What is the <=> ("spaceship", three-way comparison) operator in C++?

On 2017-11-11, the ISO C++ committee adopted Herb Sutter's proposal for the <=> "spaceship" three-way comparison operator as one of the new features that were added to C++20. In the paper titled Consistent comparison Sutter, Maurer and Brown demonstrate the concepts of the new design. For an overview of the proposal, here's an excerpt from the article:

The expression a <=> b returns an object that compares <0 if a < b, compares >0 if a > b, and compares ==0 if a and b are equal/equivalent.

Common case: To write all comparisons for your type X with type Y, with memberwise semantics, just write:

auto X::operator<=>(const Y&) =default;

Advanced cases: To write all comparisons for your type X with type Y, just write operator<=> that takes a Y, can use =default to get memberwise semantics if desired, and returns the appropriate category type:

• Return an _ordering if your type naturally supports <, and we’ll efficiently generate symmetric <, >, <=, >=, ==, and !=; otherwise return an _equality, and we’ll efficiently generate symmetric == and !=.
• Return strong_ if for your type a == b implies f(a) == f(b) (substitutability, where f reads only comparison-salient state that is accessible using the public const members), otherwise return weak_.

Comparison Categories

Five comparison categories are defined as std:: types, each having the following predefined values:

+--------------------------------------------------------------------+
|                  |          Numeric  values          | Non-numeric |
|     Category     +-----------------------------------+             |
|                  | -1   | 0          | +1            |   values    |
+------------------+------+------------+---------------+-------------+
| strong_ordering  | less | equal      | greater       |             |
| weak_ordering    | less | equivalent | greater       |             |
| partial_ordering | less | equivalent | greater       | unordered   |
| strong_equality  |      | equal      | nonequal      |             |
| weak_equality    |      | equivalent | nonequivalent |             |
+------------------+------+------------+---------------+-------------+

Implicit conversions between these types are defined as follows:

• strong_ordering with values {less, equal, greater} implicitly converts to:
  • weak_ordering with values {less, equivalent, greater}
  • partial_ordering with values {less, equivalent, greater}
  • strong_equality with values {unequal, equal, unequal}
  • weak_equality with values {nonequivalent, equivalent, nonequivalent}
• weak_ordering with values {less, equivalent, greater} implicitly converts to:
  • partial_ordering with values {less, equivalent, greater}
  • weak_equality with values {nonequivalent, equivalent, nonequivalent}
• partial_ordering with values {less, equivalent, greater, unordered} implicitly converts to:
  • weak_equality with values {nonequivalent, equivalent, nonequivalent, nonequivalent}
• strong_equality with values {equal, unequal} implicitly converts to:
  • weak_equality with values {equivalent, nonequivalent}

Three-way comparison

The<=>token is introduced. The character sequence<=>tokenizes to<= >, in old source code. For example,X<&Y::operator<=>needs to add a space to retain its meaning.

The overloadable operator<=>is a three-way comparison function and has precedence higher than< and lower than<<. It returns a type that can be compared against literal0but other return types are allowed such as to support expression templates. All<=>operators defined in the language and in the standard library return one of the 5 aforementionedstd::comparison category types.

For language types, the following built-in<=>same-type comparisons are provided. All are constexpr, except where noted otherwise. These comparisons cannot be invoked heterogeneously using scalar promotions/conversions.

• Forbool, integral, and pointer types,<=>returnsstrong_ordering.
• For pointer types, the different cv-qualifications and derived-to-base conversions are allowed to invoke a homogeneous built-in<=>, and there are built-in heterogeneousoperator<=>(T*, nullptr_t). Only comparisons of pointers to the same object/allocation are constant expressions.
• For fundamental floating point types,<=> returnspartial_ordering, and can be invoked heterogeneously by widening arguments to a larger floating point type.
• For enumerations,<=> returns the same as the enumeration's underlying type's<=>.
• Fornullptr_t,<=> returnsstrong_orderingand always yieldsequal.
• For copyable arrays,T[N] <=> T[N]returns the same type asT's<=>and performs lexicographical elementwise comparison. There is no<=>for other arrays.
• Forvoidthere is no<=>.

_{To better understand the inner workings of this operator, please read the original paper. This is just what I've found out using search engines.}

This is called the three-way comparison operator.

According to the P0515 paper proposal:

There’s a new three-way comparison operator, <=>. The expression a <=> b returns an object that compares <0 if a < b, compares >0 if a > b, and compares ==0 if a and b are equal/equivalent.

To write all comparisons for your type, just write operator<=> that returns the appropriate category type:

• Return an _ordering if your type naturally supports <, and we’ll efficiently generate <, >, <=, >=, ==, and !=; otherwise return an _equality, and we’ll efficiently generate == and !=.

• Return strong if for your type a == b implies f(a) == f(b) (substitutability, where f reads only comparison-salient state accessible using the nonprivate const interface), otherwise return weak.

The cppreference says:

The three-way comparison operator expressions have the form

lhs <=> rhs   (1)  

The expression returns an object that

• compares <0 if lhs < rhs
• compares >0 if lhs > rhs
• and compares ==0 if lhs and rhs are equal/equivalent.