#include <cmath>
using namespace llvm;
-STATISTIC(NumBruteForceEvaluations,
- "Number of brute force evaluations needed to "
- "calculate high-order polynomial exit values");
STATISTIC(NumArrayLenItCounts,
"Number of trip counts computed with array length");
STATISTIC(NumTripCountsComputed,
SCEV::~SCEV() {}
void SCEV::dump() const {
print(cerr);
+ cerr << '\n';
}
uint32_t SCEV::getBitWidth() const {
return LHS->getType();
}
+
+// SCEVSDivs - Only allow the creation of one SCEVSDivExpr for any particular
+// input. Don't use a SCEVHandle here, or else the object will never be
+// deleted!
+static ManagedStatic<std::map<std::pair<SCEV*, SCEV*>,
+ SCEVSDivExpr*> > SCEVSDivs;
+
+SCEVSDivExpr::~SCEVSDivExpr() {
+ SCEVSDivs->erase(std::make_pair(LHS, RHS));
+}
+
+void SCEVSDivExpr::print(std::ostream &OS) const {
+ OS << "(" << *LHS << " /s " << *RHS << ")";
+}
+
+const Type *SCEVSDivExpr::getType() const {
+ return LHS->getType();
+}
+
+
// SCEVAddRecExprs - Only allow the creation of one SCEVAddRecExpr for any
// particular input. Don't use a SCEVHandle here, or else the object will never
// be deleted!
}
-/// BinomialCoefficient - Compute BC(It, K). The result is of the same type as
-/// It. Assume, K > 0.
+/// BinomialCoefficient - Compute BC(It, K). The result has width W.
+// Assume, K > 0.
static SCEVHandle BinomialCoefficient(SCEVHandle It, unsigned K,
- ScalarEvolution &SE) {
+ ScalarEvolution &SE,
+ const IntegerType* ResultTy) {
+ // Handle the simplest case efficiently.
+ if (K == 1)
+ return SE.getTruncateOrZeroExtend(It, ResultTy);
+
// We are using the following formula for BC(It, K):
//
// BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / K!
//
- // Suppose, W is the bitwidth of It (and of the return value as well). We
- // must be prepared for overflow. Hence, we must assure that the result of
- // our computation is equal to the accurate one modulo 2^W. Unfortunately,
- // division isn't safe in modular arithmetic. This means we must perform the
- // whole computation accurately and then truncate the result to W bits.
+ // Suppose, W is the bitwidth of the return value. We must be prepared for
+ // overflow. Hence, we must assure that the result of our computation is
+ // equal to the accurate one modulo 2^W. Unfortunately, division isn't
+ // safe in modular arithmetic.
//
- // The dividend of the formula is a multiplication of K integers of bitwidth
- // W. K*W bits suffice to compute it accurately.
+ // However, this code doesn't use exactly that formula; the formula it uses
+ // is something like the following, where T is the number of factors of 2 in
+ // K! (i.e. trailing zeros in the binary representation of K!), and ^ is
+ // exponentiation:
//
- // FIXME: We assume the divisor can be accurately computed using 16-bit
- // unsigned integer type. It is true up to K = 8 (AddRecs of length 9). In
- // future we may use APInt to use the minimum number of bits necessary to
- // compute it accurately.
+ // BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / 2^T / (K! / 2^T)
//
- // It is safe to use unsigned division here: the dividend is nonnegative and
- // the divisor is positive.
-
- // Handle the simplest case efficiently.
- if (K == 1)
- return It;
-
- assert(K < 9 && "We cannot handle such long AddRecs yet.");
-
- // FIXME: A temporary hack to remove in future. Arbitrary precision integers
- // aren't supported by the code generator yet. For the dividend, the bitwidth
- // we use is the smallest power of 2 greater or equal to K*W and less or equal
- // to 64. Note that setting the upper bound for bitwidth may still lead to
- // miscompilation in some cases.
- unsigned DividendBits = 1U << Log2_32_Ceil(K * It->getBitWidth());
- if (DividendBits > 64)
- DividendBits = 64;
-#if 0 // Waiting for the APInt support in the code generator...
- unsigned DividendBits = K * It->getBitWidth();
-#endif
-
- const IntegerType *DividendTy = IntegerType::get(DividendBits);
- const SCEVHandle ExIt = SE.getTruncateOrZeroExtend(It, DividendTy);
+ // This formula is trivially equivalent to the previous formula. However,
+ // this formula can be implemented much more efficiently. The trick is that
+ // K! / 2^T is odd, and exact division by an odd number *is* safe in modular
+ // arithmetic. To do exact division in modular arithmetic, all we have
+ // to do is multiply by the inverse. Therefore, this step can be done at
+ // width W.
+ //
+ // The next issue is how to safely do the division by 2^T. The way this
+ // is done is by doing the multiplication step at a width of at least W + T
+ // bits. This way, the bottom W+T bits of the product are accurate. Then,
+ // when we perform the division by 2^T (which is equivalent to a right shift
+ // by T), the bottom W bits are accurate. Extra bits are okay; they'll get
+ // truncated out after the division by 2^T.
+ //
+ // In comparison to just directly using the first formula, this technique
+ // is much more efficient; using the first formula requires W * K bits,
+ // but this formula less than W + K bits. Also, the first formula requires
+ // a division step, whereas this formula only requires multiplies and shifts.
+ //
+ // It doesn't matter whether the subtraction step is done in the calculation
+ // width or the input iteration count's width; if the subtraction overflows,
+ // the result must be zero anyway. We prefer here to do it in the width of
+ // the induction variable because it helps a lot for certain cases; CodeGen
+ // isn't smart enough to ignore the overflow, which leads to much less
+ // efficient code if the width of the subtraction is wider than the native
+ // register width.
+ //
+ // (It's possible to not widen at all by pulling out factors of 2 before
+ // the multiplication; for example, K=2 can be calculated as
+ // It/2*(It+(It*INT_MIN/INT_MIN)+-1). However, it requires
+ // extra arithmetic, so it's not an obvious win, and it gets
+ // much more complicated for K > 3.)
+
+ // Protection from insane SCEVs; this bound is conservative,
+ // but it probably doesn't matter.
+ if (K > 1000)
+ return new SCEVCouldNotCompute();
- // The final number of bits we need to perform the division is the maximum of
- // dividend and divisor bitwidths.
- const IntegerType *DivisionTy =
- IntegerType::get(std::max(DividendBits, 16U));
+ unsigned W = ResultTy->getBitWidth();
+
+ // Calculate K! / 2^T and T; we divide out the factors of two before
+ // multiplying for calculating K! / 2^T to avoid overflow.
+ // Other overflow doesn't matter because we only care about the bottom
+ // W bits of the result.
+ APInt OddFactorial(W, 1);
+ unsigned T = 1;
+ for (unsigned i = 3; i <= K; ++i) {
+ APInt Mult(W, i);
+ unsigned TwoFactors = Mult.countTrailingZeros();
+ T += TwoFactors;
+ Mult = Mult.lshr(TwoFactors);
+ OddFactorial *= Mult;
+ }
- // Compute K! We know K >= 2 here.
- unsigned F = 2;
- for (unsigned i = 3; i <= K; ++i)
- F *= i;
- APInt Divisor(DivisionTy->getBitWidth(), F);
+ // We need at least W + T bits for the multiplication step
+ // FIXME: A temporary hack; we round up the bitwidths
+ // to the nearest power of 2 to be nice to the code generator.
+ unsigned CalculationBits = 1U << Log2_32_Ceil(W + T);
+ // FIXME: Temporary hack to avoid generating integers that are too wide.
+ // Although, it's not completely clear how to determine how much
+ // widening is safe; for example, on X86, we can't really widen
+ // beyond 64 because we need to be able to do multiplication
+ // that's CalculationBits wide, but on X86-64, we can safely widen up to
+ // 128 bits.
+ if (CalculationBits > 64)
+ return new SCEVCouldNotCompute();
- // Handle this case efficiently, it is common to have constant iteration
- // counts while computing loop exit values.
- if (SCEVConstant *SC = dyn_cast<SCEVConstant>(ExIt)) {
- const APInt& N = SC->getValue()->getValue();
- APInt Dividend(N.getBitWidth(), 1);
- for (; K; --K)
- Dividend *= N-(K-1);
- if (DividendTy != DivisionTy)
- Dividend = Dividend.zext(DivisionTy->getBitWidth());
+ // Calcuate 2^T, at width T+W.
+ APInt DivFactor = APInt(CalculationBits, 1).shl(T);
+
+ // Calculate the multiplicative inverse of K! / 2^T;
+ // this multiplication factor will perform the exact division by
+ // K! / 2^T.
+ APInt Mod = APInt::getSignedMinValue(W+1);
+ APInt MultiplyFactor = OddFactorial.zext(W+1);
+ MultiplyFactor = MultiplyFactor.multiplicativeInverse(Mod);
+ MultiplyFactor = MultiplyFactor.trunc(W);
+
+ // Calculate the product, at width T+W
+ const IntegerType *CalculationTy = IntegerType::get(CalculationBits);
+ SCEVHandle Dividend = SE.getTruncateOrZeroExtend(It, CalculationTy);
+ for (unsigned i = 1; i != K; ++i) {
+ SCEVHandle S = SE.getMinusSCEV(It, SE.getIntegerSCEV(i, It->getType()));
+ Dividend = SE.getMulExpr(Dividend,
+ SE.getTruncateOrZeroExtend(S, CalculationTy));
+ }
- APInt Result = Dividend.udiv(Divisor);
- if (Result.getBitWidth() != It->getBitWidth())
- Result = Result.trunc(It->getBitWidth());
+ // Divide by 2^T
+ SCEVHandle DivResult = SE.getUDivExpr(Dividend, SE.getConstant(DivFactor));
- return SE.getConstant(Result);
- }
-
- SCEVHandle Dividend = ExIt;
- for (unsigned i = 1; i != K; ++i)
- Dividend =
- SE.getMulExpr(Dividend,
- SE.getMinusSCEV(ExIt, SE.getIntegerSCEV(i, DividendTy)));
+ // Truncate the result, and divide by K! / 2^T.
- return SE.getTruncateOrZeroExtend(
- SE.getUDivExpr(
- SE.getTruncateOrZeroExtend(Dividend, DivisionTy),
- SE.getConstant(Divisor)
- ), It->getType());
+ return SE.getMulExpr(SE.getConstant(MultiplyFactor),
+ SE.getTruncateOrZeroExtend(DivResult, ResultTy));
}
/// evaluateAtIteration - Return the value of this chain of recurrences at
// The computation is correct in the face of overflow provided that the
// multiplication is performed _after_ the evaluation of the binomial
// coefficient.
- SCEVHandle Val = SE.getMulExpr(getOperand(i),
- BinomialCoefficient(It, i, SE));
- Result = SE.getAddExpr(Result, Val);
+ SCEVHandle Coeff = BinomialCoefficient(It, i, SE,
+ cast<IntegerType>(getType()));
+ if (isa<SCEVCouldNotCompute>(Coeff))
+ return Coeff;
+
+ Result = SE.getAddExpr(Result, SE.getMulExpr(getOperand(i), Coeff));
}
return Result;
}
// If we found some loop invariants, fold them into the recurrence.
if (!LIOps.empty()) {
- // NLI + LI + { Start,+,Step} --> NLI + { LI+Start,+,Step }
+ // NLI + LI + {Start,+,Step} --> NLI + {LI+Start,+,Step}
LIOps.push_back(AddRec->getStart());
std::vector<SCEVHandle> AddRecOps(AddRec->op_begin(), AddRec->op_end());
// If we found some loop invariants, fold them into the recurrence.
if (!LIOps.empty()) {
- // NLI * LI * { Start,+,Step} --> NLI * { LI*Start,+,LI*Step }
+ // NLI * LI * {Start,+,Step} --> NLI * {LI*Start,+,LI*Step}
std::vector<SCEVHandle> NewOps;
NewOps.reserve(AddRec->getNumOperands());
if (LIOps.size() == 1) {
}
SCEVHandle ScalarEvolution::getUDivExpr(const SCEVHandle &LHS, const SCEVHandle &RHS) {
+ if (LHS == RHS)
+ return getIntegerSCEV(1, LHS->getType()); // X udiv X --> 1
+
if (SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
if (RHSC->getValue()->equalsInt(1))
- return LHS; // X udiv 1 --> x
+ return LHS; // X udiv 1 --> X
if (SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS)) {
Constant *LHSCV = LHSC->getValue();
}
}
- // FIXME: implement folding of (X*4)/4 when we know X*4 doesn't overflow.
-
SCEVUDivExpr *&Result = (*SCEVUDivs)[std::make_pair(LHS, RHS)];
if (Result == 0) Result = new SCEVUDivExpr(LHS, RHS);
return Result;
}
+SCEVHandle ScalarEvolution::getSDivExpr(const SCEVHandle &LHS, const SCEVHandle &RHS) {
+ if (LHS == RHS)
+ return getIntegerSCEV(1, LHS->getType()); // X sdiv X --> 1
+
+ if (SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
+ if (RHSC->getValue()->equalsInt(1))
+ return LHS; // X sdiv 1 --> X
+
+ if (RHSC->getValue()->isAllOnesValue())
+ return getNegativeSCEV(LHS); // X sdiv -1 --> -X
+
+ if (SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS)) {
+ Constant *LHSCV = LHSC->getValue();
+ Constant *RHSCV = RHSC->getValue();
+ return getUnknown(ConstantExpr::getSDiv(LHSCV, RHSCV));
+ }
+ }
+
+ SCEVSDivExpr *&Result = (*SCEVSDivs)[std::make_pair(LHS, RHS)];
+ if (Result == 0) Result = new SCEVSDivExpr(LHS, RHS);
+ return Result;
+}
+
/// SCEVAddRecExpr::get - Get a add recurrence expression for the
/// specified loop. Simplify the expression as much as possible.
if (Operands.back()->isZero()) {
Operands.pop_back();
- return getAddRecExpr(Operands, L); // { X,+,0 } --> X
+ return getAddRecExpr(Operands, L); // {X,+,0} --> X
+ }
+
+ // Canonicalize nested AddRecs in by nesting them in order of loop depth.
+ if (SCEVAddRecExpr *NestedAR = dyn_cast<SCEVAddRecExpr>(Operands[0])) {
+ const Loop* NestedLoop = NestedAR->getLoop();
+ if (L->getLoopDepth() < NestedLoop->getLoopDepth()) {
+ std::vector<SCEVHandle> NestedOperands(NestedAR->op_begin(),
+ NestedAR->op_end());
+ SCEVHandle NestedARHandle(NestedAR);
+ Operands[0] = NestedAR->getStart();
+ NestedOperands[0] = getAddRecExpr(Operands, L);
+ return getAddRecExpr(NestedOperands, NestedLoop);
+ }
}
SCEVAddRecExpr *&Result =
void setSCEV(Value *V, const SCEVHandle &H) {
bool isNew = Scalars.insert(std::make_pair(V, H)).second;
assert(isNew && "This entry already existed!");
+ isNew = false;
}
/// specified less-than comparison will execute. If not computable, return
/// UnknownValue. isSigned specifies whether the less-than is signed.
SCEVHandle HowManyLessThans(SCEV *LHS, SCEV *RHS, const Loop *L,
- bool isSigned);
+ bool isSigned, bool trueWhenEqual);
+
+ /// getPredecessorWithUniqueSuccessorForBB - Return a predecessor of BB
+ /// (which may not be an immediate predecessor) which has exactly one
+ /// successor from which BB is reachable, or null if no such block is
+ /// found.
+ BasicBlock* getPredecessorWithUniqueSuccessorForBB(BasicBlock *BB);
/// executesAtLeastOnce - Test whether entry to the loop is protected by
/// a conditional between LHS and RHS.
- bool executesAtLeastOnce(const Loop *L, bool isSigned, SCEV *LHS, SCEV *RHS);
+ bool executesAtLeastOnce(const Loop *L, bool isSigned, bool trueWhenEqual,
+ SCEV *LHS, SCEV *RHS);
+
+ /// potentialInfiniteLoop - Test whether the loop might jump over the exit value
+ /// due to wrapping.
+ bool potentialInfiniteLoop(SCEV *Stride, SCEV *RHS, bool isSigned,
+ bool trueWhenEqual);
/// getConstantEvolutionLoopExitValue - If we know that the specified Phi is
/// in the header of its containing loop, we know the loop executes a
return MinOpRes;
}
- // SCEVUDivExpr, SCEVUnknown
+ // SCEVUDivExpr, SCEVSDivExpr, SCEVUnknown
return 0;
}
case Instruction::UDiv:
return SE.getUDivExpr(getSCEV(U->getOperand(0)),
getSCEV(U->getOperand(1)));
+ case Instruction::SDiv:
+ return SE.getSDivExpr(getSCEV(U->getOperand(0)),
+ getSCEV(U->getOperand(1)));
case Instruction::Sub:
return SE.getMinusSCEV(getSCEV(U->getOperand(0)),
getSCEV(U->getOperand(1)));
break;
case Instruction::LShr:
- // Turn logical shift right of a constant into a unsigned divide.
+ // Turn logical shift right of a constant into an unsigned divide.
if (ConstantInt *SA = dyn_cast<ConstantInt>(U->getOperand(1))) {
uint32_t BitWidth = cast<IntegerType>(V->getType())->getBitWidth();
Constant *X = ConstantInt::get(
if (LHS == U->getOperand(1) && RHS == U->getOperand(2))
return SE.getSMaxExpr(getSCEV(LHS), getSCEV(RHS));
else if (LHS == U->getOperand(2) && RHS == U->getOperand(1))
- // -smax(-x, -y) == smin(x, y).
- return SE.getNegativeSCEV(SE.getSMaxExpr(
- SE.getNegativeSCEV(getSCEV(LHS)),
- SE.getNegativeSCEV(getSCEV(RHS))));
+ // ~smax(~x, ~y) == smin(x, y).
+ return SE.getNotSCEV(SE.getSMaxExpr(
+ SE.getNotSCEV(getSCEV(LHS)),
+ SE.getNotSCEV(getSCEV(RHS))));
break;
case ICmpInst::ICMP_ULT:
case ICmpInst::ICMP_ULE:
// At this point, we would like to compute how many iterations of the
// loop the predicate will return true for these inputs.
- if (isa<SCEVConstant>(LHS) && !isa<SCEVConstant>(RHS)) {
- // If there is a constant, force it into the RHS.
+ if (LHS->isLoopInvariant(L) && !RHS->isLoopInvariant(L)) {
+ // If there is a loop-invariant, force it into the RHS.
std::swap(LHS, RHS);
Cond = ICmpInst::getSwappedPredicate(Cond);
}
break;
}
case ICmpInst::ICMP_SLT: {
- SCEVHandle TC = HowManyLessThans(LHS, RHS, L, true);
+ SCEVHandle TC = HowManyLessThans(LHS, RHS, L, true, false);
if (!isa<SCEVCouldNotCompute>(TC)) return TC;
break;
}
case ICmpInst::ICMP_SGT: {
- SCEVHandle TC = HowManyLessThans(SE.getNegativeSCEV(LHS),
- SE.getNegativeSCEV(RHS), L, true);
+ SCEVHandle TC = HowManyLessThans(SE.getNotSCEV(LHS),
+ SE.getNotSCEV(RHS), L, true, false);
if (!isa<SCEVCouldNotCompute>(TC)) return TC;
break;
}
case ICmpInst::ICMP_ULT: {
- SCEVHandle TC = HowManyLessThans(LHS, RHS, L, false);
+ SCEVHandle TC = HowManyLessThans(LHS, RHS, L, false, false);
if (!isa<SCEVCouldNotCompute>(TC)) return TC;
break;
}
case ICmpInst::ICMP_UGT: {
SCEVHandle TC = HowManyLessThans(SE.getNotSCEV(LHS),
- SE.getNotSCEV(RHS), L, false);
+ SE.getNotSCEV(RHS), L, false, false);
+ if (!isa<SCEVCouldNotCompute>(TC)) return TC;
+ break;
+ }
+ case ICmpInst::ICMP_SLE: {
+ SCEVHandle TC = HowManyLessThans(LHS, RHS, L, true, true);
+ if (!isa<SCEVCouldNotCompute>(TC)) return TC;
+ break;
+ }
+ case ICmpInst::ICMP_SGE: {
+ SCEVHandle TC = HowManyLessThans(SE.getNotSCEV(LHS),
+ SE.getNotSCEV(RHS), L, true, true);
+ if (!isa<SCEVCouldNotCompute>(TC)) return TC;
+ break;
+ }
+ case ICmpInst::ICMP_ULE: {
+ SCEVHandle TC = HowManyLessThans(LHS, RHS, L, false, true);
+ if (!isa<SCEVCouldNotCompute>(TC)) return TC;
+ break;
+ }
+ case ICmpInst::ICMP_UGE: {
+ SCEVHandle TC = HowManyLessThans(SE.getNotSCEV(LHS),
+ SE.getNotSCEV(RHS), L, false, true);
if (!isa<SCEVCouldNotCompute>(TC)) return TC;
break;
}
return Comm;
}
- if (SCEVUDivExpr *Div = dyn_cast<SCEVUDivExpr>(V)) {
- SCEVHandle LHS = getSCEVAtScope(Div->getLHS(), L);
+ if (SCEVUDivExpr *UDiv = dyn_cast<SCEVUDivExpr>(V)) {
+ SCEVHandle LHS = getSCEVAtScope(UDiv->getLHS(), L);
if (LHS == UnknownValue) return LHS;
- SCEVHandle RHS = getSCEVAtScope(Div->getRHS(), L);
+ SCEVHandle RHS = getSCEVAtScope(UDiv->getRHS(), L);
if (RHS == UnknownValue) return RHS;
- if (LHS == Div->getLHS() && RHS == Div->getRHS())
- return Div; // must be loop invariant
+ if (LHS == UDiv->getLHS() && RHS == UDiv->getRHS())
+ return UDiv; // must be loop invariant
return SE.getUDivExpr(LHS, RHS);
}
+ if (SCEVSDivExpr *SDiv = dyn_cast<SCEVSDivExpr>(V)) {
+ SCEVHandle LHS = getSCEVAtScope(SDiv->getLHS(), L);
+ if (LHS == UnknownValue) return LHS;
+ SCEVHandle RHS = getSCEVAtScope(SDiv->getRHS(), L);
+ if (RHS == UnknownValue) return RHS;
+ if (LHS == SDiv->getLHS() && RHS == SDiv->getRHS())
+ return SDiv; // must be loop invariant
+ return SE.getSDivExpr(LHS, RHS);
+ }
+
// If this is a loop recurrence for a loop that does not contain L, then we
// are dealing with the final value computed by the loop.
if (SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(V)) {
// loop iterates. Compute this now.
SCEVHandle IterationCount = getIterationCount(AddRec->getLoop());
if (IterationCount == UnknownValue) return UnknownValue;
- IterationCount = SE.getTruncateOrZeroExtend(IterationCount,
- AddRec->getType());
- // If the value is affine, simplify the expression evaluation to just
- // Start + Step*IterationCount.
- if (AddRec->isAffine())
- return SE.getAddExpr(AddRec->getStart(),
- SE.getMulExpr(IterationCount,
- AddRec->getOperand(1)));
-
- // Otherwise, evaluate it the hard way.
+ // Then, evaluate the AddRec.
return AddRec->evaluateAtIteration(IterationCount, SE);
}
return UnknownValue;
return UnknownValue;
}
+/// SolveLinEquationWithOverflow - Finds the minimum unsigned root of the
+/// following equation:
+///
+/// A * X = B (mod N)
+///
+/// where N = 2^BW and BW is the common bit width of A and B. The signedness of
+/// A and B isn't important.
+///
+/// If the equation does not have a solution, SCEVCouldNotCompute is returned.
+static SCEVHandle SolveLinEquationWithOverflow(const APInt &A, const APInt &B,
+ ScalarEvolution &SE) {
+ uint32_t BW = A.getBitWidth();
+ assert(BW == B.getBitWidth() && "Bit widths must be the same.");
+ assert(A != 0 && "A must be non-zero.");
+
+ // 1. D = gcd(A, N)
+ //
+ // The gcd of A and N may have only one prime factor: 2. The number of
+ // trailing zeros in A is its multiplicity
+ uint32_t Mult2 = A.countTrailingZeros();
+ // D = 2^Mult2
+
+ // 2. Check if B is divisible by D.
+ //
+ // B is divisible by D if and only if the multiplicity of prime factor 2 for B
+ // is not less than multiplicity of this prime factor for D.
+ if (B.countTrailingZeros() < Mult2)
+ return new SCEVCouldNotCompute();
+
+ // 3. Compute I: the multiplicative inverse of (A / D) in arithmetic
+ // modulo (N / D).
+ //
+ // (N / D) may need BW+1 bits in its representation. Hence, we'll use this
+ // bit width during computations.
+ APInt AD = A.lshr(Mult2).zext(BW + 1); // AD = A / D
+ APInt Mod(BW + 1, 0);
+ Mod.set(BW - Mult2); // Mod = N / D
+ APInt I = AD.multiplicativeInverse(Mod);
+
+ // 4. Compute the minimum unsigned root of the equation:
+ // I * (B / D) mod (N / D)
+ APInt Result = (I * B.lshr(Mult2).zext(BW + 1)).urem(Mod);
+
+ // The result is guaranteed to be less than 2^BW so we may truncate it to BW
+ // bits.
+ return SE.getConstant(Result.trunc(BW));
+}
/// SolveQuadraticEquation - Find the roots of the quadratic equation for the
/// given quadratic chrec {L,+,M,+,N}. This returns either the two roots (which
// The divisions must be performed as signed divisions.
APInt NegB(-B);
APInt TwoA( A << 1 );
+ if (TwoA.isMinValue()) {
+ SCEV *CNC = new SCEVCouldNotCompute();
+ return std::make_pair(CNC, CNC);
+ }
+
ConstantInt *Solution1 = ConstantInt::get((NegB + SqrtVal).sdiv(TwoA));
ConstantInt *Solution2 = ConstantInt::get((NegB - SqrtVal).sdiv(TwoA));
return UnknownValue;
if (AddRec->isAffine()) {
- // If this is an affine expression the execution count of this branch is
- // equal to:
+ // If this is an affine expression, the execution count of this branch is
+ // the minimum unsigned root of the following equation:
+ //
+ // Start + Step*N = 0 (mod 2^BW)
//
- // (0 - Start/Step) iff Start % Step == 0
+ // equivalent to:
//
+ // Step*N = -Start (mod 2^BW)
+ //
+ // where BW is the common bit width of Start and Step.
+
// Get the initial value for the loop.
SCEVHandle Start = getSCEVAtScope(AddRec->getStart(), L->getParentLoop());
if (isa<SCEVCouldNotCompute>(Start)) return UnknownValue;
- SCEVHandle Step = AddRec->getOperand(1);
- Step = getSCEVAtScope(Step, L->getParentLoop());
+ SCEVHandle Step = getSCEVAtScope(AddRec->getOperand(1), L->getParentLoop());
- // Figure out if Start % Step == 0.
- // FIXME: We should add DivExpr and RemExpr operations to our AST.
if (SCEVConstant *StepC = dyn_cast<SCEVConstant>(Step)) {
- if (StepC->getValue()->equalsInt(1)) // N % 1 == 0
- return SE.getNegativeSCEV(Start); // 0 - Start/1 == -Start
- if (StepC->getValue()->isAllOnesValue()) // N % -1 == 0
- return Start; // 0 - Start/-1 == Start
-
- // Check to see if Start is divisible by SC with no remainder.
- if (SCEVConstant *StartC = dyn_cast<SCEVConstant>(Start)) {
- ConstantInt *StartCC = StartC->getValue();
- Constant *StartNegC = ConstantExpr::getNeg(StartCC);
- Constant *Rem = ConstantExpr::getURem(StartNegC, StepC->getValue());
- if (Rem->isNullValue()) {
- Constant *Result = ConstantExpr::getUDiv(StartNegC,StepC->getValue());
- return SE.getUnknown(Result);
- }
- }
+ // For now we handle only constant steps.
+
+ // First, handle unitary steps.
+ if (StepC->getValue()->equalsInt(1)) // 1*N = -Start (mod 2^BW), so:
+ return SE.getNegativeSCEV(Start); // N = -Start (as unsigned)
+ if (StepC->getValue()->isAllOnesValue()) // -1*N = -Start (mod 2^BW), so:
+ return Start; // N = Start (as unsigned)
+
+ // Then, try to solve the above equation provided that Start is constant.
+ if (SCEVConstant *StartC = dyn_cast<SCEVConstant>(Start))
+ return SolveLinEquationWithOverflow(StepC->getValue()->getValue(),
+ -StartC->getValue()->getValue(),SE);
}
} else if (AddRec->isQuadratic() && AddRec->getType()->isInteger()) {
// If this is a quadratic (3-term) AddRec {L,+,M,+,N}, find the roots of
return UnknownValue;
}
+/// getPredecessorWithUniqueSuccessorForBB - Return a predecessor of BB
+/// (which may not be an immediate predecessor) which has exactly one
+/// successor from which BB is reachable, or null if no such block is
+/// found.
+///
+BasicBlock *
+ScalarEvolutionsImpl::getPredecessorWithUniqueSuccessorForBB(BasicBlock *BB) {
+ // If the block has a unique predecessor, the predecessor must have
+ // no other successors from which BB is reachable.
+ if (BasicBlock *Pred = BB->getSinglePredecessor())
+ return Pred;
+
+ // A loop's header is defined to be a block that dominates the loop.
+ // If the loop has a preheader, it must be a block that has exactly
+ // one successor that can reach BB. This is slightly more strict
+ // than necessary, but works if critical edges are split.
+ if (Loop *L = LI.getLoopFor(BB))
+ return L->getLoopPreheader();
+
+ return 0;
+}
+
/// executesAtLeastOnce - Test whether entry to the loop is protected by
/// a conditional between LHS and RHS.
bool ScalarEvolutionsImpl::executesAtLeastOnce(const Loop *L, bool isSigned,
+ bool trueWhenEqual,
SCEV *LHS, SCEV *RHS) {
BasicBlock *Preheader = L->getLoopPreheader();
BasicBlock *PreheaderDest = L->getHeader();
- if (Preheader == 0) return false;
-
- BranchInst *LoopEntryPredicate =
- dyn_cast<BranchInst>(Preheader->getTerminator());
- if (!LoopEntryPredicate) return false;
-
- // This might be a critical edge broken out. If the loop preheader ends in
- // an unconditional branch to the loop, check to see if the preheader has a
- // single predecessor, and if so, look for its terminator.
- while (LoopEntryPredicate->isUnconditional()) {
- PreheaderDest = Preheader;
- Preheader = Preheader->getSinglePredecessor();
- if (!Preheader) return false; // Multiple preds.
-
- LoopEntryPredicate =
+
+ // Starting at the preheader, climb up the predecessor chain, as long as
+ // there are predecessors that can be found that have unique successors
+ // leading to the original header.
+ for (; Preheader;
+ PreheaderDest = Preheader,
+ Preheader = getPredecessorWithUniqueSuccessorForBB(Preheader)) {
+
+ BranchInst *LoopEntryPredicate =
dyn_cast<BranchInst>(Preheader->getTerminator());
- if (!LoopEntryPredicate) return false;
+ if (!LoopEntryPredicate ||
+ LoopEntryPredicate->isUnconditional())
+ continue;
+
+ ICmpInst *ICI = dyn_cast<ICmpInst>(LoopEntryPredicate->getCondition());
+ if (!ICI) continue;
+
+ // Now that we found a conditional branch that dominates the loop, check to
+ // see if it is the comparison we are looking for.
+ Value *PreCondLHS = ICI->getOperand(0);
+ Value *PreCondRHS = ICI->getOperand(1);
+ ICmpInst::Predicate Cond;
+ if (LoopEntryPredicate->getSuccessor(0) == PreheaderDest)
+ Cond = ICI->getPredicate();
+ else
+ Cond = ICI->getInversePredicate();
+
+ switch (Cond) {
+ case ICmpInst::ICMP_UGT:
+ if (isSigned || trueWhenEqual) continue;
+ std::swap(PreCondLHS, PreCondRHS);
+ Cond = ICmpInst::ICMP_ULT;
+ break;
+ case ICmpInst::ICMP_SGT:
+ if (!isSigned || trueWhenEqual) continue;
+ std::swap(PreCondLHS, PreCondRHS);
+ Cond = ICmpInst::ICMP_SLT;
+ break;
+ case ICmpInst::ICMP_ULT:
+ if (isSigned || trueWhenEqual) continue;
+ break;
+ case ICmpInst::ICMP_SLT:
+ if (!isSigned || trueWhenEqual) continue;
+ break;
+ case ICmpInst::ICMP_UGE:
+ if (isSigned || !trueWhenEqual) continue;
+ std::swap(PreCondLHS, PreCondRHS);
+ Cond = ICmpInst::ICMP_ULE;
+ break;
+ case ICmpInst::ICMP_SGE:
+ if (!isSigned || !trueWhenEqual) continue;
+ std::swap(PreCondLHS, PreCondRHS);
+ Cond = ICmpInst::ICMP_SLE;
+ break;
+ case ICmpInst::ICMP_ULE:
+ if (isSigned || !trueWhenEqual) continue;
+ break;
+ case ICmpInst::ICMP_SLE:
+ if (!isSigned || !trueWhenEqual) continue;
+ break;
+ default:
+ continue;
+ }
+
+ if (!PreCondLHS->getType()->isInteger()) continue;
+
+ SCEVHandle PreCondLHSSCEV = getSCEV(PreCondLHS);
+ SCEVHandle PreCondRHSSCEV = getSCEV(PreCondRHS);
+ if ((LHS == PreCondLHSSCEV && RHS == PreCondRHSSCEV) ||
+ (LHS == SE.getNotSCEV(PreCondRHSSCEV) &&
+ RHS == SE.getNotSCEV(PreCondLHSSCEV)))
+ return true;
}
- ICmpInst *ICI = dyn_cast<ICmpInst>(LoopEntryPredicate->getCondition());
- if (!ICI) return false;
+ return false;
+}
- // Now that we found a conditional branch that dominates the loop, check to
- // see if it is the comparison we are looking for.
- Value *PreCondLHS = ICI->getOperand(0);
- Value *PreCondRHS = ICI->getOperand(1);
- ICmpInst::Predicate Cond;
- if (LoopEntryPredicate->getSuccessor(0) == PreheaderDest)
- Cond = ICI->getPredicate();
- else
- Cond = ICI->getInversePredicate();
+/// potentialInfiniteLoop - Test whether the loop might jump over the exit value
+/// due to wrapping around 2^n.
+bool ScalarEvolutionsImpl::potentialInfiniteLoop(SCEV *Stride, SCEV *RHS,
+ bool isSigned, bool trueWhenEqual) {
+ // Return true when the distance from RHS to maxint > Stride.
- switch (Cond) {
- case ICmpInst::ICMP_UGT:
- if (isSigned) return false;
- std::swap(PreCondLHS, PreCondRHS);
- Cond = ICmpInst::ICMP_ULT;
- break;
- case ICmpInst::ICMP_SGT:
- if (!isSigned) return false;
- std::swap(PreCondLHS, PreCondRHS);
- Cond = ICmpInst::ICMP_SLT;
- break;
- case ICmpInst::ICMP_ULT:
- if (isSigned) return false;
- break;
- case ICmpInst::ICMP_SLT:
- if (!isSigned) return false;
- break;
- default:
+ SCEVConstant *SC = dyn_cast<SCEVConstant>(Stride);
+ if (!SC)
+ return true;
+
+ if (SC->getValue()->isZero())
+ return true;
+ if (!trueWhenEqual && SC->getValue()->isOne())
return false;
- }
- if (!PreCondLHS->getType()->isInteger()) return false;
+ SCEVConstant *R = dyn_cast<SCEVConstant>(RHS);
+ if (!R)
+ return true;
+
+ if (isSigned)
+ return true; // XXX: because we don't have an sdiv scev.
+
+ // If negative, it wraps around every iteration, but we don't care about that.
+ APInt S = SC->getValue()->getValue().abs();
+
+ APInt Dist = APInt::getMaxValue(R->getValue()->getBitWidth()) -
+ R->getValue()->getValue();
- return LHS == getSCEV(PreCondLHS) && RHS == getSCEV(PreCondRHS);
+ if (trueWhenEqual)
+ return !S.ult(Dist);
+ else
+ return !S.ule(Dist);
}
/// HowManyLessThans - Return the number of times a backedge containing the
/// specified less-than comparison will execute. If not computable, return
/// UnknownValue.
SCEVHandle ScalarEvolutionsImpl::
-HowManyLessThans(SCEV *LHS, SCEV *RHS, const Loop *L, bool isSigned) {
+HowManyLessThans(SCEV *LHS, SCEV *RHS, const Loop *L,
+ bool isSigned, bool trueWhenEqual) {
// Only handle: "ADDREC < LoopInvariant".
if (!RHS->isLoopInvariant(L)) return UnknownValue;
return UnknownValue;
if (AddRec->isAffine()) {
- // FORNOW: We only support unit strides.
- SCEVHandle One = SE.getIntegerSCEV(1, RHS->getType());
- if (AddRec->getOperand(1) != One)
+ SCEVHandle Stride = AddRec->getOperand(1);
+ if (potentialInfiniteLoop(Stride, RHS, isSigned, trueWhenEqual))
return UnknownValue;
- // We know the LHS is of the form {n,+,1} and the RHS is some loop-invariant
- // m. So, we count the number of iterations in which {n,+,1} < m is true.
- // Note that we cannot simply return max(m-n,0) because it's not safe to
+ // We know the LHS is of the form {n,+,s} and the RHS is some loop-invariant
+ // m. So, we count the number of iterations in which {n,+,s} < m is true.
+ // Note that we cannot simply return max(m-n,0)/s because it's not safe to
// treat m-n as signed nor unsigned due to overflow possibility.
// First, we get the value of the LHS in the first iteration: n
SCEVHandle Start = AddRec->getOperand(0);
- if (executesAtLeastOnce(L, isSigned,
- SE.getMinusSCEV(AddRec->getOperand(0), One), RHS)) {
- // Since we know that the condition is true in order to enter the loop,
- // we know that it will run exactly m-n times.
- return SE.getMinusSCEV(RHS, Start);
- } else {
- // Then, we get the value of the LHS in the first iteration in which the
- // above condition doesn't hold. This equals to max(m,n).
- SCEVHandle End = isSigned ? SE.getSMaxExpr(RHS, Start)
- : SE.getUMaxExpr(RHS, Start);
-
- // Finally, we subtract these two values to get the number of times the
- // backedge is executed: max(m,n)-n.
- return SE.getMinusSCEV(End, Start);
+ SCEVHandle One = SE.getIntegerSCEV(1, RHS->getType());
+
+ // Assuming that the loop will run at least once, we know that it will
+ // run (m-n)/s times.
+ SCEVHandle End = RHS;
+
+ if (!executesAtLeastOnce(L, isSigned, trueWhenEqual,
+ SE.getMinusSCEV(Start, One), RHS)) {
+ // If not, we get the value of the LHS in the first iteration in which
+ // the above condition doesn't hold. This equals to max(m,n).
+ End = isSigned ? SE.getSMaxExpr(RHS, Start)
+ : SE.getUMaxExpr(RHS, Start);
}
+
+ // If the expression is less-than-or-equal to, we need to extend the
+ // loop by one iteration.
+ //
+ // The loop won't actually run (m-n)/s times because the loop iterations
+ // might not divide cleanly. For example, if you have {2,+,5} u< 10 the
+ // division would equal one, but the loop runs twice putting the
+ // induction variable at 12.
+
+ if (!trueWhenEqual)
+ // (Stride - 1) is correct only because we know it's unsigned.
+ // What we really want is to decrease the magnitude of Stride by one.
+ Start = SE.getMinusSCEV(Start, SE.getMinusSCEV(Stride, One));
+ else
+ Start = SE.getMinusSCEV(Start, Stride);
+
+ // Finally, we subtract these two values to get the number of times the
+ // backedge is executed: max(m,n)-n.
+ return SE.getUDivExpr(SE.getMinusSCEV(End, Start), Stride);
}
return UnknownValue;
}
}
- // Fallback, if this is a general polynomial, figure out the progression
- // through brute force: evaluate until we find an iteration that fails the
- // test. This is likely to be slow, but getting an accurate trip count is
- // incredibly important, we will be able to simplify the exit test a lot, and
- // we are almost guaranteed to get a trip count in this case.
- ConstantInt *TestVal = ConstantInt::get(getType(), 0);
- ConstantInt *EndVal = TestVal; // Stop when we wrap around.
- do {
- ++NumBruteForceEvaluations;
- SCEVHandle Val = evaluateAtIteration(SE.getConstant(TestVal), SE);
- if (!isa<SCEVConstant>(Val)) // This shouldn't happen.
- return new SCEVCouldNotCompute();
-
- // Check to see if we found the value!
- if (!Range.contains(cast<SCEVConstant>(Val)->getValue()->getValue()))
- return SE.getConstant(TestVal);
-
- // Increment to test the next index.
- TestVal = ConstantInt::get(TestVal->getValue()+1);
- } while (TestVal != EndVal);
-
return new SCEVCouldNotCompute();
}
for (inst_iterator I = inst_begin(F), E = inst_end(F); I != E; ++I)
if (I->getType()->isInteger()) {
OS << *I;
- OS << " --> ";
+ OS << " --> ";
SCEVHandle SV = getSCEV(&*I);
SV->print(OS);
OS << "\t\t";