diff --git a/flang/include/flang/Optimizer/Support/InitFIR.h b/flang/include/flang/Optimizer/Support/InitFIR.h
--- a/flang/include/flang/Optimizer/Support/InitFIR.h
+++ b/flang/include/flang/Optimizer/Support/InitFIR.h
@@ -17,6 +17,7 @@
 #include "flang/Optimizer/HLFIR/HLFIRDialect.h"
 #include "mlir/Conversion/Passes.h"
 #include "mlir/Dialect/Affine/Passes.h"
+#include "mlir/Dialect/Complex/IR/Complex.h"
 #include "mlir/InitAllDialects.h"
 #include "mlir/Pass/Pass.h"
 #include "mlir/Pass/PassRegistry.h"
@@ -30,7 +31,8 @@
       mlir::acc::OpenACCDialect, mlir::omp::OpenMPDialect,                     \
       mlir::scf::SCFDialect, mlir::arith::ArithDialect,                        \
       mlir::cf::ControlFlowDialect, mlir::func::FuncDialect,                   \
-      mlir::vector::VectorDialect, mlir::math::MathDialect
+      mlir::vector::VectorDialect, mlir::math::MathDialect,                    \
+      mlir::complex::ComplexDialect
 
 // The definitive list of dialects used by flang.
 #define FLANG_DIALECT_LIST                                                     \
diff --git a/flang/lib/Lower/IntrinsicCall.cpp b/flang/lib/Lower/IntrinsicCall.cpp
--- a/flang/lib/Lower/IntrinsicCall.cpp
+++ b/flang/lib/Lower/IntrinsicCall.cpp
@@ -35,6 +35,7 @@
 #include "flang/Optimizer/Dialect/FIROpsSupport.h"
 #include "flang/Optimizer/Support/FatalError.h"
 #include "flang/Runtime/entry-names.h"
+#include "mlir/Dialect/Complex/IR/Complex.h"
 #include "mlir/Dialect/LLVMIR/LLVMDialect.h"
 #include "mlir/Dialect/Math/IR/Math.h"
 #include "llvm/Support/CommandLine.h"
@@ -1072,6 +1073,12 @@
         clEnumValN(preciseVersion, "precise", "use precise runtime behavior")),
     llvm::cl::init(fastVersion));
 
+static llvm::cl::opt<bool>
+    disableMlirComplex("disable-mlir-complex",
+                       llvm::cl::desc("Use libm instead of the MLIR complex "
+                                      "dialect to lower complex operations"),
+                       llvm::cl::init(false));
+
 struct RuntimeFunction {
   // llvm::StringRef comparison operator are not constexpr, so use string_view.
   using Key = std::string_view;
@@ -1308,6 +1315,49 @@
   return result;
 }
 
+template <typename T>
+static mlir::Value genComplexMathOp(fir::FirOpBuilder &builder,
+                                    mlir::Location loc,
+                                    llvm::StringRef mathLibFuncName,
+                                    mlir::FunctionType mathLibFuncType,
+                                    llvm::ArrayRef<mlir::Value> args) {
+  mlir::Value result;
+  if (disableMlirComplex ||
+      (mathRuntimeVersion == preciseVersion && !mathLibFuncName.empty())) {
+    result = genLibCall(builder, loc, mathLibFuncName, mathLibFuncType, args);
+    LLVM_DEBUG(result.dump(); llvm::dbgs() << "\n");
+    return result;
+  }
+
+  LLVM_DEBUG(llvm::dbgs() << "Generating '" << mathLibFuncName
+                          << "' operation with type ";
+             mathLibFuncType.dump(); llvm::dbgs() << "\n");
+  auto type = mathLibFuncType.getInput(0).cast<fir::ComplexType>();
+  auto kind = type.getElementType().cast<fir::RealType>().getFKind();
+  auto realTy = builder.getRealType(kind);
+  auto mComplexTy = mlir::ComplexType::get(realTy);
+
+  llvm::SmallVector<mlir::Value, 2> cargs;
+  for (const mlir::Value &arg : args) {
+    // Convert the fir.complex to a mlir::complex
+    cargs.push_back(builder.createConvert(loc, mComplexTy, arg));
+  }
+
+  // Builder expects an extra return type to be provided if different to
+  // the argument types for an operation
+  if constexpr (T::template hasTrait<
+                    mlir::OpTrait::SameOperandsAndResultType>()) {
+    result = builder.create<T>(loc, cargs);
+    result = builder.createConvert(loc, mathLibFuncType.getResult(0), result);
+  } else {
+    result = builder.create<T>(loc, realTy, cargs);
+    result = builder.createConvert(loc, mathLibFuncType.getResult(0), result);
+  }
+
+  LLVM_DEBUG(result.dump(); llvm::dbgs() << "\n");
+  return result;
+}
+
 /// Mapping between mathematical intrinsic operations and MLIR operations
 /// of some appropriate dialect (math, complex, etc.) or libm calls.
 /// TODO: support remaining Fortran math intrinsics.
@@ -1318,8 +1368,10 @@
     {"abs", "fabs", genF64F64FuncType, genMathOp<mlir::math::AbsFOp>},
     {"abs", "llvm.fabs.f128", genF128F128FuncType,
      genMathOp<mlir::math::AbsFOp>},
-    {"abs", "cabsf", genF32ComplexFuncType, genLibCall},
-    {"abs", "cabs", genF64ComplexFuncType, genLibCall},
+    {"abs", "cabsf", genF32ComplexFuncType,
+     genComplexMathOp<mlir::complex::AbsOp>},
+    {"abs", "cabs", genF64ComplexFuncType,
+     genComplexMathOp<mlir::complex::AbsOp>},
     {"acos", "acosf", genF32F32FuncType, genLibCall},
     {"acos", "acos", genF64F64FuncType, genLibCall},
     {"acos", "cacosf", genComplexComplexFuncType<4>, genLibCall},
@@ -1374,8 +1426,10 @@
     {"ceil", "ceil", genF64F64FuncType, genMathOp<mlir::math::CeilOp>},
     {"cos", "cosf", genF32F32FuncType, genMathOp<mlir::math::CosOp>},
     {"cos", "cos", genF64F64FuncType, genMathOp<mlir::math::CosOp>},
-    {"cos", "ccosf", genComplexComplexFuncType<4>, genLibCall},
-    {"cos", "ccos", genComplexComplexFuncType<8>, genLibCall},
+    {"cos", "ccosf", genComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::CosOp>},
+    {"cos", "ccos", genComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::CosOp>},
     {"cosh", "coshf", genF32F32FuncType, genLibCall},
     {"cosh", "cosh", genF64F64FuncType, genLibCall},
     {"cosh", "ccoshf", genComplexComplexFuncType<4>, genLibCall},
@@ -1386,8 +1440,10 @@
     {"erfc", "erfc", genF64F64FuncType, genLibCall},
     {"exp", "expf", genF32F32FuncType, genMathOp<mlir::math::ExpOp>},
     {"exp", "exp", genF64F64FuncType, genMathOp<mlir::math::ExpOp>},
-    {"exp", "cexpf", genComplexComplexFuncType<4>, genLibCall},
-    {"exp", "cexp", genComplexComplexFuncType<8>, genLibCall},
+    {"exp", "cexpf", genComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::ExpOp>},
+    {"exp", "cexp", genComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::ExpOp>},
     // math::FloorOp returns a real, while Fortran FLOOR returns integer.
     {"floor", "floorf", genF32F32FuncType, genMathOp<mlir::math::FloorOp>},
     {"floor", "floor", genF64F64FuncType, genMathOp<mlir::math::FloorOp>},
@@ -1397,8 +1453,10 @@
     {"hypot", "hypot", genF64F64F64FuncType, genLibCall},
     {"log", "logf", genF32F32FuncType, genMathOp<mlir::math::LogOp>},
     {"log", "log", genF64F64FuncType, genMathOp<mlir::math::LogOp>},
-    {"log", "clogf", genComplexComplexFuncType<4>, genLibCall},
-    {"log", "clog", genComplexComplexFuncType<8>, genLibCall},
+    {"log", "clogf", genComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::LogOp>},
+    {"log", "clog", genComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::LogOp>},
     {"log10", "log10f", genF32F32FuncType, genMathOp<mlir::math::Log10Op>},
     {"log10", "log10", genF64F64FuncType, genMathOp<mlir::math::Log10Op>},
     {"log_gamma", "lgammaf", genF32F32FuncType, genLibCall},
@@ -1414,8 +1472,10 @@
     {"pow", {}, genIntIntIntFuncType<64>, genMathOp<mlir::math::IPowIOp>},
     {"pow", "powf", genF32F32F32FuncType, genMathOp<mlir::math::PowFOp>},
     {"pow", "pow", genF64F64F64FuncType, genMathOp<mlir::math::PowFOp>},
-    {"pow", "cpowf", genComplexComplexComplexFuncType<4>, genLibCall},
-    {"pow", "cpow", genComplexComplexComplexFuncType<8>, genLibCall},
+    {"pow", "cpowf", genComplexComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::PowOp>},
+    {"pow", "cpow", genComplexComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::PowOp>},
     // TODO: add PowIOp in math and complex dialects.
     {"pow", "llvm.powi.f32.i32", genF32F32IntFuncType<32>, genLibCall},
     {"pow", "llvm.powi.f64.i32", genF64F64IntFuncType<32>, genLibCall},
@@ -1437,24 +1497,32 @@
      genMathOp<mlir::math::CopySignOp>},
     {"sin", "sinf", genF32F32FuncType, genMathOp<mlir::math::SinOp>},
     {"sin", "sin", genF64F64FuncType, genMathOp<mlir::math::SinOp>},
-    {"sin", "csinf", genComplexComplexFuncType<4>, genLibCall},
-    {"sin", "csin", genComplexComplexFuncType<8>, genLibCall},
+    {"sin", "csinf", genComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::SinOp>},
+    {"sin", "csin", genComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::SinOp>},
     {"sinh", "sinhf", genF32F32FuncType, genLibCall},
     {"sinh", "sinh", genF64F64FuncType, genLibCall},
     {"sinh", "csinhf", genComplexComplexFuncType<4>, genLibCall},
     {"sinh", "csinh", genComplexComplexFuncType<8>, genLibCall},
     {"sqrt", "sqrtf", genF32F32FuncType, genMathOp<mlir::math::SqrtOp>},
     {"sqrt", "sqrt", genF64F64FuncType, genMathOp<mlir::math::SqrtOp>},
-    {"sqrt", "csqrtf", genComplexComplexFuncType<4>, genLibCall},
-    {"sqrt", "csqrt", genComplexComplexFuncType<8>, genLibCall},
+    {"sqrt", "csqrtf", genComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::SqrtOp>},
+    {"sqrt", "csqrt", genComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::SqrtOp>},
     {"tan", "tanf", genF32F32FuncType, genMathOp<mlir::math::TanOp>},
     {"tan", "tan", genF64F64FuncType, genMathOp<mlir::math::TanOp>},
-    {"tan", "ctanf", genComplexComplexFuncType<4>, genLibCall},
-    {"tan", "ctan", genComplexComplexFuncType<8>, genLibCall},
+    {"tan", "ctanf", genComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::TanOp>},
+    {"tan", "ctan", genComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::TanOp>},
     {"tanh", "tanhf", genF32F32FuncType, genMathOp<mlir::math::TanhOp>},
     {"tanh", "tanh", genF64F64FuncType, genMathOp<mlir::math::TanhOp>},
-    {"tanh", "ctanhf", genComplexComplexFuncType<4>, genLibCall},
-    {"tanh", "ctanh", genComplexComplexFuncType<8>, genLibCall},
+    {"tanh", "ctanhf", genComplexComplexFuncType<4>,
+     genComplexMathOp<mlir::complex::TanhOp>},
+    {"tanh", "ctanh", genComplexComplexFuncType<8>,
+     genComplexMathOp<mlir::complex::TanhOp>},
 };
 
 // This helper class computes a "distance" between two function types.
diff --git a/flang/lib/Optimizer/CodeGen/CodeGen.cpp b/flang/lib/Optimizer/CodeGen/CodeGen.cpp
--- a/flang/lib/Optimizer/CodeGen/CodeGen.cpp
+++ b/flang/lib/Optimizer/CodeGen/CodeGen.cpp
@@ -20,6 +20,8 @@
 #include "flang/Optimizer/Support/TypeCode.h"
 #include "flang/Semantics/runtime-type-info.h"
 #include "mlir/Conversion/ArithToLLVM/ArithToLLVM.h"
+#include "mlir/Conversion/ComplexToLLVM/ComplexToLLVM.h"
+#include "mlir/Conversion/ComplexToStandard/ComplexToStandard.h"
 #include "mlir/Conversion/ControlFlowToLLVM/ControlFlowToLLVM.h"
 #include "mlir/Conversion/FuncToLLVM/ConvertFuncToLLVM.h"
 #include "mlir/Conversion/LLVMCommon/Pattern.h"
@@ -3509,6 +3511,7 @@
     // as passes here.
     mlir::OpPassManager mathConvertionPM("builtin.module");
     mathConvertionPM.addPass(mlir::createConvertMathToFuncsPass());
+    mathConvertionPM.addPass(mlir::createConvertComplexToStandardPass());
     if (mlir::failed(runPipeline(mathConvertionPM, mod)))
       return signalPassFailure();
 
@@ -3573,6 +3576,7 @@
     // when late math lowering mode is used, into llvm dialect.
     mlir::populateMathToLLVMConversionPatterns(typeConverter, pattern);
     mlir::populateMathToLibmConversionPatterns(pattern, /*benefit=*/0);
+    mlir::populateComplexToLLVMConversionPatterns(typeConverter, pattern);
     mlir::ConversionTarget target{*context};
     target.addLegalDialect<mlir::LLVM::LLVMDialect>();
     // The OpenMP dialect is legal for Operations without regions, for those
diff --git a/flang/test/Lower/Intrinsics/abs.f90 b/flang/test/Lower/Intrinsics/abs.f90
--- a/flang/test/Lower/Intrinsics/abs.f90
+++ b/flang/test/Lower/Intrinsics/abs.f90
@@ -1,5 +1,9 @@
-! RUN: bbc -emit-fir %s -o - | FileCheck %s
-! RUN: %flang_fc1 -emit-fir %s -o - | FileCheck %s
+! RUN: bbc -emit-fir %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX,CMPLX-FAST"
+! RUN: bbc -emit-fir --math-runtime=precise %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: bbc --disable-mlir-complex -emit-fir %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX,CMPLX-FAST"
+! RUN: %flang_fc1 -emit-fir -mllvm --math-runtime=precise %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm --disable-mlir-complex %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
 
 ! Test abs intrinsic for various types (int, float, complex)
 
@@ -90,25 +94,29 @@
   b = abs(a)
 end subroutine
 
-! CHECK-LABEL: func @_QPabs_testzr(
-! CHECK-SAME:  %[[VAL_0:.*]]: !fir.ref<!fir.complex<4>>{{.*}}, %[[VAL_1:.*]]: !fir.ref<f32>{{.*}}) {
+! CMPLX-LABEL: func @_QPabs_testzr(
+! CMPLX-SAME:  %[[VAL_0:.*]]: !fir.ref<!fir.complex<4>>{{.*}}, %[[VAL_1:.*]]: !fir.ref<f32>{{.*}}) {
 subroutine abs_testzr(a, b)
-! CHECK:  %[[VAL_2:.*]] = fir.load %[[VAL_0]] : !fir.ref<!fir.complex<4>>
-! CHECK:  %[[VAL_3:.*]] = fir.call @cabsf(%[[VAL_2]]) : (!fir.complex<4>) -> f32
-! CHECK:  fir.store %[[VAL_3]] to %[[VAL_1]] : !fir.ref<f32>
-! CHECK:  return
+! CMPLX:  %[[VAL_2:.*]] = fir.load %[[VAL_0]] : !fir.ref<!fir.complex<4>>
+! CMPLX-FAST: %[[VAL_3:.*]] = fir.convert %[[VAL_2]] : (!fir.complex<4>) -> complex<f32>
+! CMPLX-FAST: %[[VAL_4:.*]] = complex.abs %[[VAL_3]] : complex<f32>
+! CMPLX-PRECISE:  %[[VAL_4:.*]] = fir.call @cabsf(%[[VAL_2]]) : (!fir.complex<4>) -> f32
+! CMPLX:  fir.store %[[VAL_4]] to %[[VAL_1]] : !fir.ref<f32>
+! CMPLX:  return
   complex :: a
   real :: b
   b = abs(a)
 end subroutine abs_testzr
 
-! CHECK-LABEL: func @_QPabs_testzd(
-! CHECK-SAME:  %[[VAL_0:.*]]: !fir.ref<!fir.complex<8>>{{.*}}, %[[VAL_1:.*]]: !fir.ref<f64>{{.*}}) {
+! CMPLX-LABEL: func @_QPabs_testzd(
+! CMPLX-SAME:  %[[VAL_0:.*]]: !fir.ref<!fir.complex<8>>{{.*}}, %[[VAL_1:.*]]: !fir.ref<f64>{{.*}}) {
 subroutine abs_testzd(a, b)
-! CHECK:  %[[VAL_2:.*]] = fir.load %[[VAL_0]] : !fir.ref<!fir.complex<8>>
-! CHECK:  %[[VAL_3:.*]] = fir.call @cabs(%[[VAL_2]]) : (!fir.complex<8>) -> f64
-! CHECK:  fir.store %[[VAL_3]] to %[[VAL_1]] : !fir.ref<f64>
-! CHECK:  return
+! CMPLX:  %[[VAL_2:.*]] = fir.load %[[VAL_0]] : !fir.ref<!fir.complex<8>>
+! CMPLX-FAST: %[[VAL_3:.*]] = fir.convert %[[VAL_2]] : (!fir.complex<8>) -> complex<f64>
+! CMPLX-FAST: %[[VAL_4:.*]] = complex.abs %[[VAL_3]] : complex<f64>
+! CMPLX-PRECISE:  %[[VAL_4:.*]] = fir.call @cabs(%[[VAL_2]]) : (!fir.complex<8>) -> f64
+! CMPLX:  fir.store %[[VAL_4]] to %[[VAL_1]] : !fir.ref<f64>
+! CMPLX:  return
   complex(kind=8) :: a
   real(kind=8) :: b
   b = abs(a)
diff --git a/flang/test/Lower/Intrinsics/exp.f90 b/flang/test/Lower/Intrinsics/exp.f90
--- a/flang/test/Lower/Intrinsics/exp.f90
+++ b/flang/test/Lower/Intrinsics/exp.f90
@@ -1,5 +1,9 @@
-! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s
-! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics %s -o - | FileCheck %s
+! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX,CMPLX-FAST"
+! RUN: bbc -emit-fir --math-runtime=precise -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: bbc -emit-fir --disable-mlir-complex -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX,CMPLX-FAST"
+! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics -mllvm --math-runtime=precise %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics -mllvm --disable-mlir-complex %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
 
 ! CHECK-LABEL: exp_testr
 ! CHECK-SAME: (%[[AREF:.*]]: !fir.ref<f32> {{.*}}, %[[BREF:.*]]: !fir.ref<f32> {{.*}})
@@ -51,12 +55,18 @@
 ! CHECK: %[[RESULT64_OUTLINE:.*]] = math.exp %[[ARG64_OUTLINE]] : f64
 ! CHECK: return %[[RESULT64_OUTLINE]] : f64
 
-! CHECK-LABEL: private @fir.exp.z4.z4
-! CHECK-SAME: (%[[ARG32_OUTLINE]]: !fir.complex<4>) -> !fir.complex<4>
-! CHECK: %[[RESULT32_OUTLINE]] = fir.call @cexpf(%[[ARG32_OUTLINE]]) : (!fir.complex<4>) -> !fir.complex<4>
-! CHECK: return %[[RESULT32_OUTLINE]] : !fir.complex<4>
+! CMPLX-LABEL: private @fir.exp.z4.z4
+! CMPLX-SAME: (%[[ARG32_OUTLINE:.*]]: !fir.complex<4>) -> !fir.complex<4>
+! CMPLX-FAST: %[[C:.*]] = fir.convert %[[ARG32_OUTLINE]] : (!fir.complex<4>) -> complex<f32>
+! CMPLX-FAST: %[[E:.*]] = complex.exp %[[C]] : complex<f32>
+! CMPLX-FAST: %[[RESULT32_OUTLINE:.*]] = fir.convert %[[E]] : (complex<f32>) -> !fir.complex<4>
+! CMPLX-PRECISE: %[[RESULT32_OUTLINE:.*]] = fir.call @cexpf(%[[ARG32_OUTLINE]]) : (!fir.complex<4>) -> !fir.complex<4>
+! CMPLX: return %[[RESULT32_OUTLINE]] : !fir.complex<4>
 
-! CHECK-LABEL: private @fir.exp.z8.z8
-! CHECK-SAME: (%[[ARG64_OUTLINE]]: !fir.complex<8>) -> !fir.complex<8>
-! CHECK: %[[RESULT64_OUTLINE]] = fir.call @cexp(%[[ARG64_OUTLINE]]) : (!fir.complex<8>) -> !fir.complex<8>
-! CHECK: return %[[RESULT64_OUTLINE]] : !fir.complex<8>
+! CMPLX-LABEL: private @fir.exp.z8.z8
+! CMPLX-SAME: (%[[ARG64_OUTLINE:.*]]: !fir.complex<8>) -> !fir.complex<8>
+! CMPLX-FAST: %[[C:.*]] = fir.convert %[[ARG64_OUTLINE]] : (!fir.complex<8>) -> complex<f64>
+! CMPLX-FAST: %[[E:.*]] = complex.exp %[[C]] : complex<f64>
+! CMPLX-FAST: %[[RESULT64_OUTLINE:.*]] = fir.convert %[[E]] : (complex<f64>) -> !fir.complex<8>
+! CMPLX-PRECISE: %[[RESULT64_OUTLINE:.*]] = fir.call @cexp(%[[ARG64_OUTLINE]]) : (!fir.complex<8>) -> !fir.complex<8>
+! CMPLX: return %[[RESULT64_OUTLINE]] : !fir.complex<8>
diff --git a/flang/test/Lower/Intrinsics/log.f90 b/flang/test/Lower/Intrinsics/log.f90
--- a/flang/test/Lower/Intrinsics/log.f90
+++ b/flang/test/Lower/Intrinsics/log.f90
@@ -1,5 +1,9 @@
-! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s
-! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics %s -o - | FileCheck %s
+! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX,CMPLX-FAST"
+! RUN: bbc -emit-fir --math-runtime=precise -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: bbc -emit-fir --disable-mlir-complex -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX,CMPLX-FAST"
+! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics -mllvm --math-runtime=precise %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics -mllvm --disable-mlir-complex %s -o - | FileCheck %s --check-prefixes="CMPLX,CMPLX-PRECISE"
 
 ! CHECK-LABEL: log_testr
 ! CHECK-SAME: (%[[AREF:.*]]: !fir.ref<f32> {{.*}}, %[[BREF:.*]]: !fir.ref<f32> {{.*}})
@@ -71,15 +75,21 @@
 ! CHECK: %[[RESULT64_OUTLINE:.*]] = math.log %[[ARG64_OUTLINE]] : f64
 ! CHECK: return %[[RESULT64_OUTLINE]] : f64
 
-! CHECK-LABEL: private @fir.log.z4.z4
-! CHECK-SAME: (%[[ARG32_OUTLINE]]: !fir.complex<4>) -> !fir.complex<4>
-! CHECK: %[[RESULT32_OUTLINE]] = fir.call @clogf(%[[ARG32_OUTLINE]]) : (!fir.complex<4>) -> !fir.complex<4>
-! CHECK: return %[[RESULT32_OUTLINE]] : !fir.complex<4>
+! CMPLX-LABEL: private @fir.log.z4.z4
+! CMPLX-SAME: (%[[ARG32_OUTLINE:.*]]: !fir.complex<4>) -> !fir.complex<4>
+! CMPLX-FAST: %[[C:.*]] = fir.convert %[[ARG32_OUTLINE]] : (!fir.complex<4>) -> complex<f32>
+! CMPLX-FAST: %[[E:.*]] = complex.log %[[C]] : complex<f32>
+! CMPLX-FAST: %[[RESULT32_OUTLINE:.*]] = fir.convert %[[E]] : (complex<f32>) -> !fir.complex<4>
+! CMPLX-PRECISE: %[[RESULT32_OUTLINE:.*]] = fir.call @clogf(%[[ARG32_OUTLINE]]) : (!fir.complex<4>) -> !fir.complex<4>
+! CMPLX: return %[[RESULT32_OUTLINE]] : !fir.complex<4>
 
-! CHECK-LABEL: private @fir.log.z8.z8
-! CHECK-SAME: (%[[ARG64_OUTLINE]]: !fir.complex<8>) -> !fir.complex<8>
-! CHECK: %[[RESULT64_OUTLINE]] = fir.call @clog(%[[ARG64_OUTLINE]]) : (!fir.complex<8>) -> !fir.complex<8>
-! CHECK: return %[[RESULT64_OUTLINE]] : !fir.complex<8>
+! CMPLX-LABEL: private @fir.log.z8.z8
+! CMPLX-SAME: (%[[ARG64_OUTLINE:.*]]: !fir.complex<8>) -> !fir.complex<8>
+! CMPLX-FAST: %[[C:.*]] = fir.convert %[[ARG64_OUTLINE]] : (!fir.complex<8>) -> complex<f64>
+! CMPLX-FAST: %[[E:.*]] = complex.log %[[C]] : complex<f64>
+! CMPLX-FAST: %[[RESULT64_OUTLINE:.*]] = fir.convert %[[E]] : (complex<f64>) -> !fir.complex<8>
+! CMPLX-PRECISE: %[[RESULT64_OUTLINE:.*]] = fir.call @clog(%[[ARG64_OUTLINE]]) : (!fir.complex<8>) -> !fir.complex<8>
+! CMPLX: return %[[RESULT64_OUTLINE]] : !fir.complex<8>
 
 ! CHECK-LABEL: private @fir.log10.f32.f32
 ! CHECK-SAME: (%[[ARG32_OUTLINE:.*]]: f32) -> f32
diff --git a/flang/test/Lower/math-lowering.f90 b/flang/test/Lower/math-lowering.f90
--- a/flang/test/Lower/math-lowering.f90
+++ b/flang/test/Lower/math-lowering.f90
@@ -43,7 +43,9 @@
 end function
 
 ! ALL-LABEL: @_QPtest_complex4
-! ALL: {{%[A-Za-z0-9._]+}} = fir.call @cabsf({{%[A-Za-z0-9._]+}}) : (!fir.complex<4>) -> f32
+! FAST: {{%[A-Za-z0-9._]+}} = complex.abs {{%[A-Za-z0-9._]+}} : complex<f32>
+! RELAXED: {{%[A-Za-z0-9._]+}} = complex.abs {{%[A-Za-z0-9._]+}} : complex<f32>
+! PRECISE: {{%[A-Za-z0-9._]+}} = fir.call @cabsf({{%[A-Za-z0-9._]+}}) : (!fir.complex<4>) -> f32
 
 function test_complex8(c)
   complex(8) :: c, test_complex8
@@ -51,7 +53,9 @@
 end function
 
 ! ALL-LABEL: @_QPtest_complex8
-! ALL: {{%[A-Za-z0-9._]+}} = fir.call @cabs({{%[A-Za-z0-9._]+}}) : (!fir.complex<8>) -> f64
+! FAST: {{%[A-Za-z0-9._]+}} = complex.abs {{%[A-Za-z0-9._]+}} : complex<f64>
+! RELAXED: {{%[A-Za-z0-9._]+}} = complex.abs {{%[A-Za-z0-9._]+}} : complex<f64>
+! PRECISE: {{%[A-Za-z0-9._]+}} = fir.call @cabs({{%[A-Za-z0-9._]+}}) : (!fir.complex<8>) -> f64
 
 //--- aint.f90
 ! RUN: bbc -emit-fir %t/aint.f90 -o - --math-runtime=fast | FileCheck --check-prefixes=ALL %t/aint.f90
diff --git a/flang/test/Lower/power-operator.f90 b/flang/test/Lower/power-operator.f90
--- a/flang/test/Lower/power-operator.f90
+++ b/flang/test/Lower/power-operator.f90
@@ -1,4 +1,9 @@
-! RUN: bbc -emit-fir %s -o - | FileCheck %s
+! RUN: bbc -emit-fir %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX-FAST"
+! RUN: bbc --math-runtime=precise -emit-fir %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
+! RUN: bbc --disable-mlir-complex -emit-fir %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX-FAST"
+! RUN: %flang_fc1 -emit-fir -mllvm --math-runtime=precise %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm --disable-mlir-complex %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
 
 ! Test power operation lowering
 
@@ -121,13 +126,15 @@
 subroutine pow_c4_c4(x, y, z)
   complex :: x, y, z
   z = x ** y
-  ! CHECK: call @cpowf
+  ! CMPLX-FAST: complex.pow %{{.*}}, %{{.*}} : complex<f32>
+  ! CMPLX-PRECISE: call @cpowf
 end subroutine
 
 ! CHECK-LABEL: pow_c8_c8
 subroutine pow_c8_c8(x, y, z)
   complex(8) :: x, y, z
   z = x ** y
-  ! CHECK: call @cpow
+  ! CMPLX-FAST: complex.pow %{{.*}}, %{{.*}} : complex<f64>
+  ! CMPLX-PRECISE: call @cpow
 end subroutine
 
diff --git a/flang/test/Lower/sqrt.f90 b/flang/test/Lower/sqrt.f90
--- a/flang/test/Lower/sqrt.f90
+++ b/flang/test/Lower/sqrt.f90
@@ -1,5 +1,9 @@
-! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s
-! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics %s -o - | FileCheck %s
+! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX-FAST"
+! RUN: bbc --math-runtime=precise -emit-fir -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
+! RUN: bbc --disable-mlir-complex -emit-fir -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX-FAST"
+! RUN: %flang_fc1 -emit-fir -mllvm --math-runtime=precise -mllvm -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
+! RUN: %flang_fc1 -emit-fir -mllvm --disable-mlir-complex -mllvm -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
 
 ! CHECK-LABEL: sqrt_testr
 subroutine sqrt_testr(a, b)
@@ -36,7 +40,9 @@
 ! CHECK: math.sqrt %{{.*}} : f64
 
 ! CHECK-LABEL: func private @fir.sqrt.z4.z4
-! CHECK: fir.call @csqrtf
+! CMPLX-FAST: complex.sqrt %{{.*}} : complex<f32>
+! CMPLX-PRECISE: fir.call @csqrtf
 
 ! CHECK-LABEL: @fir.sqrt.z8.z8
-! CHECK: fir.call @csqrt
+! CMPLX-FAST: complex.sqrt %{{.*}} : complex<f64>
+! CMPLX-PRECISE: fir.call @csqrt
diff --git a/flang/test/Lower/trigonometric-intrinsics.f90 b/flang/test/Lower/trigonometric-intrinsics.f90
--- a/flang/test/Lower/trigonometric-intrinsics.f90
+++ b/flang/test/Lower/trigonometric-intrinsics.f90
@@ -1,4 +1,5 @@
-! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s
+! RUN: bbc -emit-fir -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CHECK,CMPLX-FAST"
+! RUN: bbc --math-runtime=precise -emit-fir -outline-intrinsics %s -o - | FileCheck %s --check-prefixes="CMPLX-PRECISE"
 ! RUN: %flang_fc1 -emit-fir -mllvm -outline-intrinsics %s -o - | FileCheck %s
 
 ! CHECK-LABEL: tan_testr
@@ -176,10 +177,12 @@
 ! CHECK: math.tan %{{.*}} : f64
 
 ! CHECK-LABEL: @fir.tan.z4.z4
-! CHECK: fir.call @ctanf
+! CMPLX-FAST: complex.tan %{{.*}} : complex<f32>
+! CMPLX-PRECISE: fir.call @ctanf
 
 ! CHECK-LABEL: @fir.tan.z8.z8
-! CHECK: fir.call @ctan
+! CMPLX-FAST: complex.tan %{{.*}} : complex<f64>
+! CMPLX-PRECISE: fir.call @ctan
 
 ! CHECK-LABEL: @fir.atan.f32.f32
 ! CHECK: math.atan %{{.*}} : f32
@@ -200,10 +203,12 @@
 ! CHECK: math.cos %{{.*}} : f64
 
 ! CHECK-LABEL: @fir.cos.z4.z4
-! CHECK: fir.call @ccosf
+! CMPLX-FAST: complex.cos %{{.*}} : complex<f32>
+! CMPLX-PRECISE: fir.call @ccosf
 
 ! CHECK-LABEL: @fir.cos.z8.z8
-! CHECK: fir.call @ccos
+! CMPLX-FAST: complex.cos %{{.*}} : complex<f64>
+! CMPLX-PRECISE: fir.call @ccos
 
 ! CHECK-LABEL: @fir.cosh.f32.f32
 ! CHECK: fir.call {{.*}}cosh
@@ -224,10 +229,12 @@
 ! CHECK: math.sin %{{.*}} : f64
 
 ! CHECK-LABEL: @fir.sin.z4.z4
-! CHECK: fir.call @csinf
+! CMPLX-FAST: complex.sin %{{.*}} : complex<f32>
+! CMPLX-PRECISE: fir.call @csinf
 
 ! CHECK-LABEL: @fir.sin.z8.z8
-! CHECK: fir.call @csin
+! CMPLX-FAST: complex.sin %{{.*}} : complex<f64>
+! CMPLX-PRECISE: fir.call @csin
 
 ! CHECK-LABEL: @fir.sinh.f32.f32
 ! CHECK: fir.call {{.*}}sinh