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### 6.11 Complex Numbers

ISO C99 supports complex floating data types, and as an extension GCC supports them in C90 mode and in C++. GCC also supports complex integer data types which are not part of ISO C99. You can declare complex types using the keyword `_Complex`. As an extension, the older GNU keyword `__complex__` is also supported.

For example, ‘`_Complex double x;`’ declares `x` as a variable whose real part and imaginary part are both of type `double`. ‘`_Complex short int y;`’ declares `y` to have real and imaginary parts of type `short int`; this is not likely to be useful, but it shows that the set of complex types is complete.

To write a constant with a complex data type, use the suffix ‘`i`’ or ‘`j`’ (either one; they are equivalent). For example, `2.5fi` has type `_Complex float` and `3i` has type `_Complex int`. Such a constant always has a pure imaginary value, but you can form any complex value you like by adding one to a real constant. This is a GNU extension; if you have an ISO C99 conforming C library (such as the GNU C Library), and want to construct complex constants of floating type, you should include `<complex.h>` and use the macros `I` or `_Complex_I` instead.

The ISO C++14 library also defines the ‘`i`’ suffix, so C++14 code that includes the ‘`<complex>`’ header cannot use ‘`i`’ for the GNU extension. The ‘`j`’ suffix still has the GNU meaning.

GCC can handle both implicit and explicit casts between the `_Complex` types and other `_Complex` types as casting both the real and imaginary parts to the scalar type. GCC can handle implicit and explicit casts from a scalar type to a `_Complex` type and where the imaginary part will be considered zero. The C front-end can handle implicit and explicit casts from a `_Complex` type to a scalar type where the imaginary part will be ignored. In C++ code, this cast is considered illformed and G++ will error out.

GCC provides a built-in function `__builtin_complex` will can be used to construct a complex value.

GCC has a few extensions which can be used to extract the real and the imaginary part of the complex-valued expression. Note these expressions are lvalues if the exp is an lvalue. These expressions operands have the type of a complex type which might get prompoted to a complex type from a scalar type. E.g. `__real__ (int)x` is the same as casting to `_Complex int` before `__real__` is done.

Expression | Description
--- | ---
`__real__ exp` | Extract the real part of exp.
`__imag__ exp` | Extract the imaginary part of exp.

For values of floating point, you should use the ISO C99 functions, declared in `<complex.h>` and also provided as built-in functions by GCC.

Expression | float | double | long double
--- | --- | --- | ---
`__real__ exp` | `crealf` | `creal` | `creall`
`__imag__ exp` | `cimagf` | `cimag` | `cimagl`

The operator ‘`~`’ performs complex conjugation when used on a value with a complex type. This is a GNU extension; for values of floating type, you should use the ISO C99 functions `conjf`, `conj` and `conjl`, declared in `<complex.h>` and also provided as built-in functions by GCC. Note unlike the `__real__` and `__imag__` operators, this operator will not do an implicit cast to the complex type because the ‘`~`’ is already a normal operator.

GCC can allocate complex automatic variables in a noncontiguous fashion; it’s even possible for the real part to be in a register while the imaginary part is on the stack (or vice versa). Only the DWARF debug info format can represent this, so use of DWARF is recommended. If you are using the stabs debug info format, GCC describes a noncontiguous complex variable as if it were two separate variables of noncomplex type. If the variable’s actual name is `foo`, the two fictitious variables are named `foo$real` and `foo$imag`. You can examine and set these two fictitious variables with your debugger.

Built-in Function: type __builtin_complex (real, imag)

  The built-in function `__builtin_complex` is provided for use in implementing the ISO C11 macros `CMPLXF`, `CMPLX` and `CMPLXL`. real and imag must have the same type, a real binary floating-point type, and the result has the corresponding complex type with real and imaginary parts real and imag. Unlike ‘`real + I * imag`’, this works even when infinities, NaNs and negative zeros are involved.

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