied when the
switch {\tt PRIMAT} is turned {\tt ON}. The latter is used to produce
an internal list of pairs, consisting of the left hand side and the right 
hand side of assignment statements in prefix notation, and  defining the 
output of the optimization process in sequential order. This prefix list is 
delivered to GENTRAN or REDUCE to make the results visible in the user 
prefered form. The intial version of this list, created directly 
after the optimization process, is improved using a collection of functions, 
also grouped together in the module {\tt codpri}. These improvements may 
for instance be necessary to remove temporarily introduced names, internally 
employed as a result of a data dependency analysis.
\item {\tt codgen.red} consists of the module {\tt codgen}. The interface
between GENTRAN and SCOPE 1.5, introduced in section ~\ref{SCOPE:gopt} of 
this manual is defined in this module.
\item {\tt codhrn.red} is formed by the module {\tt ghorner}. It defines the
facilities, presented in section ~\ref{SSF:Hr} of this manual.
\item {\tt codstr.red} contains the module {\tt gstructr}. This module defines
the possibilities for expression structure recognition, as presented in section
~\ref{SSF:sr} of this manual.
\item {\tt coddom.red} finally, consists of one module, called {\tt coddom}.
It covers additional coefficient domain functions, needed to make the
extended Breuere algorithm and the additional functions, collected in the
modules {\tt codad1} and {\tt codad2} for instance, applicable for 
(multiple presicion) floating point coefficients.
\end{itemize}


A few additional remarks:

\begin{itemize}
\item GENTRAN plays an important role when creating declarations and output.
The package is automatically loaded when executing one of the first 
instructions in the module {\tt codctl}. Hence it may be necessary to 
look critically to the load instruction in {\tt codctl} before installing 
SCOPE 1.5. By changing this load instruction we easily created a {\tt fortran90} 
compatable SCOPE version ~\cite{Borst:94}. At present it is only available for our own 
internal and experimental use.  
\item We believe the code to be almost version independent. Over the past 
years all uses of {\tt nil} have been critically reviewed. However 
the {\tt coddom} module may require version based maintenance when 
installing SCOPE 1.5.
\item The present size of the source code is given in the table below.
Comment is included in these figures
\begin{center}
\begin{tabular}{| c | r | r |} \hline
File naam & number of lines & number of characters \\ \hline \hline
{\tt cosmac.red} & 172 & 4761 \\
{\tt codctl.red} & 1439 & 61466 \\
{\tt codmat.red} & 1488 & 72733 \\
{\tt codopt.red} & 1243 & 68809\\
{\tt codad1.red} & 801 & 39175 \\
{\tt codad2.red} & 1314 & 59217 \\
{\tt coddec.red} & 928 & 41069\\
{\tt codpri.red} & 1371 & 62600\\
{\tt codgen.red} & 214 & 9120\\
{\tt codhrn.red} & 752 & 30549\\
{\tt codstr.red} & 308 & 11199\\
{\tt coddom.red} & 204 & 6638\\ \hline
\end{tabular}
\end{center}
\item The modules {\tt ghorner} and {\tt gstructr} can be left out without
harming the other facilities, presented in this manual.
\end{itemize}
\newpage
\addcontentsline{toc}{section}{References}
\bibliography{scope}
\bibliographystyle{plain}
\newpage
\addcontentsline{toc}{section}{Index}
\printindex
\end{document}