aboutsummaryrefslogtreecommitdiffstats
path: root/contrib/libs/llvm12/lib/Analysis/IVDescriptors.cpp
blob: 94a24ccf2155696b15b08bababb386c7675c0c1e (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
//===- llvm/Analysis/IVDescriptors.cpp - IndVar Descriptors -----*- C++ -*-===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
//
// This file "describes" induction and recurrence variables.
//
//===----------------------------------------------------------------------===//

#include "llvm/Analysis/IVDescriptors.h"
#include "llvm/ADT/ScopeExit.h"
#include "llvm/Analysis/BasicAliasAnalysis.h"
#include "llvm/Analysis/DemandedBits.h"
#include "llvm/Analysis/DomTreeUpdater.h"
#include "llvm/Analysis/GlobalsModRef.h"
#include "llvm/Analysis/InstructionSimplify.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/LoopPass.h"
#include "llvm/Analysis/MustExecute.h"
#include "llvm/Analysis/ScalarEvolution.h"
#include "llvm/Analysis/ScalarEvolutionAliasAnalysis.h"
#include "llvm/Analysis/ScalarEvolutionExpressions.h"
#include "llvm/Analysis/TargetTransformInfo.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/IR/Dominators.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/Module.h"
#include "llvm/IR/PatternMatch.h"
#include "llvm/IR/ValueHandle.h"
#include "llvm/Pass.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/KnownBits.h"

using namespace llvm;
using namespace llvm::PatternMatch;

#define DEBUG_TYPE "iv-descriptors"

bool RecurrenceDescriptor::areAllUsesIn(Instruction *I,
                                        SmallPtrSetImpl<Instruction *> &Set) {
  for (User::op_iterator Use = I->op_begin(), E = I->op_end(); Use != E; ++Use)
    if (!Set.count(dyn_cast<Instruction>(*Use)))
      return false;
  return true;
}

bool RecurrenceDescriptor::isIntegerRecurrenceKind(RecurKind Kind) {
  switch (Kind) {
  default:
    break;
  case RecurKind::Add:
  case RecurKind::Mul:
  case RecurKind::Or:
  case RecurKind::And:
  case RecurKind::Xor:
  case RecurKind::SMax:
  case RecurKind::SMin:
  case RecurKind::UMax:
  case RecurKind::UMin:
    return true;
  }
  return false;
}

bool RecurrenceDescriptor::isFloatingPointRecurrenceKind(RecurKind Kind) {
  return (Kind != RecurKind::None) && !isIntegerRecurrenceKind(Kind);
}

bool RecurrenceDescriptor::isArithmeticRecurrenceKind(RecurKind Kind) {
  switch (Kind) {
  default:
    break;
  case RecurKind::Add:
  case RecurKind::Mul:
  case RecurKind::FAdd:
  case RecurKind::FMul:
    return true;
  }
  return false;
}

/// Determines if Phi may have been type-promoted. If Phi has a single user
/// that ANDs the Phi with a type mask, return the user. RT is updated to
/// account for the narrower bit width represented by the mask, and the AND
/// instruction is added to CI.
static Instruction *lookThroughAnd(PHINode *Phi, Type *&RT,
                                   SmallPtrSetImpl<Instruction *> &Visited,
                                   SmallPtrSetImpl<Instruction *> &CI) {
  if (!Phi->hasOneUse())
    return Phi;

  const APInt *M = nullptr;
  Instruction *I, *J = cast<Instruction>(Phi->use_begin()->getUser());

  // Matches either I & 2^x-1 or 2^x-1 & I. If we find a match, we update RT
  // with a new integer type of the corresponding bit width.
  if (match(J, m_c_And(m_Instruction(I), m_APInt(M)))) {
    int32_t Bits = (*M + 1).exactLogBase2();
    if (Bits > 0) {
      RT = IntegerType::get(Phi->getContext(), Bits);
      Visited.insert(Phi);
      CI.insert(J);
      return J;
    }
  }
  return Phi;
}

/// Compute the minimal bit width needed to represent a reduction whose exit
/// instruction is given by Exit.
static std::pair<Type *, bool> computeRecurrenceType(Instruction *Exit,
                                                     DemandedBits *DB,
                                                     AssumptionCache *AC,
                                                     DominatorTree *DT) {
  bool IsSigned = false;
  const DataLayout &DL = Exit->getModule()->getDataLayout();
  uint64_t MaxBitWidth = DL.getTypeSizeInBits(Exit->getType());

  if (DB) {
    // Use the demanded bits analysis to determine the bits that are live out
    // of the exit instruction, rounding up to the nearest power of two. If the
    // use of demanded bits results in a smaller bit width, we know the value
    // must be positive (i.e., IsSigned = false), because if this were not the
    // case, the sign bit would have been demanded.
    auto Mask = DB->getDemandedBits(Exit);
    MaxBitWidth = Mask.getBitWidth() - Mask.countLeadingZeros();
  }

  if (MaxBitWidth == DL.getTypeSizeInBits(Exit->getType()) && AC && DT) {
    // If demanded bits wasn't able to limit the bit width, we can try to use
    // value tracking instead. This can be the case, for example, if the value
    // may be negative.
    auto NumSignBits = ComputeNumSignBits(Exit, DL, 0, AC, nullptr, DT);
    auto NumTypeBits = DL.getTypeSizeInBits(Exit->getType());
    MaxBitWidth = NumTypeBits - NumSignBits;
    KnownBits Bits = computeKnownBits(Exit, DL);
    if (!Bits.isNonNegative()) {
      // If the value is not known to be non-negative, we set IsSigned to true,
      // meaning that we will use sext instructions instead of zext
      // instructions to restore the original type.
      IsSigned = true;
      if (!Bits.isNegative())
        // If the value is not known to be negative, we don't known what the
        // upper bit is, and therefore, we don't know what kind of extend we
        // will need. In this case, just increase the bit width by one bit and
        // use sext.
        ++MaxBitWidth;
    }
  }
  if (!isPowerOf2_64(MaxBitWidth))
    MaxBitWidth = NextPowerOf2(MaxBitWidth);

  return std::make_pair(Type::getIntNTy(Exit->getContext(), MaxBitWidth),
                        IsSigned);
}

/// Collect cast instructions that can be ignored in the vectorizer's cost
/// model, given a reduction exit value and the minimal type in which the
/// reduction can be represented.
static void collectCastsToIgnore(Loop *TheLoop, Instruction *Exit,
                                 Type *RecurrenceType,
                                 SmallPtrSetImpl<Instruction *> &Casts) {

  SmallVector<Instruction *, 8> Worklist;
  SmallPtrSet<Instruction *, 8> Visited;
  Worklist.push_back(Exit);

  while (!Worklist.empty()) {
    Instruction *Val = Worklist.pop_back_val();
    Visited.insert(Val);
    if (auto *Cast = dyn_cast<CastInst>(Val))
      if (Cast->getSrcTy() == RecurrenceType) {
        // If the source type of a cast instruction is equal to the recurrence
        // type, it will be eliminated, and should be ignored in the vectorizer
        // cost model.
        Casts.insert(Cast);
        continue;
      }

    // Add all operands to the work list if they are loop-varying values that
    // we haven't yet visited.
    for (Value *O : cast<User>(Val)->operands())
      if (auto *I = dyn_cast<Instruction>(O))
        if (TheLoop->contains(I) && !Visited.count(I))
          Worklist.push_back(I);
  }
}

bool RecurrenceDescriptor::AddReductionVar(PHINode *Phi, RecurKind Kind,
                                           Loop *TheLoop, bool HasFunNoNaNAttr,
                                           RecurrenceDescriptor &RedDes,
                                           DemandedBits *DB,
                                           AssumptionCache *AC,
                                           DominatorTree *DT) {
  if (Phi->getNumIncomingValues() != 2)
    return false;

  // Reduction variables are only found in the loop header block.
  if (Phi->getParent() != TheLoop->getHeader())
    return false;

  // Obtain the reduction start value from the value that comes from the loop
  // preheader.
  Value *RdxStart = Phi->getIncomingValueForBlock(TheLoop->getLoopPreheader());

  // ExitInstruction is the single value which is used outside the loop.
  // We only allow for a single reduction value to be used outside the loop.
  // This includes users of the reduction, variables (which form a cycle
  // which ends in the phi node).
  Instruction *ExitInstruction = nullptr;
  // Indicates that we found a reduction operation in our scan.
  bool FoundReduxOp = false;

  // We start with the PHI node and scan for all of the users of this
  // instruction. All users must be instructions that can be used as reduction
  // variables (such as ADD). We must have a single out-of-block user. The cycle
  // must include the original PHI.
  bool FoundStartPHI = false;

  // To recognize min/max patterns formed by a icmp select sequence, we store
  // the number of instruction we saw from the recognized min/max pattern,
  //  to make sure we only see exactly the two instructions.
  unsigned NumCmpSelectPatternInst = 0;
  InstDesc ReduxDesc(false, nullptr);

  // Data used for determining if the recurrence has been type-promoted.
  Type *RecurrenceType = Phi->getType();
  SmallPtrSet<Instruction *, 4> CastInsts;
  Instruction *Start = Phi;
  bool IsSigned = false;

  SmallPtrSet<Instruction *, 8> VisitedInsts;
  SmallVector<Instruction *, 8> Worklist;

  // Return early if the recurrence kind does not match the type of Phi. If the
  // recurrence kind is arithmetic, we attempt to look through AND operations
  // resulting from the type promotion performed by InstCombine.  Vector
  // operations are not limited to the legal integer widths, so we may be able
  // to evaluate the reduction in the narrower width.
  if (RecurrenceType->isFloatingPointTy()) {
    if (!isFloatingPointRecurrenceKind(Kind))
      return false;
  } else if (RecurrenceType->isIntegerTy()) {
    if (!isIntegerRecurrenceKind(Kind))
      return false;
    if (isArithmeticRecurrenceKind(Kind))
      Start = lookThroughAnd(Phi, RecurrenceType, VisitedInsts, CastInsts);
  } else {
    // Pointer min/max may exist, but it is not supported as a reduction op.
    return false;
  }

  Worklist.push_back(Start);
  VisitedInsts.insert(Start);

  // Start with all flags set because we will intersect this with the reduction
  // flags from all the reduction operations.
  FastMathFlags FMF = FastMathFlags::getFast();

  // A value in the reduction can be used:
  //  - By the reduction:
  //      - Reduction operation:
  //        - One use of reduction value (safe).
  //        - Multiple use of reduction value (not safe).
  //      - PHI:
  //        - All uses of the PHI must be the reduction (safe).
  //        - Otherwise, not safe.
  //  - By instructions outside of the loop (safe).
  //      * One value may have several outside users, but all outside
  //        uses must be of the same value.
  //  - By an instruction that is not part of the reduction (not safe).
  //    This is either:
  //      * An instruction type other than PHI or the reduction operation.
  //      * A PHI in the header other than the initial PHI.
  while (!Worklist.empty()) {
    Instruction *Cur = Worklist.pop_back_val();

    // No Users.
    // If the instruction has no users then this is a broken chain and can't be
    // a reduction variable.
    if (Cur->use_empty())
      return false;

    bool IsAPhi = isa<PHINode>(Cur);

    // A header PHI use other than the original PHI.
    if (Cur != Phi && IsAPhi && Cur->getParent() == Phi->getParent())
      return false;

    // Reductions of instructions such as Div, and Sub is only possible if the
    // LHS is the reduction variable.
    if (!Cur->isCommutative() && !IsAPhi && !isa<SelectInst>(Cur) &&
        !isa<ICmpInst>(Cur) && !isa<FCmpInst>(Cur) &&
        !VisitedInsts.count(dyn_cast<Instruction>(Cur->getOperand(0))))
      return false;

    // Any reduction instruction must be of one of the allowed kinds. We ignore
    // the starting value (the Phi or an AND instruction if the Phi has been
    // type-promoted).
    if (Cur != Start) {
      ReduxDesc = isRecurrenceInstr(Cur, Kind, ReduxDesc, HasFunNoNaNAttr);
      if (!ReduxDesc.isRecurrence())
        return false;
      // FIXME: FMF is allowed on phi, but propagation is not handled correctly.
      if (isa<FPMathOperator>(ReduxDesc.getPatternInst()) && !IsAPhi)
        FMF &= ReduxDesc.getPatternInst()->getFastMathFlags();
      // Update this reduction kind if we matched a new instruction.
      // TODO: Can we eliminate the need for a 2nd InstDesc by keeping 'Kind'
      //       state accurate while processing the worklist?
      if (ReduxDesc.getRecKind() != RecurKind::None)
        Kind = ReduxDesc.getRecKind();
    }

    bool IsASelect = isa<SelectInst>(Cur);

    // A conditional reduction operation must only have 2 or less uses in
    // VisitedInsts.
    if (IsASelect && (Kind == RecurKind::FAdd || Kind == RecurKind::FMul) &&
        hasMultipleUsesOf(Cur, VisitedInsts, 2))
      return false;

    // A reduction operation must only have one use of the reduction value.
    if (!IsAPhi && !IsASelect && !isMinMaxRecurrenceKind(Kind) &&
        hasMultipleUsesOf(Cur, VisitedInsts, 1))
      return false;

    // All inputs to a PHI node must be a reduction value.
    if (IsAPhi && Cur != Phi && !areAllUsesIn(Cur, VisitedInsts))
      return false;

    if (isIntMinMaxRecurrenceKind(Kind) &&
        (isa<ICmpInst>(Cur) || isa<SelectInst>(Cur)))
      ++NumCmpSelectPatternInst;
    if (isFPMinMaxRecurrenceKind(Kind) &&
        (isa<FCmpInst>(Cur) || isa<SelectInst>(Cur)))
      ++NumCmpSelectPatternInst;

    // Check  whether we found a reduction operator.
    FoundReduxOp |= !IsAPhi && Cur != Start;

    // Process users of current instruction. Push non-PHI nodes after PHI nodes
    // onto the stack. This way we are going to have seen all inputs to PHI
    // nodes once we get to them.
    SmallVector<Instruction *, 8> NonPHIs;
    SmallVector<Instruction *, 8> PHIs;
    for (User *U : Cur->users()) {
      Instruction *UI = cast<Instruction>(U);

      // Check if we found the exit user.
      BasicBlock *Parent = UI->getParent();
      if (!TheLoop->contains(Parent)) {
        // If we already know this instruction is used externally, move on to
        // the next user.
        if (ExitInstruction == Cur)
          continue;

        // Exit if you find multiple values used outside or if the header phi
        // node is being used. In this case the user uses the value of the
        // previous iteration, in which case we would loose "VF-1" iterations of
        // the reduction operation if we vectorize.
        if (ExitInstruction != nullptr || Cur == Phi)
          return false;

        // The instruction used by an outside user must be the last instruction
        // before we feed back to the reduction phi. Otherwise, we loose VF-1
        // operations on the value.
        if (!is_contained(Phi->operands(), Cur))
          return false;

        ExitInstruction = Cur;
        continue;
      }

      // Process instructions only once (termination). Each reduction cycle
      // value must only be used once, except by phi nodes and min/max
      // reductions which are represented as a cmp followed by a select.
      InstDesc IgnoredVal(false, nullptr);
      if (VisitedInsts.insert(UI).second) {
        if (isa<PHINode>(UI))
          PHIs.push_back(UI);
        else
          NonPHIs.push_back(UI);
      } else if (!isa<PHINode>(UI) &&
                 ((!isa<FCmpInst>(UI) && !isa<ICmpInst>(UI) &&
                   !isa<SelectInst>(UI)) ||
                  (!isConditionalRdxPattern(Kind, UI).isRecurrence() &&
                   !isMinMaxSelectCmpPattern(UI, IgnoredVal).isRecurrence())))
        return false;

      // Remember that we completed the cycle.
      if (UI == Phi)
        FoundStartPHI = true;
    }
    Worklist.append(PHIs.begin(), PHIs.end());
    Worklist.append(NonPHIs.begin(), NonPHIs.end());
  }

  // This means we have seen one but not the other instruction of the
  // pattern or more than just a select and cmp.
  if (isMinMaxRecurrenceKind(Kind) && NumCmpSelectPatternInst != 2)
    return false;

  if (!FoundStartPHI || !FoundReduxOp || !ExitInstruction)
    return false;

  if (Start != Phi) {
    // If the starting value is not the same as the phi node, we speculatively
    // looked through an 'and' instruction when evaluating a potential
    // arithmetic reduction to determine if it may have been type-promoted.
    //
    // We now compute the minimal bit width that is required to represent the
    // reduction. If this is the same width that was indicated by the 'and', we
    // can represent the reduction in the smaller type. The 'and' instruction
    // will be eliminated since it will essentially be a cast instruction that
    // can be ignore in the cost model. If we compute a different type than we
    // did when evaluating the 'and', the 'and' will not be eliminated, and we
    // will end up with different kinds of operations in the recurrence
    // expression (e.g., IntegerAND, IntegerADD). We give up if this is
    // the case.
    //
    // The vectorizer relies on InstCombine to perform the actual
    // type-shrinking. It does this by inserting instructions to truncate the
    // exit value of the reduction to the width indicated by RecurrenceType and
    // then extend this value back to the original width. If IsSigned is false,
    // a 'zext' instruction will be generated; otherwise, a 'sext' will be
    // used.
    //
    // TODO: We should not rely on InstCombine to rewrite the reduction in the
    //       smaller type. We should just generate a correctly typed expression
    //       to begin with.
    Type *ComputedType;
    std::tie(ComputedType, IsSigned) =
        computeRecurrenceType(ExitInstruction, DB, AC, DT);
    if (ComputedType != RecurrenceType)
      return false;

    // The recurrence expression will be represented in a narrower type. If
    // there are any cast instructions that will be unnecessary, collect them
    // in CastInsts. Note that the 'and' instruction was already included in
    // this list.
    //
    // TODO: A better way to represent this may be to tag in some way all the
    //       instructions that are a part of the reduction. The vectorizer cost
    //       model could then apply the recurrence type to these instructions,
    //       without needing a white list of instructions to ignore.
    //       This may also be useful for the inloop reductions, if it can be
    //       kept simple enough.
    collectCastsToIgnore(TheLoop, ExitInstruction, RecurrenceType, CastInsts);
  }

  // We found a reduction var if we have reached the original phi node and we
  // only have a single instruction with out-of-loop users.

  // The ExitInstruction(Instruction which is allowed to have out-of-loop users)
  // is saved as part of the RecurrenceDescriptor.

  // Save the description of this reduction variable.
  RecurrenceDescriptor RD(RdxStart, ExitInstruction, Kind, FMF,
                          ReduxDesc.getUnsafeAlgebraInst(), RecurrenceType,
                          IsSigned, CastInsts);
  RedDes = RD;

  return true;
}

RecurrenceDescriptor::InstDesc
RecurrenceDescriptor::isMinMaxSelectCmpPattern(Instruction *I,
                                               const InstDesc &Prev) {
  assert((isa<CmpInst>(I) || isa<SelectInst>(I)) &&
         "Expected a cmp or select instruction");

  // We must handle the select(cmp()) as a single instruction. Advance to the
  // select.
  CmpInst::Predicate Pred;
  if (match(I, m_OneUse(m_Cmp(Pred, m_Value(), m_Value())))) {
    if (auto *Select = dyn_cast<SelectInst>(*I->user_begin()))
      return InstDesc(Select, Prev.getRecKind());
  }

  // Only match select with single use cmp condition.
  if (!match(I, m_Select(m_OneUse(m_Cmp(Pred, m_Value(), m_Value())), m_Value(),
                         m_Value())))
    return InstDesc(false, I);

  // Look for a min/max pattern.
  if (match(I, m_UMin(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::UMin);
  if (match(I, m_UMax(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::UMax);
  if (match(I, m_SMax(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::SMax);
  if (match(I, m_SMin(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::SMin);
  if (match(I, m_OrdFMin(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::FMin);
  if (match(I, m_OrdFMax(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::FMax);
  if (match(I, m_UnordFMin(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::FMin);
  if (match(I, m_UnordFMax(m_Value(), m_Value())))
    return InstDesc(I, RecurKind::FMax);

  return InstDesc(false, I);
}

/// Returns true if the select instruction has users in the compare-and-add
/// reduction pattern below. The select instruction argument is the last one
/// in the sequence.
///
/// %sum.1 = phi ...
/// ...
/// %cmp = fcmp pred %0, %CFP
/// %add = fadd %0, %sum.1
/// %sum.2 = select %cmp, %add, %sum.1
RecurrenceDescriptor::InstDesc
RecurrenceDescriptor::isConditionalRdxPattern(RecurKind Kind, Instruction *I) {
  SelectInst *SI = dyn_cast<SelectInst>(I);
  if (!SI)
    return InstDesc(false, I);

  CmpInst *CI = dyn_cast<CmpInst>(SI->getCondition());
  // Only handle single use cases for now.
  if (!CI || !CI->hasOneUse())
    return InstDesc(false, I);

  Value *TrueVal = SI->getTrueValue();
  Value *FalseVal = SI->getFalseValue();
  // Handle only when either of operands of select instruction is a PHI
  // node for now.
  if ((isa<PHINode>(*TrueVal) && isa<PHINode>(*FalseVal)) ||
      (!isa<PHINode>(*TrueVal) && !isa<PHINode>(*FalseVal)))
    return InstDesc(false, I);

  Instruction *I1 =
      isa<PHINode>(*TrueVal) ? dyn_cast<Instruction>(FalseVal)
                             : dyn_cast<Instruction>(TrueVal);
  if (!I1 || !I1->isBinaryOp())
    return InstDesc(false, I);

  Value *Op1, *Op2;
  if ((m_FAdd(m_Value(Op1), m_Value(Op2)).match(I1)  ||
       m_FSub(m_Value(Op1), m_Value(Op2)).match(I1)) &&
      I1->isFast())
    return InstDesc(Kind == RecurKind::FAdd, SI);

  if (m_FMul(m_Value(Op1), m_Value(Op2)).match(I1) && (I1->isFast()))
    return InstDesc(Kind == RecurKind::FMul, SI);

  return InstDesc(false, I);
}

RecurrenceDescriptor::InstDesc
RecurrenceDescriptor::isRecurrenceInstr(Instruction *I, RecurKind Kind,
                                        InstDesc &Prev, bool HasFunNoNaNAttr) {
  Instruction *UAI = Prev.getUnsafeAlgebraInst();
  if (!UAI && isa<FPMathOperator>(I) && !I->hasAllowReassoc())
    UAI = I; // Found an unsafe (unvectorizable) algebra instruction.

  switch (I->getOpcode()) {
  default:
    return InstDesc(false, I);
  case Instruction::PHI:
    return InstDesc(I, Prev.getRecKind(), Prev.getUnsafeAlgebraInst());
  case Instruction::Sub:
  case Instruction::Add:
    return InstDesc(Kind == RecurKind::Add, I);
  case Instruction::Mul:
    return InstDesc(Kind == RecurKind::Mul, I);
  case Instruction::And:
    return InstDesc(Kind == RecurKind::And, I);
  case Instruction::Or:
    return InstDesc(Kind == RecurKind::Or, I);
  case Instruction::Xor:
    return InstDesc(Kind == RecurKind::Xor, I);
  case Instruction::FDiv:
  case Instruction::FMul:
    return InstDesc(Kind == RecurKind::FMul, I, UAI);
  case Instruction::FSub:
  case Instruction::FAdd:
    return InstDesc(Kind == RecurKind::FAdd, I, UAI);
  case Instruction::Select:
    if (Kind == RecurKind::FAdd || Kind == RecurKind::FMul)
      return isConditionalRdxPattern(Kind, I);
    LLVM_FALLTHROUGH;
  case Instruction::FCmp:
  case Instruction::ICmp:
    if (!isIntMinMaxRecurrenceKind(Kind) &&
        (!HasFunNoNaNAttr || !isFPMinMaxRecurrenceKind(Kind)))
      return InstDesc(false, I);
    return isMinMaxSelectCmpPattern(I, Prev);
  }
}

bool RecurrenceDescriptor::hasMultipleUsesOf(
    Instruction *I, SmallPtrSetImpl<Instruction *> &Insts,
    unsigned MaxNumUses) {
  unsigned NumUses = 0;
  for (User::op_iterator Use = I->op_begin(), E = I->op_end(); Use != E;
       ++Use) {
    if (Insts.count(dyn_cast<Instruction>(*Use)))
      ++NumUses;
    if (NumUses > MaxNumUses)
      return true;
  }

  return false;
}
bool RecurrenceDescriptor::isReductionPHI(PHINode *Phi, Loop *TheLoop,
                                          RecurrenceDescriptor &RedDes,
                                          DemandedBits *DB, AssumptionCache *AC,
                                          DominatorTree *DT) {

  BasicBlock *Header = TheLoop->getHeader();
  Function &F = *Header->getParent();
  bool HasFunNoNaNAttr =
      F.getFnAttribute("no-nans-fp-math").getValueAsString() == "true";

  if (AddReductionVar(Phi, RecurKind::Add, TheLoop, HasFunNoNaNAttr, RedDes, DB,
                      AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found an ADD reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::Mul, TheLoop, HasFunNoNaNAttr, RedDes, DB,
                      AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a MUL reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::Or, TheLoop, HasFunNoNaNAttr, RedDes, DB,
                      AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found an OR reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::And, TheLoop, HasFunNoNaNAttr, RedDes, DB,
                      AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found an AND reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::Xor, TheLoop, HasFunNoNaNAttr, RedDes, DB,
                      AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a XOR reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::SMax, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a SMAX reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::SMin, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a SMIN reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::UMax, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a UMAX reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::UMin, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a UMIN reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::FMul, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found an FMult reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::FAdd, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found an FAdd reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::FMax, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a float MAX reduction PHI." << *Phi << "\n");
    return true;
  }
  if (AddReductionVar(Phi, RecurKind::FMin, TheLoop, HasFunNoNaNAttr, RedDes,
                      DB, AC, DT)) {
    LLVM_DEBUG(dbgs() << "Found a float MIN reduction PHI." << *Phi << "\n");
    return true;
  }
  // Not a reduction of known type.
  return false;
}

bool RecurrenceDescriptor::isFirstOrderRecurrence(
    PHINode *Phi, Loop *TheLoop,
    DenseMap<Instruction *, Instruction *> &SinkAfter, DominatorTree *DT) {

  // Ensure the phi node is in the loop header and has two incoming values.
  if (Phi->getParent() != TheLoop->getHeader() ||
      Phi->getNumIncomingValues() != 2)
    return false;

  // Ensure the loop has a preheader and a single latch block. The loop
  // vectorizer will need the latch to set up the next iteration of the loop.
  auto *Preheader = TheLoop->getLoopPreheader();
  auto *Latch = TheLoop->getLoopLatch();
  if (!Preheader || !Latch)
    return false;

  // Ensure the phi node's incoming blocks are the loop preheader and latch.
  if (Phi->getBasicBlockIndex(Preheader) < 0 ||
      Phi->getBasicBlockIndex(Latch) < 0)
    return false;

  // Get the previous value. The previous value comes from the latch edge while
  // the initial value comes form the preheader edge.
  auto *Previous = dyn_cast<Instruction>(Phi->getIncomingValueForBlock(Latch));
  if (!Previous || !TheLoop->contains(Previous) || isa<PHINode>(Previous) ||
      SinkAfter.count(Previous)) // Cannot rely on dominance due to motion.
    return false;

  // Ensure every user of the phi node is dominated by the previous value.
  // The dominance requirement ensures the loop vectorizer will not need to
  // vectorize the initial value prior to the first iteration of the loop.
  // TODO: Consider extending this sinking to handle memory instructions and
  // phis with multiple users.

  // Returns true, if all users of I are dominated by DominatedBy.
  auto allUsesDominatedBy = [DT](Instruction *I, Instruction *DominatedBy) {
    return all_of(I->uses(), [DT, DominatedBy](Use &U) {
      return DT->dominates(DominatedBy, U);
    });
  };

  if (Phi->hasOneUse()) {
    Instruction *I = Phi->user_back();

    // If the user of the PHI is also the incoming value, we potentially have a
    // reduction and which cannot be handled by sinking.
    if (Previous == I)
      return false;

    // We cannot sink terminator instructions.
    if (I->getParent()->getTerminator() == I)
      return false;

    // Do not try to sink an instruction multiple times (if multiple operands
    // are first order recurrences).
    // TODO: We can support this case, by sinking the instruction after the
    // 'deepest' previous instruction.
    if (SinkAfter.find(I) != SinkAfter.end())
      return false;

    if (DT->dominates(Previous, I)) // We already are good w/o sinking.
      return true;

    // We can sink any instruction without side effects, as long as all users
    // are dominated by the instruction we are sinking after.
    if (I->getParent() == Phi->getParent() && !I->mayHaveSideEffects() &&
        allUsesDominatedBy(I, Previous)) {
      SinkAfter[I] = Previous;
      return true;
    }
  }

  return allUsesDominatedBy(Phi, Previous);
}

/// This function returns the identity element (or neutral element) for
/// the operation K.
Constant *RecurrenceDescriptor::getRecurrenceIdentity(RecurKind K, Type *Tp) {
  switch (K) {
  case RecurKind::Xor:
  case RecurKind::Add:
  case RecurKind::Or:
    // Adding, Xoring, Oring zero to a number does not change it.
    return ConstantInt::get(Tp, 0);
  case RecurKind::Mul:
    // Multiplying a number by 1 does not change it.
    return ConstantInt::get(Tp, 1);
  case RecurKind::And:
    // AND-ing a number with an all-1 value does not change it.
    return ConstantInt::get(Tp, -1, true);
  case RecurKind::FMul:
    // Multiplying a number by 1 does not change it.
    return ConstantFP::get(Tp, 1.0L);
  case RecurKind::FAdd:
    // Adding zero to a number does not change it.
    return ConstantFP::get(Tp, 0.0L);
  case RecurKind::UMin:
    return ConstantInt::get(Tp, -1);
  case RecurKind::UMax:
    return ConstantInt::get(Tp, 0);
  case RecurKind::SMin:
    return ConstantInt::get(Tp,
                            APInt::getSignedMaxValue(Tp->getIntegerBitWidth()));
  case RecurKind::SMax:
    return ConstantInt::get(Tp,
                            APInt::getSignedMinValue(Tp->getIntegerBitWidth()));
  case RecurKind::FMin:
    return ConstantFP::getInfinity(Tp, true);
  case RecurKind::FMax:
    return ConstantFP::getInfinity(Tp, false);
  default:
    llvm_unreachable("Unknown recurrence kind");
  }
}

unsigned RecurrenceDescriptor::getOpcode(RecurKind Kind) {
  switch (Kind) {
  case RecurKind::Add:
    return Instruction::Add;
  case RecurKind::Mul:
    return Instruction::Mul;
  case RecurKind::Or:
    return Instruction::Or;
  case RecurKind::And:
    return Instruction::And;
  case RecurKind::Xor:
    return Instruction::Xor;
  case RecurKind::FMul:
    return Instruction::FMul;
  case RecurKind::FAdd:
    return Instruction::FAdd;
  case RecurKind::SMax:
  case RecurKind::SMin:
  case RecurKind::UMax:
  case RecurKind::UMin:
    return Instruction::ICmp;
  case RecurKind::FMax:
  case RecurKind::FMin:
    return Instruction::FCmp;
  default:
    llvm_unreachable("Unknown recurrence operation");
  }
}

SmallVector<Instruction *, 4>
RecurrenceDescriptor::getReductionOpChain(PHINode *Phi, Loop *L) const {
  SmallVector<Instruction *, 4> ReductionOperations;
  unsigned RedOp = getOpcode(Kind);

  // Search down from the Phi to the LoopExitInstr, looking for instructions
  // with a single user of the correct type for the reduction.

  // Note that we check that the type of the operand is correct for each item in
  // the chain, including the last (the loop exit value). This can come up from
  // sub, which would otherwise be treated as an add reduction. MinMax also need
  // to check for a pair of icmp/select, for which we use getNextInstruction and
  // isCorrectOpcode functions to step the right number of instruction, and
  // check the icmp/select pair.
  // FIXME: We also do not attempt to look through Phi/Select's yet, which might
  // be part of the reduction chain, or attempt to looks through And's to find a
  // smaller bitwidth. Subs are also currently not allowed (which are usually
  // treated as part of a add reduction) as they are expected to generally be
  // more expensive than out-of-loop reductions, and need to be costed more
  // carefully.
  unsigned ExpectedUses = 1;
  if (RedOp == Instruction::ICmp || RedOp == Instruction::FCmp)
    ExpectedUses = 2;

  auto getNextInstruction = [&](Instruction *Cur) {
    if (RedOp == Instruction::ICmp || RedOp == Instruction::FCmp) {
      // We are expecting a icmp/select pair, which we go to the next select
      // instruction if we can. We already know that Cur has 2 uses.
      if (isa<SelectInst>(*Cur->user_begin()))
        return cast<Instruction>(*Cur->user_begin());
      else
        return cast<Instruction>(*std::next(Cur->user_begin()));
    }
    return cast<Instruction>(*Cur->user_begin());
  };
  auto isCorrectOpcode = [&](Instruction *Cur) {
    if (RedOp == Instruction::ICmp || RedOp == Instruction::FCmp) {
      Value *LHS, *RHS;
      return SelectPatternResult::isMinOrMax(
          matchSelectPattern(Cur, LHS, RHS).Flavor);
    }
    return Cur->getOpcode() == RedOp;
  };

  // The loop exit instruction we check first (as a quick test) but add last. We
  // check the opcode is correct (and dont allow them to be Subs) and that they
  // have expected to have the expected number of uses. They will have one use
  // from the phi and one from a LCSSA value, no matter the type.
  if (!isCorrectOpcode(LoopExitInstr) || !LoopExitInstr->hasNUses(2))
    return {};

  // Check that the Phi has one (or two for min/max) uses.
  if (!Phi->hasNUses(ExpectedUses))
    return {};
  Instruction *Cur = getNextInstruction(Phi);

  // Each other instruction in the chain should have the expected number of uses
  // and be the correct opcode.
  while (Cur != LoopExitInstr) {
    if (!isCorrectOpcode(Cur) || !Cur->hasNUses(ExpectedUses))
      return {};

    ReductionOperations.push_back(Cur);
    Cur = getNextInstruction(Cur);
  }

  ReductionOperations.push_back(Cur);
  return ReductionOperations;
}

InductionDescriptor::InductionDescriptor(Value *Start, InductionKind K,
                                         const SCEV *Step, BinaryOperator *BOp,
                                         SmallVectorImpl<Instruction *> *Casts)
    : StartValue(Start), IK(K), Step(Step), InductionBinOp(BOp) {
  assert(IK != IK_NoInduction && "Not an induction");

  // Start value type should match the induction kind and the value
  // itself should not be null.
  assert(StartValue && "StartValue is null");
  assert((IK != IK_PtrInduction || StartValue->getType()->isPointerTy()) &&
         "StartValue is not a pointer for pointer induction");
  assert((IK != IK_IntInduction || StartValue->getType()->isIntegerTy()) &&
         "StartValue is not an integer for integer induction");

  // Check the Step Value. It should be non-zero integer value.
  assert((!getConstIntStepValue() || !getConstIntStepValue()->isZero()) &&
         "Step value is zero");

  assert((IK != IK_PtrInduction || getConstIntStepValue()) &&
         "Step value should be constant for pointer induction");
  assert((IK == IK_FpInduction || Step->getType()->isIntegerTy()) &&
         "StepValue is not an integer");

  assert((IK != IK_FpInduction || Step->getType()->isFloatingPointTy()) &&
         "StepValue is not FP for FpInduction");
  assert((IK != IK_FpInduction ||
          (InductionBinOp &&
           (InductionBinOp->getOpcode() == Instruction::FAdd ||
            InductionBinOp->getOpcode() == Instruction::FSub))) &&
         "Binary opcode should be specified for FP induction");

  if (Casts) {
    for (auto &Inst : *Casts) {
      RedundantCasts.push_back(Inst);
    }
  }
}

ConstantInt *InductionDescriptor::getConstIntStepValue() const {
  if (isa<SCEVConstant>(Step))
    return dyn_cast<ConstantInt>(cast<SCEVConstant>(Step)->getValue());
  return nullptr;
}

bool InductionDescriptor::isFPInductionPHI(PHINode *Phi, const Loop *TheLoop,
                                           ScalarEvolution *SE,
                                           InductionDescriptor &D) {

  // Here we only handle FP induction variables.
  assert(Phi->getType()->isFloatingPointTy() && "Unexpected Phi type");

  if (TheLoop->getHeader() != Phi->getParent())
    return false;

  // The loop may have multiple entrances or multiple exits; we can analyze
  // this phi if it has a unique entry value and a unique backedge value.
  if (Phi->getNumIncomingValues() != 2)
    return false;
  Value *BEValue = nullptr, *StartValue = nullptr;
  if (TheLoop->contains(Phi->getIncomingBlock(0))) {
    BEValue = Phi->getIncomingValue(0);
    StartValue = Phi->getIncomingValue(1);
  } else {
    assert(TheLoop->contains(Phi->getIncomingBlock(1)) &&
           "Unexpected Phi node in the loop");
    BEValue = Phi->getIncomingValue(1);
    StartValue = Phi->getIncomingValue(0);
  }

  BinaryOperator *BOp = dyn_cast<BinaryOperator>(BEValue);
  if (!BOp)
    return false;

  Value *Addend = nullptr;
  if (BOp->getOpcode() == Instruction::FAdd) {
    if (BOp->getOperand(0) == Phi)
      Addend = BOp->getOperand(1);
    else if (BOp->getOperand(1) == Phi)
      Addend = BOp->getOperand(0);
  } else if (BOp->getOpcode() == Instruction::FSub)
    if (BOp->getOperand(0) == Phi)
      Addend = BOp->getOperand(1);

  if (!Addend)
    return false;

  // The addend should be loop invariant
  if (auto *I = dyn_cast<Instruction>(Addend))
    if (TheLoop->contains(I))
      return false;

  // FP Step has unknown SCEV
  const SCEV *Step = SE->getUnknown(Addend);
  D = InductionDescriptor(StartValue, IK_FpInduction, Step, BOp);
  return true;
}

/// This function is called when we suspect that the update-chain of a phi node
/// (whose symbolic SCEV expression sin \p PhiScev) contains redundant casts,
/// that can be ignored. (This can happen when the PSCEV rewriter adds a runtime
/// predicate P under which the SCEV expression for the phi can be the
/// AddRecurrence \p AR; See createAddRecFromPHIWithCast). We want to find the
/// cast instructions that are involved in the update-chain of this induction.
/// A caller that adds the required runtime predicate can be free to drop these
/// cast instructions, and compute the phi using \p AR (instead of some scev
/// expression with casts).
///
/// For example, without a predicate the scev expression can take the following
/// form:
///      (Ext ix (Trunc iy ( Start + i*Step ) to ix) to iy)
///
/// It corresponds to the following IR sequence:
/// %for.body:
///   %x = phi i64 [ 0, %ph ], [ %add, %for.body ]
///   %casted_phi = "ExtTrunc i64 %x"
///   %add = add i64 %casted_phi, %step
///
/// where %x is given in \p PN,
/// PSE.getSCEV(%x) is equal to PSE.getSCEV(%casted_phi) under a predicate,
/// and the IR sequence that "ExtTrunc i64 %x" represents can take one of
/// several forms, for example, such as:
///   ExtTrunc1:    %casted_phi = and  %x, 2^n-1
/// or:
///   ExtTrunc2:    %t = shl %x, m
///                 %casted_phi = ashr %t, m
///
/// If we are able to find such sequence, we return the instructions
/// we found, namely %casted_phi and the instructions on its use-def chain up
/// to the phi (not including the phi).
static bool getCastsForInductionPHI(PredicatedScalarEvolution &PSE,
                                    const SCEVUnknown *PhiScev,
                                    const SCEVAddRecExpr *AR,
                                    SmallVectorImpl<Instruction *> &CastInsts) {

  assert(CastInsts.empty() && "CastInsts is expected to be empty.");
  auto *PN = cast<PHINode>(PhiScev->getValue());
  assert(PSE.getSCEV(PN) == AR && "Unexpected phi node SCEV expression");
  const Loop *L = AR->getLoop();

  // Find any cast instructions that participate in the def-use chain of
  // PhiScev in the loop.
  // FORNOW/TODO: We currently expect the def-use chain to include only
  // two-operand instructions, where one of the operands is an invariant.
  // createAddRecFromPHIWithCasts() currently does not support anything more
  // involved than that, so we keep the search simple. This can be
  // extended/generalized as needed.

  auto getDef = [&](const Value *Val) -> Value * {
    const BinaryOperator *BinOp = dyn_cast<BinaryOperator>(Val);
    if (!BinOp)
      return nullptr;
    Value *Op0 = BinOp->getOperand(0);
    Value *Op1 = BinOp->getOperand(1);
    Value *Def = nullptr;
    if (L->isLoopInvariant(Op0))
      Def = Op1;
    else if (L->isLoopInvariant(Op1))
      Def = Op0;
    return Def;
  };

  // Look for the instruction that defines the induction via the
  // loop backedge.
  BasicBlock *Latch = L->getLoopLatch();
  if (!Latch)
    return false;
  Value *Val = PN->getIncomingValueForBlock(Latch);
  if (!Val)
    return false;

  // Follow the def-use chain until the induction phi is reached.
  // If on the way we encounter a Value that has the same SCEV Expr as the
  // phi node, we can consider the instructions we visit from that point
  // as part of the cast-sequence that can be ignored.
  bool InCastSequence = false;
  auto *Inst = dyn_cast<Instruction>(Val);
  while (Val != PN) {
    // If we encountered a phi node other than PN, or if we left the loop,
    // we bail out.
    if (!Inst || !L->contains(Inst)) {
      return false;
    }
    auto *AddRec = dyn_cast<SCEVAddRecExpr>(PSE.getSCEV(Val));
    if (AddRec && PSE.areAddRecsEqualWithPreds(AddRec, AR))
      InCastSequence = true;
    if (InCastSequence) {
      // Only the last instruction in the cast sequence is expected to have
      // uses outside the induction def-use chain.
      if (!CastInsts.empty())
        if (!Inst->hasOneUse())
          return false;
      CastInsts.push_back(Inst);
    }
    Val = getDef(Val);
    if (!Val)
      return false;
    Inst = dyn_cast<Instruction>(Val);
  }

  return InCastSequence;
}

bool InductionDescriptor::isInductionPHI(PHINode *Phi, const Loop *TheLoop,
                                         PredicatedScalarEvolution &PSE,
                                         InductionDescriptor &D, bool Assume) {
  Type *PhiTy = Phi->getType();

  // Handle integer and pointer inductions variables.
  // Now we handle also FP induction but not trying to make a
  // recurrent expression from the PHI node in-place.

  if (!PhiTy->isIntegerTy() && !PhiTy->isPointerTy() && !PhiTy->isFloatTy() &&
      !PhiTy->isDoubleTy() && !PhiTy->isHalfTy())
    return false;

  if (PhiTy->isFloatingPointTy())
    return isFPInductionPHI(Phi, TheLoop, PSE.getSE(), D);

  const SCEV *PhiScev = PSE.getSCEV(Phi);
  const auto *AR = dyn_cast<SCEVAddRecExpr>(PhiScev);

  // We need this expression to be an AddRecExpr.
  if (Assume && !AR)
    AR = PSE.getAsAddRec(Phi);

  if (!AR) {
    LLVM_DEBUG(dbgs() << "LV: PHI is not a poly recurrence.\n");
    return false;
  }

  // Record any Cast instructions that participate in the induction update
  const auto *SymbolicPhi = dyn_cast<SCEVUnknown>(PhiScev);
  // If we started from an UnknownSCEV, and managed to build an addRecurrence
  // only after enabling Assume with PSCEV, this means we may have encountered
  // cast instructions that required adding a runtime check in order to
  // guarantee the correctness of the AddRecurrence respresentation of the
  // induction.
  if (PhiScev != AR && SymbolicPhi) {
    SmallVector<Instruction *, 2> Casts;
    if (getCastsForInductionPHI(PSE, SymbolicPhi, AR, Casts))
      return isInductionPHI(Phi, TheLoop, PSE.getSE(), D, AR, &Casts);
  }

  return isInductionPHI(Phi, TheLoop, PSE.getSE(), D, AR);
}

bool InductionDescriptor::isInductionPHI(
    PHINode *Phi, const Loop *TheLoop, ScalarEvolution *SE,
    InductionDescriptor &D, const SCEV *Expr,
    SmallVectorImpl<Instruction *> *CastsToIgnore) {
  Type *PhiTy = Phi->getType();
  // We only handle integer and pointer inductions variables.
  if (!PhiTy->isIntegerTy() && !PhiTy->isPointerTy())
    return false;

  // Check that the PHI is consecutive.
  const SCEV *PhiScev = Expr ? Expr : SE->getSCEV(Phi);
  const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(PhiScev);

  if (!AR) {
    LLVM_DEBUG(dbgs() << "LV: PHI is not a poly recurrence.\n");
    return false;
  }

  if (AR->getLoop() != TheLoop) {
    // FIXME: We should treat this as a uniform. Unfortunately, we
    // don't currently know how to handled uniform PHIs.
    LLVM_DEBUG(
        dbgs() << "LV: PHI is a recurrence with respect to an outer loop.\n");
    return false;
  }

  Value *StartValue =
      Phi->getIncomingValueForBlock(AR->getLoop()->getLoopPreheader());

  BasicBlock *Latch = AR->getLoop()->getLoopLatch();
  if (!Latch)
    return false;
  BinaryOperator *BOp =
      dyn_cast<BinaryOperator>(Phi->getIncomingValueForBlock(Latch));

  const SCEV *Step = AR->getStepRecurrence(*SE);
  // Calculate the pointer stride and check if it is consecutive.
  // The stride may be a constant or a loop invariant integer value.
  const SCEVConstant *ConstStep = dyn_cast<SCEVConstant>(Step);
  if (!ConstStep && !SE->isLoopInvariant(Step, TheLoop))
    return false;

  if (PhiTy->isIntegerTy()) {
    D = InductionDescriptor(StartValue, IK_IntInduction, Step, BOp,
                            CastsToIgnore);
    return true;
  }

  assert(PhiTy->isPointerTy() && "The PHI must be a pointer");
  // Pointer induction should be a constant.
  if (!ConstStep)
    return false;

  ConstantInt *CV = ConstStep->getValue();
  Type *PointerElementType = PhiTy->getPointerElementType();
  // The pointer stride cannot be determined if the pointer element type is not
  // sized.
  if (!PointerElementType->isSized())
    return false;

  const DataLayout &DL = Phi->getModule()->getDataLayout();
  int64_t Size = static_cast<int64_t>(DL.getTypeAllocSize(PointerElementType));
  if (!Size)
    return false;

  int64_t CVSize = CV->getSExtValue();
  if (CVSize % Size)
    return false;
  auto *StepValue =
      SE->getConstant(CV->getType(), CVSize / Size, true /* signed */);
  D = InductionDescriptor(StartValue, IK_PtrInduction, StepValue, BOp);
  return true;
}