217 lines
6.4 KiB
Text
217 lines
6.4 KiB
Text
.NH 2
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The model of strength reduction
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.PP
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In this section we will describe
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the transformations performed by
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Strength Reduction (SR).
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Before doing so, we will introduce the
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central notion of an induction variable.
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.NH 3
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Induction variables
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.PP
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SR looks for variables whose
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values form an arithmetic progression
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at the beginning of a loop.
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These variables are called induction variables.
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The most frequently occurring example of such
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a variable is a loop-variable in a high-order
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programming language.
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Several quite sophisticated models of strength
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reduction can be found in the literature.
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.[
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cocke reduction strength cacm
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.]
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.[
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allen cocke kennedy reduction strength
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.]
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.[
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lowry medlock cacm
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.]
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.[
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aho compiler design
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.]
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In these models the notion of an induction variable
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is far more general than the intuitive notion
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of a loop-variable.
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The definition of an induction variable we present here
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is more restricted,
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yielding a simpler model and simpler transformations.
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We think the principle source for strength reduction lies in
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expressions using a loop-variable,
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i.e. a variable that is incremented or decremented
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by the same amount after every loop iteration,
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and that cannot be changed in any other way.
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.PP
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Of course, the EM code does not contain high level constructs
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such as for-statements.
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We will define an induction variable in terms
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of the Intermediate Code of the optimizer.
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Note that the notions of a loop in the
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EM text and of a firm basic block
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were defined in section 3.3.5.
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.sp
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.UL definition
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.sp 0
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An induction variable i of a loop L is a local variable
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that is never accessed indirectly,
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whose size is the word size of the target machine, and
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that is assigned exactly once within L,
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the assignment:
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.IP -
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being of the form i := i + c or i := c +i,
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c is a constant
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called the \fIstep value\fR of i.
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.IP -
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occurring in a firm block of L.
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.LP
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(Note that the first restriction on the assignment
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is not described in terms of the Intermediate Code;
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we will give such a description later; the current
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definition is easier to understand however).
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.NH 3
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Recognized expressions
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.PP
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SR recognizes certain expressions using
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an induction variable and replaces
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them by cheaper ones.
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Two kinds of expensive operations are recognized:
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multiplication and array address computations.
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The expressions that are simplified must
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use an induction variable
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as an operand of
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a multiplication or as index in an array expression.
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.PP
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Often a linear function of an induction variable is used,
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rather than the variable itself.
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In these cases optimization is still possible.
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We call such expressions \fIiv-expressions\fR.
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.sp
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.UL definition:
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.sp 0
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An iv-expression of an induction variable i of a loop L is
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an expression that:
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.IP -
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uses only the operators + and - (unary as well as binary)
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.IP -
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uses i as operand exactly once
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.IP -
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uses (besides i) only constants or variables that are
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never changed in L as operands.
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.LP
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.PP
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The expressions recognized by SR are of the following forms:
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.IP (1)
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iv_expression * constant
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.IP (2)
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constant * iv_expression
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.IP (3)
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A[iv-expression] := (assign to array element)
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.IP (4)
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A[iv-expression] (use array element)
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.IP (5)
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& A[iv-expression] (take address of array element)
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.LP
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(Note that EM has different instructions to use an array element,
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store into one, or take the address of one, resp. LAR, SAR, and AAR).
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.sp 0
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The size of the elements of A must
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be known statically.
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In cases (3) and (4) this size
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must equal the word size of the
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target machine.
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.NH 3
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Transformations
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.PP
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With every recognized expression we associate
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a new temporary local variable TMP,
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allocated in the stack frame of the
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procedure containing the expression.
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At any program point within the loop, TMP will
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contain the following value:
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.IP multiplication: 18
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the current value of iv-expression * constant
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.IP arrays:
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the current value of &A[iv-expression].
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.LP
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In the second case, TMP essentially is a pointer variable,
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pointing to the element of A that is currently in use.
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.sp 0
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If the same expression occurs several times in the loop,
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the same temporary local is used each time.
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.PP
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Three transformations are applied to the EM text:
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.IP (1)
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TMP is initialized with the right value.
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This initialization takes place just
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before the loop.
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.IP (2)
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The recognized expression is simplified.
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.IP (3)
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TMP is incremented; this takes place just
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after the induction variable is incremented.
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.LP
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For multiplication, the initial value of TMP
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is the value of the recognized expression at
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the program point immediately before the loop.
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For arrays, TMP is initialized with the address
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of the first array element that is accessed.
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So the initialization code is:
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.DS
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TMP := iv-expression * constant; or
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TMP := &A[iv-expression]
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.DE
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At the point immediately before the loop,
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the induction variable will already have been
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initialized,
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so the value used in the code above will be the
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value it has during the first iteration.
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.PP
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For multiplication, the recognized expression can simply be
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replaced by TMP.
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For array optimizations, the replacement
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depends on the form:
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.DS
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\fIform\fR \fIreplacement\fR
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(3) A[iv-expr] := *TMP := (assign indirect)
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(4) A[iv-expr] *TMP (use indirect)
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(5) &A[iv-expr] TMP
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.DE
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The '*' denotes the indirect operator. (Note that
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EM has different instructions to do
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an assign-indirect and a use-indirect).
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As the size of the array elements is restricted
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to be the word size in case (3) and (4),
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only one EM instruction needs to
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be generated in all cases.
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.PP
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The amount by which TMP is incremented is:
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.IP multiplication: 18
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step value * constant
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.IP arrays:
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step value * element size
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.LP
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Note that the step value (see definition of induction variable above),
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the constant, and the element size (see previous section) can all
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be determined statically.
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If the sign of the induction variable in the
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iv-expression is negative, the amount
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must be negated.
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.PP
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The transformations are demonstrated by an example.
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.DS
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i := 100; i := 100;
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while i > 1 loop TMP := (6-i) * 5;
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X := (6-i) * 5 + 2; while i > 1 loop
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Y := (6-i) * 5 - 8; --> X := TMP + 2;
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i := i - 3; Y := TMP - 8;
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end loop; i := i - 3;
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TMP := TMP + 15;
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end loop;
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Fig. 6.2 Example of complex Strength Reduction transformations
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.DE
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The expression '(6-i)*5' is recognized twice. The constant
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is 5.
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The step value is -3.
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The sign of i in the recognized expression is '-'.
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So the increment value of TMP is -(-3*5) = +15.
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