numeric

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// <numeric> -*- C++ -*-

// Copyright (C) 2001-2019 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.

// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.

// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.

// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
// <http://www.gnu.org/licenses/>.

/*
 *
 * Copyright (c) 1994
 * Hewlett-Packard Company
 *
 * Permission to use, copy, modify, distribute and sell this software
 * and its documentation for any purpose is hereby granted without fee,
 * provided that the above copyright notice appear in all copies and
 * that both that copyright notice and this permission notice appear
 * in supporting documentation.  Hewlett-Packard Company makes no
 * representations about the suitability of this software for any
 * purpose.  It is provided "as is" without express or implied warranty.
 *
 *
 * Copyright (c) 1996,1997
 * Silicon Graphics Computer Systems, Inc.
 *
 * Permission to use, copy, modify, distribute and sell this software
 * and its documentation for any purpose is hereby granted without fee,
 * provided that the above copyright notice appear in all copies and
 * that both that copyright notice and this permission notice appear
 * in supporting documentation.  Silicon Graphics makes no
 * representations about the suitability of this software for any
 * purpose.  It is provided "as is" without express or implied warranty.
 */

/** @file include/numeric
 *  This is a Standard C++ Library header.
 */

#ifndef _GLIBCXX_NUMERIC
#define _GLIBCXX_NUMERIC 1

#pragma GCC system_header

#include <bits/c++config.h>
#include <bits/stl_iterator_base_types.h>
#include <bits/stl_numeric.h>

#ifdef _GLIBCXX_PARALLEL
# include <parallel/numeric>
#endif

/**
 * @defgroup numerics Numerics
 *
 * Components for performing numeric operations. Includes support for
 * complex number types, random number generation, numeric (n-at-a-time)
 * arrays, generalized numeric algorithms, and mathematical special functions.
 */

#if __cplusplus >= 201402L
#include <type_traits>

namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION

namespace __detail
{
  // std::abs is not constexpr and doesn't support unsigned integers.
  template<typename _Tp>
    constexpr
    enable_if_t<__and_<is_integral<_Tp>, is_signed<_Tp>>::value, _Tp>
    __abs_integral(_Tp __val)
    { return __val < 0 ? -__val : __val; }

  template<typename _Tp>
    constexpr
    enable_if_t<__and_<is_integral<_Tp>, is_unsigned<_Tp>>::value, _Tp>
    __abs_integral(_Tp __val)
    { return __val; }

  void __abs_integral(bool) = delete;

  template<typename _Mn, typename _Nn>
    constexpr common_type_t<_Mn, _Nn>
    __gcd(_Mn __m, _Nn __n)
    {
      return __m == 0 ? __detail::__abs_integral(__n)
   : __n == 0 ? __detail::__abs_integral(__m)
   : __detail::__gcd(__n, __m % __n);
    }

  /// Least common multiple
  template<typename _Mn, typename _Nn>
    constexpr common_type_t<_Mn, _Nn>
    __lcm(_Mn __m, _Nn __n)
    {
      return (__m != 0 && __n != 0)
   ? (__detail::__abs_integral(__m) / __detail::__gcd(__m, __n))
     * __detail::__abs_integral(__n)
   : 0;
    }
} // namespace __detail

#if __cplusplus >= 201703L

#define __cpp_lib_gcd_lcm 201606
// These were used in drafts of SD-6:
#define __cpp_lib_gcd 201606
#define __cpp_lib_lcm 201606

  /// Greatest common divisor
  template<typename _Mn, typename _Nn>
    constexpr common_type_t<_Mn, _Nn>
    gcd(_Mn __m, _Nn __n)
    {
      static_assert(is_integral_v<_Mn>, "gcd arguments are integers");
      static_assert(is_integral_v<_Nn>, "gcd arguments are integers");
      static_assert(!is_same_v<remove_cv_t<_Mn>, bool>,
               "gcd arguments are not bools");
      static_assert(!is_same_v<remove_cv_t<_Nn>, bool>,
               "gcd arguments are not bools");
      return __detail::__gcd(__m, __n);
    }

  /// Least common multiple
  template<typename _Mn, typename _Nn>
    constexpr common_type_t<_Mn, _Nn>
    lcm(_Mn __m, _Nn __n)
    {
      static_assert(is_integral_v<_Mn>, "lcm arguments are integers");
      static_assert(is_integral_v<_Nn>, "lcm arguments are integers");
      static_assert(!is_same_v<remove_cv_t<_Mn>, bool>,
               "lcm arguments are not bools");
      static_assert(!is_same_v<remove_cv_t<_Nn>, bool>,
               "lcm arguments are not bools");
      return __detail::__lcm(__m, __n);
    }

#endif // C++17

_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std

#endif // C++14

#if __cplusplus > 201703L
#include <limits>

namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
  // midpoint
# define __cpp_lib_interpolate 201902L

  template<typename _Tp>
    constexpr
    enable_if_t<__and_v<is_arithmetic<_Tp>, is_same<remove_cv_t<_Tp>, _Tp>,
                   __not_<is_same<_Tp, bool>>>,
           _Tp>
    midpoint(_Tp __a, _Tp __b) noexcept
    {
      if constexpr (is_integral_v<_Tp>)
   {
     using _Up = make_unsigned_t<_Tp>;

     int __k = 1;
     _Up __m = __a;
     _Up __M = __b;
     if (__a > __b)
       {
         __k = -1;
         __m = __b;
         __M = __a;
       }
     return __a + __k * _Tp(_Up(__M - __m) / 2);
   }
      else // is_floating
   {
     constexpr _Tp __lo = numeric_limits<_Tp>::min() * 2;
     constexpr _Tp __hi = numeric_limits<_Tp>::max() / 2;
     const _Tp __abs_a = __a < 0 ? -__a : __a;
     const _Tp __abs_b = __b < 0 ? -__b : __b;
     if (__abs_a <= __hi && __abs_b <= __hi) [[likely]]
       return (__a + __b) / 2; // always correctly rounded
     if (__abs_a < __lo) // not safe to halve __a
       return __a + __b/2;
     if (__abs_b < __lo) // not safe to halve __b
       return __a/2 + __b;
     return __a/2 + __b/2;     // otherwise correctly rounded
   }
    }

  template<typename _Tp>
    constexpr
    enable_if_t<__and_v<is_object<_Tp>, bool_constant<sizeof(_Tp) != 0>>, _Tp*>
    midpoint(_Tp* __a, _Tp* __b) noexcept
    {
      return __a  + (__b - __a) / 2;
    }
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std

#endif // C++20

#if __cplusplus > 201402L
#include <bits/stl_function.h>

namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION

  /// @addtogroup numeric_ops
  /// @{

  /// @cond undocumented
  template<typename _It, typename _Traits = iterator_traits<_It>,
      typename _Cat = typename _Traits::iterator_category>
    using __is_random_access_iter
      = is_base_of<random_access_iterator_tag, _Cat>;
  /// @endcond

  /**
   *  @brief  Calculate reduction of values in a range.
   *
   *  @param  __first  Start of range.
   *  @param  __last  End of range.
   *  @param  __init  Starting value to add other values to.
   *  @param  __binary_op A binary function object.
   *  @return  The final sum.
   *
   *  Reduce the values in the range `[first,last)` using a binary operation.
   *  The initial value is `init`.  The values are not necessarily processed
   *  in order.
   *
   *  This algorithm is similar to `std::accumulate` but is not required to
   *  perform the operations in order from first to last. For operations
   *  that are commutative and associative the result will be the same as
   *  for `std::accumulate`, but for other operations (such as floating point
   *  arithmetic) the result can be different.
   */
  template<typename _InputIterator, typename _Tp, typename _BinaryOperation>
    _Tp
    reduce(_InputIterator __first, _InputIterator __last, _Tp __init,
      _BinaryOperation __binary_op)
    {
      using value_type = typename iterator_traits<_InputIterator>::value_type;
      static_assert(is_invocable_r_v<_Tp, _BinaryOperation&, _Tp&, _Tp&>);
      static_assert(is_convertible_v<value_type, _Tp>);
      if constexpr (__is_random_access_iter<_InputIterator>::value)
   {
     while ((__last - __first) >= 4)
       {
         _Tp __v1 = __binary_op(__first[0], __first[1]);
         _Tp __v2 = __binary_op(__first[2], __first[3]);
         _Tp __v3 = __binary_op(__v1, __v2);
         __init = __binary_op(__init, __v3);
         __first += 4;
       }
   }
      for (; __first != __last; ++__first)
   __init = __binary_op(__init, *__first);
      return __init;
    }

 /**
   *  @brief  Calculate reduction of values in a range.
   *
   *  @param  __first  Start of range.
   *  @param  __last  End of range.
   *  @param  __init  Starting value to add other values to.
   *  @return  The final sum.
   *
   *  Reduce the values in the range `[first,last)` using addition.
   *  Equivalent to calling `std::reduce(first, last, init, std::plus<>())`.
   */
  template<typename _InputIterator, typename _Tp>
    inline _Tp
    reduce(_InputIterator __first, _InputIterator __last, _Tp __init)
    { return std::reduce(__first, __last, std::move(__init), plus<>()); }

  /**
   *  @brief  Calculate reduction of values in a range.
   *
   *  @param  __first  Start of range.
   *  @param  __last  End of range.
   *  @return  The final sum.
   *
   *  Reduce the values in the range `[first,last)` using addition, with
   *  an initial value of `T{}`, where `T` is the iterator's value type.
   *  Equivalent to calling `std::reduce(first, last, T{}, std::plus<>())`.
   */
  template<typename _InputIterator>
    inline typename iterator_traits<_InputIterator>::value_type
    reduce(_InputIterator __first, _InputIterator __last)
    {
      using value_type = typename iterator_traits<_InputIterator>::value_type;
      return std::reduce(__first, __last, value_type{}, plus<>());
    }

  /**
   *  @brief  Combine elements from two ranges and reduce
   *
   *  @param  __first1  Start of first range.
   *  @param  __last1  End of first range.
   *  @param  __first2  Start of second range.
   *  @param  __init  Starting value to add other values to.
   *  @param  __binary_op1 The function used to perform reduction.
   *  @param  __binary_op2 The function used to combine values from the ranges.
   *  @return  The final sum.
   *
   *  Call `binary_op2(first1[n],first2[n])` for each `n` in `[0,last1-first1)`
   *  and then use `binary_op1` to reduce the values returned by `binary_op2`
   *  to a single value of type `T`.
   *
   *  The range beginning at `first2` must contain at least `last1-first1`
   *  elements.
   */
  template<typename _InputIterator1, typename _InputIterator2, typename _Tp,
      typename _BinaryOperation1, typename _BinaryOperation2>
    _Tp
    transform_reduce(_InputIterator1 __first1, _InputIterator1 __last1,
                _InputIterator2 __first2, _Tp __init,
                _BinaryOperation1 __binary_op1,
                _BinaryOperation2 __binary_op2)
    {
      if constexpr (__and_v<__is_random_access_iter<_InputIterator1>,
                       __is_random_access_iter<_InputIterator2>>)
   {
     while ((__last1 - __first1) >= 4)
       {
         _Tp __v1 = __binary_op1(__binary_op2(__first1[0], __first2[0]),
                                 __binary_op2(__first1[1], __first2[1]));
         _Tp __v2 = __binary_op1(__binary_op2(__first1[2], __first2[2]),
                                 __binary_op2(__first1[3], __first2[3]));
         _Tp __v3 = __binary_op1(__v1, __v2);
         __init = __binary_op1(__init, __v3);
         __first1 += 4;
         __first2 += 4;
       }
   }
      for (; __first1 != __last1; ++__first1, (void) ++__first2)
   __init = __binary_op1(__init, __binary_op2(*__first1, *__first2));
      return __init;
    }

  /**
   *  @brief  Combine elements from two ranges and reduce
   *
   *  @param  __first1  Start of first range.
   *  @param  __last1  End of first range.
   *  @param  __first2  Start of second range.
   *  @param  __init  Starting value to add other values to.
   *  @return  The final sum.
   *
   *  Call `first1[n]*first2[n]` for each `n` in `[0,last1-first1)` and then
   *  use addition to sum those products to a single value of type `T`.
   *
   *  The range beginning at `first2` must contain at least `last1-first1`
   *  elements.
   */
  template<typename _InputIterator1, typename _InputIterator2, typename _Tp>
    inline _Tp
    transform_reduce(_InputIterator1 __first1, _InputIterator1 __last1,
                _InputIterator2 __first2, _Tp __init)
    {
      return std::transform_reduce(__first1, __last1, __first2,
                              std::move(__init),
                              plus<>(), multiplies<>());
    }

  /**
   *  @brief  Transform the elements of a range and reduce
   *
   *  @param  __first  Start of range.
   *  @param  __last  End of range.
   *  @param  __init  Starting value to add other values to.
   *  @param  __binary_op The function used to perform reduction.
   *  @param  __unary_op The function used to transform values from the range.
   *  @return  The final sum.
   *
   *  Call `unary_op(first[n])` for each `n` in `[0,last-first)` and then
   *  use `binary_op` to reduce the values returned by `unary_op`
   *  to a single value of type `T`.
   */
  template<typename _InputIterator, typename _Tp,
      typename _BinaryOperation, typename _UnaryOperation>
    _Tp
    transform_reduce(_InputIterator __first, _InputIterator __last, _Tp __init,
                _BinaryOperation __binary_op, _UnaryOperation __unary_op)
    {
      if constexpr (__is_random_access_iter<_InputIterator>::value)
   {
     while ((__last - __first) >= 4)
       {
         _Tp __v1 = __binary_op(__unary_op(__first[0]),
                                __unary_op(__first[1]));
         _Tp __v2 = __binary_op(__unary_op(__first[2]),
                                __unary_op(__first[3]));
         _Tp __v3 = __binary_op(__v1, __v2);
         __init = __binary_op(__init, __v3);
         __first += 4;
       }
   }
      for (; __first != __last; ++__first)
   __init = __binary_op(__init, __unary_op(*__first));
      return __init;
    }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @param __init   Initial value.
   *  @param __binary_op Function to perform summation.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements (and the initial value),
   *  using `binary_op` for summation.
   *
   *  This function generates an "exclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N-1 input elements,
   *  so the Nth input element is not included.
   */
  template<typename _InputIterator, typename _OutputIterator, typename _Tp,
      typename _BinaryOperation>
    _OutputIterator
    exclusive_scan(_InputIterator __first, _InputIterator __last,
              _OutputIterator __result, _Tp __init,
              _BinaryOperation __binary_op)
    {
      while (__first != __last)
   {
     auto __v = __init;
     __init = __binary_op(__init, *__first);
     ++__first;
     *__result++ = std::move(__v);
   }
      return __result;
    }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @param __init   Initial value.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements (and the initial value),
   *  using `std::plus<>` for summation.
   *
   *  This function generates an "exclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N-1 input elements,
   *  so the Nth input element is not included.
   */
  template<typename _InputIterator, typename _OutputIterator, typename _Tp>
    inline _OutputIterator
    exclusive_scan(_InputIterator __first, _InputIterator __last,
              _OutputIterator __result, _Tp __init)
    {
      return std::exclusive_scan(__first, __last, __result, std::move(__init),
                            plus<>());
    }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @param __binary_op Function to perform summation.
   *  @param __init   Initial value.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements (and the initial value),
   *  using `binary_op` for summation.
   *
   *  This function generates an "inclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N input elements,
   *  so the Nth input element is included.
   */
  template<typename _InputIterator, typename _OutputIterator,
      typename _BinaryOperation, typename _Tp>
    _OutputIterator
    inclusive_scan(_InputIterator __first, _InputIterator __last,
              _OutputIterator __result, _BinaryOperation __binary_op,
              _Tp __init)
    {
      for (; __first != __last; ++__first)
   *__result++ = __init = __binary_op(__init, *__first);
      return __result;
    }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @param __binary_op Function to perform summation.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements, using `binary_op` for summation.
   *
   *  This function generates an "inclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N input elements,
   *  so the Nth input element is included.
   */
  template<typename _InputIterator, typename _OutputIterator,
      typename _BinaryOperation>
    _OutputIterator
    inclusive_scan(_InputIterator __first, _InputIterator __last,
              _OutputIterator __result, _BinaryOperation __binary_op)
    {
      if (__first != __last)
   {
     auto __init = *__first;
     *__result++ = __init;
     ++__first;
     if (__first != __last)
       __result = std::inclusive_scan(__first, __last, __result,
                                      __binary_op, std::move(__init));
   }
      return __result;
    }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements, using `std::plus<>` for summation.
   *
   *  This function generates an "inclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N input elements,
   *  so the Nth input element is included.
   */
  template<typename _InputIterator, typename _OutputIterator>
    inline _OutputIterator
    inclusive_scan(_InputIterator __first, _InputIterator __last,
              _OutputIterator __result)
    { return std::inclusive_scan(__first, __last, __result, plus<>()); }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @param __init   Initial value.
   *  @param __binary_op Function to perform summation.
   *  @param __unary_op Function to transform elements of the input range.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements (and the initial value),
   *  using `__unary_op` to transform the input elements
   *  and using `__binary_op` for summation.
   *
   *  This function generates an "exclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N-1 input elements,
   *  so the Nth input element is not included.
   */
  template<typename _InputIterator, typename _OutputIterator, typename _Tp,
      typename _BinaryOperation, typename _UnaryOperation>
    _OutputIterator
    transform_exclusive_scan(_InputIterator __first, _InputIterator __last,
                        _OutputIterator __result, _Tp __init,
                        _BinaryOperation __binary_op,
                        _UnaryOperation __unary_op)
    {
      while (__first != __last)
   {
     auto __v = __init;
     __init = __binary_op(__init, __unary_op(*__first));
     ++__first;
     *__result++ = std::move(__v);
   }
      return __result;
    }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @param __binary_op Function to perform summation.
   *  @param __unary_op Function to transform elements of the input range.
   *  @param __init   Initial value.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements (and the initial value),
   *  using `__unary_op` to transform the input elements
   *  and using `__binary_op` for summation.
   *
   *  This function generates an "inclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N input elements,
   *  so the Nth input element is included.
   */
  template<typename _InputIterator, typename _OutputIterator,
      typename _BinaryOperation, typename _UnaryOperation, typename _Tp>
    _OutputIterator
    transform_inclusive_scan(_InputIterator __first, _InputIterator __last,
                        _OutputIterator __result,
                        _BinaryOperation __binary_op,
                        _UnaryOperation __unary_op,
                        _Tp __init)
    {
      for (; __first != __last; ++__first)
   *__result++ = __init = __binary_op(__init, __unary_op(*__first));
      return __result;
    }

  /** @brief Output the cumulative sum of one range to a second range
   *
   *  @param __first  Start of input range.
   *  @param __last   End of input range.
   *  @param __result Start of output range.
   *  @param __binary_op Function to perform summation.
   *  @param __unary_op Function to transform elements of the input range.
   *  @return The end of the output range.
   *
   *  Write the cumulative sum (aka prefix sum, aka scan) of the input range
   *  to the output range. Each element of the output range contains the
   *  running total of all earlier elements,
   *  using `__unary_op` to transform the input elements
   *  and using `__binary_op` for summation.
   *
   *  This function generates an "inclusive" scan, meaning the Nth element
   *  of the output range is the sum of the first N input elements,
   *  so the Nth input element is included.
   */
  template<typename _InputIterator, typename _OutputIterator,
     typename _BinaryOperation, typename _UnaryOperation>
    _OutputIterator
    transform_inclusive_scan(_InputIterator __first, _InputIterator __last,
                        _OutputIterator __result,
                        _BinaryOperation __binary_op,
                        _UnaryOperation __unary_op)
    {
      if (__first != __last)
   {
     auto __init = __unary_op(*__first);
     *__result++ = __init;
     ++__first;
     if (__first != __last)
       __result = std::transform_inclusive_scan(__first, __last, __result,
                                                __binary_op, __unary_op,
                                                std::move(__init));
   }
      return __result;
    }

  // @} group numeric_ops

_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std

// Parallel STL algorithms
# if __PSTL_EXECUTION_POLICIES_DEFINED
// If <execution> has already been included, pull in implementations
#  include <pstl/glue_numeric_impl.h>
# else
// Otherwise just pull in forward declarations
#  include <pstl/glue_numeric_defs.h>
#  define __PSTL_NUMERIC_FORWARD_DECLARED 1
# endif

// Feature test macro for parallel algorithms
# define __cpp_lib_parallel_algorithm 201603L
#endif // C++17

#endif /* _GLIBCXX_NUMERIC */