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1 //===-- DependenceAnalysis.cpp - DA Implementation --------------*- C++ -*-===//
2 //
3 //                     The LLVM Compiler Infrastructure
4 //
5 // This file is distributed under the University of Illinois Open Source
6 // License. See LICENSE.TXT for details.
7 //
8 //===----------------------------------------------------------------------===//
9 //
10 // DependenceAnalysis is an LLVM pass that analyses dependences between memory
11 // accesses. Currently, it is an (incomplete) implementation of the approach
12 // described in
13 //
14 //            Practical Dependence Testing
15 //            Goff, Kennedy, Tseng
16 //            PLDI 1991
17 //
18 // There's a single entry point that analyzes the dependence between a pair
19 // of memory references in a function, returning either NULL, for no dependence,
20 // or a more-or-less detailed description of the dependence between them.
21 //
22 // Currently, the implementation cannot propagate constraints between
23 // coupled RDIV subscripts and lacks a multi-subscript MIV test.
24 // Both of these are conservative weaknesses;
25 // that is, not a source of correctness problems.
26 //
27 // The implementation depends on the GEP instruction to differentiate
28 // subscripts. Since Clang linearizes some array subscripts, the dependence
29 // analysis is using SCEV->delinearize to recover the representation of multiple
30 // subscripts, and thus avoid the more expensive and less precise MIV tests. The
31 // delinearization is controlled by the flag -da-delinearize.
32 //
33 // We should pay some careful attention to the possibility of integer overflow
34 // in the implementation of the various tests. This could happen with Add,
35 // Subtract, or Multiply, with both APInt's and SCEV's.
36 //
37 // Some non-linear subscript pairs can be handled by the GCD test
38 // (and perhaps other tests).
39 // Should explore how often these things occur.
40 //
41 // Finally, it seems like certain test cases expose weaknesses in the SCEV
42 // simplification, especially in the handling of sign and zero extensions.
43 // It could be useful to spend time exploring these.
44 //
45 // Please note that this is work in progress and the interface is subject to
46 // change.
47 //
48 //===----------------------------------------------------------------------===//
49 //                                                                            //
50 //                   In memory of Ken Kennedy, 1945 - 2007                    //
51 //                                                                            //
52 //===----------------------------------------------------------------------===//
53 
54 #include "llvm/Analysis/DependenceAnalysis.h"
55 #include "llvm/ADT/STLExtras.h"
56 #include "llvm/ADT/Statistic.h"
57 #include "llvm/Analysis/AliasAnalysis.h"
58 #include "llvm/Analysis/LoopInfo.h"
59 #include "llvm/Analysis/ScalarEvolution.h"
60 #include "llvm/Analysis/ScalarEvolutionExpressions.h"
61 #include "llvm/Analysis/ValueTracking.h"
62 #include "llvm/IR/InstIterator.h"
63 #include "llvm/IR/Module.h"
64 #include "llvm/IR/Operator.h"
65 #include "llvm/Support/CommandLine.h"
66 #include "llvm/Support/Debug.h"
67 #include "llvm/Support/ErrorHandling.h"
68 #include "llvm/Support/raw_ostream.h"
69 
70 using namespace llvm;
71 
72 #define DEBUG_TYPE "da"
73 
74 //===----------------------------------------------------------------------===//
75 // statistics
76 
77 STATISTIC(TotalArrayPairs, "Array pairs tested");
78 STATISTIC(SeparableSubscriptPairs, "Separable subscript pairs");
79 STATISTIC(CoupledSubscriptPairs, "Coupled subscript pairs");
80 STATISTIC(NonlinearSubscriptPairs, "Nonlinear subscript pairs");
81 STATISTIC(ZIVapplications, "ZIV applications");
82 STATISTIC(ZIVindependence, "ZIV independence");
83 STATISTIC(StrongSIVapplications, "Strong SIV applications");
84 STATISTIC(StrongSIVsuccesses, "Strong SIV successes");
85 STATISTIC(StrongSIVindependence, "Strong SIV independence");
86 STATISTIC(WeakCrossingSIVapplications, "Weak-Crossing SIV applications");
87 STATISTIC(WeakCrossingSIVsuccesses, "Weak-Crossing SIV successes");
88 STATISTIC(WeakCrossingSIVindependence, "Weak-Crossing SIV independence");
89 STATISTIC(ExactSIVapplications, "Exact SIV applications");
90 STATISTIC(ExactSIVsuccesses, "Exact SIV successes");
91 STATISTIC(ExactSIVindependence, "Exact SIV independence");
92 STATISTIC(WeakZeroSIVapplications, "Weak-Zero SIV applications");
93 STATISTIC(WeakZeroSIVsuccesses, "Weak-Zero SIV successes");
94 STATISTIC(WeakZeroSIVindependence, "Weak-Zero SIV independence");
95 STATISTIC(ExactRDIVapplications, "Exact RDIV applications");
96 STATISTIC(ExactRDIVindependence, "Exact RDIV independence");
97 STATISTIC(SymbolicRDIVapplications, "Symbolic RDIV applications");
98 STATISTIC(SymbolicRDIVindependence, "Symbolic RDIV independence");
99 STATISTIC(DeltaApplications, "Delta applications");
100 STATISTIC(DeltaSuccesses, "Delta successes");
101 STATISTIC(DeltaIndependence, "Delta independence");
102 STATISTIC(DeltaPropagations, "Delta propagations");
103 STATISTIC(GCDapplications, "GCD applications");
104 STATISTIC(GCDsuccesses, "GCD successes");
105 STATISTIC(GCDindependence, "GCD independence");
106 STATISTIC(BanerjeeApplications, "Banerjee applications");
107 STATISTIC(BanerjeeIndependence, "Banerjee independence");
108 STATISTIC(BanerjeeSuccesses, "Banerjee successes");
109 
110 static cl::opt<bool>
111 Delinearize("da-delinearize", cl::init(false), cl::Hidden, cl::ZeroOrMore,
112             cl::desc("Try to delinearize array references."));
113 
114 //===----------------------------------------------------------------------===//
115 // basics
116 
117 DependenceAnalysis::Result
run(Function & F,FunctionAnalysisManager & FAM)118 DependenceAnalysis::run(Function &F, FunctionAnalysisManager &FAM) {
119   auto &AA = FAM.getResult<AAManager>(F);
120   auto &SE = FAM.getResult<ScalarEvolutionAnalysis>(F);
121   auto &LI = FAM.getResult<LoopAnalysis>(F);
122   return DependenceInfo(&F, &AA, &SE, &LI);
123 }
124 
125 char DependenceAnalysis::PassID;
126 
127 INITIALIZE_PASS_BEGIN(DependenceAnalysisWrapperPass, "da",
128                       "Dependence Analysis", true, true)
129 INITIALIZE_PASS_DEPENDENCY(LoopInfoWrapperPass)
130 INITIALIZE_PASS_DEPENDENCY(ScalarEvolutionWrapperPass)
131 INITIALIZE_PASS_DEPENDENCY(AAResultsWrapperPass)
132 INITIALIZE_PASS_END(DependenceAnalysisWrapperPass, "da", "Dependence Analysis",
133                     true, true)
134 
135 char DependenceAnalysisWrapperPass::ID = 0;
136 
createDependenceAnalysisWrapperPass()137 FunctionPass *llvm::createDependenceAnalysisWrapperPass() {
138   return new DependenceAnalysisWrapperPass();
139 }
140 
runOnFunction(Function & F)141 bool DependenceAnalysisWrapperPass::runOnFunction(Function &F) {
142   auto &AA = getAnalysis<AAResultsWrapperPass>().getAAResults();
143   auto &SE = getAnalysis<ScalarEvolutionWrapperPass>().getSE();
144   auto &LI = getAnalysis<LoopInfoWrapperPass>().getLoopInfo();
145   info.reset(new DependenceInfo(&F, &AA, &SE, &LI));
146   return false;
147 }
148 
getDI() const149 DependenceInfo &DependenceAnalysisWrapperPass::getDI() const { return *info; }
150 
releaseMemory()151 void DependenceAnalysisWrapperPass::releaseMemory() { info.reset(); }
152 
getAnalysisUsage(AnalysisUsage & AU) const153 void DependenceAnalysisWrapperPass::getAnalysisUsage(AnalysisUsage &AU) const {
154   AU.setPreservesAll();
155   AU.addRequiredTransitive<AAResultsWrapperPass>();
156   AU.addRequiredTransitive<ScalarEvolutionWrapperPass>();
157   AU.addRequiredTransitive<LoopInfoWrapperPass>();
158 }
159 
160 
161 // Used to test the dependence analyzer.
162 // Looks through the function, noting loads and stores.
163 // Calls depends() on every possible pair and prints out the result.
164 // Ignores all other instructions.
dumpExampleDependence(raw_ostream & OS,DependenceInfo * DA)165 static void dumpExampleDependence(raw_ostream &OS, DependenceInfo *DA) {
166   auto *F = DA->getFunction();
167   for (inst_iterator SrcI = inst_begin(F), SrcE = inst_end(F); SrcI != SrcE;
168        ++SrcI) {
169     if (isa<StoreInst>(*SrcI) || isa<LoadInst>(*SrcI)) {
170       for (inst_iterator DstI = SrcI, DstE = inst_end(F);
171            DstI != DstE; ++DstI) {
172         if (isa<StoreInst>(*DstI) || isa<LoadInst>(*DstI)) {
173           OS << "da analyze - ";
174           if (auto D = DA->depends(&*SrcI, &*DstI, true)) {
175             D->dump(OS);
176             for (unsigned Level = 1; Level <= D->getLevels(); Level++) {
177               if (D->isSplitable(Level)) {
178                 OS << "da analyze - split level = " << Level;
179                 OS << ", iteration = " << *DA->getSplitIteration(*D, Level);
180                 OS << "!\n";
181               }
182             }
183           }
184           else
185             OS << "none!\n";
186         }
187       }
188     }
189   }
190 }
191 
print(raw_ostream & OS,const Module *) const192 void DependenceAnalysisWrapperPass::print(raw_ostream &OS,
193                                           const Module *) const {
194   dumpExampleDependence(OS, info.get());
195 }
196 
197 //===----------------------------------------------------------------------===//
198 // Dependence methods
199 
200 // Returns true if this is an input dependence.
isInput() const201 bool Dependence::isInput() const {
202   return Src->mayReadFromMemory() && Dst->mayReadFromMemory();
203 }
204 
205 
206 // Returns true if this is an output dependence.
isOutput() const207 bool Dependence::isOutput() const {
208   return Src->mayWriteToMemory() && Dst->mayWriteToMemory();
209 }
210 
211 
212 // Returns true if this is an flow (aka true)  dependence.
isFlow() const213 bool Dependence::isFlow() const {
214   return Src->mayWriteToMemory() && Dst->mayReadFromMemory();
215 }
216 
217 
218 // Returns true if this is an anti dependence.
isAnti() const219 bool Dependence::isAnti() const {
220   return Src->mayReadFromMemory() && Dst->mayWriteToMemory();
221 }
222 
223 
224 // Returns true if a particular level is scalar; that is,
225 // if no subscript in the source or destination mention the induction
226 // variable associated with the loop at this level.
227 // Leave this out of line, so it will serve as a virtual method anchor
isScalar(unsigned level) const228 bool Dependence::isScalar(unsigned level) const {
229   return false;
230 }
231 
232 
233 //===----------------------------------------------------------------------===//
234 // FullDependence methods
235 
FullDependence(Instruction * Source,Instruction * Destination,bool PossiblyLoopIndependent,unsigned CommonLevels)236 FullDependence::FullDependence(Instruction *Source, Instruction *Destination,
237                                bool PossiblyLoopIndependent,
238                                unsigned CommonLevels)
239     : Dependence(Source, Destination), Levels(CommonLevels),
240       LoopIndependent(PossiblyLoopIndependent) {
241   Consistent = true;
242   if (CommonLevels)
243     DV = make_unique<DVEntry[]>(CommonLevels);
244 }
245 
246 // The rest are simple getters that hide the implementation.
247 
248 // getDirection - Returns the direction associated with a particular level.
getDirection(unsigned Level) const249 unsigned FullDependence::getDirection(unsigned Level) const {
250   assert(0 < Level && Level <= Levels && "Level out of range");
251   return DV[Level - 1].Direction;
252 }
253 
254 
255 // Returns the distance (or NULL) associated with a particular level.
getDistance(unsigned Level) const256 const SCEV *FullDependence::getDistance(unsigned Level) const {
257   assert(0 < Level && Level <= Levels && "Level out of range");
258   return DV[Level - 1].Distance;
259 }
260 
261 
262 // Returns true if a particular level is scalar; that is,
263 // if no subscript in the source or destination mention the induction
264 // variable associated with the loop at this level.
isScalar(unsigned Level) const265 bool FullDependence::isScalar(unsigned Level) const {
266   assert(0 < Level && Level <= Levels && "Level out of range");
267   return DV[Level - 1].Scalar;
268 }
269 
270 
271 // Returns true if peeling the first iteration from this loop
272 // will break this dependence.
isPeelFirst(unsigned Level) const273 bool FullDependence::isPeelFirst(unsigned Level) const {
274   assert(0 < Level && Level <= Levels && "Level out of range");
275   return DV[Level - 1].PeelFirst;
276 }
277 
278 
279 // Returns true if peeling the last iteration from this loop
280 // will break this dependence.
isPeelLast(unsigned Level) const281 bool FullDependence::isPeelLast(unsigned Level) const {
282   assert(0 < Level && Level <= Levels && "Level out of range");
283   return DV[Level - 1].PeelLast;
284 }
285 
286 
287 // Returns true if splitting this loop will break the dependence.
isSplitable(unsigned Level) const288 bool FullDependence::isSplitable(unsigned Level) const {
289   assert(0 < Level && Level <= Levels && "Level out of range");
290   return DV[Level - 1].Splitable;
291 }
292 
293 
294 //===----------------------------------------------------------------------===//
295 // DependenceInfo::Constraint methods
296 
297 // If constraint is a point <X, Y>, returns X.
298 // Otherwise assert.
getX() const299 const SCEV *DependenceInfo::Constraint::getX() const {
300   assert(Kind == Point && "Kind should be Point");
301   return A;
302 }
303 
304 
305 // If constraint is a point <X, Y>, returns Y.
306 // Otherwise assert.
getY() const307 const SCEV *DependenceInfo::Constraint::getY() const {
308   assert(Kind == Point && "Kind should be Point");
309   return B;
310 }
311 
312 
313 // If constraint is a line AX + BY = C, returns A.
314 // Otherwise assert.
getA() const315 const SCEV *DependenceInfo::Constraint::getA() const {
316   assert((Kind == Line || Kind == Distance) &&
317          "Kind should be Line (or Distance)");
318   return A;
319 }
320 
321 
322 // If constraint is a line AX + BY = C, returns B.
323 // Otherwise assert.
getB() const324 const SCEV *DependenceInfo::Constraint::getB() const {
325   assert((Kind == Line || Kind == Distance) &&
326          "Kind should be Line (or Distance)");
327   return B;
328 }
329 
330 
331 // If constraint is a line AX + BY = C, returns C.
332 // Otherwise assert.
getC() const333 const SCEV *DependenceInfo::Constraint::getC() const {
334   assert((Kind == Line || Kind == Distance) &&
335          "Kind should be Line (or Distance)");
336   return C;
337 }
338 
339 
340 // If constraint is a distance, returns D.
341 // Otherwise assert.
getD() const342 const SCEV *DependenceInfo::Constraint::getD() const {
343   assert(Kind == Distance && "Kind should be Distance");
344   return SE->getNegativeSCEV(C);
345 }
346 
347 
348 // Returns the loop associated with this constraint.
getAssociatedLoop() const349 const Loop *DependenceInfo::Constraint::getAssociatedLoop() const {
350   assert((Kind == Distance || Kind == Line || Kind == Point) &&
351          "Kind should be Distance, Line, or Point");
352   return AssociatedLoop;
353 }
354 
setPoint(const SCEV * X,const SCEV * Y,const Loop * CurLoop)355 void DependenceInfo::Constraint::setPoint(const SCEV *X, const SCEV *Y,
356                                           const Loop *CurLoop) {
357   Kind = Point;
358   A = X;
359   B = Y;
360   AssociatedLoop = CurLoop;
361 }
362 
setLine(const SCEV * AA,const SCEV * BB,const SCEV * CC,const Loop * CurLoop)363 void DependenceInfo::Constraint::setLine(const SCEV *AA, const SCEV *BB,
364                                          const SCEV *CC, const Loop *CurLoop) {
365   Kind = Line;
366   A = AA;
367   B = BB;
368   C = CC;
369   AssociatedLoop = CurLoop;
370 }
371 
setDistance(const SCEV * D,const Loop * CurLoop)372 void DependenceInfo::Constraint::setDistance(const SCEV *D,
373                                              const Loop *CurLoop) {
374   Kind = Distance;
375   A = SE->getOne(D->getType());
376   B = SE->getNegativeSCEV(A);
377   C = SE->getNegativeSCEV(D);
378   AssociatedLoop = CurLoop;
379 }
380 
setEmpty()381 void DependenceInfo::Constraint::setEmpty() { Kind = Empty; }
382 
setAny(ScalarEvolution * NewSE)383 void DependenceInfo::Constraint::setAny(ScalarEvolution *NewSE) {
384   SE = NewSE;
385   Kind = Any;
386 }
387 
388 
389 // For debugging purposes. Dumps the constraint out to OS.
dump(raw_ostream & OS) const390 void DependenceInfo::Constraint::dump(raw_ostream &OS) const {
391   if (isEmpty())
392     OS << " Empty\n";
393   else if (isAny())
394     OS << " Any\n";
395   else if (isPoint())
396     OS << " Point is <" << *getX() << ", " << *getY() << ">\n";
397   else if (isDistance())
398     OS << " Distance is " << *getD() <<
399       " (" << *getA() << "*X + " << *getB() << "*Y = " << *getC() << ")\n";
400   else if (isLine())
401     OS << " Line is " << *getA() << "*X + " <<
402       *getB() << "*Y = " << *getC() << "\n";
403   else
404     llvm_unreachable("unknown constraint type in Constraint::dump");
405 }
406 
407 
408 // Updates X with the intersection
409 // of the Constraints X and Y. Returns true if X has changed.
410 // Corresponds to Figure 4 from the paper
411 //
412 //            Practical Dependence Testing
413 //            Goff, Kennedy, Tseng
414 //            PLDI 1991
intersectConstraints(Constraint * X,const Constraint * Y)415 bool DependenceInfo::intersectConstraints(Constraint *X, const Constraint *Y) {
416   ++DeltaApplications;
417   DEBUG(dbgs() << "\tintersect constraints\n");
418   DEBUG(dbgs() << "\t    X ="; X->dump(dbgs()));
419   DEBUG(dbgs() << "\t    Y ="; Y->dump(dbgs()));
420   assert(!Y->isPoint() && "Y must not be a Point");
421   if (X->isAny()) {
422     if (Y->isAny())
423       return false;
424     *X = *Y;
425     return true;
426   }
427   if (X->isEmpty())
428     return false;
429   if (Y->isEmpty()) {
430     X->setEmpty();
431     return true;
432   }
433 
434   if (X->isDistance() && Y->isDistance()) {
435     DEBUG(dbgs() << "\t    intersect 2 distances\n");
436     if (isKnownPredicate(CmpInst::ICMP_EQ, X->getD(), Y->getD()))
437       return false;
438     if (isKnownPredicate(CmpInst::ICMP_NE, X->getD(), Y->getD())) {
439       X->setEmpty();
440       ++DeltaSuccesses;
441       return true;
442     }
443     // Hmmm, interesting situation.
444     // I guess if either is constant, keep it and ignore the other.
445     if (isa<SCEVConstant>(Y->getD())) {
446       *X = *Y;
447       return true;
448     }
449     return false;
450   }
451 
452   // At this point, the pseudo-code in Figure 4 of the paper
453   // checks if (X->isPoint() && Y->isPoint()).
454   // This case can't occur in our implementation,
455   // since a Point can only arise as the result of intersecting
456   // two Line constraints, and the right-hand value, Y, is never
457   // the result of an intersection.
458   assert(!(X->isPoint() && Y->isPoint()) &&
459          "We shouldn't ever see X->isPoint() && Y->isPoint()");
460 
461   if (X->isLine() && Y->isLine()) {
462     DEBUG(dbgs() << "\t    intersect 2 lines\n");
463     const SCEV *Prod1 = SE->getMulExpr(X->getA(), Y->getB());
464     const SCEV *Prod2 = SE->getMulExpr(X->getB(), Y->getA());
465     if (isKnownPredicate(CmpInst::ICMP_EQ, Prod1, Prod2)) {
466       // slopes are equal, so lines are parallel
467       DEBUG(dbgs() << "\t\tsame slope\n");
468       Prod1 = SE->getMulExpr(X->getC(), Y->getB());
469       Prod2 = SE->getMulExpr(X->getB(), Y->getC());
470       if (isKnownPredicate(CmpInst::ICMP_EQ, Prod1, Prod2))
471         return false;
472       if (isKnownPredicate(CmpInst::ICMP_NE, Prod1, Prod2)) {
473         X->setEmpty();
474         ++DeltaSuccesses;
475         return true;
476       }
477       return false;
478     }
479     if (isKnownPredicate(CmpInst::ICMP_NE, Prod1, Prod2)) {
480       // slopes differ, so lines intersect
481       DEBUG(dbgs() << "\t\tdifferent slopes\n");
482       const SCEV *C1B2 = SE->getMulExpr(X->getC(), Y->getB());
483       const SCEV *C1A2 = SE->getMulExpr(X->getC(), Y->getA());
484       const SCEV *C2B1 = SE->getMulExpr(Y->getC(), X->getB());
485       const SCEV *C2A1 = SE->getMulExpr(Y->getC(), X->getA());
486       const SCEV *A1B2 = SE->getMulExpr(X->getA(), Y->getB());
487       const SCEV *A2B1 = SE->getMulExpr(Y->getA(), X->getB());
488       const SCEVConstant *C1A2_C2A1 =
489         dyn_cast<SCEVConstant>(SE->getMinusSCEV(C1A2, C2A1));
490       const SCEVConstant *C1B2_C2B1 =
491         dyn_cast<SCEVConstant>(SE->getMinusSCEV(C1B2, C2B1));
492       const SCEVConstant *A1B2_A2B1 =
493         dyn_cast<SCEVConstant>(SE->getMinusSCEV(A1B2, A2B1));
494       const SCEVConstant *A2B1_A1B2 =
495         dyn_cast<SCEVConstant>(SE->getMinusSCEV(A2B1, A1B2));
496       if (!C1B2_C2B1 || !C1A2_C2A1 ||
497           !A1B2_A2B1 || !A2B1_A1B2)
498         return false;
499       APInt Xtop = C1B2_C2B1->getAPInt();
500       APInt Xbot = A1B2_A2B1->getAPInt();
501       APInt Ytop = C1A2_C2A1->getAPInt();
502       APInt Ybot = A2B1_A1B2->getAPInt();
503       DEBUG(dbgs() << "\t\tXtop = " << Xtop << "\n");
504       DEBUG(dbgs() << "\t\tXbot = " << Xbot << "\n");
505       DEBUG(dbgs() << "\t\tYtop = " << Ytop << "\n");
506       DEBUG(dbgs() << "\t\tYbot = " << Ybot << "\n");
507       APInt Xq = Xtop; // these need to be initialized, even
508       APInt Xr = Xtop; // though they're just going to be overwritten
509       APInt::sdivrem(Xtop, Xbot, Xq, Xr);
510       APInt Yq = Ytop;
511       APInt Yr = Ytop;
512       APInt::sdivrem(Ytop, Ybot, Yq, Yr);
513       if (Xr != 0 || Yr != 0) {
514         X->setEmpty();
515         ++DeltaSuccesses;
516         return true;
517       }
518       DEBUG(dbgs() << "\t\tX = " << Xq << ", Y = " << Yq << "\n");
519       if (Xq.slt(0) || Yq.slt(0)) {
520         X->setEmpty();
521         ++DeltaSuccesses;
522         return true;
523       }
524       if (const SCEVConstant *CUB =
525           collectConstantUpperBound(X->getAssociatedLoop(), Prod1->getType())) {
526         const APInt &UpperBound = CUB->getAPInt();
527         DEBUG(dbgs() << "\t\tupper bound = " << UpperBound << "\n");
528         if (Xq.sgt(UpperBound) || Yq.sgt(UpperBound)) {
529           X->setEmpty();
530           ++DeltaSuccesses;
531           return true;
532         }
533       }
534       X->setPoint(SE->getConstant(Xq),
535                   SE->getConstant(Yq),
536                   X->getAssociatedLoop());
537       ++DeltaSuccesses;
538       return true;
539     }
540     return false;
541   }
542 
543   // if (X->isLine() && Y->isPoint()) This case can't occur.
544   assert(!(X->isLine() && Y->isPoint()) && "This case should never occur");
545 
546   if (X->isPoint() && Y->isLine()) {
547     DEBUG(dbgs() << "\t    intersect Point and Line\n");
548     const SCEV *A1X1 = SE->getMulExpr(Y->getA(), X->getX());
549     const SCEV *B1Y1 = SE->getMulExpr(Y->getB(), X->getY());
550     const SCEV *Sum = SE->getAddExpr(A1X1, B1Y1);
551     if (isKnownPredicate(CmpInst::ICMP_EQ, Sum, Y->getC()))
552       return false;
553     if (isKnownPredicate(CmpInst::ICMP_NE, Sum, Y->getC())) {
554       X->setEmpty();
555       ++DeltaSuccesses;
556       return true;
557     }
558     return false;
559   }
560 
561   llvm_unreachable("shouldn't reach the end of Constraint intersection");
562   return false;
563 }
564 
565 
566 //===----------------------------------------------------------------------===//
567 // DependenceInfo methods
568 
569 // For debugging purposes. Dumps a dependence to OS.
dump(raw_ostream & OS) const570 void Dependence::dump(raw_ostream &OS) const {
571   bool Splitable = false;
572   if (isConfused())
573     OS << "confused";
574   else {
575     if (isConsistent())
576       OS << "consistent ";
577     if (isFlow())
578       OS << "flow";
579     else if (isOutput())
580       OS << "output";
581     else if (isAnti())
582       OS << "anti";
583     else if (isInput())
584       OS << "input";
585     unsigned Levels = getLevels();
586     OS << " [";
587     for (unsigned II = 1; II <= Levels; ++II) {
588       if (isSplitable(II))
589         Splitable = true;
590       if (isPeelFirst(II))
591         OS << 'p';
592       const SCEV *Distance = getDistance(II);
593       if (Distance)
594         OS << *Distance;
595       else if (isScalar(II))
596         OS << "S";
597       else {
598         unsigned Direction = getDirection(II);
599         if (Direction == DVEntry::ALL)
600           OS << "*";
601         else {
602           if (Direction & DVEntry::LT)
603             OS << "<";
604           if (Direction & DVEntry::EQ)
605             OS << "=";
606           if (Direction & DVEntry::GT)
607             OS << ">";
608         }
609       }
610       if (isPeelLast(II))
611         OS << 'p';
612       if (II < Levels)
613         OS << " ";
614     }
615     if (isLoopIndependent())
616       OS << "|<";
617     OS << "]";
618     if (Splitable)
619       OS << " splitable";
620   }
621   OS << "!\n";
622 }
623 
underlyingObjectsAlias(AliasAnalysis * AA,const DataLayout & DL,const Value * A,const Value * B)624 static AliasResult underlyingObjectsAlias(AliasAnalysis *AA,
625                                           const DataLayout &DL, const Value *A,
626                                           const Value *B) {
627   const Value *AObj = GetUnderlyingObject(A, DL);
628   const Value *BObj = GetUnderlyingObject(B, DL);
629   return AA->alias(AObj, DL.getTypeStoreSize(AObj->getType()),
630                    BObj, DL.getTypeStoreSize(BObj->getType()));
631 }
632 
633 
634 // Returns true if the load or store can be analyzed. Atomic and volatile
635 // operations have properties which this analysis does not understand.
636 static
isLoadOrStore(const Instruction * I)637 bool isLoadOrStore(const Instruction *I) {
638   if (const LoadInst *LI = dyn_cast<LoadInst>(I))
639     return LI->isUnordered();
640   else if (const StoreInst *SI = dyn_cast<StoreInst>(I))
641     return SI->isUnordered();
642   return false;
643 }
644 
645 
646 static
getPointerOperand(Instruction * I)647 Value *getPointerOperand(Instruction *I) {
648   if (LoadInst *LI = dyn_cast<LoadInst>(I))
649     return LI->getPointerOperand();
650   if (StoreInst *SI = dyn_cast<StoreInst>(I))
651     return SI->getPointerOperand();
652   llvm_unreachable("Value is not load or store instruction");
653   return nullptr;
654 }
655 
656 
657 // Examines the loop nesting of the Src and Dst
658 // instructions and establishes their shared loops. Sets the variables
659 // CommonLevels, SrcLevels, and MaxLevels.
660 // The source and destination instructions needn't be contained in the same
661 // loop. The routine establishNestingLevels finds the level of most deeply
662 // nested loop that contains them both, CommonLevels. An instruction that's
663 // not contained in a loop is at level = 0. MaxLevels is equal to the level
664 // of the source plus the level of the destination, minus CommonLevels.
665 // This lets us allocate vectors MaxLevels in length, with room for every
666 // distinct loop referenced in both the source and destination subscripts.
667 // The variable SrcLevels is the nesting depth of the source instruction.
668 // It's used to help calculate distinct loops referenced by the destination.
669 // Here's the map from loops to levels:
670 //            0 - unused
671 //            1 - outermost common loop
672 //          ... - other common loops
673 // CommonLevels - innermost common loop
674 //          ... - loops containing Src but not Dst
675 //    SrcLevels - innermost loop containing Src but not Dst
676 //          ... - loops containing Dst but not Src
677 //    MaxLevels - innermost loops containing Dst but not Src
678 // Consider the follow code fragment:
679 //   for (a = ...) {
680 //     for (b = ...) {
681 //       for (c = ...) {
682 //         for (d = ...) {
683 //           A[] = ...;
684 //         }
685 //       }
686 //       for (e = ...) {
687 //         for (f = ...) {
688 //           for (g = ...) {
689 //             ... = A[];
690 //           }
691 //         }
692 //       }
693 //     }
694 //   }
695 // If we're looking at the possibility of a dependence between the store
696 // to A (the Src) and the load from A (the Dst), we'll note that they
697 // have 2 loops in common, so CommonLevels will equal 2 and the direction
698 // vector for Result will have 2 entries. SrcLevels = 4 and MaxLevels = 7.
699 // A map from loop names to loop numbers would look like
700 //     a - 1
701 //     b - 2 = CommonLevels
702 //     c - 3
703 //     d - 4 = SrcLevels
704 //     e - 5
705 //     f - 6
706 //     g - 7 = MaxLevels
establishNestingLevels(const Instruction * Src,const Instruction * Dst)707 void DependenceInfo::establishNestingLevels(const Instruction *Src,
708                                             const Instruction *Dst) {
709   const BasicBlock *SrcBlock = Src->getParent();
710   const BasicBlock *DstBlock = Dst->getParent();
711   unsigned SrcLevel = LI->getLoopDepth(SrcBlock);
712   unsigned DstLevel = LI->getLoopDepth(DstBlock);
713   const Loop *SrcLoop = LI->getLoopFor(SrcBlock);
714   const Loop *DstLoop = LI->getLoopFor(DstBlock);
715   SrcLevels = SrcLevel;
716   MaxLevels = SrcLevel + DstLevel;
717   while (SrcLevel > DstLevel) {
718     SrcLoop = SrcLoop->getParentLoop();
719     SrcLevel--;
720   }
721   while (DstLevel > SrcLevel) {
722     DstLoop = DstLoop->getParentLoop();
723     DstLevel--;
724   }
725   while (SrcLoop != DstLoop) {
726     SrcLoop = SrcLoop->getParentLoop();
727     DstLoop = DstLoop->getParentLoop();
728     SrcLevel--;
729   }
730   CommonLevels = SrcLevel;
731   MaxLevels -= CommonLevels;
732 }
733 
734 
735 // Given one of the loops containing the source, return
736 // its level index in our numbering scheme.
mapSrcLoop(const Loop * SrcLoop) const737 unsigned DependenceInfo::mapSrcLoop(const Loop *SrcLoop) const {
738   return SrcLoop->getLoopDepth();
739 }
740 
741 
742 // Given one of the loops containing the destination,
743 // return its level index in our numbering scheme.
mapDstLoop(const Loop * DstLoop) const744 unsigned DependenceInfo::mapDstLoop(const Loop *DstLoop) const {
745   unsigned D = DstLoop->getLoopDepth();
746   if (D > CommonLevels)
747     return D - CommonLevels + SrcLevels;
748   else
749     return D;
750 }
751 
752 
753 // Returns true if Expression is loop invariant in LoopNest.
isLoopInvariant(const SCEV * Expression,const Loop * LoopNest) const754 bool DependenceInfo::isLoopInvariant(const SCEV *Expression,
755                                      const Loop *LoopNest) const {
756   if (!LoopNest)
757     return true;
758   return SE->isLoopInvariant(Expression, LoopNest) &&
759     isLoopInvariant(Expression, LoopNest->getParentLoop());
760 }
761 
762 
763 
764 // Finds the set of loops from the LoopNest that
765 // have a level <= CommonLevels and are referred to by the SCEV Expression.
collectCommonLoops(const SCEV * Expression,const Loop * LoopNest,SmallBitVector & Loops) const766 void DependenceInfo::collectCommonLoops(const SCEV *Expression,
767                                         const Loop *LoopNest,
768                                         SmallBitVector &Loops) const {
769   while (LoopNest) {
770     unsigned Level = LoopNest->getLoopDepth();
771     if (Level <= CommonLevels && !SE->isLoopInvariant(Expression, LoopNest))
772       Loops.set(Level);
773     LoopNest = LoopNest->getParentLoop();
774   }
775 }
776 
unifySubscriptType(ArrayRef<Subscript * > Pairs)777 void DependenceInfo::unifySubscriptType(ArrayRef<Subscript *> Pairs) {
778 
779   unsigned widestWidthSeen = 0;
780   Type *widestType;
781 
782   // Go through each pair and find the widest bit to which we need
783   // to extend all of them.
784   for (Subscript *Pair : Pairs) {
785     const SCEV *Src = Pair->Src;
786     const SCEV *Dst = Pair->Dst;
787     IntegerType *SrcTy = dyn_cast<IntegerType>(Src->getType());
788     IntegerType *DstTy = dyn_cast<IntegerType>(Dst->getType());
789     if (SrcTy == nullptr || DstTy == nullptr) {
790       assert(SrcTy == DstTy && "This function only unify integer types and "
791              "expect Src and Dst share the same type "
792              "otherwise.");
793       continue;
794     }
795     if (SrcTy->getBitWidth() > widestWidthSeen) {
796       widestWidthSeen = SrcTy->getBitWidth();
797       widestType = SrcTy;
798     }
799     if (DstTy->getBitWidth() > widestWidthSeen) {
800       widestWidthSeen = DstTy->getBitWidth();
801       widestType = DstTy;
802     }
803   }
804 
805 
806   assert(widestWidthSeen > 0);
807 
808   // Now extend each pair to the widest seen.
809   for (Subscript *Pair : Pairs) {
810     const SCEV *Src = Pair->Src;
811     const SCEV *Dst = Pair->Dst;
812     IntegerType *SrcTy = dyn_cast<IntegerType>(Src->getType());
813     IntegerType *DstTy = dyn_cast<IntegerType>(Dst->getType());
814     if (SrcTy == nullptr || DstTy == nullptr) {
815       assert(SrcTy == DstTy && "This function only unify integer types and "
816              "expect Src and Dst share the same type "
817              "otherwise.");
818       continue;
819     }
820     if (SrcTy->getBitWidth() < widestWidthSeen)
821       // Sign-extend Src to widestType
822       Pair->Src = SE->getSignExtendExpr(Src, widestType);
823     if (DstTy->getBitWidth() < widestWidthSeen) {
824       // Sign-extend Dst to widestType
825       Pair->Dst = SE->getSignExtendExpr(Dst, widestType);
826     }
827   }
828 }
829 
830 // removeMatchingExtensions - Examines a subscript pair.
831 // If the source and destination are identically sign (or zero)
832 // extended, it strips off the extension in an effect to simplify
833 // the actual analysis.
removeMatchingExtensions(Subscript * Pair)834 void DependenceInfo::removeMatchingExtensions(Subscript *Pair) {
835   const SCEV *Src = Pair->Src;
836   const SCEV *Dst = Pair->Dst;
837   if ((isa<SCEVZeroExtendExpr>(Src) && isa<SCEVZeroExtendExpr>(Dst)) ||
838       (isa<SCEVSignExtendExpr>(Src) && isa<SCEVSignExtendExpr>(Dst))) {
839     const SCEVCastExpr *SrcCast = cast<SCEVCastExpr>(Src);
840     const SCEVCastExpr *DstCast = cast<SCEVCastExpr>(Dst);
841     const SCEV *SrcCastOp = SrcCast->getOperand();
842     const SCEV *DstCastOp = DstCast->getOperand();
843     if (SrcCastOp->getType() == DstCastOp->getType()) {
844       Pair->Src = SrcCastOp;
845       Pair->Dst = DstCastOp;
846     }
847   }
848 }
849 
850 
851 // Examine the scev and return true iff it's linear.
852 // Collect any loops mentioned in the set of "Loops".
checkSrcSubscript(const SCEV * Src,const Loop * LoopNest,SmallBitVector & Loops)853 bool DependenceInfo::checkSrcSubscript(const SCEV *Src, const Loop *LoopNest,
854                                        SmallBitVector &Loops) {
855   const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Src);
856   if (!AddRec)
857     return isLoopInvariant(Src, LoopNest);
858   const SCEV *Start = AddRec->getStart();
859   const SCEV *Step = AddRec->getStepRecurrence(*SE);
860   const SCEV *UB = SE->getBackedgeTakenCount(AddRec->getLoop());
861   if (!isa<SCEVCouldNotCompute>(UB)) {
862     if (SE->getTypeSizeInBits(Start->getType()) <
863         SE->getTypeSizeInBits(UB->getType())) {
864       if (!AddRec->getNoWrapFlags())
865         return false;
866     }
867   }
868   if (!isLoopInvariant(Step, LoopNest))
869     return false;
870   Loops.set(mapSrcLoop(AddRec->getLoop()));
871   return checkSrcSubscript(Start, LoopNest, Loops);
872 }
873 
874 
875 
876 // Examine the scev and return true iff it's linear.
877 // Collect any loops mentioned in the set of "Loops".
checkDstSubscript(const SCEV * Dst,const Loop * LoopNest,SmallBitVector & Loops)878 bool DependenceInfo::checkDstSubscript(const SCEV *Dst, const Loop *LoopNest,
879                                        SmallBitVector &Loops) {
880   const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Dst);
881   if (!AddRec)
882     return isLoopInvariant(Dst, LoopNest);
883   const SCEV *Start = AddRec->getStart();
884   const SCEV *Step = AddRec->getStepRecurrence(*SE);
885   const SCEV *UB = SE->getBackedgeTakenCount(AddRec->getLoop());
886   if (!isa<SCEVCouldNotCompute>(UB)) {
887     if (SE->getTypeSizeInBits(Start->getType()) <
888         SE->getTypeSizeInBits(UB->getType())) {
889       if (!AddRec->getNoWrapFlags())
890         return false;
891     }
892   }
893   if (!isLoopInvariant(Step, LoopNest))
894     return false;
895   Loops.set(mapDstLoop(AddRec->getLoop()));
896   return checkDstSubscript(Start, LoopNest, Loops);
897 }
898 
899 
900 // Examines the subscript pair (the Src and Dst SCEVs)
901 // and classifies it as either ZIV, SIV, RDIV, MIV, or Nonlinear.
902 // Collects the associated loops in a set.
903 DependenceInfo::Subscript::ClassificationKind
classifyPair(const SCEV * Src,const Loop * SrcLoopNest,const SCEV * Dst,const Loop * DstLoopNest,SmallBitVector & Loops)904 DependenceInfo::classifyPair(const SCEV *Src, const Loop *SrcLoopNest,
905                              const SCEV *Dst, const Loop *DstLoopNest,
906                              SmallBitVector &Loops) {
907   SmallBitVector SrcLoops(MaxLevels + 1);
908   SmallBitVector DstLoops(MaxLevels + 1);
909   if (!checkSrcSubscript(Src, SrcLoopNest, SrcLoops))
910     return Subscript::NonLinear;
911   if (!checkDstSubscript(Dst, DstLoopNest, DstLoops))
912     return Subscript::NonLinear;
913   Loops = SrcLoops;
914   Loops |= DstLoops;
915   unsigned N = Loops.count();
916   if (N == 0)
917     return Subscript::ZIV;
918   if (N == 1)
919     return Subscript::SIV;
920   if (N == 2 && (SrcLoops.count() == 0 ||
921                  DstLoops.count() == 0 ||
922                  (SrcLoops.count() == 1 && DstLoops.count() == 1)))
923     return Subscript::RDIV;
924   return Subscript::MIV;
925 }
926 
927 
928 // A wrapper around SCEV::isKnownPredicate.
929 // Looks for cases where we're interested in comparing for equality.
930 // If both X and Y have been identically sign or zero extended,
931 // it strips off the (confusing) extensions before invoking
932 // SCEV::isKnownPredicate. Perhaps, someday, the ScalarEvolution package
933 // will be similarly updated.
934 //
935 // If SCEV::isKnownPredicate can't prove the predicate,
936 // we try simple subtraction, which seems to help in some cases
937 // involving symbolics.
isKnownPredicate(ICmpInst::Predicate Pred,const SCEV * X,const SCEV * Y) const938 bool DependenceInfo::isKnownPredicate(ICmpInst::Predicate Pred, const SCEV *X,
939                                       const SCEV *Y) const {
940   if (Pred == CmpInst::ICMP_EQ ||
941       Pred == CmpInst::ICMP_NE) {
942     if ((isa<SCEVSignExtendExpr>(X) &&
943          isa<SCEVSignExtendExpr>(Y)) ||
944         (isa<SCEVZeroExtendExpr>(X) &&
945          isa<SCEVZeroExtendExpr>(Y))) {
946       const SCEVCastExpr *CX = cast<SCEVCastExpr>(X);
947       const SCEVCastExpr *CY = cast<SCEVCastExpr>(Y);
948       const SCEV *Xop = CX->getOperand();
949       const SCEV *Yop = CY->getOperand();
950       if (Xop->getType() == Yop->getType()) {
951         X = Xop;
952         Y = Yop;
953       }
954     }
955   }
956   if (SE->isKnownPredicate(Pred, X, Y))
957     return true;
958   // If SE->isKnownPredicate can't prove the condition,
959   // we try the brute-force approach of subtracting
960   // and testing the difference.
961   // By testing with SE->isKnownPredicate first, we avoid
962   // the possibility of overflow when the arguments are constants.
963   const SCEV *Delta = SE->getMinusSCEV(X, Y);
964   switch (Pred) {
965   case CmpInst::ICMP_EQ:
966     return Delta->isZero();
967   case CmpInst::ICMP_NE:
968     return SE->isKnownNonZero(Delta);
969   case CmpInst::ICMP_SGE:
970     return SE->isKnownNonNegative(Delta);
971   case CmpInst::ICMP_SLE:
972     return SE->isKnownNonPositive(Delta);
973   case CmpInst::ICMP_SGT:
974     return SE->isKnownPositive(Delta);
975   case CmpInst::ICMP_SLT:
976     return SE->isKnownNegative(Delta);
977   default:
978     llvm_unreachable("unexpected predicate in isKnownPredicate");
979   }
980 }
981 
982 
983 // All subscripts are all the same type.
984 // Loop bound may be smaller (e.g., a char).
985 // Should zero extend loop bound, since it's always >= 0.
986 // This routine collects upper bound and extends or truncates if needed.
987 // Truncating is safe when subscripts are known not to wrap. Cases without
988 // nowrap flags should have been rejected earlier.
989 // Return null if no bound available.
collectUpperBound(const Loop * L,Type * T) const990 const SCEV *DependenceInfo::collectUpperBound(const Loop *L, Type *T) const {
991   if (SE->hasLoopInvariantBackedgeTakenCount(L)) {
992     const SCEV *UB = SE->getBackedgeTakenCount(L);
993     return SE->getTruncateOrZeroExtend(UB, T);
994   }
995   return nullptr;
996 }
997 
998 
999 // Calls collectUpperBound(), then attempts to cast it to SCEVConstant.
1000 // If the cast fails, returns NULL.
collectConstantUpperBound(const Loop * L,Type * T) const1001 const SCEVConstant *DependenceInfo::collectConstantUpperBound(const Loop *L,
1002                                                               Type *T) const {
1003   if (const SCEV *UB = collectUpperBound(L, T))
1004     return dyn_cast<SCEVConstant>(UB);
1005   return nullptr;
1006 }
1007 
1008 
1009 // testZIV -
1010 // When we have a pair of subscripts of the form [c1] and [c2],
1011 // where c1 and c2 are both loop invariant, we attack it using
1012 // the ZIV test. Basically, we test by comparing the two values,
1013 // but there are actually three possible results:
1014 // 1) the values are equal, so there's a dependence
1015 // 2) the values are different, so there's no dependence
1016 // 3) the values might be equal, so we have to assume a dependence.
1017 //
1018 // Return true if dependence disproved.
testZIV(const SCEV * Src,const SCEV * Dst,FullDependence & Result) const1019 bool DependenceInfo::testZIV(const SCEV *Src, const SCEV *Dst,
1020                              FullDependence &Result) const {
1021   DEBUG(dbgs() << "    src = " << *Src << "\n");
1022   DEBUG(dbgs() << "    dst = " << *Dst << "\n");
1023   ++ZIVapplications;
1024   if (isKnownPredicate(CmpInst::ICMP_EQ, Src, Dst)) {
1025     DEBUG(dbgs() << "    provably dependent\n");
1026     return false; // provably dependent
1027   }
1028   if (isKnownPredicate(CmpInst::ICMP_NE, Src, Dst)) {
1029     DEBUG(dbgs() << "    provably independent\n");
1030     ++ZIVindependence;
1031     return true; // provably independent
1032   }
1033   DEBUG(dbgs() << "    possibly dependent\n");
1034   Result.Consistent = false;
1035   return false; // possibly dependent
1036 }
1037 
1038 
1039 // strongSIVtest -
1040 // From the paper, Practical Dependence Testing, Section 4.2.1
1041 //
1042 // When we have a pair of subscripts of the form [c1 + a*i] and [c2 + a*i],
1043 // where i is an induction variable, c1 and c2 are loop invariant,
1044 //  and a is a constant, we can solve it exactly using the Strong SIV test.
1045 //
1046 // Can prove independence. Failing that, can compute distance (and direction).
1047 // In the presence of symbolic terms, we can sometimes make progress.
1048 //
1049 // If there's a dependence,
1050 //
1051 //    c1 + a*i = c2 + a*i'
1052 //
1053 // The dependence distance is
1054 //
1055 //    d = i' - i = (c1 - c2)/a
1056 //
1057 // A dependence only exists if d is an integer and abs(d) <= U, where U is the
1058 // loop's upper bound. If a dependence exists, the dependence direction is
1059 // defined as
1060 //
1061 //                { < if d > 0
1062 //    direction = { = if d = 0
1063 //                { > if d < 0
1064 //
1065 // Return true if dependence disproved.
strongSIVtest(const SCEV * Coeff,const SCEV * SrcConst,const SCEV * DstConst,const Loop * CurLoop,unsigned Level,FullDependence & Result,Constraint & NewConstraint) const1066 bool DependenceInfo::strongSIVtest(const SCEV *Coeff, const SCEV *SrcConst,
1067                                    const SCEV *DstConst, const Loop *CurLoop,
1068                                    unsigned Level, FullDependence &Result,
1069                                    Constraint &NewConstraint) const {
1070   DEBUG(dbgs() << "\tStrong SIV test\n");
1071   DEBUG(dbgs() << "\t    Coeff = " << *Coeff);
1072   DEBUG(dbgs() << ", " << *Coeff->getType() << "\n");
1073   DEBUG(dbgs() << "\t    SrcConst = " << *SrcConst);
1074   DEBUG(dbgs() << ", " << *SrcConst->getType() << "\n");
1075   DEBUG(dbgs() << "\t    DstConst = " << *DstConst);
1076   DEBUG(dbgs() << ", " << *DstConst->getType() << "\n");
1077   ++StrongSIVapplications;
1078   assert(0 < Level && Level <= CommonLevels && "level out of range");
1079   Level--;
1080 
1081   const SCEV *Delta = SE->getMinusSCEV(SrcConst, DstConst);
1082   DEBUG(dbgs() << "\t    Delta = " << *Delta);
1083   DEBUG(dbgs() << ", " << *Delta->getType() << "\n");
1084 
1085   // check that |Delta| < iteration count
1086   if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
1087     DEBUG(dbgs() << "\t    UpperBound = " << *UpperBound);
1088     DEBUG(dbgs() << ", " << *UpperBound->getType() << "\n");
1089     const SCEV *AbsDelta =
1090       SE->isKnownNonNegative(Delta) ? Delta : SE->getNegativeSCEV(Delta);
1091     const SCEV *AbsCoeff =
1092       SE->isKnownNonNegative(Coeff) ? Coeff : SE->getNegativeSCEV(Coeff);
1093     const SCEV *Product = SE->getMulExpr(UpperBound, AbsCoeff);
1094     if (isKnownPredicate(CmpInst::ICMP_SGT, AbsDelta, Product)) {
1095       // Distance greater than trip count - no dependence
1096       ++StrongSIVindependence;
1097       ++StrongSIVsuccesses;
1098       return true;
1099     }
1100   }
1101 
1102   // Can we compute distance?
1103   if (isa<SCEVConstant>(Delta) && isa<SCEVConstant>(Coeff)) {
1104     APInt ConstDelta = cast<SCEVConstant>(Delta)->getAPInt();
1105     APInt ConstCoeff = cast<SCEVConstant>(Coeff)->getAPInt();
1106     APInt Distance  = ConstDelta; // these need to be initialized
1107     APInt Remainder = ConstDelta;
1108     APInt::sdivrem(ConstDelta, ConstCoeff, Distance, Remainder);
1109     DEBUG(dbgs() << "\t    Distance = " << Distance << "\n");
1110     DEBUG(dbgs() << "\t    Remainder = " << Remainder << "\n");
1111     // Make sure Coeff divides Delta exactly
1112     if (Remainder != 0) {
1113       // Coeff doesn't divide Distance, no dependence
1114       ++StrongSIVindependence;
1115       ++StrongSIVsuccesses;
1116       return true;
1117     }
1118     Result.DV[Level].Distance = SE->getConstant(Distance);
1119     NewConstraint.setDistance(SE->getConstant(Distance), CurLoop);
1120     if (Distance.sgt(0))
1121       Result.DV[Level].Direction &= Dependence::DVEntry::LT;
1122     else if (Distance.slt(0))
1123       Result.DV[Level].Direction &= Dependence::DVEntry::GT;
1124     else
1125       Result.DV[Level].Direction &= Dependence::DVEntry::EQ;
1126     ++StrongSIVsuccesses;
1127   }
1128   else if (Delta->isZero()) {
1129     // since 0/X == 0
1130     Result.DV[Level].Distance = Delta;
1131     NewConstraint.setDistance(Delta, CurLoop);
1132     Result.DV[Level].Direction &= Dependence::DVEntry::EQ;
1133     ++StrongSIVsuccesses;
1134   }
1135   else {
1136     if (Coeff->isOne()) {
1137       DEBUG(dbgs() << "\t    Distance = " << *Delta << "\n");
1138       Result.DV[Level].Distance = Delta; // since X/1 == X
1139       NewConstraint.setDistance(Delta, CurLoop);
1140     }
1141     else {
1142       Result.Consistent = false;
1143       NewConstraint.setLine(Coeff,
1144                             SE->getNegativeSCEV(Coeff),
1145                             SE->getNegativeSCEV(Delta), CurLoop);
1146     }
1147 
1148     // maybe we can get a useful direction
1149     bool DeltaMaybeZero     = !SE->isKnownNonZero(Delta);
1150     bool DeltaMaybePositive = !SE->isKnownNonPositive(Delta);
1151     bool DeltaMaybeNegative = !SE->isKnownNonNegative(Delta);
1152     bool CoeffMaybePositive = !SE->isKnownNonPositive(Coeff);
1153     bool CoeffMaybeNegative = !SE->isKnownNonNegative(Coeff);
1154     // The double negatives above are confusing.
1155     // It helps to read !SE->isKnownNonZero(Delta)
1156     // as "Delta might be Zero"
1157     unsigned NewDirection = Dependence::DVEntry::NONE;
1158     if ((DeltaMaybePositive && CoeffMaybePositive) ||
1159         (DeltaMaybeNegative && CoeffMaybeNegative))
1160       NewDirection = Dependence::DVEntry::LT;
1161     if (DeltaMaybeZero)
1162       NewDirection |= Dependence::DVEntry::EQ;
1163     if ((DeltaMaybeNegative && CoeffMaybePositive) ||
1164         (DeltaMaybePositive && CoeffMaybeNegative))
1165       NewDirection |= Dependence::DVEntry::GT;
1166     if (NewDirection < Result.DV[Level].Direction)
1167       ++StrongSIVsuccesses;
1168     Result.DV[Level].Direction &= NewDirection;
1169   }
1170   return false;
1171 }
1172 
1173 
1174 // weakCrossingSIVtest -
1175 // From the paper, Practical Dependence Testing, Section 4.2.2
1176 //
1177 // When we have a pair of subscripts of the form [c1 + a*i] and [c2 - a*i],
1178 // where i is an induction variable, c1 and c2 are loop invariant,
1179 // and a is a constant, we can solve it exactly using the
1180 // Weak-Crossing SIV test.
1181 //
1182 // Given c1 + a*i = c2 - a*i', we can look for the intersection of
1183 // the two lines, where i = i', yielding
1184 //
1185 //    c1 + a*i = c2 - a*i
1186 //    2a*i = c2 - c1
1187 //    i = (c2 - c1)/2a
1188 //
1189 // If i < 0, there is no dependence.
1190 // If i > upperbound, there is no dependence.
1191 // If i = 0 (i.e., if c1 = c2), there's a dependence with distance = 0.
1192 // If i = upperbound, there's a dependence with distance = 0.
1193 // If i is integral, there's a dependence (all directions).
1194 // If the non-integer part = 1/2, there's a dependence (<> directions).
1195 // Otherwise, there's no dependence.
1196 //
1197 // Can prove independence. Failing that,
1198 // can sometimes refine the directions.
1199 // Can determine iteration for splitting.
1200 //
1201 // Return true if dependence disproved.
weakCrossingSIVtest(const SCEV * Coeff,const SCEV * SrcConst,const SCEV * DstConst,const Loop * CurLoop,unsigned Level,FullDependence & Result,Constraint & NewConstraint,const SCEV * & SplitIter) const1202 bool DependenceInfo::weakCrossingSIVtest(
1203     const SCEV *Coeff, const SCEV *SrcConst, const SCEV *DstConst,
1204     const Loop *CurLoop, unsigned Level, FullDependence &Result,
1205     Constraint &NewConstraint, const SCEV *&SplitIter) const {
1206   DEBUG(dbgs() << "\tWeak-Crossing SIV test\n");
1207   DEBUG(dbgs() << "\t    Coeff = " << *Coeff << "\n");
1208   DEBUG(dbgs() << "\t    SrcConst = " << *SrcConst << "\n");
1209   DEBUG(dbgs() << "\t    DstConst = " << *DstConst << "\n");
1210   ++WeakCrossingSIVapplications;
1211   assert(0 < Level && Level <= CommonLevels && "Level out of range");
1212   Level--;
1213   Result.Consistent = false;
1214   const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
1215   DEBUG(dbgs() << "\t    Delta = " << *Delta << "\n");
1216   NewConstraint.setLine(Coeff, Coeff, Delta, CurLoop);
1217   if (Delta->isZero()) {
1218     Result.DV[Level].Direction &= unsigned(~Dependence::DVEntry::LT);
1219     Result.DV[Level].Direction &= unsigned(~Dependence::DVEntry::GT);
1220     ++WeakCrossingSIVsuccesses;
1221     if (!Result.DV[Level].Direction) {
1222       ++WeakCrossingSIVindependence;
1223       return true;
1224     }
1225     Result.DV[Level].Distance = Delta; // = 0
1226     return false;
1227   }
1228   const SCEVConstant *ConstCoeff = dyn_cast<SCEVConstant>(Coeff);
1229   if (!ConstCoeff)
1230     return false;
1231 
1232   Result.DV[Level].Splitable = true;
1233   if (SE->isKnownNegative(ConstCoeff)) {
1234     ConstCoeff = dyn_cast<SCEVConstant>(SE->getNegativeSCEV(ConstCoeff));
1235     assert(ConstCoeff &&
1236            "dynamic cast of negative of ConstCoeff should yield constant");
1237     Delta = SE->getNegativeSCEV(Delta);
1238   }
1239   assert(SE->isKnownPositive(ConstCoeff) && "ConstCoeff should be positive");
1240 
1241   // compute SplitIter for use by DependenceInfo::getSplitIteration()
1242   SplitIter = SE->getUDivExpr(
1243       SE->getSMaxExpr(SE->getZero(Delta->getType()), Delta),
1244       SE->getMulExpr(SE->getConstant(Delta->getType(), 2), ConstCoeff));
1245   DEBUG(dbgs() << "\t    Split iter = " << *SplitIter << "\n");
1246 
1247   const SCEVConstant *ConstDelta = dyn_cast<SCEVConstant>(Delta);
1248   if (!ConstDelta)
1249     return false;
1250 
1251   // We're certain that ConstCoeff > 0; therefore,
1252   // if Delta < 0, then no dependence.
1253   DEBUG(dbgs() << "\t    Delta = " << *Delta << "\n");
1254   DEBUG(dbgs() << "\t    ConstCoeff = " << *ConstCoeff << "\n");
1255   if (SE->isKnownNegative(Delta)) {
1256     // No dependence, Delta < 0
1257     ++WeakCrossingSIVindependence;
1258     ++WeakCrossingSIVsuccesses;
1259     return true;
1260   }
1261 
1262   // We're certain that Delta > 0 and ConstCoeff > 0.
1263   // Check Delta/(2*ConstCoeff) against upper loop bound
1264   if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
1265     DEBUG(dbgs() << "\t    UpperBound = " << *UpperBound << "\n");
1266     const SCEV *ConstantTwo = SE->getConstant(UpperBound->getType(), 2);
1267     const SCEV *ML = SE->getMulExpr(SE->getMulExpr(ConstCoeff, UpperBound),
1268                                     ConstantTwo);
1269     DEBUG(dbgs() << "\t    ML = " << *ML << "\n");
1270     if (isKnownPredicate(CmpInst::ICMP_SGT, Delta, ML)) {
1271       // Delta too big, no dependence
1272       ++WeakCrossingSIVindependence;
1273       ++WeakCrossingSIVsuccesses;
1274       return true;
1275     }
1276     if (isKnownPredicate(CmpInst::ICMP_EQ, Delta, ML)) {
1277       // i = i' = UB
1278       Result.DV[Level].Direction &= unsigned(~Dependence::DVEntry::LT);
1279       Result.DV[Level].Direction &= unsigned(~Dependence::DVEntry::GT);
1280       ++WeakCrossingSIVsuccesses;
1281       if (!Result.DV[Level].Direction) {
1282         ++WeakCrossingSIVindependence;
1283         return true;
1284       }
1285       Result.DV[Level].Splitable = false;
1286       Result.DV[Level].Distance = SE->getZero(Delta->getType());
1287       return false;
1288     }
1289   }
1290 
1291   // check that Coeff divides Delta
1292   APInt APDelta = ConstDelta->getAPInt();
1293   APInt APCoeff = ConstCoeff->getAPInt();
1294   APInt Distance = APDelta; // these need to be initialzed
1295   APInt Remainder = APDelta;
1296   APInt::sdivrem(APDelta, APCoeff, Distance, Remainder);
1297   DEBUG(dbgs() << "\t    Remainder = " << Remainder << "\n");
1298   if (Remainder != 0) {
1299     // Coeff doesn't divide Delta, no dependence
1300     ++WeakCrossingSIVindependence;
1301     ++WeakCrossingSIVsuccesses;
1302     return true;
1303   }
1304   DEBUG(dbgs() << "\t    Distance = " << Distance << "\n");
1305 
1306   // if 2*Coeff doesn't divide Delta, then the equal direction isn't possible
1307   APInt Two = APInt(Distance.getBitWidth(), 2, true);
1308   Remainder = Distance.srem(Two);
1309   DEBUG(dbgs() << "\t    Remainder = " << Remainder << "\n");
1310   if (Remainder != 0) {
1311     // Equal direction isn't possible
1312     Result.DV[Level].Direction &= unsigned(~Dependence::DVEntry::EQ);
1313     ++WeakCrossingSIVsuccesses;
1314   }
1315   return false;
1316 }
1317 
1318 
1319 // Kirch's algorithm, from
1320 //
1321 //        Optimizing Supercompilers for Supercomputers
1322 //        Michael Wolfe
1323 //        MIT Press, 1989
1324 //
1325 // Program 2.1, page 29.
1326 // Computes the GCD of AM and BM.
1327 // Also finds a solution to the equation ax - by = gcd(a, b).
1328 // Returns true if dependence disproved; i.e., gcd does not divide Delta.
findGCD(unsigned Bits,const APInt & AM,const APInt & BM,const APInt & Delta,APInt & G,APInt & X,APInt & Y)1329 static bool findGCD(unsigned Bits, const APInt &AM, const APInt &BM,
1330                     const APInt &Delta, APInt &G, APInt &X, APInt &Y) {
1331   APInt A0(Bits, 1, true), A1(Bits, 0, true);
1332   APInt B0(Bits, 0, true), B1(Bits, 1, true);
1333   APInt G0 = AM.abs();
1334   APInt G1 = BM.abs();
1335   APInt Q = G0; // these need to be initialized
1336   APInt R = G0;
1337   APInt::sdivrem(G0, G1, Q, R);
1338   while (R != 0) {
1339     APInt A2 = A0 - Q*A1; A0 = A1; A1 = A2;
1340     APInt B2 = B0 - Q*B1; B0 = B1; B1 = B2;
1341     G0 = G1; G1 = R;
1342     APInt::sdivrem(G0, G1, Q, R);
1343   }
1344   G = G1;
1345   DEBUG(dbgs() << "\t    GCD = " << G << "\n");
1346   X = AM.slt(0) ? -A1 : A1;
1347   Y = BM.slt(0) ? B1 : -B1;
1348 
1349   // make sure gcd divides Delta
1350   R = Delta.srem(G);
1351   if (R != 0)
1352     return true; // gcd doesn't divide Delta, no dependence
1353   Q = Delta.sdiv(G);
1354   X *= Q;
1355   Y *= Q;
1356   return false;
1357 }
1358 
floorOfQuotient(const APInt & A,const APInt & B)1359 static APInt floorOfQuotient(const APInt &A, const APInt &B) {
1360   APInt Q = A; // these need to be initialized
1361   APInt R = A;
1362   APInt::sdivrem(A, B, Q, R);
1363   if (R == 0)
1364     return Q;
1365   if ((A.sgt(0) && B.sgt(0)) ||
1366       (A.slt(0) && B.slt(0)))
1367     return Q;
1368   else
1369     return Q - 1;
1370 }
1371 
ceilingOfQuotient(const APInt & A,const APInt & B)1372 static APInt ceilingOfQuotient(const APInt &A, const APInt &B) {
1373   APInt Q = A; // these need to be initialized
1374   APInt R = A;
1375   APInt::sdivrem(A, B, Q, R);
1376   if (R == 0)
1377     return Q;
1378   if ((A.sgt(0) && B.sgt(0)) ||
1379       (A.slt(0) && B.slt(0)))
1380     return Q + 1;
1381   else
1382     return Q;
1383 }
1384 
1385 
1386 static
maxAPInt(APInt A,APInt B)1387 APInt maxAPInt(APInt A, APInt B) {
1388   return A.sgt(B) ? A : B;
1389 }
1390 
1391 
1392 static
minAPInt(APInt A,APInt B)1393 APInt minAPInt(APInt A, APInt B) {
1394   return A.slt(B) ? A : B;
1395 }
1396 
1397 
1398 // exactSIVtest -
1399 // When we have a pair of subscripts of the form [c1 + a1*i] and [c2 + a2*i],
1400 // where i is an induction variable, c1 and c2 are loop invariant, and a1
1401 // and a2 are constant, we can solve it exactly using an algorithm developed
1402 // by Banerjee and Wolfe. See Section 2.5.3 in
1403 //
1404 //        Optimizing Supercompilers for Supercomputers
1405 //        Michael Wolfe
1406 //        MIT Press, 1989
1407 //
1408 // It's slower than the specialized tests (strong SIV, weak-zero SIV, etc),
1409 // so use them if possible. They're also a bit better with symbolics and,
1410 // in the case of the strong SIV test, can compute Distances.
1411 //
1412 // Return true if dependence disproved.
exactSIVtest(const SCEV * SrcCoeff,const SCEV * DstCoeff,const SCEV * SrcConst,const SCEV * DstConst,const Loop * CurLoop,unsigned Level,FullDependence & Result,Constraint & NewConstraint) const1413 bool DependenceInfo::exactSIVtest(const SCEV *SrcCoeff, const SCEV *DstCoeff,
1414                                   const SCEV *SrcConst, const SCEV *DstConst,
1415                                   const Loop *CurLoop, unsigned Level,
1416                                   FullDependence &Result,
1417                                   Constraint &NewConstraint) const {
1418   DEBUG(dbgs() << "\tExact SIV test\n");
1419   DEBUG(dbgs() << "\t    SrcCoeff = " << *SrcCoeff << " = AM\n");
1420   DEBUG(dbgs() << "\t    DstCoeff = " << *DstCoeff << " = BM\n");
1421   DEBUG(dbgs() << "\t    SrcConst = " << *SrcConst << "\n");
1422   DEBUG(dbgs() << "\t    DstConst = " << *DstConst << "\n");
1423   ++ExactSIVapplications;
1424   assert(0 < Level && Level <= CommonLevels && "Level out of range");
1425   Level--;
1426   Result.Consistent = false;
1427   const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
1428   DEBUG(dbgs() << "\t    Delta = " << *Delta << "\n");
1429   NewConstraint.setLine(SrcCoeff, SE->getNegativeSCEV(DstCoeff),
1430                         Delta, CurLoop);
1431   const SCEVConstant *ConstDelta = dyn_cast<SCEVConstant>(Delta);
1432   const SCEVConstant *ConstSrcCoeff = dyn_cast<SCEVConstant>(SrcCoeff);
1433   const SCEVConstant *ConstDstCoeff = dyn_cast<SCEVConstant>(DstCoeff);
1434   if (!ConstDelta || !ConstSrcCoeff || !ConstDstCoeff)
1435     return false;
1436 
1437   // find gcd
1438   APInt G, X, Y;
1439   APInt AM = ConstSrcCoeff->getAPInt();
1440   APInt BM = ConstDstCoeff->getAPInt();
1441   unsigned Bits = AM.getBitWidth();
1442   if (findGCD(Bits, AM, BM, ConstDelta->getAPInt(), G, X, Y)) {
1443     // gcd doesn't divide Delta, no dependence
1444     ++ExactSIVindependence;
1445     ++ExactSIVsuccesses;
1446     return true;
1447   }
1448 
1449   DEBUG(dbgs() << "\t    X = " << X << ", Y = " << Y << "\n");
1450 
1451   // since SCEV construction normalizes, LM = 0
1452   APInt UM(Bits, 1, true);
1453   bool UMvalid = false;
1454   // UM is perhaps unavailable, let's check
1455   if (const SCEVConstant *CUB =
1456       collectConstantUpperBound(CurLoop, Delta->getType())) {
1457     UM = CUB->getAPInt();
1458     DEBUG(dbgs() << "\t    UM = " << UM << "\n");
1459     UMvalid = true;
1460   }
1461 
1462   APInt TU(APInt::getSignedMaxValue(Bits));
1463   APInt TL(APInt::getSignedMinValue(Bits));
1464 
1465   // test(BM/G, LM-X) and test(-BM/G, X-UM)
1466   APInt TMUL = BM.sdiv(G);
1467   if (TMUL.sgt(0)) {
1468     TL = maxAPInt(TL, ceilingOfQuotient(-X, TMUL));
1469     DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1470     if (UMvalid) {
1471       TU = minAPInt(TU, floorOfQuotient(UM - X, TMUL));
1472       DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1473     }
1474   }
1475   else {
1476     TU = minAPInt(TU, floorOfQuotient(-X, TMUL));
1477     DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1478     if (UMvalid) {
1479       TL = maxAPInt(TL, ceilingOfQuotient(UM - X, TMUL));
1480       DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1481     }
1482   }
1483 
1484   // test(AM/G, LM-Y) and test(-AM/G, Y-UM)
1485   TMUL = AM.sdiv(G);
1486   if (TMUL.sgt(0)) {
1487     TL = maxAPInt(TL, ceilingOfQuotient(-Y, TMUL));
1488     DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1489     if (UMvalid) {
1490       TU = minAPInt(TU, floorOfQuotient(UM - Y, TMUL));
1491       DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1492     }
1493   }
1494   else {
1495     TU = minAPInt(TU, floorOfQuotient(-Y, TMUL));
1496     DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1497     if (UMvalid) {
1498       TL = maxAPInt(TL, ceilingOfQuotient(UM - Y, TMUL));
1499       DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1500     }
1501   }
1502   if (TL.sgt(TU)) {
1503     ++ExactSIVindependence;
1504     ++ExactSIVsuccesses;
1505     return true;
1506   }
1507 
1508   // explore directions
1509   unsigned NewDirection = Dependence::DVEntry::NONE;
1510 
1511   // less than
1512   APInt SaveTU(TU); // save these
1513   APInt SaveTL(TL);
1514   DEBUG(dbgs() << "\t    exploring LT direction\n");
1515   TMUL = AM - BM;
1516   if (TMUL.sgt(0)) {
1517     TL = maxAPInt(TL, ceilingOfQuotient(X - Y + 1, TMUL));
1518     DEBUG(dbgs() << "\t\t    TL = " << TL << "\n");
1519   }
1520   else {
1521     TU = minAPInt(TU, floorOfQuotient(X - Y + 1, TMUL));
1522     DEBUG(dbgs() << "\t\t    TU = " << TU << "\n");
1523   }
1524   if (TL.sle(TU)) {
1525     NewDirection |= Dependence::DVEntry::LT;
1526     ++ExactSIVsuccesses;
1527   }
1528 
1529   // equal
1530   TU = SaveTU; // restore
1531   TL = SaveTL;
1532   DEBUG(dbgs() << "\t    exploring EQ direction\n");
1533   if (TMUL.sgt(0)) {
1534     TL = maxAPInt(TL, ceilingOfQuotient(X - Y, TMUL));
1535     DEBUG(dbgs() << "\t\t    TL = " << TL << "\n");
1536   }
1537   else {
1538     TU = minAPInt(TU, floorOfQuotient(X - Y, TMUL));
1539     DEBUG(dbgs() << "\t\t    TU = " << TU << "\n");
1540   }
1541   TMUL = BM - AM;
1542   if (TMUL.sgt(0)) {
1543     TL = maxAPInt(TL, ceilingOfQuotient(Y - X, TMUL));
1544     DEBUG(dbgs() << "\t\t    TL = " << TL << "\n");
1545   }
1546   else {
1547     TU = minAPInt(TU, floorOfQuotient(Y - X, TMUL));
1548     DEBUG(dbgs() << "\t\t    TU = " << TU << "\n");
1549   }
1550   if (TL.sle(TU)) {
1551     NewDirection |= Dependence::DVEntry::EQ;
1552     ++ExactSIVsuccesses;
1553   }
1554 
1555   // greater than
1556   TU = SaveTU; // restore
1557   TL = SaveTL;
1558   DEBUG(dbgs() << "\t    exploring GT direction\n");
1559   if (TMUL.sgt(0)) {
1560     TL = maxAPInt(TL, ceilingOfQuotient(Y - X + 1, TMUL));
1561     DEBUG(dbgs() << "\t\t    TL = " << TL << "\n");
1562   }
1563   else {
1564     TU = minAPInt(TU, floorOfQuotient(Y - X + 1, TMUL));
1565     DEBUG(dbgs() << "\t\t    TU = " << TU << "\n");
1566   }
1567   if (TL.sle(TU)) {
1568     NewDirection |= Dependence::DVEntry::GT;
1569     ++ExactSIVsuccesses;
1570   }
1571 
1572   // finished
1573   Result.DV[Level].Direction &= NewDirection;
1574   if (Result.DV[Level].Direction == Dependence::DVEntry::NONE)
1575     ++ExactSIVindependence;
1576   return Result.DV[Level].Direction == Dependence::DVEntry::NONE;
1577 }
1578 
1579 
1580 
1581 // Return true if the divisor evenly divides the dividend.
1582 static
isRemainderZero(const SCEVConstant * Dividend,const SCEVConstant * Divisor)1583 bool isRemainderZero(const SCEVConstant *Dividend,
1584                      const SCEVConstant *Divisor) {
1585   const APInt &ConstDividend = Dividend->getAPInt();
1586   const APInt &ConstDivisor = Divisor->getAPInt();
1587   return ConstDividend.srem(ConstDivisor) == 0;
1588 }
1589 
1590 
1591 // weakZeroSrcSIVtest -
1592 // From the paper, Practical Dependence Testing, Section 4.2.2
1593 //
1594 // When we have a pair of subscripts of the form [c1] and [c2 + a*i],
1595 // where i is an induction variable, c1 and c2 are loop invariant,
1596 // and a is a constant, we can solve it exactly using the
1597 // Weak-Zero SIV test.
1598 //
1599 // Given
1600 //
1601 //    c1 = c2 + a*i
1602 //
1603 // we get
1604 //
1605 //    (c1 - c2)/a = i
1606 //
1607 // If i is not an integer, there's no dependence.
1608 // If i < 0 or > UB, there's no dependence.
1609 // If i = 0, the direction is <= and peeling the
1610 // 1st iteration will break the dependence.
1611 // If i = UB, the direction is >= and peeling the
1612 // last iteration will break the dependence.
1613 // Otherwise, the direction is *.
1614 //
1615 // Can prove independence. Failing that, we can sometimes refine
1616 // the directions. Can sometimes show that first or last
1617 // iteration carries all the dependences (so worth peeling).
1618 //
1619 // (see also weakZeroDstSIVtest)
1620 //
1621 // Return true if dependence disproved.
weakZeroSrcSIVtest(const SCEV * DstCoeff,const SCEV * SrcConst,const SCEV * DstConst,const Loop * CurLoop,unsigned Level,FullDependence & Result,Constraint & NewConstraint) const1622 bool DependenceInfo::weakZeroSrcSIVtest(const SCEV *DstCoeff,
1623                                         const SCEV *SrcConst,
1624                                         const SCEV *DstConst,
1625                                         const Loop *CurLoop, unsigned Level,
1626                                         FullDependence &Result,
1627                                         Constraint &NewConstraint) const {
1628   // For the WeakSIV test, it's possible the loop isn't common to
1629   // the Src and Dst loops. If it isn't, then there's no need to
1630   // record a direction.
1631   DEBUG(dbgs() << "\tWeak-Zero (src) SIV test\n");
1632   DEBUG(dbgs() << "\t    DstCoeff = " << *DstCoeff << "\n");
1633   DEBUG(dbgs() << "\t    SrcConst = " << *SrcConst << "\n");
1634   DEBUG(dbgs() << "\t    DstConst = " << *DstConst << "\n");
1635   ++WeakZeroSIVapplications;
1636   assert(0 < Level && Level <= MaxLevels && "Level out of range");
1637   Level--;
1638   Result.Consistent = false;
1639   const SCEV *Delta = SE->getMinusSCEV(SrcConst, DstConst);
1640   NewConstraint.setLine(SE->getZero(Delta->getType()), DstCoeff, Delta,
1641                         CurLoop);
1642   DEBUG(dbgs() << "\t    Delta = " << *Delta << "\n");
1643   if (isKnownPredicate(CmpInst::ICMP_EQ, SrcConst, DstConst)) {
1644     if (Level < CommonLevels) {
1645       Result.DV[Level].Direction &= Dependence::DVEntry::LE;
1646       Result.DV[Level].PeelFirst = true;
1647       ++WeakZeroSIVsuccesses;
1648     }
1649     return false; // dependences caused by first iteration
1650   }
1651   const SCEVConstant *ConstCoeff = dyn_cast<SCEVConstant>(DstCoeff);
1652   if (!ConstCoeff)
1653     return false;
1654   const SCEV *AbsCoeff =
1655     SE->isKnownNegative(ConstCoeff) ?
1656     SE->getNegativeSCEV(ConstCoeff) : ConstCoeff;
1657   const SCEV *NewDelta =
1658     SE->isKnownNegative(ConstCoeff) ? SE->getNegativeSCEV(Delta) : Delta;
1659 
1660   // check that Delta/SrcCoeff < iteration count
1661   // really check NewDelta < count*AbsCoeff
1662   if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
1663     DEBUG(dbgs() << "\t    UpperBound = " << *UpperBound << "\n");
1664     const SCEV *Product = SE->getMulExpr(AbsCoeff, UpperBound);
1665     if (isKnownPredicate(CmpInst::ICMP_SGT, NewDelta, Product)) {
1666       ++WeakZeroSIVindependence;
1667       ++WeakZeroSIVsuccesses;
1668       return true;
1669     }
1670     if (isKnownPredicate(CmpInst::ICMP_EQ, NewDelta, Product)) {
1671       // dependences caused by last iteration
1672       if (Level < CommonLevels) {
1673         Result.DV[Level].Direction &= Dependence::DVEntry::GE;
1674         Result.DV[Level].PeelLast = true;
1675         ++WeakZeroSIVsuccesses;
1676       }
1677       return false;
1678     }
1679   }
1680 
1681   // check that Delta/SrcCoeff >= 0
1682   // really check that NewDelta >= 0
1683   if (SE->isKnownNegative(NewDelta)) {
1684     // No dependence, newDelta < 0
1685     ++WeakZeroSIVindependence;
1686     ++WeakZeroSIVsuccesses;
1687     return true;
1688   }
1689 
1690   // if SrcCoeff doesn't divide Delta, then no dependence
1691   if (isa<SCEVConstant>(Delta) &&
1692       !isRemainderZero(cast<SCEVConstant>(Delta), ConstCoeff)) {
1693     ++WeakZeroSIVindependence;
1694     ++WeakZeroSIVsuccesses;
1695     return true;
1696   }
1697   return false;
1698 }
1699 
1700 
1701 // weakZeroDstSIVtest -
1702 // From the paper, Practical Dependence Testing, Section 4.2.2
1703 //
1704 // When we have a pair of subscripts of the form [c1 + a*i] and [c2],
1705 // where i is an induction variable, c1 and c2 are loop invariant,
1706 // and a is a constant, we can solve it exactly using the
1707 // Weak-Zero SIV test.
1708 //
1709 // Given
1710 //
1711 //    c1 + a*i = c2
1712 //
1713 // we get
1714 //
1715 //    i = (c2 - c1)/a
1716 //
1717 // If i is not an integer, there's no dependence.
1718 // If i < 0 or > UB, there's no dependence.
1719 // If i = 0, the direction is <= and peeling the
1720 // 1st iteration will break the dependence.
1721 // If i = UB, the direction is >= and peeling the
1722 // last iteration will break the dependence.
1723 // Otherwise, the direction is *.
1724 //
1725 // Can prove independence. Failing that, we can sometimes refine
1726 // the directions. Can sometimes show that first or last
1727 // iteration carries all the dependences (so worth peeling).
1728 //
1729 // (see also weakZeroSrcSIVtest)
1730 //
1731 // Return true if dependence disproved.
weakZeroDstSIVtest(const SCEV * SrcCoeff,const SCEV * SrcConst,const SCEV * DstConst,const Loop * CurLoop,unsigned Level,FullDependence & Result,Constraint & NewConstraint) const1732 bool DependenceInfo::weakZeroDstSIVtest(const SCEV *SrcCoeff,
1733                                         const SCEV *SrcConst,
1734                                         const SCEV *DstConst,
1735                                         const Loop *CurLoop, unsigned Level,
1736                                         FullDependence &Result,
1737                                         Constraint &NewConstraint) const {
1738   // For the WeakSIV test, it's possible the loop isn't common to the
1739   // Src and Dst loops. If it isn't, then there's no need to record a direction.
1740   DEBUG(dbgs() << "\tWeak-Zero (dst) SIV test\n");
1741   DEBUG(dbgs() << "\t    SrcCoeff = " << *SrcCoeff << "\n");
1742   DEBUG(dbgs() << "\t    SrcConst = " << *SrcConst << "\n");
1743   DEBUG(dbgs() << "\t    DstConst = " << *DstConst << "\n");
1744   ++WeakZeroSIVapplications;
1745   assert(0 < Level && Level <= SrcLevels && "Level out of range");
1746   Level--;
1747   Result.Consistent = false;
1748   const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
1749   NewConstraint.setLine(SrcCoeff, SE->getZero(Delta->getType()), Delta,
1750                         CurLoop);
1751   DEBUG(dbgs() << "\t    Delta = " << *Delta << "\n");
1752   if (isKnownPredicate(CmpInst::ICMP_EQ, DstConst, SrcConst)) {
1753     if (Level < CommonLevels) {
1754       Result.DV[Level].Direction &= Dependence::DVEntry::LE;
1755       Result.DV[Level].PeelFirst = true;
1756       ++WeakZeroSIVsuccesses;
1757     }
1758     return false; // dependences caused by first iteration
1759   }
1760   const SCEVConstant *ConstCoeff = dyn_cast<SCEVConstant>(SrcCoeff);
1761   if (!ConstCoeff)
1762     return false;
1763   const SCEV *AbsCoeff =
1764     SE->isKnownNegative(ConstCoeff) ?
1765     SE->getNegativeSCEV(ConstCoeff) : ConstCoeff;
1766   const SCEV *NewDelta =
1767     SE->isKnownNegative(ConstCoeff) ? SE->getNegativeSCEV(Delta) : Delta;
1768 
1769   // check that Delta/SrcCoeff < iteration count
1770   // really check NewDelta < count*AbsCoeff
1771   if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
1772     DEBUG(dbgs() << "\t    UpperBound = " << *UpperBound << "\n");
1773     const SCEV *Product = SE->getMulExpr(AbsCoeff, UpperBound);
1774     if (isKnownPredicate(CmpInst::ICMP_SGT, NewDelta, Product)) {
1775       ++WeakZeroSIVindependence;
1776       ++WeakZeroSIVsuccesses;
1777       return true;
1778     }
1779     if (isKnownPredicate(CmpInst::ICMP_EQ, NewDelta, Product)) {
1780       // dependences caused by last iteration
1781       if (Level < CommonLevels) {
1782         Result.DV[Level].Direction &= Dependence::DVEntry::GE;
1783         Result.DV[Level].PeelLast = true;
1784         ++WeakZeroSIVsuccesses;
1785       }
1786       return false;
1787     }
1788   }
1789 
1790   // check that Delta/SrcCoeff >= 0
1791   // really check that NewDelta >= 0
1792   if (SE->isKnownNegative(NewDelta)) {
1793     // No dependence, newDelta < 0
1794     ++WeakZeroSIVindependence;
1795     ++WeakZeroSIVsuccesses;
1796     return true;
1797   }
1798 
1799   // if SrcCoeff doesn't divide Delta, then no dependence
1800   if (isa<SCEVConstant>(Delta) &&
1801       !isRemainderZero(cast<SCEVConstant>(Delta), ConstCoeff)) {
1802     ++WeakZeroSIVindependence;
1803     ++WeakZeroSIVsuccesses;
1804     return true;
1805   }
1806   return false;
1807 }
1808 
1809 
1810 // exactRDIVtest - Tests the RDIV subscript pair for dependence.
1811 // Things of the form [c1 + a*i] and [c2 + b*j],
1812 // where i and j are induction variable, c1 and c2 are loop invariant,
1813 // and a and b are constants.
1814 // Returns true if any possible dependence is disproved.
1815 // Marks the result as inconsistent.
1816 // Works in some cases that symbolicRDIVtest doesn't, and vice versa.
exactRDIVtest(const SCEV * SrcCoeff,const SCEV * DstCoeff,const SCEV * SrcConst,const SCEV * DstConst,const Loop * SrcLoop,const Loop * DstLoop,FullDependence & Result) const1817 bool DependenceInfo::exactRDIVtest(const SCEV *SrcCoeff, const SCEV *DstCoeff,
1818                                    const SCEV *SrcConst, const SCEV *DstConst,
1819                                    const Loop *SrcLoop, const Loop *DstLoop,
1820                                    FullDependence &Result) const {
1821   DEBUG(dbgs() << "\tExact RDIV test\n");
1822   DEBUG(dbgs() << "\t    SrcCoeff = " << *SrcCoeff << " = AM\n");
1823   DEBUG(dbgs() << "\t    DstCoeff = " << *DstCoeff << " = BM\n");
1824   DEBUG(dbgs() << "\t    SrcConst = " << *SrcConst << "\n");
1825   DEBUG(dbgs() << "\t    DstConst = " << *DstConst << "\n");
1826   ++ExactRDIVapplications;
1827   Result.Consistent = false;
1828   const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
1829   DEBUG(dbgs() << "\t    Delta = " << *Delta << "\n");
1830   const SCEVConstant *ConstDelta = dyn_cast<SCEVConstant>(Delta);
1831   const SCEVConstant *ConstSrcCoeff = dyn_cast<SCEVConstant>(SrcCoeff);
1832   const SCEVConstant *ConstDstCoeff = dyn_cast<SCEVConstant>(DstCoeff);
1833   if (!ConstDelta || !ConstSrcCoeff || !ConstDstCoeff)
1834     return false;
1835 
1836   // find gcd
1837   APInt G, X, Y;
1838   APInt AM = ConstSrcCoeff->getAPInt();
1839   APInt BM = ConstDstCoeff->getAPInt();
1840   unsigned Bits = AM.getBitWidth();
1841   if (findGCD(Bits, AM, BM, ConstDelta->getAPInt(), G, X, Y)) {
1842     // gcd doesn't divide Delta, no dependence
1843     ++ExactRDIVindependence;
1844     return true;
1845   }
1846 
1847   DEBUG(dbgs() << "\t    X = " << X << ", Y = " << Y << "\n");
1848 
1849   // since SCEV construction seems to normalize, LM = 0
1850   APInt SrcUM(Bits, 1, true);
1851   bool SrcUMvalid = false;
1852   // SrcUM is perhaps unavailable, let's check
1853   if (const SCEVConstant *UpperBound =
1854       collectConstantUpperBound(SrcLoop, Delta->getType())) {
1855     SrcUM = UpperBound->getAPInt();
1856     DEBUG(dbgs() << "\t    SrcUM = " << SrcUM << "\n");
1857     SrcUMvalid = true;
1858   }
1859 
1860   APInt DstUM(Bits, 1, true);
1861   bool DstUMvalid = false;
1862   // UM is perhaps unavailable, let's check
1863   if (const SCEVConstant *UpperBound =
1864       collectConstantUpperBound(DstLoop, Delta->getType())) {
1865     DstUM = UpperBound->getAPInt();
1866     DEBUG(dbgs() << "\t    DstUM = " << DstUM << "\n");
1867     DstUMvalid = true;
1868   }
1869 
1870   APInt TU(APInt::getSignedMaxValue(Bits));
1871   APInt TL(APInt::getSignedMinValue(Bits));
1872 
1873   // test(BM/G, LM-X) and test(-BM/G, X-UM)
1874   APInt TMUL = BM.sdiv(G);
1875   if (TMUL.sgt(0)) {
1876     TL = maxAPInt(TL, ceilingOfQuotient(-X, TMUL));
1877     DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1878     if (SrcUMvalid) {
1879       TU = minAPInt(TU, floorOfQuotient(SrcUM - X, TMUL));
1880       DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1881     }
1882   }
1883   else {
1884     TU = minAPInt(TU, floorOfQuotient(-X, TMUL));
1885     DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1886     if (SrcUMvalid) {
1887       TL = maxAPInt(TL, ceilingOfQuotient(SrcUM - X, TMUL));
1888       DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1889     }
1890   }
1891 
1892   // test(AM/G, LM-Y) and test(-AM/G, Y-UM)
1893   TMUL = AM.sdiv(G);
1894   if (TMUL.sgt(0)) {
1895     TL = maxAPInt(TL, ceilingOfQuotient(-Y, TMUL));
1896     DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1897     if (DstUMvalid) {
1898       TU = minAPInt(TU, floorOfQuotient(DstUM - Y, TMUL));
1899       DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1900     }
1901   }
1902   else {
1903     TU = minAPInt(TU, floorOfQuotient(-Y, TMUL));
1904     DEBUG(dbgs() << "\t    TU = " << TU << "\n");
1905     if (DstUMvalid) {
1906       TL = maxAPInt(TL, ceilingOfQuotient(DstUM - Y, TMUL));
1907       DEBUG(dbgs() << "\t    TL = " << TL << "\n");
1908     }
1909   }
1910   if (TL.sgt(TU))
1911     ++ExactRDIVindependence;
1912   return TL.sgt(TU);
1913 }
1914 
1915 
1916 // symbolicRDIVtest -
1917 // In Section 4.5 of the Practical Dependence Testing paper,the authors
1918 // introduce a special case of Banerjee's Inequalities (also called the
1919 // Extreme-Value Test) that can handle some of the SIV and RDIV cases,
1920 // particularly cases with symbolics. Since it's only able to disprove
1921 // dependence (not compute distances or directions), we'll use it as a
1922 // fall back for the other tests.
1923 //
1924 // When we have a pair of subscripts of the form [c1 + a1*i] and [c2 + a2*j]
1925 // where i and j are induction variables and c1 and c2 are loop invariants,
1926 // we can use the symbolic tests to disprove some dependences, serving as a
1927 // backup for the RDIV test. Note that i and j can be the same variable,
1928 // letting this test serve as a backup for the various SIV tests.
1929 //
1930 // For a dependence to exist, c1 + a1*i must equal c2 + a2*j for some
1931 //  0 <= i <= N1 and some 0 <= j <= N2, where N1 and N2 are the (normalized)
1932 // loop bounds for the i and j loops, respectively. So, ...
1933 //
1934 // c1 + a1*i = c2 + a2*j
1935 // a1*i - a2*j = c2 - c1
1936 //
1937 // To test for a dependence, we compute c2 - c1 and make sure it's in the
1938 // range of the maximum and minimum possible values of a1*i - a2*j.
1939 // Considering the signs of a1 and a2, we have 4 possible cases:
1940 //
1941 // 1) If a1 >= 0 and a2 >= 0, then
1942 //        a1*0 - a2*N2 <= c2 - c1 <= a1*N1 - a2*0
1943 //              -a2*N2 <= c2 - c1 <= a1*N1
1944 //
1945 // 2) If a1 >= 0 and a2 <= 0, then
1946 //        a1*0 - a2*0 <= c2 - c1 <= a1*N1 - a2*N2
1947 //                  0 <= c2 - c1 <= a1*N1 - a2*N2
1948 //
1949 // 3) If a1 <= 0 and a2 >= 0, then
1950 //        a1*N1 - a2*N2 <= c2 - c1 <= a1*0 - a2*0
1951 //        a1*N1 - a2*N2 <= c2 - c1 <= 0
1952 //
1953 // 4) If a1 <= 0 and a2 <= 0, then
1954 //        a1*N1 - a2*0  <= c2 - c1 <= a1*0 - a2*N2
1955 //        a1*N1         <= c2 - c1 <=       -a2*N2
1956 //
1957 // return true if dependence disproved
symbolicRDIVtest(const SCEV * A1,const SCEV * A2,const SCEV * C1,const SCEV * C2,const Loop * Loop1,const Loop * Loop2) const1958 bool DependenceInfo::symbolicRDIVtest(const SCEV *A1, const SCEV *A2,
1959                                       const SCEV *C1, const SCEV *C2,
1960                                       const Loop *Loop1,
1961                                       const Loop *Loop2) const {
1962   ++SymbolicRDIVapplications;
1963   DEBUG(dbgs() << "\ttry symbolic RDIV test\n");
1964   DEBUG(dbgs() << "\t    A1 = " << *A1);
1965   DEBUG(dbgs() << ", type = " << *A1->getType() << "\n");
1966   DEBUG(dbgs() << "\t    A2 = " << *A2 << "\n");
1967   DEBUG(dbgs() << "\t    C1 = " << *C1 << "\n");
1968   DEBUG(dbgs() << "\t    C2 = " << *C2 << "\n");
1969   const SCEV *N1 = collectUpperBound(Loop1, A1->getType());
1970   const SCEV *N2 = collectUpperBound(Loop2, A1->getType());
1971   DEBUG(if (N1) dbgs() << "\t    N1 = " << *N1 << "\n");
1972   DEBUG(if (N2) dbgs() << "\t    N2 = " << *N2 << "\n");
1973   const SCEV *C2_C1 = SE->getMinusSCEV(C2, C1);
1974   const SCEV *C1_C2 = SE->getMinusSCEV(C1, C2);
1975   DEBUG(dbgs() << "\t    C2 - C1 = " << *C2_C1 << "\n");
1976   DEBUG(dbgs() << "\t    C1 - C2 = " << *C1_C2 << "\n");
1977   if (SE->isKnownNonNegative(A1)) {
1978     if (SE->isKnownNonNegative(A2)) {
1979       // A1 >= 0 && A2 >= 0
1980       if (N1) {
1981         // make sure that c2 - c1 <= a1*N1
1982         const SCEV *A1N1 = SE->getMulExpr(A1, N1);
1983         DEBUG(dbgs() << "\t    A1*N1 = " << *A1N1 << "\n");
1984         if (isKnownPredicate(CmpInst::ICMP_SGT, C2_C1, A1N1)) {
1985           ++SymbolicRDIVindependence;
1986           return true;
1987         }
1988       }
1989       if (N2) {
1990         // make sure that -a2*N2 <= c2 - c1, or a2*N2 >= c1 - c2
1991         const SCEV *A2N2 = SE->getMulExpr(A2, N2);
1992         DEBUG(dbgs() << "\t    A2*N2 = " << *A2N2 << "\n");
1993         if (isKnownPredicate(CmpInst::ICMP_SLT, A2N2, C1_C2)) {
1994           ++SymbolicRDIVindependence;
1995           return true;
1996         }
1997       }
1998     }
1999     else if (SE->isKnownNonPositive(A2)) {
2000       // a1 >= 0 && a2 <= 0
2001       if (N1 && N2) {
2002         // make sure that c2 - c1 <= a1*N1 - a2*N2
2003         const SCEV *A1N1 = SE->getMulExpr(A1, N1);
2004         const SCEV *A2N2 = SE->getMulExpr(A2, N2);
2005         const SCEV *A1N1_A2N2 = SE->getMinusSCEV(A1N1, A2N2);
2006         DEBUG(dbgs() << "\t    A1*N1 - A2*N2 = " << *A1N1_A2N2 << "\n");
2007         if (isKnownPredicate(CmpInst::ICMP_SGT, C2_C1, A1N1_A2N2)) {
2008           ++SymbolicRDIVindependence;
2009           return true;
2010         }
2011       }
2012       // make sure that 0 <= c2 - c1
2013       if (SE->isKnownNegative(C2_C1)) {
2014         ++SymbolicRDIVindependence;
2015         return true;
2016       }
2017     }
2018   }
2019   else if (SE->isKnownNonPositive(A1)) {
2020     if (SE->isKnownNonNegative(A2)) {
2021       // a1 <= 0 && a2 >= 0
2022       if (N1 && N2) {
2023         // make sure that a1*N1 - a2*N2 <= c2 - c1
2024         const SCEV *A1N1 = SE->getMulExpr(A1, N1);
2025         const SCEV *A2N2 = SE->getMulExpr(A2, N2);
2026         const SCEV *A1N1_A2N2 = SE->getMinusSCEV(A1N1, A2N2);
2027         DEBUG(dbgs() << "\t    A1*N1 - A2*N2 = " << *A1N1_A2N2 << "\n");
2028         if (isKnownPredicate(CmpInst::ICMP_SGT, A1N1_A2N2, C2_C1)) {
2029           ++SymbolicRDIVindependence;
2030           return true;
2031         }
2032       }
2033       // make sure that c2 - c1 <= 0
2034       if (SE->isKnownPositive(C2_C1)) {
2035         ++SymbolicRDIVindependence;
2036         return true;
2037       }
2038     }
2039     else if (SE->isKnownNonPositive(A2)) {
2040       // a1 <= 0 && a2 <= 0
2041       if (N1) {
2042         // make sure that a1*N1 <= c2 - c1
2043         const SCEV *A1N1 = SE->getMulExpr(A1, N1);
2044         DEBUG(dbgs() << "\t    A1*N1 = " << *A1N1 << "\n");
2045         if (isKnownPredicate(CmpInst::ICMP_SGT, A1N1, C2_C1)) {
2046           ++SymbolicRDIVindependence;
2047           return true;
2048         }
2049       }
2050       if (N2) {
2051         // make sure that c2 - c1 <= -a2*N2, or c1 - c2 >= a2*N2
2052         const SCEV *A2N2 = SE->getMulExpr(A2, N2);
2053         DEBUG(dbgs() << "\t    A2*N2 = " << *A2N2 << "\n");
2054         if (isKnownPredicate(CmpInst::ICMP_SLT, C1_C2, A2N2)) {
2055           ++SymbolicRDIVindependence;
2056           return true;
2057         }
2058       }
2059     }
2060   }
2061   return false;
2062 }
2063 
2064 
2065 // testSIV -
2066 // When we have a pair of subscripts of the form [c1 + a1*i] and [c2 - a2*i]
2067 // where i is an induction variable, c1 and c2 are loop invariant, and a1 and
2068 // a2 are constant, we attack it with an SIV test. While they can all be
2069 // solved with the Exact SIV test, it's worthwhile to use simpler tests when
2070 // they apply; they're cheaper and sometimes more precise.
2071 //
2072 // Return true if dependence disproved.
testSIV(const SCEV * Src,const SCEV * Dst,unsigned & Level,FullDependence & Result,Constraint & NewConstraint,const SCEV * & SplitIter) const2073 bool DependenceInfo::testSIV(const SCEV *Src, const SCEV *Dst, unsigned &Level,
2074                              FullDependence &Result, Constraint &NewConstraint,
2075                              const SCEV *&SplitIter) const {
2076   DEBUG(dbgs() << "    src = " << *Src << "\n");
2077   DEBUG(dbgs() << "    dst = " << *Dst << "\n");
2078   const SCEVAddRecExpr *SrcAddRec = dyn_cast<SCEVAddRecExpr>(Src);
2079   const SCEVAddRecExpr *DstAddRec = dyn_cast<SCEVAddRecExpr>(Dst);
2080   if (SrcAddRec && DstAddRec) {
2081     const SCEV *SrcConst = SrcAddRec->getStart();
2082     const SCEV *DstConst = DstAddRec->getStart();
2083     const SCEV *SrcCoeff = SrcAddRec->getStepRecurrence(*SE);
2084     const SCEV *DstCoeff = DstAddRec->getStepRecurrence(*SE);
2085     const Loop *CurLoop = SrcAddRec->getLoop();
2086     assert(CurLoop == DstAddRec->getLoop() &&
2087            "both loops in SIV should be same");
2088     Level = mapSrcLoop(CurLoop);
2089     bool disproven;
2090     if (SrcCoeff == DstCoeff)
2091       disproven = strongSIVtest(SrcCoeff, SrcConst, DstConst, CurLoop,
2092                                 Level, Result, NewConstraint);
2093     else if (SrcCoeff == SE->getNegativeSCEV(DstCoeff))
2094       disproven = weakCrossingSIVtest(SrcCoeff, SrcConst, DstConst, CurLoop,
2095                                       Level, Result, NewConstraint, SplitIter);
2096     else
2097       disproven = exactSIVtest(SrcCoeff, DstCoeff, SrcConst, DstConst, CurLoop,
2098                                Level, Result, NewConstraint);
2099     return disproven ||
2100       gcdMIVtest(Src, Dst, Result) ||
2101       symbolicRDIVtest(SrcCoeff, DstCoeff, SrcConst, DstConst, CurLoop, CurLoop);
2102   }
2103   if (SrcAddRec) {
2104     const SCEV *SrcConst = SrcAddRec->getStart();
2105     const SCEV *SrcCoeff = SrcAddRec->getStepRecurrence(*SE);
2106     const SCEV *DstConst = Dst;
2107     const Loop *CurLoop = SrcAddRec->getLoop();
2108     Level = mapSrcLoop(CurLoop);
2109     return weakZeroDstSIVtest(SrcCoeff, SrcConst, DstConst, CurLoop,
2110                               Level, Result, NewConstraint) ||
2111       gcdMIVtest(Src, Dst, Result);
2112   }
2113   if (DstAddRec) {
2114     const SCEV *DstConst = DstAddRec->getStart();
2115     const SCEV *DstCoeff = DstAddRec->getStepRecurrence(*SE);
2116     const SCEV *SrcConst = Src;
2117     const Loop *CurLoop = DstAddRec->getLoop();
2118     Level = mapDstLoop(CurLoop);
2119     return weakZeroSrcSIVtest(DstCoeff, SrcConst, DstConst,
2120                               CurLoop, Level, Result, NewConstraint) ||
2121       gcdMIVtest(Src, Dst, Result);
2122   }
2123   llvm_unreachable("SIV test expected at least one AddRec");
2124   return false;
2125 }
2126 
2127 
2128 // testRDIV -
2129 // When we have a pair of subscripts of the form [c1 + a1*i] and [c2 + a2*j]
2130 // where i and j are induction variables, c1 and c2 are loop invariant,
2131 // and a1 and a2 are constant, we can solve it exactly with an easy adaptation
2132 // of the Exact SIV test, the Restricted Double Index Variable (RDIV) test.
2133 // It doesn't make sense to talk about distance or direction in this case,
2134 // so there's no point in making special versions of the Strong SIV test or
2135 // the Weak-crossing SIV test.
2136 //
2137 // With minor algebra, this test can also be used for things like
2138 // [c1 + a1*i + a2*j][c2].
2139 //
2140 // Return true if dependence disproved.
testRDIV(const SCEV * Src,const SCEV * Dst,FullDependence & Result) const2141 bool DependenceInfo::testRDIV(const SCEV *Src, const SCEV *Dst,
2142                               FullDependence &Result) const {
2143   // we have 3 possible situations here:
2144   //   1) [a*i + b] and [c*j + d]
2145   //   2) [a*i + c*j + b] and [d]
2146   //   3) [b] and [a*i + c*j + d]
2147   // We need to find what we've got and get organized
2148 
2149   const SCEV *SrcConst, *DstConst;
2150   const SCEV *SrcCoeff, *DstCoeff;
2151   const Loop *SrcLoop, *DstLoop;
2152 
2153   DEBUG(dbgs() << "    src = " << *Src << "\n");
2154   DEBUG(dbgs() << "    dst = " << *Dst << "\n");
2155   const SCEVAddRecExpr *SrcAddRec = dyn_cast<SCEVAddRecExpr>(Src);
2156   const SCEVAddRecExpr *DstAddRec = dyn_cast<SCEVAddRecExpr>(Dst);
2157   if (SrcAddRec && DstAddRec) {
2158     SrcConst = SrcAddRec->getStart();
2159     SrcCoeff = SrcAddRec->getStepRecurrence(*SE);
2160     SrcLoop = SrcAddRec->getLoop();
2161     DstConst = DstAddRec->getStart();
2162     DstCoeff = DstAddRec->getStepRecurrence(*SE);
2163     DstLoop = DstAddRec->getLoop();
2164   }
2165   else if (SrcAddRec) {
2166     if (const SCEVAddRecExpr *tmpAddRec =
2167         dyn_cast<SCEVAddRecExpr>(SrcAddRec->getStart())) {
2168       SrcConst = tmpAddRec->getStart();
2169       SrcCoeff = tmpAddRec->getStepRecurrence(*SE);
2170       SrcLoop = tmpAddRec->getLoop();
2171       DstConst = Dst;
2172       DstCoeff = SE->getNegativeSCEV(SrcAddRec->getStepRecurrence(*SE));
2173       DstLoop = SrcAddRec->getLoop();
2174     }
2175     else
2176       llvm_unreachable("RDIV reached by surprising SCEVs");
2177   }
2178   else if (DstAddRec) {
2179     if (const SCEVAddRecExpr *tmpAddRec =
2180         dyn_cast<SCEVAddRecExpr>(DstAddRec->getStart())) {
2181       DstConst = tmpAddRec->getStart();
2182       DstCoeff = tmpAddRec->getStepRecurrence(*SE);
2183       DstLoop = tmpAddRec->getLoop();
2184       SrcConst = Src;
2185       SrcCoeff = SE->getNegativeSCEV(DstAddRec->getStepRecurrence(*SE));
2186       SrcLoop = DstAddRec->getLoop();
2187     }
2188     else
2189       llvm_unreachable("RDIV reached by surprising SCEVs");
2190   }
2191   else
2192     llvm_unreachable("RDIV expected at least one AddRec");
2193   return exactRDIVtest(SrcCoeff, DstCoeff,
2194                        SrcConst, DstConst,
2195                        SrcLoop, DstLoop,
2196                        Result) ||
2197     gcdMIVtest(Src, Dst, Result) ||
2198     symbolicRDIVtest(SrcCoeff, DstCoeff,
2199                      SrcConst, DstConst,
2200                      SrcLoop, DstLoop);
2201 }
2202 
2203 
2204 // Tests the single-subscript MIV pair (Src and Dst) for dependence.
2205 // Return true if dependence disproved.
2206 // Can sometimes refine direction vectors.
testMIV(const SCEV * Src,const SCEV * Dst,const SmallBitVector & Loops,FullDependence & Result) const2207 bool DependenceInfo::testMIV(const SCEV *Src, const SCEV *Dst,
2208                              const SmallBitVector &Loops,
2209                              FullDependence &Result) const {
2210   DEBUG(dbgs() << "    src = " << *Src << "\n");
2211   DEBUG(dbgs() << "    dst = " << *Dst << "\n");
2212   Result.Consistent = false;
2213   return gcdMIVtest(Src, Dst, Result) ||
2214     banerjeeMIVtest(Src, Dst, Loops, Result);
2215 }
2216 
2217 
2218 // Given a product, e.g., 10*X*Y, returns the first constant operand,
2219 // in this case 10. If there is no constant part, returns NULL.
2220 static
getConstantPart(const SCEV * Expr)2221 const SCEVConstant *getConstantPart(const SCEV *Expr) {
2222   if (const auto *Constant = dyn_cast<SCEVConstant>(Expr))
2223     return Constant;
2224   else if (const auto *Product = dyn_cast<SCEVMulExpr>(Expr))
2225     if (const auto *Constant = dyn_cast<SCEVConstant>(Product->getOperand(0)))
2226       return Constant;
2227   return nullptr;
2228 }
2229 
2230 
2231 //===----------------------------------------------------------------------===//
2232 // gcdMIVtest -
2233 // Tests an MIV subscript pair for dependence.
2234 // Returns true if any possible dependence is disproved.
2235 // Marks the result as inconsistent.
2236 // Can sometimes disprove the equal direction for 1 or more loops,
2237 // as discussed in Michael Wolfe's book,
2238 // High Performance Compilers for Parallel Computing, page 235.
2239 //
2240 // We spend some effort (code!) to handle cases like
2241 // [10*i + 5*N*j + 15*M + 6], where i and j are induction variables,
2242 // but M and N are just loop-invariant variables.
2243 // This should help us handle linearized subscripts;
2244 // also makes this test a useful backup to the various SIV tests.
2245 //
2246 // It occurs to me that the presence of loop-invariant variables
2247 // changes the nature of the test from "greatest common divisor"
2248 // to "a common divisor".
gcdMIVtest(const SCEV * Src,const SCEV * Dst,FullDependence & Result) const2249 bool DependenceInfo::gcdMIVtest(const SCEV *Src, const SCEV *Dst,
2250                                 FullDependence &Result) const {
2251   DEBUG(dbgs() << "starting gcd\n");
2252   ++GCDapplications;
2253   unsigned BitWidth = SE->getTypeSizeInBits(Src->getType());
2254   APInt RunningGCD = APInt::getNullValue(BitWidth);
2255 
2256   // Examine Src coefficients.
2257   // Compute running GCD and record source constant.
2258   // Because we're looking for the constant at the end of the chain,
2259   // we can't quit the loop just because the GCD == 1.
2260   const SCEV *Coefficients = Src;
2261   while (const SCEVAddRecExpr *AddRec =
2262          dyn_cast<SCEVAddRecExpr>(Coefficients)) {
2263     const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
2264     // If the coefficient is the product of a constant and other stuff,
2265     // we can use the constant in the GCD computation.
2266     const auto *Constant = getConstantPart(Coeff);
2267     if (!Constant)
2268       return false;
2269     APInt ConstCoeff = Constant->getAPInt();
2270     RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
2271     Coefficients = AddRec->getStart();
2272   }
2273   const SCEV *SrcConst = Coefficients;
2274 
2275   // Examine Dst coefficients.
2276   // Compute running GCD and record destination constant.
2277   // Because we're looking for the constant at the end of the chain,
2278   // we can't quit the loop just because the GCD == 1.
2279   Coefficients = Dst;
2280   while (const SCEVAddRecExpr *AddRec =
2281          dyn_cast<SCEVAddRecExpr>(Coefficients)) {
2282     const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
2283     // If the coefficient is the product of a constant and other stuff,
2284     // we can use the constant in the GCD computation.
2285     const auto *Constant = getConstantPart(Coeff);
2286     if (!Constant)
2287       return false;
2288     APInt ConstCoeff = Constant->getAPInt();
2289     RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
2290     Coefficients = AddRec->getStart();
2291   }
2292   const SCEV *DstConst = Coefficients;
2293 
2294   APInt ExtraGCD = APInt::getNullValue(BitWidth);
2295   const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
2296   DEBUG(dbgs() << "    Delta = " << *Delta << "\n");
2297   const SCEVConstant *Constant = dyn_cast<SCEVConstant>(Delta);
2298   if (const SCEVAddExpr *Sum = dyn_cast<SCEVAddExpr>(Delta)) {
2299     // If Delta is a sum of products, we may be able to make further progress.
2300     for (unsigned Op = 0, Ops = Sum->getNumOperands(); Op < Ops; Op++) {
2301       const SCEV *Operand = Sum->getOperand(Op);
2302       if (isa<SCEVConstant>(Operand)) {
2303         assert(!Constant && "Surprised to find multiple constants");
2304         Constant = cast<SCEVConstant>(Operand);
2305       }
2306       else if (const SCEVMulExpr *Product = dyn_cast<SCEVMulExpr>(Operand)) {
2307         // Search for constant operand to participate in GCD;
2308         // If none found; return false.
2309         const SCEVConstant *ConstOp = getConstantPart(Product);
2310         if (!ConstOp)
2311           return false;
2312         APInt ConstOpValue = ConstOp->getAPInt();
2313         ExtraGCD = APIntOps::GreatestCommonDivisor(ExtraGCD,
2314                                                    ConstOpValue.abs());
2315       }
2316       else
2317         return false;
2318     }
2319   }
2320   if (!Constant)
2321     return false;
2322   APInt ConstDelta = cast<SCEVConstant>(Constant)->getAPInt();
2323   DEBUG(dbgs() << "    ConstDelta = " << ConstDelta << "\n");
2324   if (ConstDelta == 0)
2325     return false;
2326   RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ExtraGCD);
2327   DEBUG(dbgs() << "    RunningGCD = " << RunningGCD << "\n");
2328   APInt Remainder = ConstDelta.srem(RunningGCD);
2329   if (Remainder != 0) {
2330     ++GCDindependence;
2331     return true;
2332   }
2333 
2334   // Try to disprove equal directions.
2335   // For example, given a subscript pair [3*i + 2*j] and [i' + 2*j' - 1],
2336   // the code above can't disprove the dependence because the GCD = 1.
2337   // So we consider what happen if i = i' and what happens if j = j'.
2338   // If i = i', we can simplify the subscript to [2*i + 2*j] and [2*j' - 1],
2339   // which is infeasible, so we can disallow the = direction for the i level.
2340   // Setting j = j' doesn't help matters, so we end up with a direction vector
2341   // of [<>, *]
2342   //
2343   // Given A[5*i + 10*j*M + 9*M*N] and A[15*i + 20*j*M - 21*N*M + 5],
2344   // we need to remember that the constant part is 5 and the RunningGCD should
2345   // be initialized to ExtraGCD = 30.
2346   DEBUG(dbgs() << "    ExtraGCD = " << ExtraGCD << '\n');
2347 
2348   bool Improved = false;
2349   Coefficients = Src;
2350   while (const SCEVAddRecExpr *AddRec =
2351          dyn_cast<SCEVAddRecExpr>(Coefficients)) {
2352     Coefficients = AddRec->getStart();
2353     const Loop *CurLoop = AddRec->getLoop();
2354     RunningGCD = ExtraGCD;
2355     const SCEV *SrcCoeff = AddRec->getStepRecurrence(*SE);
2356     const SCEV *DstCoeff = SE->getMinusSCEV(SrcCoeff, SrcCoeff);
2357     const SCEV *Inner = Src;
2358     while (RunningGCD != 1 && isa<SCEVAddRecExpr>(Inner)) {
2359       AddRec = cast<SCEVAddRecExpr>(Inner);
2360       const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
2361       if (CurLoop == AddRec->getLoop())
2362         ; // SrcCoeff == Coeff
2363       else {
2364         // If the coefficient is the product of a constant and other stuff,
2365         // we can use the constant in the GCD computation.
2366         Constant = getConstantPart(Coeff);
2367         if (!Constant)
2368           return false;
2369         APInt ConstCoeff = Constant->getAPInt();
2370         RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
2371       }
2372       Inner = AddRec->getStart();
2373     }
2374     Inner = Dst;
2375     while (RunningGCD != 1 && isa<SCEVAddRecExpr>(Inner)) {
2376       AddRec = cast<SCEVAddRecExpr>(Inner);
2377       const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
2378       if (CurLoop == AddRec->getLoop())
2379         DstCoeff = Coeff;
2380       else {
2381         // If the coefficient is the product of a constant and other stuff,
2382         // we can use the constant in the GCD computation.
2383         Constant = getConstantPart(Coeff);
2384         if (!Constant)
2385           return false;
2386         APInt ConstCoeff = Constant->getAPInt();
2387         RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
2388       }
2389       Inner = AddRec->getStart();
2390     }
2391     Delta = SE->getMinusSCEV(SrcCoeff, DstCoeff);
2392     // If the coefficient is the product of a constant and other stuff,
2393     // we can use the constant in the GCD computation.
2394     Constant = getConstantPart(Delta);
2395     if (!Constant)
2396       // The difference of the two coefficients might not be a product
2397       // or constant, in which case we give up on this direction.
2398       continue;
2399     APInt ConstCoeff = Constant->getAPInt();
2400     RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
2401     DEBUG(dbgs() << "\tRunningGCD = " << RunningGCD << "\n");
2402     if (RunningGCD != 0) {
2403       Remainder = ConstDelta.srem(RunningGCD);
2404       DEBUG(dbgs() << "\tRemainder = " << Remainder << "\n");
2405       if (Remainder != 0) {
2406         unsigned Level = mapSrcLoop(CurLoop);
2407         Result.DV[Level - 1].Direction &= unsigned(~Dependence::DVEntry::EQ);
2408         Improved = true;
2409       }
2410     }
2411   }
2412   if (Improved)
2413     ++GCDsuccesses;
2414   DEBUG(dbgs() << "all done\n");
2415   return false;
2416 }
2417 
2418 
2419 //===----------------------------------------------------------------------===//
2420 // banerjeeMIVtest -
2421 // Use Banerjee's Inequalities to test an MIV subscript pair.
2422 // (Wolfe, in the race-car book, calls this the Extreme Value Test.)
2423 // Generally follows the discussion in Section 2.5.2 of
2424 //
2425 //    Optimizing Supercompilers for Supercomputers
2426 //    Michael Wolfe
2427 //
2428 // The inequalities given on page 25 are simplified in that loops are
2429 // normalized so that the lower bound is always 0 and the stride is always 1.
2430 // For example, Wolfe gives
2431 //
2432 //     LB^<_k = (A^-_k - B_k)^- (U_k - L_k - N_k) + (A_k - B_k)L_k - B_k N_k
2433 //
2434 // where A_k is the coefficient of the kth index in the source subscript,
2435 // B_k is the coefficient of the kth index in the destination subscript,
2436 // U_k is the upper bound of the kth index, L_k is the lower bound of the Kth
2437 // index, and N_k is the stride of the kth index. Since all loops are normalized
2438 // by the SCEV package, N_k = 1 and L_k = 0, allowing us to simplify the
2439 // equation to
2440 //
2441 //     LB^<_k = (A^-_k - B_k)^- (U_k - 0 - 1) + (A_k - B_k)0 - B_k 1
2442 //            = (A^-_k - B_k)^- (U_k - 1)  - B_k
2443 //
2444 // Similar simplifications are possible for the other equations.
2445 //
2446 // When we can't determine the number of iterations for a loop,
2447 // we use NULL as an indicator for the worst case, infinity.
2448 // When computing the upper bound, NULL denotes +inf;
2449 // for the lower bound, NULL denotes -inf.
2450 //
2451 // Return true if dependence disproved.
banerjeeMIVtest(const SCEV * Src,const SCEV * Dst,const SmallBitVector & Loops,FullDependence & Result) const2452 bool DependenceInfo::banerjeeMIVtest(const SCEV *Src, const SCEV *Dst,
2453                                      const SmallBitVector &Loops,
2454                                      FullDependence &Result) const {
2455   DEBUG(dbgs() << "starting Banerjee\n");
2456   ++BanerjeeApplications;
2457   DEBUG(dbgs() << "    Src = " << *Src << '\n');
2458   const SCEV *A0;
2459   CoefficientInfo *A = collectCoeffInfo(Src, true, A0);
2460   DEBUG(dbgs() << "    Dst = " << *Dst << '\n');
2461   const SCEV *B0;
2462   CoefficientInfo *B = collectCoeffInfo(Dst, false, B0);
2463   BoundInfo *Bound = new BoundInfo[MaxLevels + 1];
2464   const SCEV *Delta = SE->getMinusSCEV(B0, A0);
2465   DEBUG(dbgs() << "\tDelta = " << *Delta << '\n');
2466 
2467   // Compute bounds for all the * directions.
2468   DEBUG(dbgs() << "\tBounds[*]\n");
2469   for (unsigned K = 1; K <= MaxLevels; ++K) {
2470     Bound[K].Iterations = A[K].Iterations ? A[K].Iterations : B[K].Iterations;
2471     Bound[K].Direction = Dependence::DVEntry::ALL;
2472     Bound[K].DirSet = Dependence::DVEntry::NONE;
2473     findBoundsALL(A, B, Bound, K);
2474 #ifndef NDEBUG
2475     DEBUG(dbgs() << "\t    " << K << '\t');
2476     if (Bound[K].Lower[Dependence::DVEntry::ALL])
2477       DEBUG(dbgs() << *Bound[K].Lower[Dependence::DVEntry::ALL] << '\t');
2478     else
2479       DEBUG(dbgs() << "-inf\t");
2480     if (Bound[K].Upper[Dependence::DVEntry::ALL])
2481       DEBUG(dbgs() << *Bound[K].Upper[Dependence::DVEntry::ALL] << '\n');
2482     else
2483       DEBUG(dbgs() << "+inf\n");
2484 #endif
2485   }
2486 
2487   // Test the *, *, *, ... case.
2488   bool Disproved = false;
2489   if (testBounds(Dependence::DVEntry::ALL, 0, Bound, Delta)) {
2490     // Explore the direction vector hierarchy.
2491     unsigned DepthExpanded = 0;
2492     unsigned NewDeps = exploreDirections(1, A, B, Bound,
2493                                          Loops, DepthExpanded, Delta);
2494     if (NewDeps > 0) {
2495       bool Improved = false;
2496       for (unsigned K = 1; K <= CommonLevels; ++K) {
2497         if (Loops[K]) {
2498           unsigned Old = Result.DV[K - 1].Direction;
2499           Result.DV[K - 1].Direction = Old & Bound[K].DirSet;
2500           Improved |= Old != Result.DV[K - 1].Direction;
2501           if (!Result.DV[K - 1].Direction) {
2502             Improved = false;
2503             Disproved = true;
2504             break;
2505           }
2506         }
2507       }
2508       if (Improved)
2509         ++BanerjeeSuccesses;
2510     }
2511     else {
2512       ++BanerjeeIndependence;
2513       Disproved = true;
2514     }
2515   }
2516   else {
2517     ++BanerjeeIndependence;
2518     Disproved = true;
2519   }
2520   delete [] Bound;
2521   delete [] A;
2522   delete [] B;
2523   return Disproved;
2524 }
2525 
2526 
2527 // Hierarchically expands the direction vector
2528 // search space, combining the directions of discovered dependences
2529 // in the DirSet field of Bound. Returns the number of distinct
2530 // dependences discovered. If the dependence is disproved,
2531 // it will return 0.
exploreDirections(unsigned Level,CoefficientInfo * A,CoefficientInfo * B,BoundInfo * Bound,const SmallBitVector & Loops,unsigned & DepthExpanded,const SCEV * Delta) const2532 unsigned DependenceInfo::exploreDirections(unsigned Level, CoefficientInfo *A,
2533                                            CoefficientInfo *B, BoundInfo *Bound,
2534                                            const SmallBitVector &Loops,
2535                                            unsigned &DepthExpanded,
2536                                            const SCEV *Delta) const {
2537   if (Level > CommonLevels) {
2538     // record result
2539     DEBUG(dbgs() << "\t[");
2540     for (unsigned K = 1; K <= CommonLevels; ++K) {
2541       if (Loops[K]) {
2542         Bound[K].DirSet |= Bound[K].Direction;
2543 #ifndef NDEBUG
2544         switch (Bound[K].Direction) {
2545         case Dependence::DVEntry::LT:
2546           DEBUG(dbgs() << " <");
2547           break;
2548         case Dependence::DVEntry::EQ:
2549           DEBUG(dbgs() << " =");
2550           break;
2551         case Dependence::DVEntry::GT:
2552           DEBUG(dbgs() << " >");
2553           break;
2554         case Dependence::DVEntry::ALL:
2555           DEBUG(dbgs() << " *");
2556           break;
2557         default:
2558           llvm_unreachable("unexpected Bound[K].Direction");
2559         }
2560 #endif
2561       }
2562     }
2563     DEBUG(dbgs() << " ]\n");
2564     return 1;
2565   }
2566   if (Loops[Level]) {
2567     if (Level > DepthExpanded) {
2568       DepthExpanded = Level;
2569       // compute bounds for <, =, > at current level
2570       findBoundsLT(A, B, Bound, Level);
2571       findBoundsGT(A, B, Bound, Level);
2572       findBoundsEQ(A, B, Bound, Level);
2573 #ifndef NDEBUG
2574       DEBUG(dbgs() << "\tBound for level = " << Level << '\n');
2575       DEBUG(dbgs() << "\t    <\t");
2576       if (Bound[Level].Lower[Dependence::DVEntry::LT])
2577         DEBUG(dbgs() << *Bound[Level].Lower[Dependence::DVEntry::LT] << '\t');
2578       else
2579         DEBUG(dbgs() << "-inf\t");
2580       if (Bound[Level].Upper[Dependence::DVEntry::LT])
2581         DEBUG(dbgs() << *Bound[Level].Upper[Dependence::DVEntry::LT] << '\n');
2582       else
2583         DEBUG(dbgs() << "+inf\n");
2584       DEBUG(dbgs() << "\t    =\t");
2585       if (Bound[Level].Lower[Dependence::DVEntry::EQ])
2586         DEBUG(dbgs() << *Bound[Level].Lower[Dependence::DVEntry::EQ] << '\t');
2587       else
2588         DEBUG(dbgs() << "-inf\t");
2589       if (Bound[Level].Upper[Dependence::DVEntry::EQ])
2590         DEBUG(dbgs() << *Bound[Level].Upper[Dependence::DVEntry::EQ] << '\n');
2591       else
2592         DEBUG(dbgs() << "+inf\n");
2593       DEBUG(dbgs() << "\t    >\t");
2594       if (Bound[Level].Lower[Dependence::DVEntry::GT])
2595         DEBUG(dbgs() << *Bound[Level].Lower[Dependence::DVEntry::GT] << '\t');
2596       else
2597         DEBUG(dbgs() << "-inf\t");
2598       if (Bound[Level].Upper[Dependence::DVEntry::GT])
2599         DEBUG(dbgs() << *Bound[Level].Upper[Dependence::DVEntry::GT] << '\n');
2600       else
2601         DEBUG(dbgs() << "+inf\n");
2602 #endif
2603     }
2604 
2605     unsigned NewDeps = 0;
2606 
2607     // test bounds for <, *, *, ...
2608     if (testBounds(Dependence::DVEntry::LT, Level, Bound, Delta))
2609       NewDeps += exploreDirections(Level + 1, A, B, Bound,
2610                                    Loops, DepthExpanded, Delta);
2611 
2612     // Test bounds for =, *, *, ...
2613     if (testBounds(Dependence::DVEntry::EQ, Level, Bound, Delta))
2614       NewDeps += exploreDirections(Level + 1, A, B, Bound,
2615                                    Loops, DepthExpanded, Delta);
2616 
2617     // test bounds for >, *, *, ...
2618     if (testBounds(Dependence::DVEntry::GT, Level, Bound, Delta))
2619       NewDeps += exploreDirections(Level + 1, A, B, Bound,
2620                                    Loops, DepthExpanded, Delta);
2621 
2622     Bound[Level].Direction = Dependence::DVEntry::ALL;
2623     return NewDeps;
2624   }
2625   else
2626     return exploreDirections(Level + 1, A, B, Bound, Loops, DepthExpanded, Delta);
2627 }
2628 
2629 
2630 // Returns true iff the current bounds are plausible.
testBounds(unsigned char DirKind,unsigned Level,BoundInfo * Bound,const SCEV * Delta) const2631 bool DependenceInfo::testBounds(unsigned char DirKind, unsigned Level,
2632                                 BoundInfo *Bound, const SCEV *Delta) const {
2633   Bound[Level].Direction = DirKind;
2634   if (const SCEV *LowerBound = getLowerBound(Bound))
2635     if (isKnownPredicate(CmpInst::ICMP_SGT, LowerBound, Delta))
2636       return false;
2637   if (const SCEV *UpperBound = getUpperBound(Bound))
2638     if (isKnownPredicate(CmpInst::ICMP_SGT, Delta, UpperBound))
2639       return false;
2640   return true;
2641 }
2642 
2643 
2644 // Computes the upper and lower bounds for level K
2645 // using the * direction. Records them in Bound.
2646 // Wolfe gives the equations
2647 //
2648 //    LB^*_k = (A^-_k - B^+_k)(U_k - L_k) + (A_k - B_k)L_k
2649 //    UB^*_k = (A^+_k - B^-_k)(U_k - L_k) + (A_k - B_k)L_k
2650 //
2651 // Since we normalize loops, we can simplify these equations to
2652 //
2653 //    LB^*_k = (A^-_k - B^+_k)U_k
2654 //    UB^*_k = (A^+_k - B^-_k)U_k
2655 //
2656 // We must be careful to handle the case where the upper bound is unknown.
2657 // Note that the lower bound is always <= 0
2658 // and the upper bound is always >= 0.
findBoundsALL(CoefficientInfo * A,CoefficientInfo * B,BoundInfo * Bound,unsigned K) const2659 void DependenceInfo::findBoundsALL(CoefficientInfo *A, CoefficientInfo *B,
2660                                    BoundInfo *Bound, unsigned K) const {
2661   Bound[K].Lower[Dependence::DVEntry::ALL] = nullptr; // Default value = -infinity.
2662   Bound[K].Upper[Dependence::DVEntry::ALL] = nullptr; // Default value = +infinity.
2663   if (Bound[K].Iterations) {
2664     Bound[K].Lower[Dependence::DVEntry::ALL] =
2665       SE->getMulExpr(SE->getMinusSCEV(A[K].NegPart, B[K].PosPart),
2666                      Bound[K].Iterations);
2667     Bound[K].Upper[Dependence::DVEntry::ALL] =
2668       SE->getMulExpr(SE->getMinusSCEV(A[K].PosPart, B[K].NegPart),
2669                      Bound[K].Iterations);
2670   }
2671   else {
2672     // If the difference is 0, we won't need to know the number of iterations.
2673     if (isKnownPredicate(CmpInst::ICMP_EQ, A[K].NegPart, B[K].PosPart))
2674       Bound[K].Lower[Dependence::DVEntry::ALL] =
2675           SE->getZero(A[K].Coeff->getType());
2676     if (isKnownPredicate(CmpInst::ICMP_EQ, A[K].PosPart, B[K].NegPart))
2677       Bound[K].Upper[Dependence::DVEntry::ALL] =
2678           SE->getZero(A[K].Coeff->getType());
2679   }
2680 }
2681 
2682 
2683 // Computes the upper and lower bounds for level K
2684 // using the = direction. Records them in Bound.
2685 // Wolfe gives the equations
2686 //
2687 //    LB^=_k = (A_k - B_k)^- (U_k - L_k) + (A_k - B_k)L_k
2688 //    UB^=_k = (A_k - B_k)^+ (U_k - L_k) + (A_k - B_k)L_k
2689 //
2690 // Since we normalize loops, we can simplify these equations to
2691 //
2692 //    LB^=_k = (A_k - B_k)^- U_k
2693 //    UB^=_k = (A_k - B_k)^+ U_k
2694 //
2695 // We must be careful to handle the case where the upper bound is unknown.
2696 // Note that the lower bound is always <= 0
2697 // and the upper bound is always >= 0.
findBoundsEQ(CoefficientInfo * A,CoefficientInfo * B,BoundInfo * Bound,unsigned K) const2698 void DependenceInfo::findBoundsEQ(CoefficientInfo *A, CoefficientInfo *B,
2699                                   BoundInfo *Bound, unsigned K) const {
2700   Bound[K].Lower[Dependence::DVEntry::EQ] = nullptr; // Default value = -infinity.
2701   Bound[K].Upper[Dependence::DVEntry::EQ] = nullptr; // Default value = +infinity.
2702   if (Bound[K].Iterations) {
2703     const SCEV *Delta = SE->getMinusSCEV(A[K].Coeff, B[K].Coeff);
2704     const SCEV *NegativePart = getNegativePart(Delta);
2705     Bound[K].Lower[Dependence::DVEntry::EQ] =
2706       SE->getMulExpr(NegativePart, Bound[K].Iterations);
2707     const SCEV *PositivePart = getPositivePart(Delta);
2708     Bound[K].Upper[Dependence::DVEntry::EQ] =
2709       SE->getMulExpr(PositivePart, Bound[K].Iterations);
2710   }
2711   else {
2712     // If the positive/negative part of the difference is 0,
2713     // we won't need to know the number of iterations.
2714     const SCEV *Delta = SE->getMinusSCEV(A[K].Coeff, B[K].Coeff);
2715     const SCEV *NegativePart = getNegativePart(Delta);
2716     if (NegativePart->isZero())
2717       Bound[K].Lower[Dependence::DVEntry::EQ] = NegativePart; // Zero
2718     const SCEV *PositivePart = getPositivePart(Delta);
2719     if (PositivePart->isZero())
2720       Bound[K].Upper[Dependence::DVEntry::EQ] = PositivePart; // Zero
2721   }
2722 }
2723 
2724 
2725 // Computes the upper and lower bounds for level K
2726 // using the < direction. Records them in Bound.
2727 // Wolfe gives the equations
2728 //
2729 //    LB^<_k = (A^-_k - B_k)^- (U_k - L_k - N_k) + (A_k - B_k)L_k - B_k N_k
2730 //    UB^<_k = (A^+_k - B_k)^+ (U_k - L_k - N_k) + (A_k - B_k)L_k - B_k N_k
2731 //
2732 // Since we normalize loops, we can simplify these equations to
2733 //
2734 //    LB^<_k = (A^-_k - B_k)^- (U_k - 1) - B_k
2735 //    UB^<_k = (A^+_k - B_k)^+ (U_k - 1) - B_k
2736 //
2737 // We must be careful to handle the case where the upper bound is unknown.
findBoundsLT(CoefficientInfo * A,CoefficientInfo * B,BoundInfo * Bound,unsigned K) const2738 void DependenceInfo::findBoundsLT(CoefficientInfo *A, CoefficientInfo *B,
2739                                   BoundInfo *Bound, unsigned K) const {
2740   Bound[K].Lower[Dependence::DVEntry::LT] = nullptr; // Default value = -infinity.
2741   Bound[K].Upper[Dependence::DVEntry::LT] = nullptr; // Default value = +infinity.
2742   if (Bound[K].Iterations) {
2743     const SCEV *Iter_1 = SE->getMinusSCEV(
2744         Bound[K].Iterations, SE->getOne(Bound[K].Iterations->getType()));
2745     const SCEV *NegPart =
2746       getNegativePart(SE->getMinusSCEV(A[K].NegPart, B[K].Coeff));
2747     Bound[K].Lower[Dependence::DVEntry::LT] =
2748       SE->getMinusSCEV(SE->getMulExpr(NegPart, Iter_1), B[K].Coeff);
2749     const SCEV *PosPart =
2750       getPositivePart(SE->getMinusSCEV(A[K].PosPart, B[K].Coeff));
2751     Bound[K].Upper[Dependence::DVEntry::LT] =
2752       SE->getMinusSCEV(SE->getMulExpr(PosPart, Iter_1), B[K].Coeff);
2753   }
2754   else {
2755     // If the positive/negative part of the difference is 0,
2756     // we won't need to know the number of iterations.
2757     const SCEV *NegPart =
2758       getNegativePart(SE->getMinusSCEV(A[K].NegPart, B[K].Coeff));
2759     if (NegPart->isZero())
2760       Bound[K].Lower[Dependence::DVEntry::LT] = SE->getNegativeSCEV(B[K].Coeff);
2761     const SCEV *PosPart =
2762       getPositivePart(SE->getMinusSCEV(A[K].PosPart, B[K].Coeff));
2763     if (PosPart->isZero())
2764       Bound[K].Upper[Dependence::DVEntry::LT] = SE->getNegativeSCEV(B[K].Coeff);
2765   }
2766 }
2767 
2768 
2769 // Computes the upper and lower bounds for level K
2770 // using the > direction. Records them in Bound.
2771 // Wolfe gives the equations
2772 //
2773 //    LB^>_k = (A_k - B^+_k)^- (U_k - L_k - N_k) + (A_k - B_k)L_k + A_k N_k
2774 //    UB^>_k = (A_k - B^-_k)^+ (U_k - L_k - N_k) + (A_k - B_k)L_k + A_k N_k
2775 //
2776 // Since we normalize loops, we can simplify these equations to
2777 //
2778 //    LB^>_k = (A_k - B^+_k)^- (U_k - 1) + A_k
2779 //    UB^>_k = (A_k - B^-_k)^+ (U_k - 1) + A_k
2780 //
2781 // We must be careful to handle the case where the upper bound is unknown.
findBoundsGT(CoefficientInfo * A,CoefficientInfo * B,BoundInfo * Bound,unsigned K) const2782 void DependenceInfo::findBoundsGT(CoefficientInfo *A, CoefficientInfo *B,
2783                                   BoundInfo *Bound, unsigned K) const {
2784   Bound[K].Lower[Dependence::DVEntry::GT] = nullptr; // Default value = -infinity.
2785   Bound[K].Upper[Dependence::DVEntry::GT] = nullptr; // Default value = +infinity.
2786   if (Bound[K].Iterations) {
2787     const SCEV *Iter_1 = SE->getMinusSCEV(
2788         Bound[K].Iterations, SE->getOne(Bound[K].Iterations->getType()));
2789     const SCEV *NegPart =
2790       getNegativePart(SE->getMinusSCEV(A[K].Coeff, B[K].PosPart));
2791     Bound[K].Lower[Dependence::DVEntry::GT] =
2792       SE->getAddExpr(SE->getMulExpr(NegPart, Iter_1), A[K].Coeff);
2793     const SCEV *PosPart =
2794       getPositivePart(SE->getMinusSCEV(A[K].Coeff, B[K].NegPart));
2795     Bound[K].Upper[Dependence::DVEntry::GT] =
2796       SE->getAddExpr(SE->getMulExpr(PosPart, Iter_1), A[K].Coeff);
2797   }
2798   else {
2799     // If the positive/negative part of the difference is 0,
2800     // we won't need to know the number of iterations.
2801     const SCEV *NegPart = getNegativePart(SE->getMinusSCEV(A[K].Coeff, B[K].PosPart));
2802     if (NegPart->isZero())
2803       Bound[K].Lower[Dependence::DVEntry::GT] = A[K].Coeff;
2804     const SCEV *PosPart = getPositivePart(SE->getMinusSCEV(A[K].Coeff, B[K].NegPart));
2805     if (PosPart->isZero())
2806       Bound[K].Upper[Dependence::DVEntry::GT] = A[K].Coeff;
2807   }
2808 }
2809 
2810 
2811 // X^+ = max(X, 0)
getPositivePart(const SCEV * X) const2812 const SCEV *DependenceInfo::getPositivePart(const SCEV *X) const {
2813   return SE->getSMaxExpr(X, SE->getZero(X->getType()));
2814 }
2815 
2816 
2817 // X^- = min(X, 0)
getNegativePart(const SCEV * X) const2818 const SCEV *DependenceInfo::getNegativePart(const SCEV *X) const {
2819   return SE->getSMinExpr(X, SE->getZero(X->getType()));
2820 }
2821 
2822 
2823 // Walks through the subscript,
2824 // collecting each coefficient, the associated loop bounds,
2825 // and recording its positive and negative parts for later use.
2826 DependenceInfo::CoefficientInfo *
collectCoeffInfo(const SCEV * Subscript,bool SrcFlag,const SCEV * & Constant) const2827 DependenceInfo::collectCoeffInfo(const SCEV *Subscript, bool SrcFlag,
2828                                  const SCEV *&Constant) const {
2829   const SCEV *Zero = SE->getZero(Subscript->getType());
2830   CoefficientInfo *CI = new CoefficientInfo[MaxLevels + 1];
2831   for (unsigned K = 1; K <= MaxLevels; ++K) {
2832     CI[K].Coeff = Zero;
2833     CI[K].PosPart = Zero;
2834     CI[K].NegPart = Zero;
2835     CI[K].Iterations = nullptr;
2836   }
2837   while (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Subscript)) {
2838     const Loop *L = AddRec->getLoop();
2839     unsigned K = SrcFlag ? mapSrcLoop(L) : mapDstLoop(L);
2840     CI[K].Coeff = AddRec->getStepRecurrence(*SE);
2841     CI[K].PosPart = getPositivePart(CI[K].Coeff);
2842     CI[K].NegPart = getNegativePart(CI[K].Coeff);
2843     CI[K].Iterations = collectUpperBound(L, Subscript->getType());
2844     Subscript = AddRec->getStart();
2845   }
2846   Constant = Subscript;
2847 #ifndef NDEBUG
2848   DEBUG(dbgs() << "\tCoefficient Info\n");
2849   for (unsigned K = 1; K <= MaxLevels; ++K) {
2850     DEBUG(dbgs() << "\t    " << K << "\t" << *CI[K].Coeff);
2851     DEBUG(dbgs() << "\tPos Part = ");
2852     DEBUG(dbgs() << *CI[K].PosPart);
2853     DEBUG(dbgs() << "\tNeg Part = ");
2854     DEBUG(dbgs() << *CI[K].NegPart);
2855     DEBUG(dbgs() << "\tUpper Bound = ");
2856     if (CI[K].Iterations)
2857       DEBUG(dbgs() << *CI[K].Iterations);
2858     else
2859       DEBUG(dbgs() << "+inf");
2860     DEBUG(dbgs() << '\n');
2861   }
2862   DEBUG(dbgs() << "\t    Constant = " << *Subscript << '\n');
2863 #endif
2864   return CI;
2865 }
2866 
2867 
2868 // Looks through all the bounds info and
2869 // computes the lower bound given the current direction settings
2870 // at each level. If the lower bound for any level is -inf,
2871 // the result is -inf.
getLowerBound(BoundInfo * Bound) const2872 const SCEV *DependenceInfo::getLowerBound(BoundInfo *Bound) const {
2873   const SCEV *Sum = Bound[1].Lower[Bound[1].Direction];
2874   for (unsigned K = 2; Sum && K <= MaxLevels; ++K) {
2875     if (Bound[K].Lower[Bound[K].Direction])
2876       Sum = SE->getAddExpr(Sum, Bound[K].Lower[Bound[K].Direction]);
2877     else
2878       Sum = nullptr;
2879   }
2880   return Sum;
2881 }
2882 
2883 
2884 // Looks through all the bounds info and
2885 // computes the upper bound given the current direction settings
2886 // at each level. If the upper bound at any level is +inf,
2887 // the result is +inf.
getUpperBound(BoundInfo * Bound) const2888 const SCEV *DependenceInfo::getUpperBound(BoundInfo *Bound) const {
2889   const SCEV *Sum = Bound[1].Upper[Bound[1].Direction];
2890   for (unsigned K = 2; Sum && K <= MaxLevels; ++K) {
2891     if (Bound[K].Upper[Bound[K].Direction])
2892       Sum = SE->getAddExpr(Sum, Bound[K].Upper[Bound[K].Direction]);
2893     else
2894       Sum = nullptr;
2895   }
2896   return Sum;
2897 }
2898 
2899 
2900 //===----------------------------------------------------------------------===//
2901 // Constraint manipulation for Delta test.
2902 
2903 // Given a linear SCEV,
2904 // return the coefficient (the step)
2905 // corresponding to the specified loop.
2906 // If there isn't one, return 0.
2907 // For example, given a*i + b*j + c*k, finding the coefficient
2908 // corresponding to the j loop would yield b.
findCoefficient(const SCEV * Expr,const Loop * TargetLoop) const2909 const SCEV *DependenceInfo::findCoefficient(const SCEV *Expr,
2910                                             const Loop *TargetLoop) const {
2911   const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Expr);
2912   if (!AddRec)
2913     return SE->getZero(Expr->getType());
2914   if (AddRec->getLoop() == TargetLoop)
2915     return AddRec->getStepRecurrence(*SE);
2916   return findCoefficient(AddRec->getStart(), TargetLoop);
2917 }
2918 
2919 
2920 // Given a linear SCEV,
2921 // return the SCEV given by zeroing out the coefficient
2922 // corresponding to the specified loop.
2923 // For example, given a*i + b*j + c*k, zeroing the coefficient
2924 // corresponding to the j loop would yield a*i + c*k.
zeroCoefficient(const SCEV * Expr,const Loop * TargetLoop) const2925 const SCEV *DependenceInfo::zeroCoefficient(const SCEV *Expr,
2926                                             const Loop *TargetLoop) const {
2927   const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Expr);
2928   if (!AddRec)
2929     return Expr; // ignore
2930   if (AddRec->getLoop() == TargetLoop)
2931     return AddRec->getStart();
2932   return SE->getAddRecExpr(zeroCoefficient(AddRec->getStart(), TargetLoop),
2933                            AddRec->getStepRecurrence(*SE),
2934                            AddRec->getLoop(),
2935                            AddRec->getNoWrapFlags());
2936 }
2937 
2938 
2939 // Given a linear SCEV Expr,
2940 // return the SCEV given by adding some Value to the
2941 // coefficient corresponding to the specified TargetLoop.
2942 // For example, given a*i + b*j + c*k, adding 1 to the coefficient
2943 // corresponding to the j loop would yield a*i + (b+1)*j + c*k.
addToCoefficient(const SCEV * Expr,const Loop * TargetLoop,const SCEV * Value) const2944 const SCEV *DependenceInfo::addToCoefficient(const SCEV *Expr,
2945                                              const Loop *TargetLoop,
2946                                              const SCEV *Value) const {
2947   const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Expr);
2948   if (!AddRec) // create a new addRec
2949     return SE->getAddRecExpr(Expr,
2950                              Value,
2951                              TargetLoop,
2952                              SCEV::FlagAnyWrap); // Worst case, with no info.
2953   if (AddRec->getLoop() == TargetLoop) {
2954     const SCEV *Sum = SE->getAddExpr(AddRec->getStepRecurrence(*SE), Value);
2955     if (Sum->isZero())
2956       return AddRec->getStart();
2957     return SE->getAddRecExpr(AddRec->getStart(),
2958                              Sum,
2959                              AddRec->getLoop(),
2960                              AddRec->getNoWrapFlags());
2961   }
2962   if (SE->isLoopInvariant(AddRec, TargetLoop))
2963     return SE->getAddRecExpr(AddRec, Value, TargetLoop, SCEV::FlagAnyWrap);
2964   return SE->getAddRecExpr(
2965       addToCoefficient(AddRec->getStart(), TargetLoop, Value),
2966       AddRec->getStepRecurrence(*SE), AddRec->getLoop(),
2967       AddRec->getNoWrapFlags());
2968 }
2969 
2970 
2971 // Review the constraints, looking for opportunities
2972 // to simplify a subscript pair (Src and Dst).
2973 // Return true if some simplification occurs.
2974 // If the simplification isn't exact (that is, if it is conservative
2975 // in terms of dependence), set consistent to false.
2976 // Corresponds to Figure 5 from the paper
2977 //
2978 //            Practical Dependence Testing
2979 //            Goff, Kennedy, Tseng
2980 //            PLDI 1991
propagate(const SCEV * & Src,const SCEV * & Dst,SmallBitVector & Loops,SmallVectorImpl<Constraint> & Constraints,bool & Consistent)2981 bool DependenceInfo::propagate(const SCEV *&Src, const SCEV *&Dst,
2982                                SmallBitVector &Loops,
2983                                SmallVectorImpl<Constraint> &Constraints,
2984                                bool &Consistent) {
2985   bool Result = false;
2986   for (int LI = Loops.find_first(); LI >= 0; LI = Loops.find_next(LI)) {
2987     DEBUG(dbgs() << "\t    Constraint[" << LI << "] is");
2988     DEBUG(Constraints[LI].dump(dbgs()));
2989     if (Constraints[LI].isDistance())
2990       Result |= propagateDistance(Src, Dst, Constraints[LI], Consistent);
2991     else if (Constraints[LI].isLine())
2992       Result |= propagateLine(Src, Dst, Constraints[LI], Consistent);
2993     else if (Constraints[LI].isPoint())
2994       Result |= propagatePoint(Src, Dst, Constraints[LI]);
2995   }
2996   return Result;
2997 }
2998 
2999 
3000 // Attempt to propagate a distance
3001 // constraint into a subscript pair (Src and Dst).
3002 // Return true if some simplification occurs.
3003 // If the simplification isn't exact (that is, if it is conservative
3004 // in terms of dependence), set consistent to false.
propagateDistance(const SCEV * & Src,const SCEV * & Dst,Constraint & CurConstraint,bool & Consistent)3005 bool DependenceInfo::propagateDistance(const SCEV *&Src, const SCEV *&Dst,
3006                                        Constraint &CurConstraint,
3007                                        bool &Consistent) {
3008   const Loop *CurLoop = CurConstraint.getAssociatedLoop();
3009   DEBUG(dbgs() << "\t\tSrc is " << *Src << "\n");
3010   const SCEV *A_K = findCoefficient(Src, CurLoop);
3011   if (A_K->isZero())
3012     return false;
3013   const SCEV *DA_K = SE->getMulExpr(A_K, CurConstraint.getD());
3014   Src = SE->getMinusSCEV(Src, DA_K);
3015   Src = zeroCoefficient(Src, CurLoop);
3016   DEBUG(dbgs() << "\t\tnew Src is " << *Src << "\n");
3017   DEBUG(dbgs() << "\t\tDst is " << *Dst << "\n");
3018   Dst = addToCoefficient(Dst, CurLoop, SE->getNegativeSCEV(A_K));
3019   DEBUG(dbgs() << "\t\tnew Dst is " << *Dst << "\n");
3020   if (!findCoefficient(Dst, CurLoop)->isZero())
3021     Consistent = false;
3022   return true;
3023 }
3024 
3025 
3026 // Attempt to propagate a line
3027 // constraint into a subscript pair (Src and Dst).
3028 // Return true if some simplification occurs.
3029 // If the simplification isn't exact (that is, if it is conservative
3030 // in terms of dependence), set consistent to false.
propagateLine(const SCEV * & Src,const SCEV * & Dst,Constraint & CurConstraint,bool & Consistent)3031 bool DependenceInfo::propagateLine(const SCEV *&Src, const SCEV *&Dst,
3032                                    Constraint &CurConstraint,
3033                                    bool &Consistent) {
3034   const Loop *CurLoop = CurConstraint.getAssociatedLoop();
3035   const SCEV *A = CurConstraint.getA();
3036   const SCEV *B = CurConstraint.getB();
3037   const SCEV *C = CurConstraint.getC();
3038   DEBUG(dbgs() << "\t\tA = " << *A << ", B = " << *B << ", C = " << *C << "\n");
3039   DEBUG(dbgs() << "\t\tSrc = " << *Src << "\n");
3040   DEBUG(dbgs() << "\t\tDst = " << *Dst << "\n");
3041   if (A->isZero()) {
3042     const SCEVConstant *Bconst = dyn_cast<SCEVConstant>(B);
3043     const SCEVConstant *Cconst = dyn_cast<SCEVConstant>(C);
3044     if (!Bconst || !Cconst) return false;
3045     APInt Beta = Bconst->getAPInt();
3046     APInt Charlie = Cconst->getAPInt();
3047     APInt CdivB = Charlie.sdiv(Beta);
3048     assert(Charlie.srem(Beta) == 0 && "C should be evenly divisible by B");
3049     const SCEV *AP_K = findCoefficient(Dst, CurLoop);
3050     //    Src = SE->getAddExpr(Src, SE->getMulExpr(AP_K, SE->getConstant(CdivB)));
3051     Src = SE->getMinusSCEV(Src, SE->getMulExpr(AP_K, SE->getConstant(CdivB)));
3052     Dst = zeroCoefficient(Dst, CurLoop);
3053     if (!findCoefficient(Src, CurLoop)->isZero())
3054       Consistent = false;
3055   }
3056   else if (B->isZero()) {
3057     const SCEVConstant *Aconst = dyn_cast<SCEVConstant>(A);
3058     const SCEVConstant *Cconst = dyn_cast<SCEVConstant>(C);
3059     if (!Aconst || !Cconst) return false;
3060     APInt Alpha = Aconst->getAPInt();
3061     APInt Charlie = Cconst->getAPInt();
3062     APInt CdivA = Charlie.sdiv(Alpha);
3063     assert(Charlie.srem(Alpha) == 0 && "C should be evenly divisible by A");
3064     const SCEV *A_K = findCoefficient(Src, CurLoop);
3065     Src = SE->getAddExpr(Src, SE->getMulExpr(A_K, SE->getConstant(CdivA)));
3066     Src = zeroCoefficient(Src, CurLoop);
3067     if (!findCoefficient(Dst, CurLoop)->isZero())
3068       Consistent = false;
3069   }
3070   else if (isKnownPredicate(CmpInst::ICMP_EQ, A, B)) {
3071     const SCEVConstant *Aconst = dyn_cast<SCEVConstant>(A);
3072     const SCEVConstant *Cconst = dyn_cast<SCEVConstant>(C);
3073     if (!Aconst || !Cconst) return false;
3074     APInt Alpha = Aconst->getAPInt();
3075     APInt Charlie = Cconst->getAPInt();
3076     APInt CdivA = Charlie.sdiv(Alpha);
3077     assert(Charlie.srem(Alpha) == 0 && "C should be evenly divisible by A");
3078     const SCEV *A_K = findCoefficient(Src, CurLoop);
3079     Src = SE->getAddExpr(Src, SE->getMulExpr(A_K, SE->getConstant(CdivA)));
3080     Src = zeroCoefficient(Src, CurLoop);
3081     Dst = addToCoefficient(Dst, CurLoop, A_K);
3082     if (!findCoefficient(Dst, CurLoop)->isZero())
3083       Consistent = false;
3084   }
3085   else {
3086     // paper is incorrect here, or perhaps just misleading
3087     const SCEV *A_K = findCoefficient(Src, CurLoop);
3088     Src = SE->getMulExpr(Src, A);
3089     Dst = SE->getMulExpr(Dst, A);
3090     Src = SE->getAddExpr(Src, SE->getMulExpr(A_K, C));
3091     Src = zeroCoefficient(Src, CurLoop);
3092     Dst = addToCoefficient(Dst, CurLoop, SE->getMulExpr(A_K, B));
3093     if (!findCoefficient(Dst, CurLoop)->isZero())
3094       Consistent = false;
3095   }
3096   DEBUG(dbgs() << "\t\tnew Src = " << *Src << "\n");
3097   DEBUG(dbgs() << "\t\tnew Dst = " << *Dst << "\n");
3098   return true;
3099 }
3100 
3101 
3102 // Attempt to propagate a point
3103 // constraint into a subscript pair (Src and Dst).
3104 // Return true if some simplification occurs.
propagatePoint(const SCEV * & Src,const SCEV * & Dst,Constraint & CurConstraint)3105 bool DependenceInfo::propagatePoint(const SCEV *&Src, const SCEV *&Dst,
3106                                     Constraint &CurConstraint) {
3107   const Loop *CurLoop = CurConstraint.getAssociatedLoop();
3108   const SCEV *A_K = findCoefficient(Src, CurLoop);
3109   const SCEV *AP_K = findCoefficient(Dst, CurLoop);
3110   const SCEV *XA_K = SE->getMulExpr(A_K, CurConstraint.getX());
3111   const SCEV *YAP_K = SE->getMulExpr(AP_K, CurConstraint.getY());
3112   DEBUG(dbgs() << "\t\tSrc is " << *Src << "\n");
3113   Src = SE->getAddExpr(Src, SE->getMinusSCEV(XA_K, YAP_K));
3114   Src = zeroCoefficient(Src, CurLoop);
3115   DEBUG(dbgs() << "\t\tnew Src is " << *Src << "\n");
3116   DEBUG(dbgs() << "\t\tDst is " << *Dst << "\n");
3117   Dst = zeroCoefficient(Dst, CurLoop);
3118   DEBUG(dbgs() << "\t\tnew Dst is " << *Dst << "\n");
3119   return true;
3120 }
3121 
3122 
3123 // Update direction vector entry based on the current constraint.
updateDirection(Dependence::DVEntry & Level,const Constraint & CurConstraint) const3124 void DependenceInfo::updateDirection(Dependence::DVEntry &Level,
3125                                      const Constraint &CurConstraint) const {
3126   DEBUG(dbgs() << "\tUpdate direction, constraint =");
3127   DEBUG(CurConstraint.dump(dbgs()));
3128   if (CurConstraint.isAny())
3129     ; // use defaults
3130   else if (CurConstraint.isDistance()) {
3131     // this one is consistent, the others aren't
3132     Level.Scalar = false;
3133     Level.Distance = CurConstraint.getD();
3134     unsigned NewDirection = Dependence::DVEntry::NONE;
3135     if (!SE->isKnownNonZero(Level.Distance)) // if may be zero
3136       NewDirection = Dependence::DVEntry::EQ;
3137     if (!SE->isKnownNonPositive(Level.Distance)) // if may be positive
3138       NewDirection |= Dependence::DVEntry::LT;
3139     if (!SE->isKnownNonNegative(Level.Distance)) // if may be negative
3140       NewDirection |= Dependence::DVEntry::GT;
3141     Level.Direction &= NewDirection;
3142   }
3143   else if (CurConstraint.isLine()) {
3144     Level.Scalar = false;
3145     Level.Distance = nullptr;
3146     // direction should be accurate
3147   }
3148   else if (CurConstraint.isPoint()) {
3149     Level.Scalar = false;
3150     Level.Distance = nullptr;
3151     unsigned NewDirection = Dependence::DVEntry::NONE;
3152     if (!isKnownPredicate(CmpInst::ICMP_NE,
3153                           CurConstraint.getY(),
3154                           CurConstraint.getX()))
3155       // if X may be = Y
3156       NewDirection |= Dependence::DVEntry::EQ;
3157     if (!isKnownPredicate(CmpInst::ICMP_SLE,
3158                           CurConstraint.getY(),
3159                           CurConstraint.getX()))
3160       // if Y may be > X
3161       NewDirection |= Dependence::DVEntry::LT;
3162     if (!isKnownPredicate(CmpInst::ICMP_SGE,
3163                           CurConstraint.getY(),
3164                           CurConstraint.getX()))
3165       // if Y may be < X
3166       NewDirection |= Dependence::DVEntry::GT;
3167     Level.Direction &= NewDirection;
3168   }
3169   else
3170     llvm_unreachable("constraint has unexpected kind");
3171 }
3172 
3173 /// Check if we can delinearize the subscripts. If the SCEVs representing the
3174 /// source and destination array references are recurrences on a nested loop,
3175 /// this function flattens the nested recurrences into separate recurrences
3176 /// for each loop level.
tryDelinearize(Instruction * Src,Instruction * Dst,SmallVectorImpl<Subscript> & Pair)3177 bool DependenceInfo::tryDelinearize(Instruction *Src, Instruction *Dst,
3178                                     SmallVectorImpl<Subscript> &Pair) {
3179   Value *SrcPtr = getPointerOperand(Src);
3180   Value *DstPtr = getPointerOperand(Dst);
3181 
3182   Loop *SrcLoop = LI->getLoopFor(Src->getParent());
3183   Loop *DstLoop = LI->getLoopFor(Dst->getParent());
3184 
3185   // Below code mimics the code in Delinearization.cpp
3186   const SCEV *SrcAccessFn =
3187     SE->getSCEVAtScope(SrcPtr, SrcLoop);
3188   const SCEV *DstAccessFn =
3189     SE->getSCEVAtScope(DstPtr, DstLoop);
3190 
3191   const SCEVUnknown *SrcBase =
3192       dyn_cast<SCEVUnknown>(SE->getPointerBase(SrcAccessFn));
3193   const SCEVUnknown *DstBase =
3194       dyn_cast<SCEVUnknown>(SE->getPointerBase(DstAccessFn));
3195 
3196   if (!SrcBase || !DstBase || SrcBase != DstBase)
3197     return false;
3198 
3199   const SCEV *ElementSize = SE->getElementSize(Src);
3200   if (ElementSize != SE->getElementSize(Dst))
3201     return false;
3202 
3203   const SCEV *SrcSCEV = SE->getMinusSCEV(SrcAccessFn, SrcBase);
3204   const SCEV *DstSCEV = SE->getMinusSCEV(DstAccessFn, DstBase);
3205 
3206   const SCEVAddRecExpr *SrcAR = dyn_cast<SCEVAddRecExpr>(SrcSCEV);
3207   const SCEVAddRecExpr *DstAR = dyn_cast<SCEVAddRecExpr>(DstSCEV);
3208   if (!SrcAR || !DstAR || !SrcAR->isAffine() || !DstAR->isAffine())
3209     return false;
3210 
3211   // First step: collect parametric terms in both array references.
3212   SmallVector<const SCEV *, 4> Terms;
3213   SE->collectParametricTerms(SrcAR, Terms);
3214   SE->collectParametricTerms(DstAR, Terms);
3215 
3216   // Second step: find subscript sizes.
3217   SmallVector<const SCEV *, 4> Sizes;
3218   SE->findArrayDimensions(Terms, Sizes, ElementSize);
3219 
3220   // Third step: compute the access functions for each subscript.
3221   SmallVector<const SCEV *, 4> SrcSubscripts, DstSubscripts;
3222   SE->computeAccessFunctions(SrcAR, SrcSubscripts, Sizes);
3223   SE->computeAccessFunctions(DstAR, DstSubscripts, Sizes);
3224 
3225   // Fail when there is only a subscript: that's a linearized access function.
3226   if (SrcSubscripts.size() < 2 || DstSubscripts.size() < 2 ||
3227       SrcSubscripts.size() != DstSubscripts.size())
3228     return false;
3229 
3230   int size = SrcSubscripts.size();
3231 
3232   DEBUG({
3233       dbgs() << "\nSrcSubscripts: ";
3234     for (int i = 0; i < size; i++)
3235       dbgs() << *SrcSubscripts[i];
3236     dbgs() << "\nDstSubscripts: ";
3237     for (int i = 0; i < size; i++)
3238       dbgs() << *DstSubscripts[i];
3239     });
3240 
3241   // The delinearization transforms a single-subscript MIV dependence test into
3242   // a multi-subscript SIV dependence test that is easier to compute. So we
3243   // resize Pair to contain as many pairs of subscripts as the delinearization
3244   // has found, and then initialize the pairs following the delinearization.
3245   Pair.resize(size);
3246   for (int i = 0; i < size; ++i) {
3247     Pair[i].Src = SrcSubscripts[i];
3248     Pair[i].Dst = DstSubscripts[i];
3249     unifySubscriptType(&Pair[i]);
3250 
3251     // FIXME: we should record the bounds SrcSizes[i] and DstSizes[i] that the
3252     // delinearization has found, and add these constraints to the dependence
3253     // check to avoid memory accesses overflow from one dimension into another.
3254     // This is related to the problem of determining the existence of data
3255     // dependences in array accesses using a different number of subscripts: in
3256     // C one can access an array A[100][100]; as A[0][9999], *A[9999], etc.
3257   }
3258 
3259   return true;
3260 }
3261 
3262 //===----------------------------------------------------------------------===//
3263 
3264 #ifndef NDEBUG
3265 // For debugging purposes, dump a small bit vector to dbgs().
dumpSmallBitVector(SmallBitVector & BV)3266 static void dumpSmallBitVector(SmallBitVector &BV) {
3267   dbgs() << "{";
3268   for (int VI = BV.find_first(); VI >= 0; VI = BV.find_next(VI)) {
3269     dbgs() << VI;
3270     if (BV.find_next(VI) >= 0)
3271       dbgs() << ' ';
3272   }
3273   dbgs() << "}\n";
3274 }
3275 #endif
3276 
3277 // depends -
3278 // Returns NULL if there is no dependence.
3279 // Otherwise, return a Dependence with as many details as possible.
3280 // Corresponds to Section 3.1 in the paper
3281 //
3282 //            Practical Dependence Testing
3283 //            Goff, Kennedy, Tseng
3284 //            PLDI 1991
3285 //
3286 // Care is required to keep the routine below, getSplitIteration(),
3287 // up to date with respect to this routine.
3288 std::unique_ptr<Dependence>
depends(Instruction * Src,Instruction * Dst,bool PossiblyLoopIndependent)3289 DependenceInfo::depends(Instruction *Src, Instruction *Dst,
3290                         bool PossiblyLoopIndependent) {
3291   if (Src == Dst)
3292     PossiblyLoopIndependent = false;
3293 
3294   if ((!Src->mayReadFromMemory() && !Src->mayWriteToMemory()) ||
3295       (!Dst->mayReadFromMemory() && !Dst->mayWriteToMemory()))
3296     // if both instructions don't reference memory, there's no dependence
3297     return nullptr;
3298 
3299   if (!isLoadOrStore(Src) || !isLoadOrStore(Dst)) {
3300     // can only analyze simple loads and stores, i.e., no calls, invokes, etc.
3301     DEBUG(dbgs() << "can only handle simple loads and stores\n");
3302     return make_unique<Dependence>(Src, Dst);
3303   }
3304 
3305   Value *SrcPtr = getPointerOperand(Src);
3306   Value *DstPtr = getPointerOperand(Dst);
3307 
3308   switch (underlyingObjectsAlias(AA, F->getParent()->getDataLayout(), DstPtr,
3309                                  SrcPtr)) {
3310   case MayAlias:
3311   case PartialAlias:
3312     // cannot analyse objects if we don't understand their aliasing.
3313     DEBUG(dbgs() << "can't analyze may or partial alias\n");
3314     return make_unique<Dependence>(Src, Dst);
3315   case NoAlias:
3316     // If the objects noalias, they are distinct, accesses are independent.
3317     DEBUG(dbgs() << "no alias\n");
3318     return nullptr;
3319   case MustAlias:
3320     break; // The underlying objects alias; test accesses for dependence.
3321   }
3322 
3323   // establish loop nesting levels
3324   establishNestingLevels(Src, Dst);
3325   DEBUG(dbgs() << "    common nesting levels = " << CommonLevels << "\n");
3326   DEBUG(dbgs() << "    maximum nesting levels = " << MaxLevels << "\n");
3327 
3328   FullDependence Result(Src, Dst, PossiblyLoopIndependent, CommonLevels);
3329   ++TotalArrayPairs;
3330 
3331   // See if there are GEPs we can use.
3332   bool UsefulGEP = false;
3333   GEPOperator *SrcGEP = dyn_cast<GEPOperator>(SrcPtr);
3334   GEPOperator *DstGEP = dyn_cast<GEPOperator>(DstPtr);
3335   if (SrcGEP && DstGEP &&
3336       SrcGEP->getPointerOperandType() == DstGEP->getPointerOperandType()) {
3337     const SCEV *SrcPtrSCEV = SE->getSCEV(SrcGEP->getPointerOperand());
3338     const SCEV *DstPtrSCEV = SE->getSCEV(DstGEP->getPointerOperand());
3339     DEBUG(dbgs() << "    SrcPtrSCEV = " << *SrcPtrSCEV << "\n");
3340     DEBUG(dbgs() << "    DstPtrSCEV = " << *DstPtrSCEV << "\n");
3341 
3342     UsefulGEP = isLoopInvariant(SrcPtrSCEV, LI->getLoopFor(Src->getParent())) &&
3343                 isLoopInvariant(DstPtrSCEV, LI->getLoopFor(Dst->getParent())) &&
3344                 (SrcGEP->getNumOperands() == DstGEP->getNumOperands());
3345   }
3346   unsigned Pairs = UsefulGEP ? SrcGEP->idx_end() - SrcGEP->idx_begin() : 1;
3347   SmallVector<Subscript, 4> Pair(Pairs);
3348   if (UsefulGEP) {
3349     DEBUG(dbgs() << "    using GEPs\n");
3350     unsigned P = 0;
3351     for (GEPOperator::const_op_iterator SrcIdx = SrcGEP->idx_begin(),
3352            SrcEnd = SrcGEP->idx_end(),
3353            DstIdx = DstGEP->idx_begin();
3354          SrcIdx != SrcEnd;
3355          ++SrcIdx, ++DstIdx, ++P) {
3356       Pair[P].Src = SE->getSCEV(*SrcIdx);
3357       Pair[P].Dst = SE->getSCEV(*DstIdx);
3358       unifySubscriptType(&Pair[P]);
3359     }
3360   }
3361   else {
3362     DEBUG(dbgs() << "    ignoring GEPs\n");
3363     const SCEV *SrcSCEV = SE->getSCEV(SrcPtr);
3364     const SCEV *DstSCEV = SE->getSCEV(DstPtr);
3365     DEBUG(dbgs() << "    SrcSCEV = " << *SrcSCEV << "\n");
3366     DEBUG(dbgs() << "    DstSCEV = " << *DstSCEV << "\n");
3367     Pair[0].Src = SrcSCEV;
3368     Pair[0].Dst = DstSCEV;
3369   }
3370 
3371   if (Delinearize && CommonLevels > 1) {
3372     if (tryDelinearize(Src, Dst, Pair)) {
3373       DEBUG(dbgs() << "    delinerized GEP\n");
3374       Pairs = Pair.size();
3375     }
3376   }
3377 
3378   for (unsigned P = 0; P < Pairs; ++P) {
3379     Pair[P].Loops.resize(MaxLevels + 1);
3380     Pair[P].GroupLoops.resize(MaxLevels + 1);
3381     Pair[P].Group.resize(Pairs);
3382     removeMatchingExtensions(&Pair[P]);
3383     Pair[P].Classification =
3384       classifyPair(Pair[P].Src, LI->getLoopFor(Src->getParent()),
3385                    Pair[P].Dst, LI->getLoopFor(Dst->getParent()),
3386                    Pair[P].Loops);
3387     Pair[P].GroupLoops = Pair[P].Loops;
3388     Pair[P].Group.set(P);
3389     DEBUG(dbgs() << "    subscript " << P << "\n");
3390     DEBUG(dbgs() << "\tsrc = " << *Pair[P].Src << "\n");
3391     DEBUG(dbgs() << "\tdst = " << *Pair[P].Dst << "\n");
3392     DEBUG(dbgs() << "\tclass = " << Pair[P].Classification << "\n");
3393     DEBUG(dbgs() << "\tloops = ");
3394     DEBUG(dumpSmallBitVector(Pair[P].Loops));
3395   }
3396 
3397   SmallBitVector Separable(Pairs);
3398   SmallBitVector Coupled(Pairs);
3399 
3400   // Partition subscripts into separable and minimally-coupled groups
3401   // Algorithm in paper is algorithmically better;
3402   // this may be faster in practice. Check someday.
3403   //
3404   // Here's an example of how it works. Consider this code:
3405   //
3406   //   for (i = ...) {
3407   //     for (j = ...) {
3408   //       for (k = ...) {
3409   //         for (l = ...) {
3410   //           for (m = ...) {
3411   //             A[i][j][k][m] = ...;
3412   //             ... = A[0][j][l][i + j];
3413   //           }
3414   //         }
3415   //       }
3416   //     }
3417   //   }
3418   //
3419   // There are 4 subscripts here:
3420   //    0 [i] and [0]
3421   //    1 [j] and [j]
3422   //    2 [k] and [l]
3423   //    3 [m] and [i + j]
3424   //
3425   // We've already classified each subscript pair as ZIV, SIV, etc.,
3426   // and collected all the loops mentioned by pair P in Pair[P].Loops.
3427   // In addition, we've initialized Pair[P].GroupLoops to Pair[P].Loops
3428   // and set Pair[P].Group = {P}.
3429   //
3430   //      Src Dst    Classification Loops  GroupLoops Group
3431   //    0 [i] [0]         SIV       {1}      {1}        {0}
3432   //    1 [j] [j]         SIV       {2}      {2}        {1}
3433   //    2 [k] [l]         RDIV      {3,4}    {3,4}      {2}
3434   //    3 [m] [i + j]     MIV       {1,2,5}  {1,2,5}    {3}
3435   //
3436   // For each subscript SI 0 .. 3, we consider each remaining subscript, SJ.
3437   // So, 0 is compared against 1, 2, and 3; 1 is compared against 2 and 3, etc.
3438   //
3439   // We begin by comparing 0 and 1. The intersection of the GroupLoops is empty.
3440   // Next, 0 and 2. Again, the intersection of their GroupLoops is empty.
3441   // Next 0 and 3. The intersection of their GroupLoop = {1}, not empty,
3442   // so Pair[3].Group = {0,3} and Done = false (that is, 0 will not be added
3443   // to either Separable or Coupled).
3444   //
3445   // Next, we consider 1 and 2. The intersection of the GroupLoops is empty.
3446   // Next, 1 and 3. The intersectionof their GroupLoops = {2}, not empty,
3447   // so Pair[3].Group = {0, 1, 3} and Done = false.
3448   //
3449   // Next, we compare 2 against 3. The intersection of the GroupLoops is empty.
3450   // Since Done remains true, we add 2 to the set of Separable pairs.
3451   //
3452   // Finally, we consider 3. There's nothing to compare it with,
3453   // so Done remains true and we add it to the Coupled set.
3454   // Pair[3].Group = {0, 1, 3} and GroupLoops = {1, 2, 5}.
3455   //
3456   // In the end, we've got 1 separable subscript and 1 coupled group.
3457   for (unsigned SI = 0; SI < Pairs; ++SI) {
3458     if (Pair[SI].Classification == Subscript::NonLinear) {
3459       // ignore these, but collect loops for later
3460       ++NonlinearSubscriptPairs;
3461       collectCommonLoops(Pair[SI].Src,
3462                          LI->getLoopFor(Src->getParent()),
3463                          Pair[SI].Loops);
3464       collectCommonLoops(Pair[SI].Dst,
3465                          LI->getLoopFor(Dst->getParent()),
3466                          Pair[SI].Loops);
3467       Result.Consistent = false;
3468     } else if (Pair[SI].Classification == Subscript::ZIV) {
3469       // always separable
3470       Separable.set(SI);
3471     }
3472     else {
3473       // SIV, RDIV, or MIV, so check for coupled group
3474       bool Done = true;
3475       for (unsigned SJ = SI + 1; SJ < Pairs; ++SJ) {
3476         SmallBitVector Intersection = Pair[SI].GroupLoops;
3477         Intersection &= Pair[SJ].GroupLoops;
3478         if (Intersection.any()) {
3479           // accumulate set of all the loops in group
3480           Pair[SJ].GroupLoops |= Pair[SI].GroupLoops;
3481           // accumulate set of all subscripts in group
3482           Pair[SJ].Group |= Pair[SI].Group;
3483           Done = false;
3484         }
3485       }
3486       if (Done) {
3487         if (Pair[SI].Group.count() == 1) {
3488           Separable.set(SI);
3489           ++SeparableSubscriptPairs;
3490         }
3491         else {
3492           Coupled.set(SI);
3493           ++CoupledSubscriptPairs;
3494         }
3495       }
3496     }
3497   }
3498 
3499   DEBUG(dbgs() << "    Separable = ");
3500   DEBUG(dumpSmallBitVector(Separable));
3501   DEBUG(dbgs() << "    Coupled = ");
3502   DEBUG(dumpSmallBitVector(Coupled));
3503 
3504   Constraint NewConstraint;
3505   NewConstraint.setAny(SE);
3506 
3507   // test separable subscripts
3508   for (int SI = Separable.find_first(); SI >= 0; SI = Separable.find_next(SI)) {
3509     DEBUG(dbgs() << "testing subscript " << SI);
3510     switch (Pair[SI].Classification) {
3511     case Subscript::ZIV:
3512       DEBUG(dbgs() << ", ZIV\n");
3513       if (testZIV(Pair[SI].Src, Pair[SI].Dst, Result))
3514         return nullptr;
3515       break;
3516     case Subscript::SIV: {
3517       DEBUG(dbgs() << ", SIV\n");
3518       unsigned Level;
3519       const SCEV *SplitIter = nullptr;
3520       if (testSIV(Pair[SI].Src, Pair[SI].Dst, Level, Result, NewConstraint,
3521                   SplitIter))
3522         return nullptr;
3523       break;
3524     }
3525     case Subscript::RDIV:
3526       DEBUG(dbgs() << ", RDIV\n");
3527       if (testRDIV(Pair[SI].Src, Pair[SI].Dst, Result))
3528         return nullptr;
3529       break;
3530     case Subscript::MIV:
3531       DEBUG(dbgs() << ", MIV\n");
3532       if (testMIV(Pair[SI].Src, Pair[SI].Dst, Pair[SI].Loops, Result))
3533         return nullptr;
3534       break;
3535     default:
3536       llvm_unreachable("subscript has unexpected classification");
3537     }
3538   }
3539 
3540   if (Coupled.count()) {
3541     // test coupled subscript groups
3542     DEBUG(dbgs() << "starting on coupled subscripts\n");
3543     DEBUG(dbgs() << "MaxLevels + 1 = " << MaxLevels + 1 << "\n");
3544     SmallVector<Constraint, 4> Constraints(MaxLevels + 1);
3545     for (unsigned II = 0; II <= MaxLevels; ++II)
3546       Constraints[II].setAny(SE);
3547     for (int SI = Coupled.find_first(); SI >= 0; SI = Coupled.find_next(SI)) {
3548       DEBUG(dbgs() << "testing subscript group " << SI << " { ");
3549       SmallBitVector Group(Pair[SI].Group);
3550       SmallBitVector Sivs(Pairs);
3551       SmallBitVector Mivs(Pairs);
3552       SmallBitVector ConstrainedLevels(MaxLevels + 1);
3553       SmallVector<Subscript *, 4> PairsInGroup;
3554       for (int SJ = Group.find_first(); SJ >= 0; SJ = Group.find_next(SJ)) {
3555         DEBUG(dbgs() << SJ << " ");
3556         if (Pair[SJ].Classification == Subscript::SIV)
3557           Sivs.set(SJ);
3558         else
3559           Mivs.set(SJ);
3560         PairsInGroup.push_back(&Pair[SJ]);
3561       }
3562       unifySubscriptType(PairsInGroup);
3563       DEBUG(dbgs() << "}\n");
3564       while (Sivs.any()) {
3565         bool Changed = false;
3566         for (int SJ = Sivs.find_first(); SJ >= 0; SJ = Sivs.find_next(SJ)) {
3567           DEBUG(dbgs() << "testing subscript " << SJ << ", SIV\n");
3568           // SJ is an SIV subscript that's part of the current coupled group
3569           unsigned Level;
3570           const SCEV *SplitIter = nullptr;
3571           DEBUG(dbgs() << "SIV\n");
3572           if (testSIV(Pair[SJ].Src, Pair[SJ].Dst, Level, Result, NewConstraint,
3573                       SplitIter))
3574             return nullptr;
3575           ConstrainedLevels.set(Level);
3576           if (intersectConstraints(&Constraints[Level], &NewConstraint)) {
3577             if (Constraints[Level].isEmpty()) {
3578               ++DeltaIndependence;
3579               return nullptr;
3580             }
3581             Changed = true;
3582           }
3583           Sivs.reset(SJ);
3584         }
3585         if (Changed) {
3586           // propagate, possibly creating new SIVs and ZIVs
3587           DEBUG(dbgs() << "    propagating\n");
3588           DEBUG(dbgs() << "\tMivs = ");
3589           DEBUG(dumpSmallBitVector(Mivs));
3590           for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
3591             // SJ is an MIV subscript that's part of the current coupled group
3592             DEBUG(dbgs() << "\tSJ = " << SJ << "\n");
3593             if (propagate(Pair[SJ].Src, Pair[SJ].Dst, Pair[SJ].Loops,
3594                           Constraints, Result.Consistent)) {
3595               DEBUG(dbgs() << "\t    Changed\n");
3596               ++DeltaPropagations;
3597               Pair[SJ].Classification =
3598                 classifyPair(Pair[SJ].Src, LI->getLoopFor(Src->getParent()),
3599                              Pair[SJ].Dst, LI->getLoopFor(Dst->getParent()),
3600                              Pair[SJ].Loops);
3601               switch (Pair[SJ].Classification) {
3602               case Subscript::ZIV:
3603                 DEBUG(dbgs() << "ZIV\n");
3604                 if (testZIV(Pair[SJ].Src, Pair[SJ].Dst, Result))
3605                   return nullptr;
3606                 Mivs.reset(SJ);
3607                 break;
3608               case Subscript::SIV:
3609                 Sivs.set(SJ);
3610                 Mivs.reset(SJ);
3611                 break;
3612               case Subscript::RDIV:
3613               case Subscript::MIV:
3614                 break;
3615               default:
3616                 llvm_unreachable("bad subscript classification");
3617               }
3618             }
3619           }
3620         }
3621       }
3622 
3623       // test & propagate remaining RDIVs
3624       for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
3625         if (Pair[SJ].Classification == Subscript::RDIV) {
3626           DEBUG(dbgs() << "RDIV test\n");
3627           if (testRDIV(Pair[SJ].Src, Pair[SJ].Dst, Result))
3628             return nullptr;
3629           // I don't yet understand how to propagate RDIV results
3630           Mivs.reset(SJ);
3631         }
3632       }
3633 
3634       // test remaining MIVs
3635       // This code is temporary.
3636       // Better to somehow test all remaining subscripts simultaneously.
3637       for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
3638         if (Pair[SJ].Classification == Subscript::MIV) {
3639           DEBUG(dbgs() << "MIV test\n");
3640           if (testMIV(Pair[SJ].Src, Pair[SJ].Dst, Pair[SJ].Loops, Result))
3641             return nullptr;
3642         }
3643         else
3644           llvm_unreachable("expected only MIV subscripts at this point");
3645       }
3646 
3647       // update Result.DV from constraint vector
3648       DEBUG(dbgs() << "    updating\n");
3649       for (int SJ = ConstrainedLevels.find_first(); SJ >= 0;
3650            SJ = ConstrainedLevels.find_next(SJ)) {
3651         if (SJ > (int)CommonLevels)
3652           break;
3653         updateDirection(Result.DV[SJ - 1], Constraints[SJ]);
3654         if (Result.DV[SJ - 1].Direction == Dependence::DVEntry::NONE)
3655           return nullptr;
3656       }
3657     }
3658   }
3659 
3660   // Make sure the Scalar flags are set correctly.
3661   SmallBitVector CompleteLoops(MaxLevels + 1);
3662   for (unsigned SI = 0; SI < Pairs; ++SI)
3663     CompleteLoops |= Pair[SI].Loops;
3664   for (unsigned II = 1; II <= CommonLevels; ++II)
3665     if (CompleteLoops[II])
3666       Result.DV[II - 1].Scalar = false;
3667 
3668   if (PossiblyLoopIndependent) {
3669     // Make sure the LoopIndependent flag is set correctly.
3670     // All directions must include equal, otherwise no
3671     // loop-independent dependence is possible.
3672     for (unsigned II = 1; II <= CommonLevels; ++II) {
3673       if (!(Result.getDirection(II) & Dependence::DVEntry::EQ)) {
3674         Result.LoopIndependent = false;
3675         break;
3676       }
3677     }
3678   }
3679   else {
3680     // On the other hand, if all directions are equal and there's no
3681     // loop-independent dependence possible, then no dependence exists.
3682     bool AllEqual = true;
3683     for (unsigned II = 1; II <= CommonLevels; ++II) {
3684       if (Result.getDirection(II) != Dependence::DVEntry::EQ) {
3685         AllEqual = false;
3686         break;
3687       }
3688     }
3689     if (AllEqual)
3690       return nullptr;
3691   }
3692 
3693   return make_unique<FullDependence>(std::move(Result));
3694 }
3695 
3696 
3697 
3698 //===----------------------------------------------------------------------===//
3699 // getSplitIteration -
3700 // Rather than spend rarely-used space recording the splitting iteration
3701 // during the Weak-Crossing SIV test, we re-compute it on demand.
3702 // The re-computation is basically a repeat of the entire dependence test,
3703 // though simplified since we know that the dependence exists.
3704 // It's tedious, since we must go through all propagations, etc.
3705 //
3706 // Care is required to keep this code up to date with respect to the routine
3707 // above, depends().
3708 //
3709 // Generally, the dependence analyzer will be used to build
3710 // a dependence graph for a function (basically a map from instructions
3711 // to dependences). Looking for cycles in the graph shows us loops
3712 // that cannot be trivially vectorized/parallelized.
3713 //
3714 // We can try to improve the situation by examining all the dependences
3715 // that make up the cycle, looking for ones we can break.
3716 // Sometimes, peeling the first or last iteration of a loop will break
3717 // dependences, and we've got flags for those possibilities.
3718 // Sometimes, splitting a loop at some other iteration will do the trick,
3719 // and we've got a flag for that case. Rather than waste the space to
3720 // record the exact iteration (since we rarely know), we provide
3721 // a method that calculates the iteration. It's a drag that it must work
3722 // from scratch, but wonderful in that it's possible.
3723 //
3724 // Here's an example:
3725 //
3726 //    for (i = 0; i < 10; i++)
3727 //        A[i] = ...
3728 //        ... = A[11 - i]
3729 //
3730 // There's a loop-carried flow dependence from the store to the load,
3731 // found by the weak-crossing SIV test. The dependence will have a flag,
3732 // indicating that the dependence can be broken by splitting the loop.
3733 // Calling getSplitIteration will return 5.
3734 // Splitting the loop breaks the dependence, like so:
3735 //
3736 //    for (i = 0; i <= 5; i++)
3737 //        A[i] = ...
3738 //        ... = A[11 - i]
3739 //    for (i = 6; i < 10; i++)
3740 //        A[i] = ...
3741 //        ... = A[11 - i]
3742 //
3743 // breaks the dependence and allows us to vectorize/parallelize
3744 // both loops.
getSplitIteration(const Dependence & Dep,unsigned SplitLevel)3745 const SCEV *DependenceInfo::getSplitIteration(const Dependence &Dep,
3746                                               unsigned SplitLevel) {
3747   assert(Dep.isSplitable(SplitLevel) &&
3748          "Dep should be splitable at SplitLevel");
3749   Instruction *Src = Dep.getSrc();
3750   Instruction *Dst = Dep.getDst();
3751   assert(Src->mayReadFromMemory() || Src->mayWriteToMemory());
3752   assert(Dst->mayReadFromMemory() || Dst->mayWriteToMemory());
3753   assert(isLoadOrStore(Src));
3754   assert(isLoadOrStore(Dst));
3755   Value *SrcPtr = getPointerOperand(Src);
3756   Value *DstPtr = getPointerOperand(Dst);
3757   assert(underlyingObjectsAlias(AA, F->getParent()->getDataLayout(), DstPtr,
3758                                 SrcPtr) == MustAlias);
3759 
3760   // establish loop nesting levels
3761   establishNestingLevels(Src, Dst);
3762 
3763   FullDependence Result(Src, Dst, false, CommonLevels);
3764 
3765   // See if there are GEPs we can use.
3766   bool UsefulGEP = false;
3767   GEPOperator *SrcGEP = dyn_cast<GEPOperator>(SrcPtr);
3768   GEPOperator *DstGEP = dyn_cast<GEPOperator>(DstPtr);
3769   if (SrcGEP && DstGEP &&
3770       SrcGEP->getPointerOperandType() == DstGEP->getPointerOperandType()) {
3771     const SCEV *SrcPtrSCEV = SE->getSCEV(SrcGEP->getPointerOperand());
3772     const SCEV *DstPtrSCEV = SE->getSCEV(DstGEP->getPointerOperand());
3773     UsefulGEP = isLoopInvariant(SrcPtrSCEV, LI->getLoopFor(Src->getParent())) &&
3774                 isLoopInvariant(DstPtrSCEV, LI->getLoopFor(Dst->getParent())) &&
3775                 (SrcGEP->getNumOperands() == DstGEP->getNumOperands());
3776   }
3777   unsigned Pairs = UsefulGEP ? SrcGEP->idx_end() - SrcGEP->idx_begin() : 1;
3778   SmallVector<Subscript, 4> Pair(Pairs);
3779   if (UsefulGEP) {
3780     unsigned P = 0;
3781     for (GEPOperator::const_op_iterator SrcIdx = SrcGEP->idx_begin(),
3782            SrcEnd = SrcGEP->idx_end(),
3783            DstIdx = DstGEP->idx_begin();
3784          SrcIdx != SrcEnd;
3785          ++SrcIdx, ++DstIdx, ++P) {
3786       Pair[P].Src = SE->getSCEV(*SrcIdx);
3787       Pair[P].Dst = SE->getSCEV(*DstIdx);
3788     }
3789   }
3790   else {
3791     const SCEV *SrcSCEV = SE->getSCEV(SrcPtr);
3792     const SCEV *DstSCEV = SE->getSCEV(DstPtr);
3793     Pair[0].Src = SrcSCEV;
3794     Pair[0].Dst = DstSCEV;
3795   }
3796 
3797   if (Delinearize && CommonLevels > 1) {
3798     if (tryDelinearize(Src, Dst, Pair)) {
3799       DEBUG(dbgs() << "    delinerized GEP\n");
3800       Pairs = Pair.size();
3801     }
3802   }
3803 
3804   for (unsigned P = 0; P < Pairs; ++P) {
3805     Pair[P].Loops.resize(MaxLevels + 1);
3806     Pair[P].GroupLoops.resize(MaxLevels + 1);
3807     Pair[P].Group.resize(Pairs);
3808     removeMatchingExtensions(&Pair[P]);
3809     Pair[P].Classification =
3810       classifyPair(Pair[P].Src, LI->getLoopFor(Src->getParent()),
3811                    Pair[P].Dst, LI->getLoopFor(Dst->getParent()),
3812                    Pair[P].Loops);
3813     Pair[P].GroupLoops = Pair[P].Loops;
3814     Pair[P].Group.set(P);
3815   }
3816 
3817   SmallBitVector Separable(Pairs);
3818   SmallBitVector Coupled(Pairs);
3819 
3820   // partition subscripts into separable and minimally-coupled groups
3821   for (unsigned SI = 0; SI < Pairs; ++SI) {
3822     if (Pair[SI].Classification == Subscript::NonLinear) {
3823       // ignore these, but collect loops for later
3824       collectCommonLoops(Pair[SI].Src,
3825                          LI->getLoopFor(Src->getParent()),
3826                          Pair[SI].Loops);
3827       collectCommonLoops(Pair[SI].Dst,
3828                          LI->getLoopFor(Dst->getParent()),
3829                          Pair[SI].Loops);
3830       Result.Consistent = false;
3831     }
3832     else if (Pair[SI].Classification == Subscript::ZIV)
3833       Separable.set(SI);
3834     else {
3835       // SIV, RDIV, or MIV, so check for coupled group
3836       bool Done = true;
3837       for (unsigned SJ = SI + 1; SJ < Pairs; ++SJ) {
3838         SmallBitVector Intersection = Pair[SI].GroupLoops;
3839         Intersection &= Pair[SJ].GroupLoops;
3840         if (Intersection.any()) {
3841           // accumulate set of all the loops in group
3842           Pair[SJ].GroupLoops |= Pair[SI].GroupLoops;
3843           // accumulate set of all subscripts in group
3844           Pair[SJ].Group |= Pair[SI].Group;
3845           Done = false;
3846         }
3847       }
3848       if (Done) {
3849         if (Pair[SI].Group.count() == 1)
3850           Separable.set(SI);
3851         else
3852           Coupled.set(SI);
3853       }
3854     }
3855   }
3856 
3857   Constraint NewConstraint;
3858   NewConstraint.setAny(SE);
3859 
3860   // test separable subscripts
3861   for (int SI = Separable.find_first(); SI >= 0; SI = Separable.find_next(SI)) {
3862     switch (Pair[SI].Classification) {
3863     case Subscript::SIV: {
3864       unsigned Level;
3865       const SCEV *SplitIter = nullptr;
3866       (void) testSIV(Pair[SI].Src, Pair[SI].Dst, Level,
3867                      Result, NewConstraint, SplitIter);
3868       if (Level == SplitLevel) {
3869         assert(SplitIter != nullptr);
3870         return SplitIter;
3871       }
3872       break;
3873     }
3874     case Subscript::ZIV:
3875     case Subscript::RDIV:
3876     case Subscript::MIV:
3877       break;
3878     default:
3879       llvm_unreachable("subscript has unexpected classification");
3880     }
3881   }
3882 
3883   if (Coupled.count()) {
3884     // test coupled subscript groups
3885     SmallVector<Constraint, 4> Constraints(MaxLevels + 1);
3886     for (unsigned II = 0; II <= MaxLevels; ++II)
3887       Constraints[II].setAny(SE);
3888     for (int SI = Coupled.find_first(); SI >= 0; SI = Coupled.find_next(SI)) {
3889       SmallBitVector Group(Pair[SI].Group);
3890       SmallBitVector Sivs(Pairs);
3891       SmallBitVector Mivs(Pairs);
3892       SmallBitVector ConstrainedLevels(MaxLevels + 1);
3893       for (int SJ = Group.find_first(); SJ >= 0; SJ = Group.find_next(SJ)) {
3894         if (Pair[SJ].Classification == Subscript::SIV)
3895           Sivs.set(SJ);
3896         else
3897           Mivs.set(SJ);
3898       }
3899       while (Sivs.any()) {
3900         bool Changed = false;
3901         for (int SJ = Sivs.find_first(); SJ >= 0; SJ = Sivs.find_next(SJ)) {
3902           // SJ is an SIV subscript that's part of the current coupled group
3903           unsigned Level;
3904           const SCEV *SplitIter = nullptr;
3905           (void) testSIV(Pair[SJ].Src, Pair[SJ].Dst, Level,
3906                          Result, NewConstraint, SplitIter);
3907           if (Level == SplitLevel && SplitIter)
3908             return SplitIter;
3909           ConstrainedLevels.set(Level);
3910           if (intersectConstraints(&Constraints[Level], &NewConstraint))
3911             Changed = true;
3912           Sivs.reset(SJ);
3913         }
3914         if (Changed) {
3915           // propagate, possibly creating new SIVs and ZIVs
3916           for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
3917             // SJ is an MIV subscript that's part of the current coupled group
3918             if (propagate(Pair[SJ].Src, Pair[SJ].Dst,
3919                           Pair[SJ].Loops, Constraints, Result.Consistent)) {
3920               Pair[SJ].Classification =
3921                 classifyPair(Pair[SJ].Src, LI->getLoopFor(Src->getParent()),
3922                              Pair[SJ].Dst, LI->getLoopFor(Dst->getParent()),
3923                              Pair[SJ].Loops);
3924               switch (Pair[SJ].Classification) {
3925               case Subscript::ZIV:
3926                 Mivs.reset(SJ);
3927                 break;
3928               case Subscript::SIV:
3929                 Sivs.set(SJ);
3930                 Mivs.reset(SJ);
3931                 break;
3932               case Subscript::RDIV:
3933               case Subscript::MIV:
3934                 break;
3935               default:
3936                 llvm_unreachable("bad subscript classification");
3937               }
3938             }
3939           }
3940         }
3941       }
3942     }
3943   }
3944   llvm_unreachable("somehow reached end of routine");
3945   return nullptr;
3946 }
3947