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+# Complex Operations
+
+```eval_rst
+.. contents::
+   :local:
+```
+
+Fortran includes support for complex number types and a set of operators and
+intrinsics that work on these types. Some of those operations are complicated
+and require runtime function calls to implement.
+
+This document outlines a design for generating these operations using the MLIR
+complex dialect while avoiding cross-platform ABI issues.
+
+## FIR Representation
+
+MLIR contains a complex dialect, similar to the Math dialect also used for
+lowering some integer and floating point operations in Flang. Conversion between
+fir.complex types and MLIR complex types is supported.
+
+As a result at the FIR level, complex operations can be represented as
+conversions from the fir.complex type to the equivalent MLIR complex type, use
+of the MLIR operation and a conversion back.
+
+This is similar to the way the math intrinsics are lowered, as proposed [here][1]
+
+**Fortran**
+```fortran
+function pow_self(c)
+  complex, intent(in) :: c
+  complex :: pow_self
+  pow_self = c ** c
+end function pow_self
+```
+
+**FIR**
+```c
+func.func @_QPpow_self(%arg0: !fir.ref<!fir.complex<4>>) -> !fir.complex<4> {
+    %0 = fir.alloca !fir.complex<4>
+    %1 = fir.load %arg0 : !fir.ref<!fir.complex<4>>
+    %2 = fir.load %arg0 : !fir.ref<!fir.complex<4>>
+    %3 = fir.convert %1 : (!fir.complex<4>) -> complex<f32>
+    %4 = fir.convert %2 : (!fir.complex<4>) -> complex<f32>
+    %5 = complex.pow %3, %4 : complex<f32>
+    %6 = fir.convert %5 : (complex<f32>) -> !fir.complex<4>
+    fir.store %6 to %0 : !fir.ref<!fir.complex<4>>
+    %7 = fir.load %0 : !fir.ref<!fir.complex<4>>
+    return %7 : !fir.complex<4>
+  }
+```
+
+Some operations are currently missing in the MLIR complex dialect that we would
+want to use here, such as powi and the hyperbolic trigonometry functions.
+For the missing operations we call directly to libm where possible, for powi
+we provide an implementation in the flang runtime.
+
+## Lowering
+
+The MLIR complex dialect supports lowering either by emitting calls to the
+complex functions in libm (ComplexToLibm), or through lowering to the standard
+dialect (ComplexToStandard). However, as MLIR has no target awareness, the
+lowering to libm functions suffers from ABI incompatibilities on some platforms.
+As such the custom lowering to the standard dialect is used. This may be
+something to revisit in future if performance could be improved by using the
+libm functions.
+
+Similarly to the numerical lowering through the math dialect, certain MLIR
+optimisations could violate the precise floating point model, so when that is
+requested lowering manually emits calls to libm, rather than going through the 
+MLIR complex dialect.
+
+The ComplexToStandard dialect does still call into libm for some floating
+point math operations, however these don't have the same ABI issues as the
+complex libm functions.
+
+[1]: https://discourse.llvm.org/t/rfc-change-lowering-of-fortran-math-intrinsics/63971