[mlir][math] Modified the 'math.exp' lowering for higher precision

[rsuderman]

Summary

The existing lowering has lower precision for certain use cases, e.g.
tanh. Improved version should demonstrate an overall higher level of precision.

[jpienaar]

Thanks! (well and Emilio for his help here too)

mlir/lib/Dialect/Math/Transforms/PolynomialApproximation.cpp

898

Nit: clampWithNormals to follow naming convention.

900

lowerBound, upperBound

908

selectCmp

935

You can drop getType here (getElementTypeOrSelf has a Value overload)

936

MLIR style is to elide trivial braces

mlir/lib/Dialect/Math/Transforms/PolynomialApproximation.cpp

 }
 
 //----------------------------------------------------------------------------//
 // Exp approximation.
 //----------------------------------------------------------------------------//
 
 namespace {
 
+Value clampWithNormals(ImplicitLocOpBuilder &builder,
+                       const llvm::ArrayRef<int64_t> shape, Value value,
+                       float lowerBound, float upperBound) {
+  assert(!std::isnan(lowerBound));
+  assert(!std::isnan(upperBound));
+
+  auto bcast = [&](Value value) -> Value {
+    return broadcast(builder, value, shape);
+  };
+
+  auto selectCmp = [&builder](auto pred, Value value, Value bound) {
+    return builder.create<arith::SelectOp>(
+        builder.create<arith::CmpFOp>(pred, value, bound), value, bound);
+  };
+
+  // Note: prefer UGE/ULE vs. UGT/ULT, since they generate vmaxps/vminps vs.
+  // vcmpleps+vmovaps on x86_64. The latter outcome is also obtained with
+  // arith::{Max,Min}FOp.
+  value = selectCmp(arith::CmpFPredicate::UGE, value,
+                    bcast(f32Cst(builder, lowerBound)));
+  value = selectCmp(arith::CmpFPredicate::ULE, value,
+                    bcast(f32Cst(builder, upperBound)));
+  return value;
+}
+
 struct ExpApproximation : public OpRewritePattern<math::ExpOp> {
 public:
   using OpRewritePattern::OpRewritePattern;
 
   LogicalResult matchAndRewrite(math::ExpOp op,
                                 PatternRewriter &rewriter) const final;
 };
-} // namespace
 
-// Approximate exp(x) using its reduced range exp(y) where y is in the range
-// [0, ln(2)], let y = x - floor(x / ln(2)) * ln(2) = x - k * ln(2), exp(x)
-// = exp(y) * 2^k. exp(y).
 LogicalResult
 ExpApproximation::matchAndRewrite(math::ExpOp op,
                                   PatternRewriter &rewriter) const {
-  if (!getElementTypeOrSelf(op.getOperand()).isF32())
+  auto shape = vectorShape(op.getOperand().getType());
+  auto elementTy = getElementTypeOrSelf(op.getType());
+  if (!elementTy.isF32())
     return rewriter.notifyMatchFailure(op, "unsupported operand type");
 
-  ArrayRef<int64_t> shape = vectorShape(op.getOperand());
-
   ImplicitLocOpBuilder builder(op->getLoc(), rewriter);
 
-  // TODO: Consider a common pattern rewriter with all methods below to
-  // write the approximations.
+  auto add = [&](Value a, Value b) -> Value {
+    return builder.create<arith::AddFOp>(a, b);
+  };
   auto bcast = [&](Value value) -> Value {
     return broadcast(builder, value, shape);
   };
+  auto floor = [&](Value a) { return builder.create<math::FloorOp>(a); };
   auto fmla = [&](Value a, Value b, Value c) {
     return builder.create<math::FmaOp>(a, b, c);
   };
   auto mul = [&](Value a, Value b) -> Value {
     return builder.create<arith::MulFOp>(a, b);
   };
-  auto sub = [&](Value a, Value b) -> Value {
-    return builder.create<arith::SubFOp>(a, b);
-  };
-  auto floor = [&](Value a) { return builder.create<math::FloorOp>(a); };
-
-  Value cstLn2 = bcast(f32Cst(builder, static_cast<float>(LN2_VALUE)));
-  Value cstLog2E = bcast(f32Cst(builder, static_cast<float>(LOG2E_VALUE)));
 
-  // Polynomial coefficients.
-  Value cstCephesExpP0 = bcast(f32Cst(builder, 1.0));
-  Value cstCephesExpP1 = bcast(f32Cst(builder, 1.0));
-  Value cstCephesExpP2 = bcast(f32Cst(builder, 0.49970514590562437052f));
-  Value cstCephesExpP3 = bcast(f32Cst(builder, 0.16873890085469545053f));
-  Value cstCephesExpP4 = bcast(f32Cst(builder, 0.03668965196652099192f));
-  Value cstCephesExpP5 = bcast(f32Cst(builder, 0.01314350012789660196f));
+  // Polynomial approximation from Cephes.
+  //
+  // To compute e^x, we re-express it as
+  //
+  //   e^x = e^(a + b)
+  //       = e^(a + n log(2))
+  //       = e^a * 2^n.
+  //
+  // We choose n = round(x / log(2)), restricting the value of `a` to
+  // (-log(2)/2, log(2)/2).  We then use a polynomial to compute e^a. The
+  // relative error between our approximation and the true value of e^a is less
+  // than 2^-22.5 for all values of `a` within this range.
+
+  // Restrict input to a small range, including some values that evaluate to
+  // +/- inf.  Note that for our lower bound, we choose log(2^-126) instead of
+  // log(F32_EPSILON). We do so because this routine always flushes denormal
+  // floating points to 0. Therefore, we only need to worry about exponentiating
+  // up to the smallest representable non-denormal floating point, which is
+  // 2^-126.
+
+  // Constants.
+  Value cst_half = bcast(f32Cst(builder, 0.5f));
+  Value cst_one = bcast(f32Cst(builder, 1.0f));
+
+  // 1/log(2)
+  Value cst_log2ef = bcast(f32Cst(builder, 1.44269504088896341f));
+
+  Value cst_exp_c1 = bcast(f32Cst(builder, -0.693359375f));
+  Value cst_exp_c2 = bcast(f32Cst(builder, 2.12194440e-4f));
+  Value cst_exp_p0 = bcast(f32Cst(builder, 1.9875691500E-4f));
+  Value cst_exp_p1 = bcast(f32Cst(builder, 1.3981999507E-3f));
+  Value cst_exp_p2 = bcast(f32Cst(builder, 8.3334519073E-3f));
+  Value cst_exp_p3 = bcast(f32Cst(builder, 4.1665795894E-2f));
+  Value cst_exp_p4 = bcast(f32Cst(builder, 1.6666665459E-1f));
+  Value cst_exp_p5 = bcast(f32Cst(builder, 5.0000001201E-1f));
 
+  // Our computations below aren't particularly sensitive to the exact choices
+  // here, so we choose values a bit larger/smaller than
+  //
+  //   log(F32_MAX) = 88.723...
+  //   log(2^-126) = -87.337...
   Value x = op.getOperand();
+  x = clampWithNormals(builder, shape, x, -87.8f, 88.8f);
+  Value n = floor(fmla(x, cst_log2ef, cst_half));
 
-  Value isNan = builder.create<arith::CmpFOp>(arith::CmpFPredicate::UNO, x, x);
-
-  // Reduced y = x - floor(x / ln(2)) * ln(2) = x - k * ln(2)
-  Value xL2Inv = mul(x, cstLog2E);
-  Value kF32 = floor(xL2Inv);
-  Value kLn2 = mul(kF32, cstLn2);
-  Value y = sub(x, kLn2);
-
-  // Use Estrin's evaluation scheme with 3 independent parts:
-  // P(y)^y : (c0 + c1 y) + (c2 + c3 y) y^2 + (c4 + c5 y) y^4
-  Value y2 = mul(y, y);
-  Value y4 = mul(y2, y2);
-
-  Value q0 = fmla(cstCephesExpP1, y, cstCephesExpP0);
-  Value q1 = fmla(cstCephesExpP3, y, cstCephesExpP2);
-  Value q2 = fmla(cstCephesExpP5, y, cstCephesExpP4);
-  Value expY = fmla(q1, y2, q0);
-  expY = fmla(q2, y4, expY);
-
-  auto i32Vec = broadcast(builder.getI32Type(), shape);
+  // When we eventually do the multiplication in e^a * 2^n, we need to handle
+  // the case when n > 127, the max fp32 exponent (so 2^n == inf) but e^a < 1
+  // (so e^a * 2^n != inf).  There's a similar problem for n < -126, the
+  // smallest fp32 exponent.
+  //
+  // A straightforward solution would be to detect n out of range and split it
+  // up, doing
+  //
+  //   e^a * 2^n = e^a * 2^(n1 + n2)
+  //             = (2^n1 * e^a) * 2^n2.
+  //
+  // But it turns out this approach is quite slow, probably because it
+  // manipulates subnormal values.
+  //
+  // The approach we use instead is to clamp n to [-127, 127]. Let n' be the
+  // value of n clamped to [-127, 127]. In the case where n' = 127, `a` can grow
+  // up to as large as 88.8 - 127 * log(2) which is about 0.7703. Even though
+  // this value of `a` is outside our previously specified range, e^a will still
+  // only have a relative error of approximately 2^-16 at worse. In practice
+  // this seems to work well enough; it passes our exhaustive tests, breaking
+  // only one result, and by one ulp (we return exp(88.7228394) = max-float but
+  // we should return inf).
+  //
+  // In the case where n' = -127, the original input value of x is so small that
+  // e^x, our final answer, is less than 2^-126. Since 2^-126 is the smallest
+  // normal floating point, and since we flush denormals, we simply return 0. We
+  // do this in a branchless way by observing that our code for constructing 2^n
+  // produces 0 if n = -127.
+  //
+  // The proof that n' = -127 implies e^x < 2^-126 is as follows:
+  //
+  //    n' = -127 implies n <= -127
+  //              implies round(x / log(2)) <= -127
+  //              implies x/log(2) < -126.5
+  //              implies x < -126.5 * log(2)
+  //              implies e^x < e^(-126.5 * log(2))
+  //              implies e^x < 2^-126.5 < 2^-126
+  //
+  //    This proves that n' = -127 implies e^x < 2^-126.
+  n = clampWithNormals(builder, shape, n, -127.0f, 127.0f);
 
-  // exp2(k)
-  Value k = builder.create<arith::FPToSIOp>(i32Vec, kF32);
-  Value exp2KValue = exp2I32(builder, k);
-
-  // exp(x) = exp(y) * exp2(k)
-  expY = mul(expY, exp2KValue);
-
-  // Handle overflow, inf and underflow of exp(x). exp(x) range is [0, inf], its
-  // partitioned as the following:
-  // exp(x) = 0, x <= -inf
-  // exp(x) = underflow (min_float), x <= -88
-  // exp(x) = inf (min_float), x >= 88
-  // Note: |k| = 127 is the value where the 8-bits exponent saturates.
-  Value zerof32Const = bcast(f32Cst(builder, 0));
-  auto constPosInfinity =
-      bcast(f32Cst(builder, std::numeric_limits<float>::infinity()));
-  auto constNegIfinity =
-      bcast(f32Cst(builder, -std::numeric_limits<float>::infinity()));
-  auto underflow = bcast(f32Cst(builder, std::numeric_limits<float>::min()));
-
-  Value kMaxConst = bcast(i32Cst(builder, 127));
-  Value kMaxNegConst = bcast(i32Cst(builder, -127));
-  Value rightBound =
-      builder.create<arith::CmpIOp>(arith::CmpIPredicate::sle, k, kMaxConst);
-  Value leftBound =
-      builder.create<arith::CmpIOp>(arith::CmpIPredicate::sge, k, kMaxNegConst);
-
-  Value isNegInfinityX = builder.create<arith::CmpFOp>(
-      arith::CmpFPredicate::OEQ, x, constNegIfinity);
-  Value isPosInfinityX = builder.create<arith::CmpFOp>(
-      arith::CmpFPredicate::OEQ, x, constPosInfinity);
-  Value isPostiveX =
-      builder.create<arith::CmpFOp>(arith::CmpFPredicate::OGT, x, zerof32Const);
-  Value isComputable = builder.create<arith::AndIOp>(rightBound, leftBound);
+  // Computes x = x - n' * log(2), the value for `a`
+  x = fmla(cst_exp_c1, n, x);
+  x = fmla(cst_exp_c2, n, x);
+
+  // Polynomial to compute z = e^a, accurate for a in (-0.5, 0.5).
+  Value z = fmla(x, cst_exp_p0, cst_exp_p1);
+  z = fmla(z, x, cst_exp_p2);
+  z = fmla(z, x, cst_exp_p3);
+  z = fmla(z, x, cst_exp_p4);
+  z = fmla(z, x, cst_exp_p5);
+  z = fmla(z, mul(x, x), x);
+  z = add(cst_one, z);
+
+  // Convert n' to an i32.  This is safe because we clamped it above.
+  auto i32_vec = broadcast(builder.getI32Type(), shape);
+  Value n_i32 = builder.create<arith::FPToSIOp>(i32_vec, n);
 
-  expY = builder.create<arith::SelectOp>(
-      isNan, x,
-      builder.create<arith::SelectOp>(
-          isNegInfinityX, zerof32Const,
-          builder.create<arith::SelectOp>(
-              isPosInfinityX, constPosInfinity,
-              builder.create<arith::SelectOp>(
-                  isComputable, expY,
-                  builder.create<arith::SelectOp>(isPostiveX, constPosInfinity,
-                                                  underflow)))));
+  // Creates the value 2^n' if -126 <= n' <= 127 and 0 if n' = -127.
+  Value pow2 = exp2I32(builder, n_i32);
 
-  rewriter.replaceOp(op, expY);
+  // Return z * 2^n' if -126 <= n' <= 127 and 0 if n = -127.
+  Value ret = mul(z, pow2);
 
-  return success();
+  rewriter.replaceOp(op, ret);
+  return mlir::success();
 }
 
+} // namespace
+
 //----------------------------------------------------------------------------//
 // ExpM1 approximation.
 //----------------------------------------------------------------------------//
 
 namespace {
 
 struct ExpM1Approximation : public OpRewritePattern<math::ExpM1Op> {
 public:

mlir/test/Dialect/Math/polynomial-approximation.mlir

 // CHECK:         }
 func.func @erf_vector(%arg0: vector<8xf32>) -> vector<8xf32> {
   %0 = math.erf %arg0 : vector<8xf32>
   return %0 : vector<8xf32>
 }
 
 // CHECK-LABEL:   func @exp_scalar(
 // CHECK-SAME:                     %[[VAL_0:.*]]: f32) -> f32 {
-// CHECK-DAG:           %[[VAL_1:.*]] = arith.constant 0.693147182 : f32
-// CHECK-DAG:           %[[VAL_2:.*]] = arith.constant 1.44269502 : f32
-// CHECK-DAG:           %[[VAL_3:.*]] = arith.constant 1.000000e+00 : f32
-// CHECK-DAG:           %[[VAL_4:.*]] = arith.constant 0.499705136 : f32
-// CHECK-DAG:           %[[VAL_5:.*]] = arith.constant 0.168738902 : f32
-// CHECK-DAG:           %[[VAL_6:.*]] = arith.constant 0.0366896503 : f32
-// CHECK-DAG:           %[[VAL_7:.*]] = arith.constant 1.314350e-02 : f32
-// CHECK-DAG:           %[[VAL_8:.*]] = arith.constant 23 : i32
-// CHECK-DAG:           %[[VAL_9:.*]] = arith.constant 0.000000e+00 : f32
-// CHECK-DAG:           %[[VAL_10:.*]] = arith.constant 0x7F800000 : f32
-// CHECK-DAG:           %[[VAL_11:.*]] = arith.constant 0xFF800000 : f32
-// CHECK-DAG:           %[[VAL_12:.*]] = arith.constant 1.17549435E-38 : f32
-// CHECK-DAG:           %[[VAL_13:.*]] = arith.constant 127 : i32
-// CHECK-DAG:           %[[VAL_14:.*]] = arith.constant -127 : i32
-// CHECK:           %[[IS_NAN:.*]] = arith.cmpf uno, %[[VAL_0]], %[[VAL_0]] : f32
-// CHECK:           %[[VAL_15:.*]] = arith.mulf %[[VAL_0]], %[[VAL_2]] : f32
-// CHECK:           %[[VAL_16:.*]] = math.floor %[[VAL_15]] : f32
-// CHECK:           %[[VAL_17:.*]] = arith.mulf %[[VAL_16]], %[[VAL_1]] : f32
-// CHECK:           %[[VAL_18:.*]] = arith.subf %[[VAL_0]], %[[VAL_17]] : f32
-// CHECK:           %[[VAL_19:.*]] = arith.mulf %[[VAL_18]], %[[VAL_18]] : f32
-// CHECK:           %[[VAL_20:.*]] = arith.mulf %[[VAL_19]], %[[VAL_19]] : f32
-// CHECK:           %[[VAL_21:.*]] = math.fma %[[VAL_3]], %[[VAL_18]], %[[VAL_3]] : f32
-// CHECK:           %[[VAL_22:.*]] = math.fma %[[VAL_5]], %[[VAL_18]], %[[VAL_4]] : f32
-// CHECK:           %[[VAL_23:.*]] = math.fma %[[VAL_7]], %[[VAL_18]], %[[VAL_6]] : f32
-// CHECK:           %[[VAL_24:.*]] = math.fma %[[VAL_22]], %[[VAL_19]], %[[VAL_21]] : f32
-// CHECK:           %[[VAL_25:.*]] = math.fma %[[VAL_23]], %[[VAL_20]], %[[VAL_24]] : f32
-// CHECK:           %[[VAL_26:.*]] = arith.fptosi %[[VAL_16]] : f32 to i32
-// CHECK:           %[[VAL_27:.*]] = arith.addi %[[VAL_26]], %[[VAL_13]] : i32
-// CHECK:           %[[VAL_28:.*]] = arith.shli %[[VAL_27]], %[[VAL_8]] : i32
-// CHECK:           %[[VAL_29:.*]] = arith.bitcast %[[VAL_28]] : i32 to f32
-// CHECK:           %[[VAL_30:.*]] = arith.mulf %[[VAL_25]], %[[VAL_29]] : f32
-// CHECK:           %[[VAL_31:.*]] = arith.cmpi sle, %[[VAL_26]], %[[VAL_13]] : i32
-// CHECK:           %[[VAL_32:.*]] = arith.cmpi sge, %[[VAL_26]], %[[VAL_14]] : i32
-// CHECK:           %[[VAL_33:.*]] = arith.cmpf oeq, %[[VAL_0]], %[[VAL_11]] : f32
-// CHECK:           %[[VAL_34:.*]] = arith.cmpf oeq, %[[VAL_0]], %[[VAL_10]] : f32
-// CHECK:           %[[VAL_35:.*]] = arith.cmpf ogt, %[[VAL_0]], %[[VAL_9]] : f32
-// CHECK:           %[[VAL_36:.*]] = arith.andi %[[VAL_31]], %[[VAL_32]] : i1
-// CHECK:           %[[VAL_37:.*]] = arith.select %[[VAL_35]], %[[VAL_10]], %[[VAL_12]] : f32
-// CHECK:           %[[VAL_38:.*]] = arith.select %[[VAL_36]], %[[VAL_30]], %[[VAL_37]] : f32
-// CHECK:           %[[VAL_39:.*]] = arith.select %[[VAL_34]], %[[VAL_10]], %[[VAL_38]] : f32
-// CHECK:           %[[VAL_40:.*]] = arith.select %[[VAL_33]], %[[VAL_9]], %[[VAL_39]] : f32
-// CHECK:           %[[VAL_41:.*]] = arith.select %[[IS_NAN]], %[[VAL_0]], %[[VAL_40]] : f32
+// CHECK-DAG:       %[[VAL_1:.*]] = arith.constant 5.000000e-01 : f32
+// CHECK-DAG:       %[[VAL_2:.*]] = arith.constant 1.000000e+00 : f32
+// CHECK-DAG:       %[[VAL_3:.*]] = arith.constant 1.44269502 : f32
+// CHECK-DAG:       %[[VAL_4:.*]] = arith.constant -0.693359375 : f32
+// CHECK-DAG:       %[[VAL_5:.*]] = arith.constant 2.12194442E-4 : f32
+// CHECK-DAG:       %[[VAL_6:.*]] = arith.constant 1.98756912E-4 : f32
+// CHECK-DAG:       %[[VAL_7:.*]] = arith.constant 0.00139819994 : f32
+// CHECK-DAG:       %[[VAL_8:.*]] = arith.constant 0.00833345205 : f32
+// CHECK-DAG:       %[[VAL_9:.*]] = arith.constant 0.0416657962 : f32
+// CHECK-DAG:       %[[VAL_10:.*]] = arith.constant 0.166666657 : f32
+// CHECK-DAG:       %[[VAL_11:.*]] = arith.constant -8.780000e+01 : f32
+// CHECK-DAG:       %[[VAL_12:.*]] = arith.constant 8.880000e+01 : f32
+// CHECK-DAG:       %[[VAL_13:.*]] = arith.constant -1.270000e+02 : f32
+// CHECK-DAG:       %[[VAL_14:.*]] = arith.constant 1.270000e+02 : f32
+// CHECK-DAG:       %[[VAL_15:.*]] = arith.constant 23 : i32
+// CHECK-DAG:       %[[VAL_16:.*]] = arith.constant 127 : i32
+// CHECK-DAG:       %[[VAL_17:.*]] = arith.cmpf uge, %[[VAL_0]], %[[VAL_11]] : f32
+// CHECK-DAG:       %[[VAL_18:.*]] = arith.select %[[VAL_17]], %[[VAL_0]], %[[VAL_11]] : f32
+// CHECK-DAG:       %[[VAL_19:.*]] = arith.cmpf ule, %[[VAL_18]], %[[VAL_12]] : f32
+// CHECK-DAG:       %[[VAL_20:.*]] = arith.select %[[VAL_19]], %[[VAL_18]], %[[VAL_12]] : f32
+// CHECK-DAG:       %[[VAL_21:.*]] = math.fma %[[VAL_20]], %[[VAL_3]], %[[VAL_1]] : f32
+// CHECK-DAG:       %[[VAL_22:.*]] = math.floor %[[VAL_21]] : f32
+// CHECK-DAG:       %[[VAL_23:.*]] = arith.cmpf uge, %[[VAL_22]], %[[VAL_13]] : f32
+// CHECK-DAG:       %[[VAL_24:.*]] = arith.select %[[VAL_23]], %[[VAL_22]], %[[VAL_13]] : f32
+// CHECK-DAG:       %[[VAL_25:.*]] = arith.cmpf ule, %[[VAL_24]], %[[VAL_14]] : f32
+// CHECK-DAG:       %[[VAL_26:.*]] = arith.select %[[VAL_25]], %[[VAL_24]], %[[VAL_14]] : f32
+// CHECK-DAG:       %[[VAL_27:.*]] = math.fma %[[VAL_4]], %[[VAL_26]], %[[VAL_20]] : f32
+// CHECK-DAG:       %[[VAL_28:.*]] = math.fma %[[VAL_5]], %[[VAL_26]], %[[VAL_27]] : f32
+// CHECK-DAG:       %[[VAL_29:.*]] = math.fma %[[VAL_28]], %[[VAL_6]], %[[VAL_7]] : f32
+// CHECK-DAG:       %[[VAL_30:.*]] = math.fma %[[VAL_29]], %[[VAL_28]], %[[VAL_8]] : f32
+// CHECK-DAG:       %[[VAL_31:.*]] = math.fma %[[VAL_30]], %[[VAL_28]], %[[VAL_9]] : f32
+// CHECK-DAG:       %[[VAL_32:.*]] = math.fma %[[VAL_31]], %[[VAL_28]], %[[VAL_10]] : f32
+// CHECK-DAG:       %[[VAL_33:.*]] = math.fma %[[VAL_32]], %[[VAL_28]], %[[VAL_1]] : f32
+// CHECK-DAG:       %[[VAL_34:.*]] = arith.mulf %[[VAL_28]], %[[VAL_28]] : f32
+// CHECK-DAG:       %[[VAL_35:.*]] = math.fma %[[VAL_33]], %[[VAL_34]], %[[VAL_28]] : f32
+// CHECK-DAG:       %[[VAL_36:.*]] = arith.addf %[[VAL_35]], %[[VAL_2]] : f32
+// CHECK-DAG:       %[[VAL_37:.*]] = arith.fptosi %[[VAL_26]] : f32 to i32
+// CHECK-DAG:       %[[VAL_38:.*]] = arith.addi %[[VAL_37]], %[[VAL_16]] : i32
+// CHECK-DAG:       %[[VAL_39:.*]] = arith.shli %[[VAL_38]], %[[VAL_15]] : i32
+// CHECK-DAG:       %[[VAL_40:.*]] = arith.bitcast %[[VAL_39]] : i32 to f32
+// CHECK-DAG:       %[[VAL_41:.*]] = arith.mulf %[[VAL_36]], %[[VAL_40]] : f32
 // CHECK:           return %[[VAL_41]] : f32
 func.func @exp_scalar(%arg0: f32) -> f32 {
   %0 = math.exp %arg0 : f32
   return %0 : f32
 }
 
 // CHECK-LABEL:   func @exp_vector(
 // CHECK-SAME:                     %[[VAL_0:.*]]: vector<8xf32>) -> vector<8xf32> {
-// CHECK:           %[[VAL_1:.*]] = arith.constant dense<0.693147182> : vector<8xf32>
-// CHECK-NOT:       exp
-// CHECK-COUNT-4:   select
-// CHECK:           %[[VAL_40:.*]] = arith.select
-// CHECK:           return %[[VAL_40]] : vector<8xf32>
+// CHECK-NOT:   math.exp
 func.func @exp_vector(%arg0: vector<8xf32>) -> vector<8xf32> {
   %0 = math.exp %arg0 : vector<8xf32>
   return %0 : vector<8xf32>
 }
 
 // CHECK-LABEL:   func @expm1_scalar(
 // CHECK-SAME:                       %[[X:.*]]: f32) -> f32 {
-// CHECK-DAG:           %[[CST_MINUSONE:.*]] = arith.constant -1.000000e+00 : f32
-// CHECK-DAG:           %[[CST_LOG2E:.*]] = arith.constant 1.44269502 : f32
-// CHECK-DAG:           %[[CST_ONE:.*]] = arith.constant 1.000000e+00 : f32
-// CHECK:           %[[BEGIN_EXP_X:.*]] = arith.mulf %[[X]], %[[CST_LOG2E]] : f32
-// CHECK-NOT:       exp
-// CHECK-COUNT-4:   select
-// CHECK:           %[[EXP_X:.*]] = arith.select
-// CHECK:           %[[IS_ONE_OR_NAN:.*]] = arith.cmpf ueq, %[[EXP_X]], %[[CST_ONE]] : f32
-// CHECK:           %[[VAL_59:.*]] = arith.subf %[[EXP_X]], %[[CST_ONE]] : f32
-// CHECK:           %[[VAL_60:.*]] = arith.cmpf oeq, %[[VAL_59]], %[[CST_MINUSONE]] : f32
-// CHECK-NOT:       log
-// CHECK-COUNT-5:   select
-// CHECK:           %[[LOG_U:.*]] = arith.select
-// CHECK:           %[[VAL_104:.*]] = arith.cmpf oeq, %[[LOG_U]], %[[EXP_X]] : f32
-// CHECK:           %[[VAL_105:.*]] = arith.divf %[[X]], %[[LOG_U]] : f32
-// CHECK:           %[[VAL_106:.*]] = arith.mulf %[[VAL_59]], %[[VAL_105]] : f32
-// CHECK:           %[[VAL_107:.*]] = arith.select %[[VAL_104]], %[[EXP_X]], %[[VAL_106]] : f32
-// CHECK:           %[[VAL_108:.*]] = arith.select %[[VAL_60]], %[[CST_MINUSONE]], %[[VAL_107]] : f32
-// CHECK:           %[[VAL_109:.*]] = arith.select %[[IS_ONE_OR_NAN]], %[[X]], %[[VAL_108]] : f32
-// CHECK:           return %[[VAL_109]] : f32
+// CHECK-DAG:       %[[VAL_1:.*]] = arith.constant 1.000000e+00 : f32
+// CHECK-DAG:       %[[VAL_2:.*]] = arith.constant -1.000000e+00 : f32
+// CHECK-DAG:       %[[VAL_3:.*]] = arith.constant 5.000000e-01 : f32
+// CHECK-DAG:       %[[VAL_4:.*]] = arith.constant 1.44269502 : f32
+// CHECK-DAG:       %[[VAL_5:.*]] = arith.constant -0.693359375 : f32
+// CHECK-DAG:       %[[VAL_6:.*]] = arith.constant 2.12194442E-4 : f32
+// CHECK-DAG:       %[[VAL_7:.*]] = arith.constant 1.98756912E-4 : f32
+// CHECK-DAG:       %[[VAL_8:.*]] = arith.constant 0.00139819994 : f32
+// CHECK-DAG:       %[[VAL_9:.*]] = arith.constant 0.00833345205 : f32
+// CHECK-DAG:       %[[VAL_10:.*]] = arith.constant 0.0416657962 : f32
+// CHECK-DAG:       %[[VAL_11:.*]] = arith.constant 0.166666657 : f32
+// CHECK-DAG:       %[[VAL_12:.*]] = arith.constant -8.780000e+01 : f32
+// CHECK-DAG:       %[[VAL_13:.*]] = arith.constant 8.880000e+01 : f32
+// CHECK-DAG:       %[[VAL_14:.*]] = arith.constant -1.270000e+02 : f32
+// CHECK-DAG:       %[[VAL_15:.*]] = arith.constant 1.270000e+02 : f32
+// CHECK-DAG:       %[[VAL_16:.*]] = arith.constant 23 : i32
+// CHECK-DAG:       %[[VAL_17:.*]] = arith.constant 127 : i32
+// CHECK-DAG:       %[[VAL_18:.*]] = arith.constant 0.000000e+00 : f32
+// CHECK-DAG:       %[[VAL_19:.*]] = arith.constant -5.000000e-01 : f32
+// CHECK-DAG:       %[[VAL_20:.*]] = arith.constant 1.17549435E-38 : f32
+// CHECK-DAG:       %[[VAL_21:.*]] = arith.constant 0xFF800000 : f32
+// CHECK-DAG:       %[[VAL_22:.*]] = arith.constant 0x7F800000 : f32
+// CHECK-DAG:       %[[VAL_23:.*]] = arith.constant 0x7FC00000 : f32
+// CHECK-DAG:       %[[VAL_24:.*]] = arith.constant 0.707106769 : f32
+// CHECK-DAG:       %[[VAL_25:.*]] = arith.constant 0.0703768358 : f32
+// CHECK-DAG:       %[[VAL_26:.*]] = arith.constant -0.115146101 : f32
+// CHECK-DAG:       %[[VAL_27:.*]] = arith.constant 0.116769984 : f32
+// CHECK-DAG:       %[[VAL_28:.*]] = arith.constant -0.12420141 : f32
+// CHECK-DAG:       %[[VAL_29:.*]] = arith.constant 0.142493233 : f32
+// CHECK-DAG:       %[[VAL_30:.*]] = arith.constant -0.166680574 : f32
+// CHECK-DAG:       %[[VAL_31:.*]] = arith.constant 0.200007141 : f32
+// CHECK-DAG:       %[[VAL_32:.*]] = arith.constant -0.24999994 : f32
+// CHECK-DAG:       %[[VAL_33:.*]] = arith.constant 0.333333313 : f32
+// CHECK-DAG:       %[[VAL_34:.*]] = arith.constant 1.260000e+02 : f32
+// CHECK-DAG:       %[[VAL_35:.*]] = arith.constant -2139095041 : i32
+// CHECK-DAG:       %[[VAL_36:.*]] = arith.constant 1056964608 : i32
+// CHECK-DAG:       %[[VAL_37:.*]] = arith.constant 0.693147182 : f32
+// CHECK-DAG:       %[[VAL_38:.*]] = arith.cmpf uge, %[[X]], %[[VAL_12]] : f32
+// CHECK-DAG:       %[[VAL_39:.*]] = arith.select %[[VAL_38]], %[[X]], %[[VAL_12]] : f32
+// CHECK-DAG:       %[[VAL_40:.*]] = arith.cmpf ule, %[[VAL_39]], %[[VAL_13]] : f32
+// CHECK-DAG:       %[[VAL_41:.*]] = arith.select %[[VAL_40]], %[[VAL_39]], %[[VAL_13]] : f32
+// CHECK-DAG:       %[[VAL_42:.*]] = math.fma %[[VAL_41]], %[[VAL_4]], %[[VAL_3]] : f32
+// CHECK-DAG:       %[[VAL_43:.*]] = math.floor %[[VAL_42]] : f32
+// CHECK-DAG:       %[[VAL_44:.*]] = arith.cmpf uge, %[[VAL_43]], %[[VAL_14]] : f32
+// CHECK-DAG:       %[[VAL_45:.*]] = arith.select %[[VAL_44]], %[[VAL_43]], %[[VAL_14]] : f32
+// CHECK-DAG:       %[[VAL_46:.*]] = arith.cmpf ule, %[[VAL_45]], %[[VAL_15]] : f32
+// CHECK-DAG:       %[[VAL_47:.*]] = arith.select %[[VAL_46]], %[[VAL_45]], %[[VAL_15]] : f32
+// CHECK-DAG:       %[[VAL_48:.*]] = math.fma %[[VAL_5]], %[[VAL_47]], %[[VAL_41]] : f32
+// CHECK-DAG:       %[[VAL_49:.*]] = math.fma %[[VAL_6]], %[[VAL_47]], %[[VAL_48]] : f32
+// CHECK-DAG:       %[[VAL_50:.*]] = math.fma %[[VAL_49]], %[[VAL_7]], %[[VAL_8]] : f32
+// CHECK-DAG:       %[[VAL_51:.*]] = math.fma %[[VAL_50]], %[[VAL_49]], %[[VAL_9]] : f32
+// CHECK-DAG:       %[[VAL_52:.*]] = math.fma %[[VAL_51]], %[[VAL_49]], %[[VAL_10]] : f32
+// CHECK-DAG:       %[[VAL_53:.*]] = math.fma %[[VAL_52]], %[[VAL_49]], %[[VAL_11]] : f32
+// CHECK-DAG:       %[[VAL_54:.*]] = math.fma %[[VAL_53]], %[[VAL_49]], %[[VAL_3]] : f32
+// CHECK-DAG:       %[[VAL_55:.*]] = arith.mulf %[[VAL_49]], %[[VAL_49]] : f32
+// CHECK-DAG:       %[[VAL_56:.*]] = math.fma %[[VAL_54]], %[[VAL_55]], %[[VAL_49]] : f32
+// CHECK-DAG:       %[[VAL_57:.*]] = arith.addf %[[VAL_56]], %[[VAL_1]] : f32
+// CHECK-DAG:       %[[VAL_58:.*]] = arith.fptosi %[[VAL_47]] : f32 to i32
+// CHECK-DAG:       %[[VAL_59:.*]] = arith.addi %[[VAL_58]], %[[VAL_17]] : i32
+// CHECK-DAG:       %[[VAL_60:.*]] = arith.shli %[[VAL_59]], %[[VAL_16]] : i32
+// CHECK-DAG:       %[[VAL_61:.*]] = arith.bitcast %[[VAL_60]] : i32 to f32
+// CHECK-DAG:       %[[VAL_62:.*]] = arith.mulf %[[VAL_57]], %[[VAL_61]] : f32
+// CHECK-DAG:       %[[VAL_63:.*]] = arith.cmpf ueq, %[[VAL_62]], %[[VAL_1]] : f32
+// CHECK-DAG:       %[[VAL_64:.*]] = arith.subf %[[VAL_62]], %[[VAL_1]] : f32
+// CHECK-DAG:       %[[VAL_65:.*]] = arith.cmpf oeq, %[[VAL_64]], %[[VAL_2]] : f32
+// CHECK-DAG:       %[[VAL_66:.*]] = arith.cmpf ugt, %[[VAL_62]], %[[VAL_20]] : f32
+// CHECK-DAG:       %[[VAL_67:.*]] = arith.select %[[VAL_66]], %[[VAL_62]], %[[VAL_20]] : f32
+// CHECK-DAG:       %[[VAL_68:.*]] = arith.bitcast %[[VAL_67]] : f32 to i32
+// CHECK-DAG:       %[[VAL_69:.*]] = arith.andi %[[VAL_68]], %[[VAL_35]] : i32
+// CHECK-DAG:       %[[VAL_70:.*]] = arith.ori %[[VAL_69]], %[[VAL_36]] : i32
+// CHECK-DAG:       %[[VAL_71:.*]] = arith.bitcast %[[VAL_70]] : i32 to f32
+// CHECK-DAG:       %[[VAL_72:.*]] = arith.bitcast %[[VAL_67]] : f32 to i32
+// CHECK-DAG:       %[[VAL_73:.*]] = arith.shrui %[[VAL_72]], %[[VAL_16]] : i32
+// CHECK-DAG:       %[[VAL_74:.*]] = arith.sitofp %[[VAL_73]] : i32 to f32
+// CHECK-DAG:       %[[VAL_75:.*]] = arith.subf %[[VAL_74]], %[[VAL_34]] : f32
+// CHECK-DAG:       %[[VAL_76:.*]] = arith.cmpf olt, %[[VAL_71]], %[[VAL_24]] : f32
+// CHECK-DAG:       %[[VAL_77:.*]] = arith.select %[[VAL_76]], %[[VAL_71]], %[[VAL_18]] : f32
+// CHECK-DAG:       %[[VAL_78:.*]] = arith.subf %[[VAL_71]], %[[VAL_1]] : f32
+// CHECK-DAG:       %[[VAL_79:.*]] = arith.select %[[VAL_76]], %[[VAL_1]], %[[VAL_18]] : f32
+// CHECK-DAG:       %[[VAL_80:.*]] = arith.subf %[[VAL_75]], %[[VAL_79]] : f32
+// CHECK-DAG:       %[[VAL_81:.*]] = arith.addf %[[VAL_78]], %[[VAL_77]] : f32
+// CHECK-DAG:       %[[VAL_82:.*]] = arith.mulf %[[VAL_81]], %[[VAL_81]] : f32
+// CHECK-DAG:       %[[VAL_83:.*]] = arith.mulf %[[VAL_82]], %[[VAL_81]] : f32
+// CHECK-DAG:       %[[VAL_84:.*]] = math.fma %[[VAL_25]], %[[VAL_81]], %[[VAL_26]] : f32
+// CHECK-DAG:       %[[VAL_85:.*]] = math.fma %[[VAL_28]], %[[VAL_81]], %[[VAL_29]] : f32
+// CHECK-DAG:       %[[VAL_86:.*]] = math.fma %[[VAL_31]], %[[VAL_81]], %[[VAL_32]] : f32
+// CHECK-DAG:       %[[VAL_87:.*]] = math.fma %[[VAL_84]], %[[VAL_81]], %[[VAL_27]] : f32
+// CHECK-DAG:       %[[VAL_88:.*]] = math.fma %[[VAL_85]], %[[VAL_81]], %[[VAL_30]] : f32
+// CHECK-DAG:       %[[VAL_89:.*]] = math.fma %[[VAL_86]], %[[VAL_81]], %[[VAL_33]] : f32
+// CHECK-DAG:       %[[VAL_90:.*]] = math.fma %[[VAL_87]], %[[VAL_83]], %[[VAL_88]] : f32
+// CHECK-DAG:       %[[VAL_91:.*]] = math.fma %[[VAL_90]], %[[VAL_83]], %[[VAL_89]] : f32
+// CHECK-DAG:       %[[VAL_92:.*]] = arith.mulf %[[VAL_91]], %[[VAL_83]] : f32
+// CHECK-DAG:       %[[VAL_93:.*]] = math.fma %[[VAL_19]], %[[VAL_82]], %[[VAL_92]] : f32
+// CHECK-DAG:       %[[VAL_94:.*]] = arith.addf %[[VAL_81]], %[[VAL_93]] : f32
+// CHECK-DAG:       %[[VAL_95:.*]] = math.fma %[[VAL_80]], %[[VAL_37]], %[[VAL_94]] : f32
+// CHECK-DAG:       %[[VAL_96:.*]] = arith.cmpf ult, %[[VAL_62]], %[[VAL_18]] : f32
+// CHECK-DAG:       %[[VAL_97:.*]] = arith.cmpf oeq, %[[VAL_62]], %[[VAL_18]] : f32
+// CHECK-DAG:       %[[VAL_98:.*]] = arith.cmpf oeq, %[[VAL_62]], %[[VAL_22]] : f32
+// CHECK-DAG:       %[[VAL_99:.*]] = arith.select %[[VAL_98]], %[[VAL_22]], %[[VAL_95]] : f32
+// CHECK-DAG:       %[[VAL_100:.*]] = arith.select %[[VAL_96]], %[[VAL_23]], %[[VAL_99]] : f32
+// CHECK-DAG:       %[[VAL_101:.*]] = arith.select %[[VAL_97]], %[[VAL_21]], %[[VAL_100]] : f32
+// CHECK-DAG:       %[[VAL_102:.*]] = arith.cmpf oeq, %[[VAL_101]], %[[VAL_62]] : f32
+// CHECK-DAG:       %[[VAL_103:.*]] = arith.divf %[[X]], %[[VAL_101]] : f32
+// CHECK-DAG:       %[[VAL_104:.*]] = arith.mulf %[[VAL_64]], %[[VAL_103]] : f32
+// CHECK-DAG:       %[[VAL_105:.*]] = arith.select %[[VAL_102]], %[[VAL_62]], %[[VAL_104]] : f32
+// CHECK-DAG:       %[[VAL_106:.*]] = arith.select %[[VAL_65]], %[[VAL_2]], %[[VAL_105]] : f32
+// CHECK-DAG:       %[[VAL_107:.*]] = arith.select %[[VAL_63]], %[[X]], %[[VAL_106]] : f32
+// CHECK-DAG:       return %[[VAL_107]] : f32
 // CHECK:         }
 func.func @expm1_scalar(%arg0: f32) -> f32 {
   %0 = math.expm1 %arg0 : f32
   return %0 : f32
 }
 
 // CHECK-LABEL:   func @expm1_vector(
 // CHECK-SAME:                       %[[VAL_0:.*]]: vector<8x8xf32>) -> vector<8x8xf32> {
-// CHECK:           %[[VAL_1:.*]] = arith.constant dense<-1.000000e+00> : vector<8x8xf32>
 // CHECK-NOT:       exp
-// CHECK-COUNT-5:   select
 // CHECK-NOT:       log
-// CHECK-COUNT-5:   select
 // CHECK-NOT:       expm1
-// CHECK-COUNT-3:   select
-// CHECK:           %[[VAL_115:.*]] = arith.select
-// CHECK:           return %[[VAL_115]] : vector<8x8xf32>
-// CHECK:         }
 func.func @expm1_vector(%arg0: vector<8x8xf32>) -> vector<8x8xf32> {
   %0 = math.expm1 %arg0 : vector<8x8xf32>
   return %0 : vector<8x8xf32>
 }
 
 // CHECK-LABEL:   func @log_scalar(
 // CHECK-SAME:                             %[[X:.*]]: f32) -> f32 {
 // CHECK:           %[[VAL_1:.*]] = arith.constant 0.000000e+00 : f32

mlir/test/mlir-cpu-runner/math-polynomial-approx.mlir

func.func @exp_4xf32(%a : vector<4xf32>) {
   return
 }
 
 func.func @exp() {
   // CHECK: 2.71828
   %f0 = arith.constant 1.0 : f32
   call @exp_f32(%f0) : (f32) -> ()
 
-  // CHECK: 0.778802, 2.117, 2.71828, 3.85742
+  // CHECK: 0.778801, 2.117, 2.71828, 3.85743
   %v1 = arith.constant dense<[-0.25, 0.75, 1.0, 1.35]> : vector<4xf32>
   call @exp_4xf32(%v1) : (vector<4xf32>) -> ()
 
   // CHECK: 1
   %zero = arith.constant 0.0 : f32
   call @exp_f32(%zero) : (f32) -> ()
 
-  // CHECK: 1.17549e-38, 1.38879e-11, 7.20049e+10, inf
+  // CHECK: 0, 1.38879e-11, 7.20049e+10, inf
   %special_vec = arith.constant dense<[-89.0, -25.0, 25.0, 89.0]> : vector<4xf32>
   call @exp_4xf32(%special_vec) : (vector<4xf32>) -> ()
 
   // CHECK: inf
   %inf = arith.constant 0x7f800000 : f32
   call @exp_f32(%inf) : (f32) -> ()
 
   // CHECK: 0
func.func @expm1_8xf32(%a : vector<8xf32>) {
   return
 }
 
 func.func @expm1() {
   // CHECK: 1e-10
   %f0 = arith.constant 1.0e-10 : f32
   call @expm1_f32(%f0) : (f32) -> ()
 
-  // CHECK: -0.00995016, 0.0100502, 0.648721, 6.38905
+  // CHECK: -0.00995017, 0.0100502, 0.648721, 6.38906
   %v1 = arith.constant dense<[-0.01, 0.01, 0.5, 2.0]> : vector<4xf32>
   call @expm1_4xf32(%v1) : (vector<4xf32>) -> ()
 
   // CHECK: -0.181269, 0, 0.221403, 0.491825, 0.822119, 1.22554, 1.71828, 2.32012
   %v2 = arith.constant dense<[-0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2]> : vector<8xf32>
   call @expm1_8xf32(%v2) : (vector<8xf32>) -> ()
 
   // CHECK: -1
func.func @main() {
   call @cos(): () -> ()
   call @atan() : () -> ()
   call @atan2() : () -> ()
   call @cbrt() : () -> ()
   call @floorf() : () -> ()
   call @ceilf() : () -> ()
   return
 }
-
-