magnum/src/Magnum/Math/Algorithms/KahanSum.h

#ifndef Magnum_Math_Algorithms_KahanSum_h
#define Magnum_Math_Algorithms_KahanSum_h
/*
    This file is part of Magnum.

    Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
                2020, 2021, 2022, 2023 Vladimír Vondruš <mosra@centrum.cz>

    Permission is hereby granted, free of charge, to any person obtaining a
    copy of this software and associated documentation files (the "Software"),
    to deal in the Software without restriction, including without limitation
    the rights to use, copy, modify, merge, publish, distribute, sublicense,
    and/or sell copies of the Software, and to permit persons to whom the
    Software is furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included
    in all copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
    THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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*/

/** @file
 * @brief Function @ref Magnum::Math::Algorithms::kahanSum()
 */

/* std::declval() is said to be in <utility> but libstdc++, libc++ and MSVC STL
   all have it directly in <type_traits> because it just makes sense */
#include <type_traits>

namespace Magnum { namespace Math { namespace Algorithms {

/**
@brief Kahan summation algorithm
@param[in] begin            Range begin
@param[in] end              Range end
@param[in] sum              Initial value for the sum
@param[in,out] compensation Floating-point roundoff error compensation value.
    If non-@cpp nullptr @ce, value behind the pointer is used as initial
    compensation value and the resulting compensation value is saved back
    there.

Calculates a sum of a large range of floating-point numbers with roundoff error
compensation. Compared to for example @ref std::accumulate() the algorithm
significantly reduces numerical error in the total. See the
[Kahan summation algorithm](https://en.wikipedia.org/wiki/Kahan_summation_algorithm)
article on Wikipedia for an in-depth explanation. Example with summation of a
hundred million ones:

@snippet MagnumMathAlgorithms.cpp kahanSum

If required, it is also possible to use this algorithm on non-contiguous ranges
or single values (for example when calculating sum of pixel values in an image
with some row padding or when the inputs are generated / converted from other
values):

@snippet MagnumMathAlgorithms.cpp kahanSum-iterative
*/
template<class Iterator, class T = typename std::decay<decltype(*std::declval<Iterator>())>::type> T kahanSum(Iterator begin, Iterator end, T sum = T(0), T* compensation = nullptr) {
    T c = compensation ? *compensation : T(0);
    for(Iterator it = begin; it != end; ++it) {
        const T y = *it - c;
        const T t = sum + y;
        c = (t - sum) - y;
        sum = t;
    }

    if(compensation) *compensation = c;
    return sum;
}

}}}

#endif