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merged 1 commit into from
Mar 15, 2024
Merged

[mlir][math] Implement alternative decomposition for tanh #85025

merged 1 commit into from
Mar 15, 2024

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@srcarroll srcarroll commented Mar 13, 2024 •
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The previous implementation decomposes tanh(x) into
(exp(2x) - 1)/(exp(2x)+1), x < 0
(1 - exp(-2x))/(1 + exp(-2x)), x >= 0
This is fine as it avoids overflow with the exponential, but the whole decomposition is computed for both cases unconditionally, then the result is chosen based off the sign of the input. This results in doing two expensive exp computations.

The proposed change avoids doing the whole computation twice by exploiting the reflection symmetry tanh(-x) = -tanh(x). We can "normalize" the input to be positive by setting y = sign(x) * x, where the sign of x is computed as sign(x) = (float)(x > 0) * (-2) + 1. Then compute z = tanh(y) with the decomposition above for x >=0 and "denormalize" the result z * sign(x) to retain the sign. The reason it is done this way is that it is very amenable to vectorization.

This method trades the duplicate decomposition computations (which takes 5 instructions including an extra expensive exp and div) for 4 cheap instructions to compute the signs value

  1. arith.cmpf (which is a pre-existing instruction in the previous impl)
  2. arith.sitofp
  3. arith.mulf
  4. arith.addf

and 1 more instruction to get the right sign in the result
5. arith.mulf. Moreover, numerically, this implementation will yield the exact same results as the previous implementation.

@srcarroll srcarroll marked this pull request as ready for review March 13, 2024 15:40
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llvmbot commented Mar 13, 2024

@llvm/pr-subscribers-mlir-math

Author: None (srcarroll)

Changes

The previous implementation decomposes tanh(x) into
(exp(2x) - 1)/(exp(2x)+1), x &lt; 0
(1 - exp(-2x))/(1 + exp(2x)), x &gt;= 0
This is fine as it avoids overflow with the exponential, but the whole decomposition is computed for both cases unconditionally, then the result is chosen based off the sign of the input. This results in doing two expensive exp computations.

The proposed change avoids doing the whole computation twice by exploiting the reflection symmetry tanh(-x) = -tanh(x). We can "normalize" the input to be positive by setting y = sign(x) * x. Then compute z = tanh(y) and "denormalize" the result z * sign(x) to retain the sign. The reason it is done this way is that it is very amenable to vectorization.

This method trades the duplicate decomposition computations (which takes 5 instructions including an extra expensive exp and div) for 4 cheap instructions to compute the signs value

  1. arith.cmpf
  2. arith.sitofp
  3. arith.mulf
  4. arith.addf

and 1 more instruction to get the right sign in the result
5. arith.mulf. Moreover, numerically, this implementation will yield the exact same results as the previous implementation.

Full diff: https://github.com/llvm/llvm-project/pull/85025.diff

2 Files Affected:

  • (modified) mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp (+23-17)
  • (modified) mlir/test/Dialect/Math/expand-math.mlir (+9-10)
diff --git a/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp b/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp
index 989a3e5536ec66..1750171b81a10e 100644
--- a/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp
+++ b/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp
@@ -91,34 +91,40 @@ static LogicalResult convertCoshOp(math::CoshOp op, PatternRewriter &rewriter) {
 }
 
 /// Expands tanh op into
-///   1) 1-exp^{-2x} / 1+exp^{-2x}, if x => 0
-///   2) exp^{2x}-1 / exp^{2x}+1  , if x < 0
+/// 1-exp^{-2x} / 1+exp^{-2x}
+/// To avoid overflow we exploit the reflection symmetry `tanh(-x) = -tanh(x)`.
+/// We compute a "signs" value which is -1 if input is negative and +1 if input
+/// is positive.  Then multiply the input by this value, guaranteeing that the
+/// result is positive, which also guarantees `exp^{-2x * sign(x)}` is in (0,
+/// 1]. Expand the computation on the input `x * sign(x)`, then multiply the
+/// result by `sign(x)` to retain sign of the real result.
 static LogicalResult convertTanhOp(math::TanhOp op, PatternRewriter &rewriter) {
   auto floatType = op.getOperand().getType();
   Location loc = op.getLoc();
+  Value zero = createFloatConst(loc, floatType, 0.0, rewriter);
   Value one = createFloatConst(loc, floatType, 1.0, rewriter);
-  Value two = createFloatConst(loc, floatType, 2.0, rewriter);
-  Value doubledX = rewriter.create<arith::MulFOp>(loc, op.getOperand(), two);
+  Value negTwo = createFloatConst(loc, floatType, -2.0, rewriter);
+
+  // Compute sign(x) = cast<float_type>(x < 0) * (-2) + 1
+  Value sign = rewriter.create<arith::CmpFOp>(loc, arith::CmpFPredicate::OLT,
+                                              op.getOperand(), zero);
+  sign = rewriter.create<arith::SIToFPOp>(loc, floatType, sign);
+  sign = rewriter.create<arith::MulFOp>(loc, sign, negTwo);
+  sign = rewriter.create<arith::AddFOp>(loc, sign, one);
 
-  // Case 1: tanh(x) = 1-exp^{-2x} / 1+exp^{-2x}
-  Value negDoubledX = rewriter.create<arith::NegFOp>(loc, doubledX);
+  // Normalize input to positive value: y = sign(x) * x
+  Value positiveX = rewriter.create<arith::MulFOp>(loc, sign, op.getOperand());
+
+  // Decompose on normalized input
+  Value negDoubledX = rewriter.create<arith::MulFOp>(loc, negTwo, positiveX);
   Value exp2x = rewriter.create<math::ExpOp>(loc, negDoubledX);
   Value dividend = rewriter.create<arith::SubFOp>(loc, one, exp2x);
   Value divisor = rewriter.create<arith::AddFOp>(loc, one, exp2x);
   Value positiveRes = rewriter.create<arith::DivFOp>(loc, dividend, divisor);
 
-  // Case 2: tanh(x) = exp^{2x}-1 / exp^{2x}+1
-  exp2x = rewriter.create<math::ExpOp>(loc, doubledX);
-  dividend = rewriter.create<arith::SubFOp>(loc, exp2x, one);
-  divisor = rewriter.create<arith::AddFOp>(loc, exp2x, one);
-  Value negativeRes = rewriter.create<arith::DivFOp>(loc, dividend, divisor);
+  // Multiply result by sign(x) to retain signs from negative inputs
+  rewriter.replaceOpWithNewOp<arith::MulFOp>(op, sign, positiveRes);
 
-  // tanh(x) = x >= 0 ? positiveRes : negativeRes
-  Value zero = createFloatConst(loc, floatType, 0.0, rewriter);
-  Value cmpRes = rewriter.create<arith::CmpFOp>(loc, arith::CmpFPredicate::OGE,
-                                                op.getOperand(), zero);
-  rewriter.replaceOpWithNewOp<arith::SelectOp>(op, cmpRes, positiveRes,
-                                               negativeRes);
   return success();
 }
 
diff --git a/mlir/test/Dialect/Math/expand-math.mlir b/mlir/test/Dialect/Math/expand-math.mlir
index 6ee65b085dad1b..86ee5c8620472b 100644
--- a/mlir/test/Dialect/Math/expand-math.mlir
+++ b/mlir/test/Dialect/Math/expand-math.mlir
@@ -7,19 +7,18 @@ func.func @tanh(%arg: f32) -> f32 {
 }
 // CHECK-DAG: %[[ZERO:.+]] = arith.constant 0.000000e+00 : f32
 // CHECK-DAG: %[[ONE:.+]] = arith.constant 1.000000e+00 : f32
-// CHECK-DAG: %[[TWO:.+]] = arith.constant 2.000000e+00 : f32
-// CHECK: %[[DOUBLEDX:.+]] = arith.mulf %arg0, %[[TWO]] : f32
-// CHECK: %[[NEGDOUBLEDX:.+]] = arith.negf %[[DOUBLEDX]] : f32
+// CHECK-DAG: %[[TWO:.+]] = arith.constant -2.000000e+00 : f32
+// CHECK: %[[VAL0:.+]] = arith.cmpf olt, %arg0, %[[ZERO]] : f32
+// CHECK: %[[VAL1:.+]] = arith.sitofp %[[VAL0]] : i1 to f32
+// CHECK: %[[VAL2:.+]] = arith.mulf %[[VAL1]], %[[TWO]] : f32
+// CHECK: %[[SIGN:.+]] = arith.addf %[[VAL2]], %[[ONE]] : f32
+// CHECK: %[[POSX:.+]] = arith.mulf %[[SIGN]], %arg0 : f32
+// CHECK: %[[NEGDOUBLEDX:.+]] = arith.mulf %[[POSX]], %[[TWO]] : f32
 // CHECK: %[[EXP1:.+]] = math.exp %[[NEGDOUBLEDX]] : f32
 // CHECK: %[[DIVIDEND1:.+]] = arith.subf %[[ONE]], %[[EXP1]] : f32
 // CHECK: %[[DIVISOR1:.+]] = arith.addf %[[EXP1]], %[[ONE]] : f32
-// CHECK: %[[RES1:.+]] = arith.divf %[[DIVIDEND1]], %[[DIVISOR1]] : f32
-// CHECK: %[[EXP2:.+]] = math.exp %[[DOUBLEDX]] : f32
-// CHECK: %[[DIVIDEND2:.+]] = arith.subf %[[EXP2]], %[[ONE]] : f32
-// CHECK: %[[DIVISOR2:.+]] = arith.addf %[[EXP2]], %[[ONE]] : f32
-// CHECK: %[[RES2:.+]] = arith.divf %[[DIVIDEND2]], %[[DIVISOR2]] : f32
-// CHECK: %[[COND:.+]] = arith.cmpf oge, %arg0, %[[ZERO]] : f32
-// CHECK: %[[RESULT:.+]] = arith.select %[[COND]], %[[RES1]], %[[RES2]] : f32
+// CHECK: %[[POSRES:.+]] = arith.divf %[[DIVIDEND1]], %[[DIVISOR1]] : f32
+// CHECK: %[[RESULT:.+]] = arith.mulf %[[SIGN]], %[[POSRES]] : f32
 // CHECK: return %[[RESULT]]
 
 // -----

@llvmbot
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llvmbot commented Mar 13, 2024

@llvm/pr-subscribers-mlir

Author: None (srcarroll)

Changes

The previous implementation decomposes tanh(x) into
(exp(2x) - 1)/(exp(2x)+1), x &lt; 0
(1 - exp(-2x))/(1 + exp(2x)), x &gt;= 0
This is fine as it avoids overflow with the exponential, but the whole decomposition is computed for both cases unconditionally, then the result is chosen based off the sign of the input. This results in doing two expensive exp computations.

The proposed change avoids doing the whole computation twice by exploiting the reflection symmetry tanh(-x) = -tanh(x). We can "normalize" the input to be positive by setting y = sign(x) * x. Then compute z = tanh(y) and "denormalize" the result z * sign(x) to retain the sign. The reason it is done this way is that it is very amenable to vectorization.

This method trades the duplicate decomposition computations (which takes 5 instructions including an extra expensive exp and div) for 4 cheap instructions to compute the signs value

  1. arith.cmpf
  2. arith.sitofp
  3. arith.mulf
  4. arith.addf

and 1 more instruction to get the right sign in the result
5. arith.mulf. Moreover, numerically, this implementation will yield the exact same results as the previous implementation.

Full diff: https://github.com/llvm/llvm-project/pull/85025.diff

2 Files Affected:

  • (modified) mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp (+23-17)
  • (modified) mlir/test/Dialect/Math/expand-math.mlir (+9-10)
diff --git a/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp b/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp
index 989a3e5536ec66..1750171b81a10e 100644
--- a/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp
+++ b/mlir/lib/Dialect/Math/Transforms/ExpandPatterns.cpp
@@ -91,34 +91,40 @@ static LogicalResult convertCoshOp(math::CoshOp op, PatternRewriter &rewriter) {
 }
 
 /// Expands tanh op into
-///   1) 1-exp^{-2x} / 1+exp^{-2x}, if x => 0
-///   2) exp^{2x}-1 / exp^{2x}+1  , if x < 0
+/// 1-exp^{-2x} / 1+exp^{-2x}
+/// To avoid overflow we exploit the reflection symmetry `tanh(-x) = -tanh(x)`.
+/// We compute a "signs" value which is -1 if input is negative and +1 if input
+/// is positive.  Then multiply the input by this value, guaranteeing that the
+/// result is positive, which also guarantees `exp^{-2x * sign(x)}` is in (0,
+/// 1]. Expand the computation on the input `x * sign(x)`, then multiply the
+/// result by `sign(x)` to retain sign of the real result.
 static LogicalResult convertTanhOp(math::TanhOp op, PatternRewriter &rewriter) {
   auto floatType = op.getOperand().getType();
   Location loc = op.getLoc();
+  Value zero = createFloatConst(loc, floatType, 0.0, rewriter);
   Value one = createFloatConst(loc, floatType, 1.0, rewriter);
-  Value two = createFloatConst(loc, floatType, 2.0, rewriter);
-  Value doubledX = rewriter.create<arith::MulFOp>(loc, op.getOperand(), two);
+  Value negTwo = createFloatConst(loc, floatType, -2.0, rewriter);
+
+  // Compute sign(x) = cast<float_type>(x < 0) * (-2) + 1
+  Value sign = rewriter.create<arith::CmpFOp>(loc, arith::CmpFPredicate::OLT,
+                                              op.getOperand(), zero);
+  sign = rewriter.create<arith::SIToFPOp>(loc, floatType, sign);
+  sign = rewriter.create<arith::MulFOp>(loc, sign, negTwo);
+  sign = rewriter.create<arith::AddFOp>(loc, sign, one);
 
-  // Case 1: tanh(x) = 1-exp^{-2x} / 1+exp^{-2x}
-  Value negDoubledX = rewriter.create<arith::NegFOp>(loc, doubledX);
+  // Normalize input to positive value: y = sign(x) * x
+  Value positiveX = rewriter.create<arith::MulFOp>(loc, sign, op.getOperand());
+
+  // Decompose on normalized input
+  Value negDoubledX = rewriter.create<arith::MulFOp>(loc, negTwo, positiveX);
   Value exp2x = rewriter.create<math::ExpOp>(loc, negDoubledX);
   Value dividend = rewriter.create<arith::SubFOp>(loc, one, exp2x);
   Value divisor = rewriter.create<arith::AddFOp>(loc, one, exp2x);
   Value positiveRes = rewriter.create<arith::DivFOp>(loc, dividend, divisor);
 
-  // Case 2: tanh(x) = exp^{2x}-1 / exp^{2x}+1
-  exp2x = rewriter.create<math::ExpOp>(loc, doubledX);
-  dividend = rewriter.create<arith::SubFOp>(loc, exp2x, one);
-  divisor = rewriter.create<arith::AddFOp>(loc, exp2x, one);
-  Value negativeRes = rewriter.create<arith::DivFOp>(loc, dividend, divisor);
+  // Multiply result by sign(x) to retain signs from negative inputs
+  rewriter.replaceOpWithNewOp<arith::MulFOp>(op, sign, positiveRes);
 
-  // tanh(x) = x >= 0 ? positiveRes : negativeRes
-  Value zero = createFloatConst(loc, floatType, 0.0, rewriter);
-  Value cmpRes = rewriter.create<arith::CmpFOp>(loc, arith::CmpFPredicate::OGE,
-                                                op.getOperand(), zero);
-  rewriter.replaceOpWithNewOp<arith::SelectOp>(op, cmpRes, positiveRes,
-                                               negativeRes);
   return success();
 }
 
diff --git a/mlir/test/Dialect/Math/expand-math.mlir b/mlir/test/Dialect/Math/expand-math.mlir
index 6ee65b085dad1b..86ee5c8620472b 100644
--- a/mlir/test/Dialect/Math/expand-math.mlir
+++ b/mlir/test/Dialect/Math/expand-math.mlir
@@ -7,19 +7,18 @@ func.func @tanh(%arg: f32) -> f32 {
 }
 // CHECK-DAG: %[[ZERO:.+]] = arith.constant 0.000000e+00 : f32
 // CHECK-DAG: %[[ONE:.+]] = arith.constant 1.000000e+00 : f32
-// CHECK-DAG: %[[TWO:.+]] = arith.constant 2.000000e+00 : f32
-// CHECK: %[[DOUBLEDX:.+]] = arith.mulf %arg0, %[[TWO]] : f32
-// CHECK: %[[NEGDOUBLEDX:.+]] = arith.negf %[[DOUBLEDX]] : f32
+// CHECK-DAG: %[[TWO:.+]] = arith.constant -2.000000e+00 : f32
+// CHECK: %[[VAL0:.+]] = arith.cmpf olt, %arg0, %[[ZERO]] : f32
+// CHECK: %[[VAL1:.+]] = arith.sitofp %[[VAL0]] : i1 to f32
+// CHECK: %[[VAL2:.+]] = arith.mulf %[[VAL1]], %[[TWO]] : f32
+// CHECK: %[[SIGN:.+]] = arith.addf %[[VAL2]], %[[ONE]] : f32
+// CHECK: %[[POSX:.+]] = arith.mulf %[[SIGN]], %arg0 : f32
+// CHECK: %[[NEGDOUBLEDX:.+]] = arith.mulf %[[POSX]], %[[TWO]] : f32
 // CHECK: %[[EXP1:.+]] = math.exp %[[NEGDOUBLEDX]] : f32
 // CHECK: %[[DIVIDEND1:.+]] = arith.subf %[[ONE]], %[[EXP1]] : f32
 // CHECK: %[[DIVISOR1:.+]] = arith.addf %[[EXP1]], %[[ONE]] : f32
-// CHECK: %[[RES1:.+]] = arith.divf %[[DIVIDEND1]], %[[DIVISOR1]] : f32
-// CHECK: %[[EXP2:.+]] = math.exp %[[DOUBLEDX]] : f32
-// CHECK: %[[DIVIDEND2:.+]] = arith.subf %[[EXP2]], %[[ONE]] : f32
-// CHECK: %[[DIVISOR2:.+]] = arith.addf %[[EXP2]], %[[ONE]] : f32
-// CHECK: %[[RES2:.+]] = arith.divf %[[DIVIDEND2]], %[[DIVISOR2]] : f32
-// CHECK: %[[COND:.+]] = arith.cmpf oge, %arg0, %[[ZERO]] : f32
-// CHECK: %[[RESULT:.+]] = arith.select %[[COND]], %[[RES1]], %[[RES2]] : f32
+// CHECK: %[[POSRES:.+]] = arith.divf %[[DIVIDEND1]], %[[DIVISOR1]] : f32
+// CHECK: %[[RESULT:.+]] = arith.mulf %[[SIGN]], %[[POSRES]] : f32
 // CHECK: return %[[RESULT]]
 
 // -----

@srcarroll srcarroll changed the title Implement alternative decomposition for tanh [mlir][math] Implement alternative decomposition for tanh Mar 13, 2024
@srcarroll srcarroll requested a review from rsuderman March 13, 2024 20:10
Lewuathe
Lewuathe approved these changes Mar 14, 2024
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It looks good to me. We may want to have the correctness check running in CPU runner if possible. I think it would be possible to write down such a case something like math.ctlz.

https://github.com/llvm/llvm-project/blob/main/mlir/test/Integration/Dialect/Math/CPU/mathtofuncs_ctlz.mlir

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Thanks for the review. I will submit another PR for correctness check soon

@srcarroll srcarroll merged commit 58ef9be into llvm:main Mar 15, 2024
srcarroll added a commit to srcarroll/llvm-project that referenced this pull request Mar 15, 2024
@srcarroll
Revert "[mlir][math] Implement alternative decomposition for tanh (ll…
bc7dc3b 
…vm#85025)"

This reverts commit 58ef9be.
@srcarroll srcarroll deleted the improve-tanh-expansion branch June 5, 2024 02:56
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