[SCEV] Add [zs]ext{C,+,x} -> (D + [zs]ext{C-D,+,x})<nuw><nsw> transform

as well as sext(C + x + ...) -> (D + sext(C-D + x + ...))<nuw><nsw>
similar to the equivalent transformation for zext's

if the top level addition in (D + (C-D + x * n)) could be proven to
not wrap, where the choice of D also maximizes the number of trailing
zeroes of (C-D + x * n), ensuring homogeneous behaviour of the
transformation and better canonicalization of such AddRec's

(indeed, there are 2^(2w) different expressions in `B1 + ext(B2 + Y)` form for
the same Y, but only 2^(2w - k) different expressions in the resulting `B3 +
ext((B4 * 2^k) + Y)` form, where w is the bit width of the integral type)

This patch generalizes sext(C1 + C2*X) --> sext(C1) + sext(C2*X) and
sext{C1,+,C2} --> sext(C1) + sext{0,+,C2} transformations added in
r209568 relaxing the requirements the following way:

1. C2 doesn't have to be a power of 2, it's enough if it's divisible by 2
 a sufficient number of times;
2. C1 doesn't have to be less than C2, instead of extracting the entire
  C1 we can split it into 2 terms: (00...0XXX + YY...Y000), keep the
  second one that may cause wrapping within the extension operator, and
  move the first one that doesn't affect wrapping out of the extension
  operator, enabling further simplifications;
3. C1 and C2 don't have to be positive, splitting C1 like shown above
 produces a sum that is guaranteed to not wrap, signed or unsigned;
4. in AddExpr case there could be more than 2 terms, and in case of
  AddExpr the 2nd and following terms and in case of AddRecExpr the
  Step component don't have to be in the C2*X form or constant
  (respectively), they just need to have enough trailing zeros,
  which in turn could be guaranteed by means other than arithmetics,
  e.g. by a pointer alignment;
5. the extension operator doesn't have to be a sext, the same
  transformation works and profitable for zext's as well.

Apparently, optimizations like SLPVectorizer currently fail to
vectorize even rather trivial cases like the following:

 double bar(double *a, unsigned n) {
   double x = 0.0;
   double y = 0.0;
   for (unsigned i = 0; i < n; i += 2) {
     x += a[i];
     y += a[i + 1];
   }
   return x * y;
 }

If compiled with `clang -std=c11 -Wpedantic -Wall -O3 main.c -S -o - -emit-llvm`
(!{!"clang version 7.0.0 (trunk 337339) (llvm/trunk 337344)"})

it produces scalar code with the loop not unrolled with the unsigned `n` and
`i` (like shown above), but vectorized and unrolled loop with signed `n` and
`i`. With the changes made in this commit the unsigned version will be
vectorized (though not unrolled for unclear reasons).

How it all works:

Let say we have an AddExpr that looks like (C + x + y + ...), where C
is a constant and x, y, ... are arbitrary SCEVs. Let's compute the
minimum number of trailing zeroes guaranteed of that sum w/o the
constant term: (x + y + ...). If, for example, those terms look like
follows:

        i
XXXX...X000
YYYY...YY00
   ...
ZZZZ...0000

then the rightmost non-guaranteed-zero bit (a potential one at i-th
position above) can change the bits of the sum to the left (and at
i-th position itself), but it can not possibly change the bits to the
right. So we can compute the number of trailing zeroes by taking a
minimum between the numbers of trailing zeroes of the terms.

Now let's say that our original sum with the constant is effectively
just C + X, where X = x + y + .... Let's also say that we've got 2
guaranteed trailing zeros for X:

         j
CCCC...CCCC
XXXX...XX00  // this is X = (x + y + ...)

Any bit of C to the left of j may in the end cause the C + X sum to
wrap, but the rightmost 2 bits of C (at positions j and j - 1) do not
affect wrapping in any way. If the upper bits cause a wrap, it will be
a wrap regardless of the values of the 2 least significant bits of C.
If the upper bits do not cause a wrap, it won't be a wrap regardless
of the values of the 2 bits on the right (again).

So let's split C to 2 constants like follows:

0000...00CC  = D
CCCC...CC00  = (C - D)

and represent the whole sum as D + (C - D + X). The second term of
this new sum looks like this:

CCCC...CC00
XXXX...XX00
-----------  // let's add them up
YYYY...YY00

The sum above (let's call it Y)) may or may not wrap, we don't know,
so we need to keep it under a sext/zext. Adding D to that sum though
will never wrap, signed or unsigned, if performed on the original bit
width or the extended one, because all that that final add does is
setting the 2 least significant bits of Y to the bits of D:

YYYY...YY00 = Y
0000...00CC = D
-----------  <nuw><nsw>
YYYY...YYCC

Which means we can safely move that D out of the sext or zext and
claim that the top-level sum neither sign wraps nor unsigned wraps.

Let's run an example, let's say we're working in i8's and the original
expression (zext's or sext's operand) is 21 + 12x + 8y. So it goes
like this:

0001 0101  // 21
XXXX XX00  // 12x
YYYY Y000  // 8y

0001 0101  // 21
ZZZZ ZZ00  // 12x + 8y

0000 0001  // D
0001 0100  // 21 - D = 20
ZZZZ ZZ00  // 12x + 8y

0000 0001  // D
WWWW WW00  // 21 - D + 12x + 8y = 20 + 12x + 8y

therefore zext(21 + 12x + 8y) = (1 + zext(20 + 12x + 8y))<nuw><nsw>

This approach could be improved if we move away from using trailing
zeroes and use KnownBits instead. For instance, with KnownBits we could
have the following picture:

    i
10 1110...0011  // this is C
XX X1XX...XX00  // this is X = (x + y + ...)

Notice that some of the bits of X are known ones, also notice that
known bits of X are interspersed with unknown bits and not grouped on
the rigth or left.

We can see at the position i that C(i) and X(i) are both known ones,
therefore the (i + 1)th carry bit is guaranteed to be 1 regardless of
the bits of C to the right of i. For instance, the C(i - 1) bit only
affects the bits of the sum at positions i - 1 and i, and does not
influence if the sum is going to wrap or not. Therefore we could split
the constant C the following way:

    i
00 0010...0011  = D
10 1100...0000  = (C - D)

Let's compute the KnownBits of (C - D) + X:

XX1 1            = carry bit, blanks stand for known zeroes
 10 1100...0000  = (C - D)
 XX X1XX...XX00  = X
--- -----------
 XX X0XX...XX00

Will this add wrap or not essentially depends on bits of X. Adding D
to this sum, however, is guaranteed to not to wrap:

0    X
 00 0010...0011  = D
 sX X0XX...XX00  = (C - D) + X
--- -----------
 sX XXXX   XX11

As could be seen above, adding D preserves the sign bit of (C - D) +
X, if any, and has a guaranteed 0 carry out, as expected.

The more bits of (C - D) we constrain, the better the transformations
introduced here canonicalize expressions as it leaves less freedom to
what values the constant part of ((C - D) + x + y + ...) can take.

Reviewed By: mzolotukhin, efriedma

Differential Revision: https://reviews.llvm.org/D48853

llvm-svn: 337943

1 files changed, 104 insertions, 63 deletions

diff --git a/llvm/lib/Analysis/ScalarEvolution.cpp b/llvm/lib/Analysis/ScalarEvolution.cpp
index b2f12a12d1d..aa95ace9301 100644
--- a/llvm/lib/Analysis/ScalarEvolution.cpp
+++ b/llvm/lib/Analysis/ScalarEvolution.cpp
@@ -1559,6 +1559,43 @@ bool ScalarEvolution::proveNoWrapByVaryingStart(const SCEV *Start,
   return false;
 }
 
+// Finds an integer D for an expression (C + x + y + ...) such that the top
+// level addition in (D + (C - D + x + y + ...)) would not wrap (signed or
+// unsigned) and the number of trailing zeros of (C - D + x + y + ...) is
+// maximized, where C is the \p ConstantTerm, x, y, ... are arbitrary SCEVs, and
+// the (C + x + y + ...) expression is \p WholeAddExpr.
+static APInt extractConstantWithoutWrapping(ScalarEvolution &SE,
+                                            const SCEVConstant *ConstantTerm,
+                                            const SCEVAddExpr *WholeAddExpr) {
+  const APInt C = ConstantTerm->getAPInt();
+  const unsigned BitWidth = C.getBitWidth();
+  // Find number of trailing zeros of (x + y + ...) w/o the C first:
+  uint32_t TZ = BitWidth;
+  for (unsigned I = 1, E = WholeAddExpr->getNumOperands(); I < E && TZ; ++I)
+    TZ = std::min(TZ, SE.GetMinTrailingZeros(WholeAddExpr->getOperand(I)));
+  if (TZ) {
+    // Set D to be as many least significant bits of C as possible while still
+    // guaranteeing that adding D to (C - D + x + y + ...) won't cause a wrap:
+    return TZ < BitWidth ? C.trunc(TZ).zext(BitWidth) : C;
+  }
+  return APInt(BitWidth, 0);
+}
+
+// Finds an integer D for an affine AddRec expression {C,+,x} such that the top
+// level addition in (D + {C-D,+,x}) would not wrap (signed or unsigned) and the
+// number of trailing zeros of (C - D + x * n) is maximized, where C is the \p
+// ConstantStart, x is an arbitrary \p Step, and n is the loop trip count.
+static APInt extractConstantWithoutWrapping(ScalarEvolution &SE,
+                                            const APInt &ConstantStart,
+                                            const SCEV *Step) {
+  const unsigned BitWidth = ConstantStart.getBitWidth();
+  const uint32_t TZ = SE.GetMinTrailingZeros(Step);
+  if (TZ)
+    return TZ < BitWidth ? ConstantStart.trunc(TZ).zext(BitWidth)
+                         : ConstantStart;
+  return APInt(BitWidth, 0);
+}
+
 const SCEV *
 ScalarEvolution::getZeroExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
   assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
@@ -1745,6 +1782,23 @@ ScalarEvolution::getZeroExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
         }
       }
 
+      // zext({C,+,Step}) --> (zext(D) + zext({C-D,+,Step}))<nuw><nsw>
+      // if D + (C - D + Step * n) could be proven to not unsigned wrap
+      // where D maximizes the number of trailing zeros of (C - D + Step * n)
+      if (const auto *SC = dyn_cast<SCEVConstant>(Start)) {
+        const APInt &C = SC->getAPInt();
+        const APInt &D = extractConstantWithoutWrapping(*this, C, Step);
+        if (D != 0) {
+          const SCEV *SZExtD = getZeroExtendExpr(getConstant(D), Ty, Depth);
+          const SCEV *SResidual =
+              getAddRecExpr(getConstant(C - D), Step, L, AR->getNoWrapFlags());
+          const SCEV *SZExtR = getZeroExtendExpr(SResidual, Ty, Depth + 1);
+          return getAddExpr(SZExtD, SZExtR,
+                            (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
+                            Depth + 1);
+        }
+      }
+
       if (proveNoWrapByVaryingStart<SCEVZeroExtendExpr>(Start, Step, L)) {
         const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNUW);
         return getAddRecExpr(
@@ -1778,41 +1832,24 @@ ScalarEvolution::getZeroExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
       return getAddExpr(Ops, SCEV::FlagNUW, Depth + 1);
     }
 
-    // zext(C + x + y + ...) --> (zext(D) + zext((C - D) + x + y + ...))<nuw>
+    // zext(C + x + y + ...) --> (zext(D) + zext((C - D) + x + y + ...))
     // if D + (C - D + x + y + ...) could be proven to not unsigned wrap
     // where D maximizes the number of trailing zeros of (C - D + x + y + ...)
     //
-    // Useful while proving that address arithmetic expressions are equal or
-    // differ by a small constant amount, see LoadStoreVectorizer pass.
+    // Often address arithmetics contain expressions like
+    // (zext (add (shl X, C1), C2)), for instance, (zext (5 + (4 * X))).
+    // This transformation is useful while proving that such expressions are
+    // equal or differ by a small constant amount, see LoadStoreVectorizer pass.
     if (const auto *SC = dyn_cast<SCEVConstant>(SA->getOperand(0))) {
-      // Often address arithmetics contain expressions like
-      // (zext (add (shl X, C1), C2)), for instance, (zext (5 + (4 * X))).
-      // ConstantRange is unable to prove that it's possible to transform
-      // (5 + (4 * X)) to (1 + (4 + (4 * X))) w/o underflowing:
-      //
-      // |  Expression   |     ConstantRange      |       KnownBits       |
-      // |---------------|------------------------|-----------------------|
-      // | i8 4 * X      | [L: 0, U: 253)         | XXXX XX00             |
-      // |               |   => Min: 0, Max: 252  |   => Min: 0, Max: 252 |
-      // |               |                        |                       |
-      // | i8 4 * X + 5  | [L: 5, U: 2) (wrapped) | YYYY YY01             |
-      // |         (101) |   => Min: 0, Max: 255  |   => Min: 1, Max: 253 |
-      //
-      // As KnownBits are not available for SCEV expressions, use number of
-      // trailing zeroes instead:
-      APInt C = SC->getAPInt();
-      uint32_t TZ = C.getBitWidth();
-      for (unsigned I = 1, E = SA->getNumOperands(); I < E && TZ; ++I)
-        TZ = std::min(TZ, GetMinTrailingZeros(SA->getOperand(I)));
-      if (TZ) {
-        APInt D = TZ < C.getBitWidth() ? C.trunc(TZ).zext(C.getBitWidth()) : C;
-        if (D != 0) {
-          const SCEV *SZExtD = getZeroExtendExpr(getConstant(D), Ty, Depth);
-          const SCEV *SResidual =
-              getAddExpr(getConstant(-D), SA, SCEV::FlagAnyWrap, Depth);
-          const SCEV *SZExtR = getZeroExtendExpr(SResidual, Ty, Depth + 1);
-          return getAddExpr(SZExtD, SZExtR, SCEV::FlagNUW, Depth + 1);
-        }
+      const APInt &D = extractConstantWithoutWrapping(*this, SC, SA);
+      if (D != 0) {
+        const SCEV *SZExtD = getZeroExtendExpr(getConstant(D), Ty, Depth);
+        const SCEV *SResidual =
+            getAddExpr(getConstant(-D), SA, SCEV::FlagAnyWrap, Depth);
+        const SCEV *SZExtR = getZeroExtendExpr(SResidual, Ty, Depth + 1);
+        return getAddExpr(SZExtD, SZExtR,
+                          (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
+                          Depth + 1);
       }
     }
   }
@@ -1916,24 +1953,7 @@ ScalarEvolution::getSignExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
       return getTruncateOrSignExtend(X, Ty);
   }
 
-  // sext(C1 + (C2 * x)) --> C1 + sext(C2 * x) if C1 < C2
   if (auto *SA = dyn_cast<SCEVAddExpr>(Op)) {
-    if (SA->getNumOperands() == 2) {
-      auto *SC1 = dyn_cast<SCEVConstant>(SA->getOperand(0));
-      auto *SMul = dyn_cast<SCEVMulExpr>(SA->getOperand(1));
-      if (SMul && SC1) {
-        if (auto *SC2 = dyn_cast<SCEVConstant>(SMul->getOperand(0))) {
-          const APInt &C1 = SC1->getAPInt();
-          const APInt &C2 = SC2->getAPInt();
-          if (C1.isStrictlyPositive() && C2.isStrictlyPositive() &&
-              C2.ugt(C1) && C2.isPowerOf2())
-            return getAddExpr(getSignExtendExpr(SC1, Ty, Depth + 1),
-                              getSignExtendExpr(SMul, Ty, Depth + 1),
-                              SCEV::FlagAnyWrap, Depth + 1);
-        }
-      }
-    }
-
     // sext((A + B + ...)<nsw>) --> (sext(A) + sext(B) + ...)<nsw>
     if (SA->hasNoSignedWrap()) {
       // If the addition does not sign overflow then we can, by definition,
@@ -1943,6 +1963,28 @@ ScalarEvolution::getSignExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
         Ops.push_back(getSignExtendExpr(Op, Ty, Depth + 1));
       return getAddExpr(Ops, SCEV::FlagNSW, Depth + 1);
     }
+
+    // sext(C + x + y + ...) --> (sext(D) + sext((C - D) + x + y + ...))
+    // if D + (C - D + x + y + ...) could be proven to not signed wrap
+    // where D maximizes the number of trailing zeros of (C - D + x + y + ...)
+    //
+    // For instance, this will bring two seemingly different expressions:
+    //     1 + sext(5 + 20 * %x + 24 * %y)  and
+    //         sext(6 + 20 * %x + 24 * %y)
+    // to the same form:
+    //     2 + sext(4 + 20 * %x + 24 * %y)
+    if (const auto *SC = dyn_cast<SCEVConstant>(SA->getOperand(0))) {
+      const APInt &D = extractConstantWithoutWrapping(*this, SC, SA);
+      if (D != 0) {
+        const SCEV *SSExtD = getSignExtendExpr(getConstant(D), Ty, Depth);
+        const SCEV *SResidual =
+            getAddExpr(getConstant(-D), SA, SCEV::FlagAnyWrap, Depth);
+        const SCEV *SSExtR = getSignExtendExpr(SResidual, Ty, Depth + 1);
+        return getAddExpr(SSExtD, SSExtR,
+                          (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
+                          Depth + 1);
+      }
+    }
   }
   // If the input value is a chrec scev, and we can prove that the value
   // did not overflow the old, smaller, value, we can sign extend all of the
@@ -2072,21 +2114,20 @@ ScalarEvolution::getSignExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
         }
       }
 
-      // If Start and Step are constants, check if we can apply this
-      // transformation:
-      // sext{C1,+,C2} --> C1 + sext{0,+,C2} if C1 < C2
-      auto *SC1 = dyn_cast<SCEVConstant>(Start);
-      auto *SC2 = dyn_cast<SCEVConstant>(Step);
-      if (SC1 && SC2) {
-        const APInt &C1 = SC1->getAPInt();
-        const APInt &C2 = SC2->getAPInt();
-        if (C1.isStrictlyPositive() && C2.isStrictlyPositive() && C2.ugt(C1) &&
-            C2.isPowerOf2()) {
-          Start = getSignExtendExpr(Start, Ty, Depth + 1);
-          const SCEV *NewAR = getAddRecExpr(getZero(AR->getType()), Step, L,
-                                            AR->getNoWrapFlags());
-          return getAddExpr(Start, getSignExtendExpr(NewAR, Ty, Depth + 1),
-                            SCEV::FlagAnyWrap, Depth + 1);
+      // sext({C,+,Step}) --> (sext(D) + sext({C-D,+,Step}))<nuw><nsw>
+      // if D + (C - D + Step * n) could be proven to not signed wrap
+      // where D maximizes the number of trailing zeros of (C - D + Step * n)
+      if (const auto *SC = dyn_cast<SCEVConstant>(Start)) {
+        const APInt &C = SC->getAPInt();
+        const APInt &D = extractConstantWithoutWrapping(*this, C, Step);
+        if (D != 0) {
+          const SCEV *SSExtD = getSignExtendExpr(getConstant(D), Ty, Depth);
+          const SCEV *SResidual =
+              getAddRecExpr(getConstant(C - D), Step, L, AR->getNoWrapFlags());
+          const SCEV *SSExtR = getSignExtendExpr(SResidual, Ty, Depth + 1);
+          return getAddExpr(SSExtD, SSExtR,
+                            (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
+                            Depth + 1);
         }
       }