Advanced Compiler Techniques Partial Redundancy Elimination LIU Xianhua School of EECS, Peking University

“Advanced Compiler Techniques” REVIEW Foundations Data Flow Framework Lattice-Theoretic Formulation Meet-Over-Paths Solution Extensions Other DFA Methods “Advanced Compiler Techniques”

REVIEW Available Expressions Analysis Live Variables Analysis ≤ is ⊆, ∧ is ∩ ≤ is ⊇, ∧ is ∪ “Advanced Compiler Techniques”

“Advanced Compiler Techniques” REVIEW ∀i , Ini = Outi = ⊤ All possible assignments Meet Over Paths Assignment All safe assignments Maximum Fixed Point Least Fixed Point ∀i , Ini = Outi = ⊥ All fixed point solutions “Advanced Compiler Techniques”

“Advanced Compiler Techniques” REVIEW f(x) f(y) MOP considers paths independently and and combines at the last possible moment. OUT = f(x) ∧ f(y) f OUT = x OUT = y IN = x∧y OUT = f(x∧y) In MFP, Values x and y get combined too soon. B Entry Since f(x ∧ y) ≤ f(x) ∧ f(y), it is as if we added nonexistent paths, but we’re safe. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Summary of DFA Methods Method Speed Simple? Structure Both Way? Graph Class Iterative O(n2) Simple No Yes All Interval Middle Reducible Balance Tree O(nlogn) Complicated Path Comp. Semi Node List Balance Path O(nα(n,n)) ? Grammar n L(Grammar) High Level Parse Trees “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Outline Forms of redundancy global common sub-expression loop invariant partial redundancy expression Lazy Code Motion Algorithm A set of four analysis “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Role of PRE Goal: Minimize the number of expression evaluations. Keep the value of evaluation in a temporary variable and use it later. Sources of redundancy: Global common sub-expressions Loop-invariant computations True partial redundancy: an expression is sometimes available, sometimes not “Advanced Compiler Techniques”

Partial redundancy elimination One of the most complex dataflow analysis Subsumes common sub-expression elimination and loop invariant code motion Originally proposed in 1979 by Morel and Renvoise, Used a bi-directional dataflow analysis Reformulated by Knoop, Rüthing and Steffen in 1992, Uses a backward dataflow analysis followed by a forward analysis We will discuss this latter formulation “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Convention Throughout, assume that neither argument of an expression x+y is modified unless we explicitly assign to x or y. And of course, we assume x+y is the only expression anyone would ever want to compute.  Can easily extend this to multiple expressions by using a bit vector lattice. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Example: Global CSE A common expression may have different values on different paths. On every path reaching p, expression x+y has been computed x, y not overwritten after the expression a = x+y b = x+y c = x+y t = x+y a = t b = t c = t a=x+y a=x+y x=7 a=x+y x=7 f=x+y d=x+y d=x+y d=x+y “Advanced Compiler Techniques”

Example: Loop-Invariant Code Motion Given an expression (x+y) inside a loop, does the value of x+y change inside the loop? is the code executed at least once? t = x+y a = x+y a = t “Advanced Compiler Techniques”

Example: True Partial Redundancy Can we place calculations of x+y such that no path re-executes the same expression? a = x+y b = x+y t = x+y a = t b = t “Advanced Compiler Techniques”

Modifying the Flow Graph We could: Add a new block along an edge. Only necessary if the edge enters a block with several predecessors. Duplicate blocks so an expression x+y is evaluated only along paths where it is needed. “Advanced Compiler Techniques”

Example: Node Splitting = x+y t = x+y t = x+y = t “Advanced Compiler Techniques”

Can All Redundancy Be Eliminated? Critical edges source basic block has multiple successors destination basic block has multiple predecessors “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Code Duplication “Advanced Compiler Techniques”

Problem With Node-Splitting Can exponentiate the number of nodes. Our PRE algorithm needs to move code to new blocks along edges, but will not split blocks. Convention: All new instructions are either inserted at the beginning of a block or placed in a new block. “Advanced Compiler Techniques”

Lazy Code Motion Problem Desired properties of a PRE algorithm All redundant computations of expressions that can be eliminated without code duplication are eliminated. The optimized program does not perform any computation that is not in the original program execution. Expressions are computed at the latest possible time. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Full Redundancy Full redundancy at p: expression a+b redundant on all paths cutset: nodes that separate entry from p cutset contains calculation of a+b a, b, not redefined “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Partial Redundancy Partial redundancy at p: redundant on some but not all paths Add operations to create a cutset containing a+b Note: Moving operations up can eliminate redundancy “Advanced Compiler Techniques”

“Advanced Compiler Techniques” The Plan Determine for each expression the earliest place(s) it can be computed while still being sure that it will be used. Postpone the expressions as long as possible without introducing redundancy. We trade space for time --- an expression can be computed in many places, but never if it is already computed. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” The Guarantee No expression is computed at a place where its value might have been computed previously, and preserved instead. Even along a subset of the possible paths. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” The Plan – (2) 3. Determine those places where it is really necessary to store x+y in a temporary rather than compute it when needed. Example: If x+y is computed in only one place. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” More about the plan Don’t introduce/insert new operations didn’t exist originally: Anticipate the range of code motion Eliminate as many redundant calculations of an expression as possible, without duplicating code Move it up as early as possible Delay computation as much as possible to minimize register Lifetimes move it down unless it creates redundancy (lazy code motion) Remove temporary assignment “Advanced Compiler Techniques”

“Advanced Compiler Techniques” More About the Plan We use four data-flow analysis, in succession, plus some set operations on the results of these analysis. Anticipated Expressions Available Expressions Postponable Expressions Used Expressions After the first, each analysis uses the results of the previous ones. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Assumptions Assume every statement is a basic block Only place statements at the beginning of a basic block Add a basic block for every edge that leads to a basic block with multiple predecessors “Advanced Compiler Techniques”

Anticipated Expressions Expression x+y is anticipated at a point if x+y is certain to be evaluated along any computation path, before any recomputation of x or y. Copies of an expression must be placed only at program points where the expression is anticipated. The earlier an expression is placed, the more redundancy can be removed “Advanced Compiler Techniques”

Example: Anticipated Expressions = x+y x+y is anticipated here and could be computed now rather than later. = x+y x+y is anticipated here, but is also available. No computa- tion is needed. = x+y = x+y “Advanced Compiler Techniques”

Computing Anticipated Expressions Use(B) = set of expressions x+y evaluated in B before any assignment to x or y. Def(B) = set of expressions one of whose arguments is assigned in B. “Advanced Compiler Techniques”

Computing Anticipated Expressions Direction = backwards. Join (or Meet) = intersection. Boundary condition: IN[exit] = ∅. Transfer function: IN[B] = (OUT[B] – Def(B)) ∪ Use(B) “Advanced Compiler Techniques”

Example: Anticipated Expressions = x+y = x+y Backwards; Intersection; IN[B] = (OUT[B] – Def(B)) ∪ Use(B) “Advanced Compiler Techniques”

“Available” Expressions Modification of the usual AE. x+y is “available” at a point if either: It is available in the usual sense; i.e., it has been computed and not killed, or It is anticipated; i.e., it could be available if we chose to precompute it there. “Advanced Compiler Techniques”

“Available” Expressions x+y is in Kill(B) if x or y is defined, and x+y is not recomputed later in B (same as previously). Direction = Forward Meet = intersection. Transfer function: OUT[B] = (IN[B] ∪ INANTICIPATED[B]) – Kill(B) “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Earliest Placement x+y is in Earliest[B] if it is anticipated at the beginning of B but not “available” there. That is: when we compute anticipated expressions, x+y is in IN[B], but When we compute “available” expressions, x+y is not in IN[B]. I.e., x+y is anticipated at B, but not anticipated at OUT of some predecessor. “Advanced Compiler Techniques”

Example: Available/Earliest Earliest = anticipated but not available Anticipated “Available” = x+y = x+y Forward; Intersection; OUT[B] = (IN[B] ∪ INANTICIPATED[B]) – Kill(B) “Advanced Compiler Techniques”

Postponable Expressions Now, we need to delay the evaluation of expressions as long as possible, but … Not past the use of the expression. Not so far that we wind up computing an expression that is already evaluated. Note viewpoint: It is OK to use code space if we save register use. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Example t = b+c a = t d = t “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Example t = b+c a = t t = b+c d = t “Advanced Compiler Techniques”

Postponable Expressions – (2) x+y is postponable to a point p if on every path from the entry to p: There is a block B for which x+y is in earliest[B], and After that block, there is no use of x+y. “Advanced Compiler Techniques”

Postponable Expressions – (3) Computed like “available” expressions, with two differences: In place of killing an expression (assigning to one of its arguments): Use(B), the set of expressions used in block B. In place of INANTICIPATED[B]: earliest[B]. “Advanced Compiler Techniques”

Postponable Expressions – (4) Direction = forward. Meet = intersection. Transfer function: OUT[B] = (IN[B] ∪ earliest[B]) – Use(B) “Advanced Compiler Techniques”

Example: Postponable Expressions Earliest Postponable Three places to compute x+y = x+y = x+y = x+y Forward; Intersection; OUT[B] = (IN[B] ∪ earliest[B]) – Use(B) “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Latest Placement We want to postpone as far as possible. How do we compute the “winners” – the blocks such that we can postpone no further? Remember – postponing stops at a use or at a block with another predecessor where x+y is not postponable. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Latest[B] For x+y to be in latest[B]: x+y is either in earliest[B] or in INPOSTPONABLE[B]. I.e., we can place the computation at B. x+y is either used in B or there is some successor of B for which (1) does not hold. I.e., we cannot postpone further along all branches. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Example: Latest Earliest Or Postponable to beginning Latest = Blue and red. = x+y = x+y Used Or has a suc- cessor not red. = x+y “Advanced Compiler Techniques”

Final Touch – Used Expressions We’re now ready to introduce a temporary t to hold the value of expression x+y everywhere. But there is a small glitch: t may be totally unnecessary. E.g., x+y is computed in exactly one place. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Used Expressions An expression is used at point p if There exists a path leading from p that uses the expression before the value is reevaluated. Essentially liveness analysis for expressions rather than for variables. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Example: Used Recall: Latest = x+y = x+y Used = x+y Backwards; Union; IN[B] = (OUT[B] ∪ e-used[B]) – Latest(B) “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Used Expressions – (2) used[B] = expressions used along some path from the end of B. Direction = backward. Meet = union. Transfer function: IN[B] = (OUT[B] ∪ e-used[B]) – Latest(B) e-used = “expression is used in B.” “Advanced Compiler Techniques”

Rules for Introducing Temporaries If x+y is in both Latest[B] and OUTUSED[B], introduce t = x+y at the beginning of B. If x+y is used in B, but either Is not in Latest[B] or Is in OUTUSED[B], replace the use(s) of x+y by uses of t. “Advanced Compiler Techniques”

Example: Where is a Temporary Used? Recall: Latest Create temp- orary here = x+y = x+y Recall OUTUSED But not here --- x+y is in Latest and not in OUTUSED Use it here = x+y “Advanced Compiler Techniques”

Example: Here’s Where t is Used = x+y t = x+y t = x+y = t = t “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Summary Cannot execute any operations not executed originally Pass 1: Anticipation: range of code motion Eliminate as many redundant calculations of an expression as possible, without duplicating code Pass 2: Availability: move it up as early as possible Delay computation as much as possible to minimize register lifetimes Pass 3: Postponable: move it down unless it creates redundancy (lazy code motion) Pass 4: Remove temporary assignment “Advanced Compiler Techniques”

Data Flow Analysis of PRE “Advanced Compiler Techniques”

Data Flow Analysis of PRE “Advanced Compiler Techniques”

Algorithm of Lazy Code Motion INPUT: A flow graph for which e_useB and e_killB have been computed for each block B. OUTPUT: A modified flow graph satisfying the four lazy code motion conditions. METHOD: Insert an empty block along all edges entering a block with more than one predecessor Find anticipated[B] .in for all blocks B Find available[B] .in for all blocks B Compute the earliest placements for all blocks B Find postponable[B].in for all blocks B Compute the latest placements for all blocks B Find used[B] .out for all blocks B “Advanced Compiler Techniques”

Algorithm of Lazy Code Motion METHOD: For each expression, say x+y, computed by the program, do the following: Create a new temporary, say t, for x + y. For all blocks B such that x + y is in latest[B]∩ used[B].out, add t = x +y at the beginning of B. For all blocks B such that x + y is in e_use B ∩ (┐latest[B] ∪ used.out[B] ) replace every original x + y by t. “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Example “Advanced Compiler Techniques”

“Advanced Compiler Techniques” More about PRE Don’t need heuristic Dhamdhere, Drechsler-Stadel, Knoop,et.al. use restricted flow graph or allow edge placements. Data flow can be separated into unidirectional passes Dhamdhere, Knoop, et. al. Improvement still tied to accuracy of computational model Assumes performance depends only on the number of computations along any path. Ignores resource constraint issues: register alloc., etc. Knoop, et.al. give “earliest” and “latest” placement algorithms which begin to address this. Further issues: more than one expression at once, strength reduction, redundant assignments, redundant stores With GVN,SSA… “Advanced Compiler Techniques”

“Advanced Compiler Techniques” Next Time Homework 9.5.1， 9.5.2 Loops Dragon Book: §9.6 “Advanced Compiler Techniques”