//===--------- SCEVAffinator.cpp - Create Scops from LLVM IR -------------===// // // 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 // //===----------------------------------------------------------------------===// // // Create a polyhedral description for a SCEV value. // //===----------------------------------------------------------------------===// #include "polly/Support/SCEVAffinator.h" #include "polly/Options.h" #include "polly/ScopInfo.h" #include "polly/Support/GICHelper.h" #include "polly/Support/SCEVValidator.h" #include "isl/aff.h" #include "isl/local_space.h" #include "isl/set.h" #include "isl/val.h" using namespace llvm; using namespace polly; static cl::opt IgnoreIntegerWrapping( "polly-ignore-integer-wrapping", cl::desc("Do not build run-time checks to proof absence of integer " "wrapping"), cl::Hidden, cl::cat(PollyCategory)); // The maximal number of basic sets we allow during the construction of a // piecewise affine function. More complex ones will result in very high // compile time. static int const MaxDisjunctionsInPwAff = 100; // The maximal number of bits for which a general expression is modeled // precisely. static unsigned const MaxSmallBitWidth = 7; /// Add the number of basic sets in @p Domain to @p User static isl_stat addNumBasicSets(__isl_take isl_set *Domain, __isl_take isl_aff *Aff, void *User) { auto *NumBasicSets = static_cast(User); *NumBasicSets += isl_set_n_basic_set(Domain); isl_set_free(Domain); isl_aff_free(Aff); return isl_stat_ok; } /// Determine if @p PWAC is too complex to continue. static bool isTooComplex(PWACtx PWAC) { unsigned NumBasicSets = 0; isl_pw_aff_foreach_piece(PWAC.first.get(), addNumBasicSets, &NumBasicSets); if (NumBasicSets <= MaxDisjunctionsInPwAff) return false; return true; } /// Return the flag describing the possible wrapping of @p Expr. static SCEV::NoWrapFlags getNoWrapFlags(const SCEV *Expr) { if (auto *NAry = dyn_cast(Expr)) return NAry->getNoWrapFlags(); return SCEV::NoWrapMask; } static PWACtx combine(PWACtx PWAC0, PWACtx PWAC1, __isl_give isl_pw_aff *(Fn)(__isl_take isl_pw_aff *, __isl_take isl_pw_aff *)) { PWAC0.first = isl::manage(Fn(PWAC0.first.release(), PWAC1.first.release())); PWAC0.second = PWAC0.second.unite(PWAC1.second); return PWAC0; } static __isl_give isl_pw_aff *getWidthExpValOnDomain(unsigned Width, __isl_take isl_set *Dom) { auto *Ctx = isl_set_get_ctx(Dom); auto *WidthVal = isl_val_int_from_ui(Ctx, Width); auto *ExpVal = isl_val_2exp(WidthVal); return isl_pw_aff_val_on_domain(Dom, ExpVal); } SCEVAffinator::SCEVAffinator(Scop *S, LoopInfo &LI) : S(S), Ctx(S->getIslCtx().get()), SE(*S->getSE()), LI(LI), TD(S->getFunction().getParent()->getDataLayout()) {} Loop *SCEVAffinator::getScope() { return BB ? LI.getLoopFor(BB) : nullptr; } void SCEVAffinator::interpretAsUnsigned(PWACtx &PWAC, unsigned Width) { auto *NonNegDom = isl_pw_aff_nonneg_set(PWAC.first.copy()); auto *NonNegPWA = isl_pw_aff_intersect_domain(PWAC.first.copy(), isl_set_copy(NonNegDom)); auto *ExpPWA = getWidthExpValOnDomain(Width, isl_set_complement(NonNegDom)); PWAC.first = isl::manage(isl_pw_aff_union_add( NonNegPWA, isl_pw_aff_add(PWAC.first.release(), ExpPWA))); } void SCEVAffinator::takeNonNegativeAssumption( PWACtx &PWAC, RecordedAssumptionsTy *RecordedAssumptions) { this->RecordedAssumptions = RecordedAssumptions; auto *NegPWA = isl_pw_aff_neg(PWAC.first.copy()); auto *NegDom = isl_pw_aff_pos_set(NegPWA); PWAC.second = isl::manage(isl_set_union(PWAC.second.release(), isl_set_copy(NegDom))); auto *Restriction = BB ? NegDom : isl_set_params(NegDom); auto DL = BB ? BB->getTerminator()->getDebugLoc() : DebugLoc(); recordAssumption(RecordedAssumptions, UNSIGNED, isl::manage(Restriction), DL, AS_RESTRICTION, BB); } PWACtx SCEVAffinator::getPWACtxFromPWA(isl::pw_aff PWA) { return std::make_pair(PWA, isl::set::empty(isl::space(Ctx, 0, NumIterators))); } PWACtx SCEVAffinator::getPwAff(const SCEV *Expr, BasicBlock *BB, RecordedAssumptionsTy *RecordedAssumptions) { this->BB = BB; this->RecordedAssumptions = RecordedAssumptions; if (BB) { auto *DC = S->getDomainConditions(BB).release(); NumIterators = isl_set_n_dim(DC); isl_set_free(DC); } else NumIterators = 0; return visit(Expr); } PWACtx SCEVAffinator::checkForWrapping(const SCEV *Expr, PWACtx PWAC) const { // If the SCEV flags do contain NSW (no signed wrap) then PWA already // represents Expr in modulo semantic (it is not allowed to overflow), thus we // are done. Otherwise, we will compute: // PWA = ((PWA + 2^(n-1)) mod (2 ^ n)) - 2^(n-1) // whereas n is the number of bits of the Expr, hence: // n = bitwidth(ExprType) if (IgnoreIntegerWrapping || (getNoWrapFlags(Expr) & SCEV::FlagNSW)) return PWAC; isl::pw_aff PWAMod = addModuloSemantic(PWAC.first, Expr->getType()); isl::set NotEqualSet = PWAC.first.ne_set(PWAMod); PWAC.second = PWAC.second.unite(NotEqualSet).coalesce(); const DebugLoc &Loc = BB ? BB->getTerminator()->getDebugLoc() : DebugLoc(); if (!BB) NotEqualSet = NotEqualSet.params(); NotEqualSet = NotEqualSet.coalesce(); if (!NotEqualSet.is_empty()) recordAssumption(RecordedAssumptions, WRAPPING, NotEqualSet, Loc, AS_RESTRICTION, BB); return PWAC; } isl::pw_aff SCEVAffinator::addModuloSemantic(isl::pw_aff PWA, Type *ExprType) const { unsigned Width = TD.getTypeSizeInBits(ExprType); auto ModVal = isl::val::int_from_ui(Ctx, Width); ModVal = ModVal.pow2(); isl::set Domain = PWA.domain(); isl::pw_aff AddPW = isl::manage(getWidthExpValOnDomain(Width - 1, Domain.release())); return PWA.add(AddPW).mod(ModVal).sub(AddPW); } bool SCEVAffinator::hasNSWAddRecForLoop(Loop *L) const { for (const auto &CachedPair : CachedExpressions) { auto *AddRec = dyn_cast(CachedPair.first.first); if (!AddRec) continue; if (AddRec->getLoop() != L) continue; if (AddRec->getNoWrapFlags() & SCEV::FlagNSW) return true; } return false; } bool SCEVAffinator::computeModuloForExpr(const SCEV *Expr) { unsigned Width = TD.getTypeSizeInBits(Expr->getType()); // We assume nsw expressions never overflow. if (auto *NAry = dyn_cast(Expr)) if (NAry->getNoWrapFlags() & SCEV::FlagNSW) return false; return Width <= MaxSmallBitWidth; } PWACtx SCEVAffinator::visit(const SCEV *Expr) { auto Key = std::make_pair(Expr, BB); PWACtx PWAC = CachedExpressions[Key]; if (!PWAC.first.is_null()) return PWAC; auto ConstantAndLeftOverPair = extractConstantFactor(Expr, SE); auto *Factor = ConstantAndLeftOverPair.first; Expr = ConstantAndLeftOverPair.second; auto *Scope = getScope(); S->addParams(getParamsInAffineExpr(&S->getRegion(), Scope, Expr, SE)); // In case the scev is a valid parameter, we do not further analyze this // expression, but create a new parameter in the isl_pw_aff. This allows us // to treat subexpressions that we cannot translate into an piecewise affine // expression, as constant parameters of the piecewise affine expression. if (isl_id *Id = S->getIdForParam(Expr).release()) { isl_space *Space = isl_space_set_alloc(Ctx.get(), 1, NumIterators); Space = isl_space_set_dim_id(Space, isl_dim_param, 0, Id); isl_set *Domain = isl_set_universe(isl_space_copy(Space)); isl_aff *Affine = isl_aff_zero_on_domain(isl_local_space_from_space(Space)); Affine = isl_aff_add_coefficient_si(Affine, isl_dim_param, 0, 1); PWAC = getPWACtxFromPWA(isl::manage(isl_pw_aff_alloc(Domain, Affine))); } else { PWAC = SCEVVisitor::visit(Expr); if (computeModuloForExpr(Expr)) PWAC.first = addModuloSemantic(PWAC.first, Expr->getType()); else PWAC = checkForWrapping(Expr, PWAC); } if (!Factor->getType()->isIntegerTy(1)) { PWAC = combine(PWAC, visitConstant(Factor), isl_pw_aff_mul); if (computeModuloForExpr(Key.first)) PWAC.first = addModuloSemantic(PWAC.first, Expr->getType()); } // For compile time reasons we need to simplify the PWAC before we cache and // return it. PWAC.first = PWAC.first.coalesce(); if (!computeModuloForExpr(Key.first)) PWAC = checkForWrapping(Key.first, PWAC); CachedExpressions[Key] = PWAC; return PWAC; } PWACtx SCEVAffinator::visitConstant(const SCEVConstant *Expr) { ConstantInt *Value = Expr->getValue(); isl_val *v; // LLVM does not define if an integer value is interpreted as a signed or // unsigned value. Hence, without further information, it is unknown how // this value needs to be converted to GMP. At the moment, we only support // signed operations. So we just interpret it as signed. Later, there are // two options: // // 1. We always interpret any value as signed and convert the values on // demand. // 2. We pass down the signedness of the calculation and use it to interpret // this constant correctly. v = isl_valFromAPInt(Ctx.get(), Value->getValue(), /* isSigned */ true); isl_space *Space = isl_space_set_alloc(Ctx.get(), 0, NumIterators); isl_local_space *ls = isl_local_space_from_space(Space); return getPWACtxFromPWA( isl::manage(isl_pw_aff_from_aff(isl_aff_val_on_domain(ls, v)))); } PWACtx SCEVAffinator::visitPtrToIntExpr(const SCEVPtrToIntExpr *Expr) { return visit(Expr->getOperand(0)); } PWACtx SCEVAffinator::visitTruncateExpr(const SCEVTruncateExpr *Expr) { // Truncate operations are basically modulo operations, thus we can // model them that way. However, for large types we assume the operand // to fit in the new type size instead of introducing a modulo with a very // large constant. auto *Op = Expr->getOperand(); auto OpPWAC = visit(Op); unsigned Width = TD.getTypeSizeInBits(Expr->getType()); if (computeModuloForExpr(Expr)) return OpPWAC; auto *Dom = OpPWAC.first.domain().release(); auto *ExpPWA = getWidthExpValOnDomain(Width - 1, Dom); auto *GreaterDom = isl_pw_aff_ge_set(OpPWAC.first.copy(), isl_pw_aff_copy(ExpPWA)); auto *SmallerDom = isl_pw_aff_lt_set(OpPWAC.first.copy(), isl_pw_aff_neg(ExpPWA)); auto *OutOfBoundsDom = isl_set_union(SmallerDom, GreaterDom); OpPWAC.second = OpPWAC.second.unite(isl::manage_copy(OutOfBoundsDom)); if (!BB) { assert(isl_set_dim(OutOfBoundsDom, isl_dim_set) == 0 && "Expected a zero dimensional set for non-basic-block domains"); OutOfBoundsDom = isl_set_params(OutOfBoundsDom); } recordAssumption(RecordedAssumptions, UNSIGNED, isl::manage(OutOfBoundsDom), DebugLoc(), AS_RESTRICTION, BB); return OpPWAC; } PWACtx SCEVAffinator::visitZeroExtendExpr(const SCEVZeroExtendExpr *Expr) { // A zero-extended value can be interpreted as a piecewise defined signed // value. If the value was non-negative it stays the same, otherwise it // is the sum of the original value and 2^n where n is the bit-width of // the original (or operand) type. Examples: // zext i8 127 to i32 -> { [127] } // zext i8 -1 to i32 -> { [256 + (-1)] } = { [255] } // zext i8 %v to i32 -> [v] -> { [v] | v >= 0; [256 + v] | v < 0 } // // However, LLVM/Scalar Evolution uses zero-extend (potentially lead by a // truncate) to represent some forms of modulo computation. The left-hand side // of the condition in the code below would result in the SCEV // "zext i1 for.body" which is just another description // of the C expression "i & 1 != 0" or, equivalently, "i % 2 != 0". // // for (i = 0; i < N; i++) // if (i & 1 != 0 /* == i % 2 */) // /* do something */ // // If we do not make the modulo explicit but only use the mechanism described // above we will get the very restrictive assumption "N < 3", because for all // values of N >= 3 the SCEVAddRecExpr operand of the zero-extend would wrap. // Alternatively, we can make the modulo in the operand explicit in the // resulting piecewise function and thereby avoid the assumption on N. For the // example this would result in the following piecewise affine function: // { [i0] -> [(1)] : 2*floor((-1 + i0)/2) = -1 + i0; // [i0] -> [(0)] : 2*floor((i0)/2) = i0 } // To this end we can first determine if the (immediate) operand of the // zero-extend can wrap and, in case it might, we will use explicit modulo // semantic to compute the result instead of emitting non-wrapping // assumptions. // // Note that operands with large bit-widths are less likely to be negative // because it would result in a very large access offset or loop bound after // the zero-extend. To this end one can optimistically assume the operand to // be positive and avoid the piecewise definition if the bit-width is bigger // than some threshold (here MaxZextSmallBitWidth). // // We choose to go with a hybrid solution of all modeling techniques described // above. For small bit-widths (up to MaxZextSmallBitWidth) we will model the // wrapping explicitly and use a piecewise defined function. However, if the // bit-width is bigger than MaxZextSmallBitWidth we will employ overflow // assumptions and assume the "former negative" piece will not exist. auto *Op = Expr->getOperand(); auto OpPWAC = visit(Op); // If the width is to big we assume the negative part does not occur. if (!computeModuloForExpr(Op)) { takeNonNegativeAssumption(OpPWAC, RecordedAssumptions); return OpPWAC; } // If the width is small build the piece for the non-negative part and // the one for the negative part and unify them. unsigned Width = TD.getTypeSizeInBits(Op->getType()); interpretAsUnsigned(OpPWAC, Width); return OpPWAC; } PWACtx SCEVAffinator::visitSignExtendExpr(const SCEVSignExtendExpr *Expr) { // As all values are represented as signed, a sign extension is a noop. return visit(Expr->getOperand()); } PWACtx SCEVAffinator::visitAddExpr(const SCEVAddExpr *Expr) { PWACtx Sum = visit(Expr->getOperand(0)); for (int i = 1, e = Expr->getNumOperands(); i < e; ++i) { Sum = combine(Sum, visit(Expr->getOperand(i)), isl_pw_aff_add); if (isTooComplex(Sum)) return complexityBailout(); } return Sum; } PWACtx SCEVAffinator::visitMulExpr(const SCEVMulExpr *Expr) { PWACtx Prod = visit(Expr->getOperand(0)); for (int i = 1, e = Expr->getNumOperands(); i < e; ++i) { Prod = combine(Prod, visit(Expr->getOperand(i)), isl_pw_aff_mul); if (isTooComplex(Prod)) return complexityBailout(); } return Prod; } PWACtx SCEVAffinator::visitAddRecExpr(const SCEVAddRecExpr *Expr) { assert(Expr->isAffine() && "Only affine AddRecurrences allowed"); auto Flags = Expr->getNoWrapFlags(); // Directly generate isl_pw_aff for Expr if 'start' is zero. if (Expr->getStart()->isZero()) { assert(S->contains(Expr->getLoop()) && "Scop does not contain the loop referenced in this AddRec"); PWACtx Step = visit(Expr->getOperand(1)); isl_space *Space = isl_space_set_alloc(Ctx.get(), 0, NumIterators); isl_local_space *LocalSpace = isl_local_space_from_space(Space); unsigned loopDimension = S->getRelativeLoopDepth(Expr->getLoop()); isl_aff *LAff = isl_aff_set_coefficient_si( isl_aff_zero_on_domain(LocalSpace), isl_dim_in, loopDimension, 1); isl_pw_aff *LPwAff = isl_pw_aff_from_aff(LAff); Step.first = Step.first.mul(isl::manage(LPwAff)); return Step; } // Translate AddRecExpr from '{start, +, inc}' into 'start + {0, +, inc}' // if 'start' is not zero. // TODO: Using the original SCEV no-wrap flags is not always safe, however // as our code generation is reordering the expression anyway it doesn't // really matter. const SCEV *ZeroStartExpr = SE.getAddRecExpr(SE.getConstant(Expr->getStart()->getType(), 0), Expr->getStepRecurrence(SE), Expr->getLoop(), Flags); PWACtx Result = visit(ZeroStartExpr); PWACtx Start = visit(Expr->getStart()); Result = combine(Result, Start, isl_pw_aff_add); return Result; } PWACtx SCEVAffinator::visitSMaxExpr(const SCEVSMaxExpr *Expr) { PWACtx Max = visit(Expr->getOperand(0)); for (int i = 1, e = Expr->getNumOperands(); i < e; ++i) { Max = combine(Max, visit(Expr->getOperand(i)), isl_pw_aff_max); if (isTooComplex(Max)) return complexityBailout(); } return Max; } PWACtx SCEVAffinator::visitSMinExpr(const SCEVSMinExpr *Expr) { PWACtx Min = visit(Expr->getOperand(0)); for (int i = 1, e = Expr->getNumOperands(); i < e; ++i) { Min = combine(Min, visit(Expr->getOperand(i)), isl_pw_aff_min); if (isTooComplex(Min)) return complexityBailout(); } return Min; } PWACtx SCEVAffinator::visitUMaxExpr(const SCEVUMaxExpr *Expr) { llvm_unreachable("SCEVUMaxExpr not yet supported"); } PWACtx SCEVAffinator::visitUMinExpr(const SCEVUMinExpr *Expr) { llvm_unreachable("SCEVUMinExpr not yet supported"); } PWACtx SCEVAffinator::visitSequentialUMinExpr(const SCEVSequentialUMinExpr *Expr) { llvm_unreachable("SCEVSequentialUMinExpr not yet supported"); } PWACtx SCEVAffinator::visitUDivExpr(const SCEVUDivExpr *Expr) { // The handling of unsigned division is basically the same as for signed // division, except the interpretation of the operands. As the divisor // has to be constant in both cases we can simply interpret it as an // unsigned value without additional complexity in the representation. // For the dividend we could choose from the different representation // schemes introduced for zero-extend operations but for now we will // simply use an assumption. auto *Dividend = Expr->getLHS(); auto *Divisor = Expr->getRHS(); assert(isa(Divisor) && "UDiv is no parameter but has a non-constant RHS."); auto DividendPWAC = visit(Dividend); auto DivisorPWAC = visit(Divisor); if (SE.isKnownNegative(Divisor)) { // Interpret negative divisors unsigned. This is a special case of the // piece-wise defined value described for zero-extends as we already know // the actual value of the constant divisor. unsigned Width = TD.getTypeSizeInBits(Expr->getType()); auto *DivisorDom = DivisorPWAC.first.domain().release(); auto *WidthExpPWA = getWidthExpValOnDomain(Width, DivisorDom); DivisorPWAC.first = DivisorPWAC.first.add(isl::manage(WidthExpPWA)); } // TODO: One can represent the dividend as piece-wise function to be more // precise but therefor a heuristic is needed. // Assume a non-negative dividend. takeNonNegativeAssumption(DividendPWAC, RecordedAssumptions); DividendPWAC = combine(DividendPWAC, DivisorPWAC, isl_pw_aff_div); DividendPWAC.first = DividendPWAC.first.floor(); return DividendPWAC; } PWACtx SCEVAffinator::visitSDivInstruction(Instruction *SDiv) { assert(SDiv->getOpcode() == Instruction::SDiv && "Assumed SDiv instruction!"); auto *Scope = getScope(); auto *Divisor = SDiv->getOperand(1); auto *DivisorSCEV = SE.getSCEVAtScope(Divisor, Scope); auto DivisorPWAC = visit(DivisorSCEV); assert(isa(DivisorSCEV) && "SDiv is no parameter but has a non-constant RHS."); auto *Dividend = SDiv->getOperand(0); auto *DividendSCEV = SE.getSCEVAtScope(Dividend, Scope); auto DividendPWAC = visit(DividendSCEV); DividendPWAC = combine(DividendPWAC, DivisorPWAC, isl_pw_aff_tdiv_q); return DividendPWAC; } PWACtx SCEVAffinator::visitSRemInstruction(Instruction *SRem) { assert(SRem->getOpcode() == Instruction::SRem && "Assumed SRem instruction!"); auto *Scope = getScope(); auto *Divisor = SRem->getOperand(1); auto *DivisorSCEV = SE.getSCEVAtScope(Divisor, Scope); auto DivisorPWAC = visit(DivisorSCEV); assert(isa(Divisor) && "SRem is no parameter but has a non-constant RHS."); auto *Dividend = SRem->getOperand(0); auto *DividendSCEV = SE.getSCEVAtScope(Dividend, Scope); auto DividendPWAC = visit(DividendSCEV); DividendPWAC = combine(DividendPWAC, DivisorPWAC, isl_pw_aff_tdiv_r); return DividendPWAC; } PWACtx SCEVAffinator::visitUnknown(const SCEVUnknown *Expr) { if (Instruction *I = dyn_cast(Expr->getValue())) { switch (I->getOpcode()) { case Instruction::IntToPtr: return visit(SE.getSCEVAtScope(I->getOperand(0), getScope())); case Instruction::SDiv: return visitSDivInstruction(I); case Instruction::SRem: return visitSRemInstruction(I); default: break; // Fall through. } } if (isa(Expr->getValue())) { isl::val v{Ctx, 0}; isl::space Space{Ctx, 0, NumIterators}; isl::local_space ls{Space}; return getPWACtxFromPWA(isl::aff(ls, v)); } llvm_unreachable("Unknowns SCEV was neither a parameter, a constant nor a " "valid instruction."); } PWACtx SCEVAffinator::complexityBailout() { // We hit the complexity limit for affine expressions; invalidate the scop // and return a constant zero. const DebugLoc &Loc = BB ? BB->getTerminator()->getDebugLoc() : DebugLoc(); S->invalidate(COMPLEXITY, Loc); return visit(SE.getZero(Type::getInt32Ty(S->getFunction().getContext()))); }