# Standard library header <numeric>

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< cpp‎ | header

This header is part of the numeric library.

### Functions

iota (C++11) | fills a range with successive increments of the starting value (function template)
ranges::iota (C++23) | fills a range with successive increments of the starting value (algorithm function object)
accumulate | sums up or folds a range of elements (function template)
reduce (C++17) | similar to std::accumulate, except out of order (function template)
transform_reduce (C++17) | applies an invocable, then reduces out of order (function template)
inner_product | computes the inner product of two ranges of elements (function template)
adjacent_difference | computes the differences between adjacent elements in a range (function template)
partial_sum | computes the partial sum of a range of elements (function template)
inclusive_scan (C++17) | similar to std::partial_sum, includes the ith input element in the ith sum (function template)
exclusive_scan (C++17) | similar to std::partial_sum, excludes the ith input element from the ith sum (function template)
transform_inclusive_scan (C++17) | applies an invocable, then calculates inclusive scan (function template)
transform_exclusive_scan (C++17) | applies an invocable, then calculates exclusive scan (function template)
gcd (C++17) | computes the greatest common divisor of two integers (function template)
lcm (C++17) | computes the least common multiple of two integers (function template)
midpoint (C++20) | midpoint between two numbers or pointers (function template)
add_sat (C++26) | saturating addition operation on two integers (function template)
sub_sat (C++26) | saturating subtraction operation on two integers (function template)
mul_sat (C++26) | saturating multiplication operation on two integers (function template)
div_sat (C++26) | saturating division operation on two integers (function template)
saturate_cast (C++26) | returns an integer value clamped to the range of another integer type (function template)

### Synopsis

    namespace std {
      // accumulate
      template<class InputIt, class T>
        constexpr T accumulate(InputIt first, InputIt last, T init);
      template<class InputIt, class T, class BinaryOperation>
        constexpr T accumulate(InputIt first, InputIt last, T init, BinaryOperation binary_op);

      // reduce
      template<class InputIt>
        constexpr typename iterator_traits<InputIt>::value_type
          reduce(InputIt first, InputIt last);
      template<class InputIt, class T>
        constexpr T reduce(InputIt first, InputIt last, T init);
      template<class InputIt, class T, class BinaryOperation>
        constexpr T reduce(InputIt first, InputIt last, T init, BinaryOperation binary_op);
      template<class ExecutionPolicy, class ForwardIt>
        typename iterator_traits<ForwardIt>::value_type
          reduce(ExecutionPolicy&& exec,
                 ForwardIt first, ForwardIt last);
      template<class ExecutionPolicy, class ForwardIt, class T>
        T reduce(ExecutionPolicy&& exec,
                 ForwardIt first, ForwardIt last, T init);
      template<class ExecutionPolicy, class ForwardIt, class T, class BinaryOperation>
        T reduce(ExecutionPolicy&& exec,
                 ForwardIt first, ForwardIt last, T init, BinaryOperation binary_op);

      // inner product
      template<class InputIt1, class InputIt2, class T>
        constexpr T inner_product(InputIt1 first1, InputIt1 last1,
                                  InputIt2 first2, T init);
      template<class InputIt1, class InputIt2, class T,
               class BinaryOperation1, class BinaryOperation2>
        constexpr T inner_product(InputIt1 first1, InputIt1 last1,
                                  InputIt2 first2, T init,
                                  BinaryOperation1 binary_op1,
                                  BinaryOperation2 binary_op2);

      // transform reduce
      template<class InputIt1, class InputIt2, class T>
        constexpr T transform_reduce(InputIt1 first1, InputIt1 last1,
                                     InputIt2 first2,
                                     T init);
      template<class InputIt1, class InputIt2, class T,
               class BinaryOperation1, class BinaryOperation2>
        constexpr T transform_reduce(InputIt1 first1, InputIt1 last1,
                                     InputIt2 first2,
                                     T init,
                                     BinaryOperation1 binary_op1,
                                     BinaryOperation2 binary_op2);
      template<class InputIt, class T,
               class BinaryOperation, class UnaryOperation>
        constexpr T transform_reduce(InputIt first, InputIt last,
                                     T init,
                                     BinaryOperation binary_op, UnaryOperation unary_op);
      template<class ExecutionPolicy,
               class ForwardIt1, class ForwardIt2, class T>
        T transform_reduce(ExecutionPolicy&& exec,
                           ForwardIt1 first1, ForwardIt1 last1,
                           ForwardIt2 first2,
                           T init);
      template<class ExecutionPolicy,
               class ForwardIt1, class ForwardIt2, class T,
               class BinaryOperation1, class BinaryOperation2>
        T transform_reduce(ExecutionPolicy&& exec,
                           ForwardIt1 first1, ForwardIt1 last1,
                           ForwardIt2 first2,
                           T init,
                           BinaryOperation1 binary_op1,
                           BinaryOperation2 binary_op2);
      template<class ExecutionPolicy,
               class ForwardIt, class T,
               class BinaryOperation, class UnaryOperation>
        T transform_reduce(ExecutionPolicy&& exec,
                           ForwardIt first, ForwardIt last,
                           T init,
                           BinaryOperation binary_op, UnaryOperation unary_op);

      // partial sum
      template<class InputIt, class OutputIt>
        constexpr OutputIt partial_sum(InputIt first,
                                       InputIt last,
                                       OutputIt result);
      template<class InputIt, class OutputIt, class BinaryOperation>
        constexpr OutputIt partial_sum(InputIt first,
                                       InputIt last,
                                       OutputIt result,
                                       BinaryOperation binary_op);

      // exclusive scan
      template<class InputIt, class OutputIt, class T>
        constexpr OutputIt exclusive_scan(InputIt first, InputIt last,
                                          OutputIt result,
                                          T init);
      template<class InputIt, class OutputIt, class T, class BinaryOperation>
        constexpr OutputIt exclusive_scan(InputIt first, InputIt last,
                                          OutputIt result,
                                          T init, BinaryOperation binary_op);
      template<class ExecutionPolicy, class ForwardIt1, class ForwardIt2, class T>
        ForwardIt2 exclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIt1 first, ForwardIt1 last,
                                  ForwardIt2 result,
                                  T init);
      template<class ExecutionPolicy, class ForwardIt1, class ForwardIt2, class T,
               class BinaryOperation>
        ForwardIt2 exclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIt1 first, ForwardIt1 last,
                                  ForwardIt2 result,
                                  T init, BinaryOperation binary_op);

      // inclusive scan
      template<class InputIt, class OutputIt>
        constexpr OutputIt inclusive_scan(InputIt first, InputIt last, OutputIt result);
      template<class InputIt, class OutputIt, class BinaryOperation>
        constexpr OutputIt inclusive_scan(InputIt first, InputIt last,
                                          OutputIt result,
                                          BinaryOperation binary_op);
      template<class InputIt, class OutputIt, class BinaryOperation, class T>
        constexpr OutputIt inclusive_scan(InputIt first, InputIt last,
                                          OutputIt result,
                                          BinaryOperation binary_op, T init);
      template<class ExecutionPolicy, class ForwardIt1, class ForwardIt2>
        ForwardIt2 inclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIt1 first, ForwardIt1 last,
                                  ForwardIt2 result);
      template<class ExecutionPolicy, class ForwardIt1, class ForwardIt2,
               class BinaryOperation>
        ForwardIt2 inclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIt1 first, ForwardIt1 last,
                                  ForwardIt2 result,
                                  BinaryOperation binary_op);
      template<class ExecutionPolicy, class ForwardIt1, class ForwardIt2,
               class BinaryOperation, class T>
        ForwardIt2 inclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIt1 first, ForwardIt1 last,
                                  ForwardIt2 result,
                                  BinaryOperation binary_op, T init);

      // transform exclusive scan
      template<class InputIt, class OutputIt, class T,
               class BinaryOperation, class UnaryOperation>
        constexpr OutputIt transform_exclusive_scan(InputIt first, InputIt last,
                                                    OutputIt result,
                                                    T init,
                                                    BinaryOperation binary_op,
                                                    UnaryOperation unary_op);
      template<class ExecutionPolicy,
               class ForwardIt1, class ForwardIt2, class T,
               class BinaryOperation, class UnaryOperation>
        ForwardIt2 transform_exclusive_scan(ExecutionPolicy&& exec,
                                            ForwardIt1 first, ForwardIt1 last,
                                            ForwardIt2 result,
                                            T init,
                                            BinaryOperation binary_op,
                                            UnaryOperation unary_op);

      // transform inclusive scan
      template<class InputIt, class OutputIt,
               class BinaryOperation, class UnaryOperation>
        constexpr OutputIt transform_inclusive_scan(InputIt first, InputIt last,
                                                    OutputIt result,
                                                    BinaryOperation binary_op,
                                                    UnaryOperation unary_op);
      template<class InputIt, class OutputIt,
               class BinaryOperation, class UnaryOperation, class T>
        constexpr OutputIt transform_inclusive_scan(InputIt first, InputIt last,
                                                    OutputIt result,
                                                    BinaryOperation binary_op,
                                                    UnaryOperation unary_op,
                                                    T init);
      template<class ExecutionPolicy,
               class ForwardIt1, class ForwardIt2,
               class BinaryOperation, class UnaryOperation>
        ForwardIt2 transform_inclusive_scan(ExecutionPolicy&& exec,
                                            ForwardIt1 first, ForwardIt1 last,
                                            ForwardIt2 result,
                                            BinaryOperation binary_op,
                                            UnaryOperation unary_op);
      template<class ExecutionPolicy,
               class ForwardIt1, class ForwardIt2,
               class BinaryOperation, class UnaryOperation, class T>
        ForwardIt2 transform_inclusive_scan(ExecutionPolicy&& exec,
                                            ForwardIt1 first, ForwardIt1 last,
                                            ForwardIt2 result,
                                            BinaryOperation binary_op,
                                            UnaryOperation unary_op,
                                            T init);

      // adjacent difference
      template<class InputIt, class OutputIt>
        constexpr OutputIt adjacent_difference(InputIt first, InputIt last,
                                               OutputIt result);
      template<class InputIt, class OutputIt, class BinaryOperation>
        constexpr OutputIt adjacent_difference(InputIt first, InputIt last,
                                               OutputIt result,
                                               BinaryOperation binary_op);
      template<class ExecutionPolicy, class ForwardIt1, class ForwardIt2>
        ForwardIt2 adjacent_difference(ExecutionPolicy&& exec,
                                       ForwardIt1 first, ForwardIt1 last,
                                       ForwardIt2 result);
      template<class ExecutionPolicy, class ForwardIt1, class ForwardIt2,
               class BinaryOperation>
        ForwardIt2 adjacent_difference(ExecutionPolicy&& exec,
                                       ForwardIt1 first, ForwardIt1 last,
                                       ForwardIt2 result,
                                       BinaryOperation binary_op);

      // iota
      template<class ForwardIt, class T>
        constexpr void iota(ForwardIt first, ForwardIt last, T value);

      namespace ranges {
        template<class O, class T>
          using iota_result = out_value_result<O, T>;

        template<input_or_output_iterator O, sentinel_for<O> S, weakly_incrementable T>
          requires indirectly_writable<O, const T&>
          constexpr iota_result<O, T> iota(O first, S last, T value);

        template<weakly_incrementable T, output_range<const T&> R>
          constexpr iota_result<borrowed_iterator_t<R>, T> iota(R&& r, T value);
      }

      // greatest common divisor
      template<class M, class N>
        constexpr common_type_t<M, N> gcd(M m, N n);

      // least common multiple
      template<class M, class N>
        constexpr common_type_t<M, N> lcm(M m, N n);

      // midpoint
      template<class T>
        constexpr T midpoint(T a, T b) noexcept;
      template<class T>
        constexpr T* midpoint(T* a, T* b);

      // saturation arithmetic
      template<class T>
        constexpr T add_sat(T x, T y) noexcept;           // freestanding
      template<class T>
        constexpr T sub_sat(T x, T y) noexcept;           // freestanding
      template<class T>
        constexpr T mul_sat(T x, T y) noexcept;           // freestanding
      template<class T>
        constexpr T div_sat(T x, T y) noexcept;           // freestanding
      template<class T, class U>
        constexpr T saturate_cast(U x) noexcept;          // freestanding
    }

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