[PATCH] D73741: [SCEV] SCEVExpander::isHighCostExpansionHelper(): cost-model polynomial recurrence

[Roman Lebedev via Phabricator via llvm-commits]

lebedev.ri created this revision.
lebedev.ri added reviewers: reames, mkazantsev, wmi, sanjoy.
lebedev.ri added a project: LLVM.
Herald added subscribers: javed.absar, hiraditya.

So, i wouldn't call this *obviously* correct,
but i think i got it right this time :)

Roughly, we have

  Op0*x^0 + Op1*x^1 + Op2*x^2 ...

where `Op_{n} * x^{n}` is called term, and `n` the degree of term.

Due to the way they are stored internally in `SCEVAddRecExpr`,
i believe we can have `Op_{n}` to be `0`, so we should not charge for those.

I think it is most straight-forward to count the cost in 4 steps:

1. First, count it the same way we counted `scAddExpr`, but be sure to skip terms with zero constants. Much like with `add` expr we will have one less addition than number of terms.
2. Each non-constant term (term degree >= 1) requires a multiplication between the `Op_{n}` and `x^{n}`. But again, only charge for it if it is required - `Op_{n}` must not be 0 (no term) or 1 (no multiplication needed), and obviously don't charge constant terms (`x^0 == 1`).
3. We must charge for all the `x^0`..x^{poly_degree}` themselves. Since `x^{poly_degree}` is `x * x * ...  * x`, i.e. `poly_degree` `x`'es multiplied, for final `poly_degree` term we again require `poly_degree-1` multiplications. Note that all the `x^{0}`..`x^{poly_degree-1}` will be computed for the free along the way there.
4. And finally, the operands themselves.

Here, much like with add/mul exprs, we really don't look for preexisting instructions..


Repository:
  rG LLVM Github Monorepo

https://reviews.llvm.org/D73741

Files:
  llvm/lib/Analysis/ScalarEvolutionExpander.cpp


Index: llvm/lib/Analysis/ScalarEvolutionExpander.cpp
===================================================================
--- llvm/lib/Analysis/ScalarEvolutionExpander.cpp
+++ llvm/lib/Analysis/ScalarEvolutionExpander.cpp
@@ -2253,20 +2253,68 @@
     return BudgetRemaining < 0;
   }
 
+  if (const auto *NAry = dyn_cast<SCEVAddRecExpr>(S)) {
+    Type *OpType = NAry->getType();
+
+    assert(NAry->getNumOperands() >= 2 &&
+           "Polynomial should be at least linear");
+
+    int AddCost = TTI->getOperationCost(Instruction::Add, OpType);
+    int MulCost = TTI->getOperationCost(Instruction::Mul, OpType);
+
+    // In this polynominal, we may have some zero operands, and we shouldn't
+    // really charge for those. So how many non-zero coeffients are there?
+    int NumTerms = llvm::count_if(NAry->operands(), [](const SCEV *S) {
+      auto *SConst = dyn_cast<SCEVConstant>(S);
+      return !SConst || !SConst->isZero();
+    });
+    assert(NumTerms >= 1 && "Polynominal should have at least one term.");
+
+    // Much like with normal add expr, the polynominal will require
+    // one less addition than the number of it's terms.
+    BudgetRemaining -= AddCost * (NumTerms - 1);
+    if (BudgetRemaining < 0)
+      return true;
+
+    // Ignoring constant term (operand 0), how many of the coeffients are u> 1?
+    int NumNonZeroDegreeTerms =
+        llvm::count_if(make_range(std::next(NAry->op_begin()), NAry->op_end()),
+                       [](const SCEV *S) {
+                         auto *SConst = dyn_cast<SCEVConstant>(S);
+                         return !SConst || SConst->getAPInt().ugt(1);
+                       });
+    // Here, *each* one of those will require a multiplication.
+    BudgetRemaining -= MulCost * NumNonZeroDegreeTerms;
+    if (BudgetRemaining < 0)
+      return true;
+
+    // What is the degree of this polynominal?
+    int PolyDegree = NAry->getNumOperands() - 1;
+    assert(PolyDegree >= 1 && "Should be at least affine.");
+
+    // The final term will be:
+    //   Op_{PolyDegree} * x ^ {PolyDegree}
+    // Where  x ^ {PolyDegree}  will again require PolyDegree-1 mul operations.
+    // Note that  x ^ {PolyDegree} = x * x ^ {PolyDegree-1}  so charging for
+    // x ^ {PolyDegree}  will give us  x ^ {2} .. x ^ {PolyDegree-1}  for free.
+    BudgetRemaining -= MulCost * (PolyDegree - 1);
+    if (BudgetRemaining < 0)
+      return true;
+
+    // And finally, the operands themselves should fit within the budget.
+    for (const SCEV *Op : NAry->operands()) {
+      if (isHighCostExpansionHelper(Op, L, At, BudgetRemaining, TTI, Processed))
+        return true;
+    }
+
+    return BudgetRemaining < 0;
+  }
+
   // HowManyLessThans uses a Max expression whenever the loop is not guarded by
   // the exit condition.
   if (isa<SCEVMinMaxExpr>(S))
     return true;
 
-  // Recurse past nary expressions, which commonly occur in the
-  // BackedgeTakenCount. They may already exist in program code, and if not,
-  // they are not too expensive rematerialize.
-  if (const SCEVNAryExpr *NAry = dyn_cast<SCEVNAryExpr>(S)) {
-    for (auto *Op : NAry->operands())
-      if (isHighCostExpansionHelper(Op, L, At, BudgetRemaining, TTI, Processed))
-        return true;
-  }
-
   // If we haven't recognized an expensive SCEV pattern, assume it's an
   // expression produced by program code.
   return false;


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