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/**
* \file Accumulator.hpp
* \brief Header for GeographicLib::Accumulator class
*
* Copyright (c) Charles Karney (2010-2020) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_ACCUMULATOR_HPP)
#define GEOGRAPHICLIB_ACCUMULATOR_HPP 1
#include <GeographicLib/Constants.hpp>
namespace GeographicLib {
/**
* \brief An accumulator for sums
*
* This allows many numbers of floating point type \e T to be added together
* with twice the normal precision. Thus if \e T is double, the effective
* precision of the sum is 106 bits or about 32 decimal places.
*
* The implementation follows J. R. Shewchuk,
* <a href="https://doi.org/10.1007/PL00009321"> Adaptive Precision
* Floating-Point Arithmetic and Fast Robust Geometric Predicates</a>,
* Discrete & Computational Geometry 18(3) 305--363 (1997).
*
* Approximate timings (summing a vector<double>)
* - double: 2ns
* - Accumulator<double>: 23ns
*
* In the documentation of the member functions, \e sum stands for the value
* currently held in the accumulator.
*
* Example of use:
* \include example-Accumulator.cpp
**********************************************************************/
template<typename T = Math::real>
class GEOGRAPHICLIB_EXPORT Accumulator {
private:
// _s + _t accumulators for the sum.
T _s, _t;
// Same as Math::sum, but requires abs(u) >= abs(v). This isn't currently
// used.
static T fastsum(T u, T v, T& t) {
GEOGRAPHICLIB_VOLATILE T s = u + v;
GEOGRAPHICLIB_VOLATILE T vp = s - u;
t = v - vp;
return s;
}
void Add(T y) {
// Here's Shewchuk's solution...
T u; // hold exact sum as [s, t, u]
// Accumulate starting at least significant end
y = Math::sum(y, _t, u);
_s = Math::sum(y, _s, _t);
// Start is _s, _t decreasing and non-adjacent. Sum is now (s + t + u)
// exactly with s, t, u non-adjacent and in decreasing order (except for
// possible zeros). The following code tries to normalize the result.
// Ideally, we want _s = round(s+t+u) and _u = round(s+t+u - _s). The
// following does an approximate job (and maintains the decreasing
// non-adjacent property). Here are two "failures" using 3-bit floats:
//
// Case 1: _s is not equal to round(s+t+u) -- off by 1 ulp
// [12, -1] - 8 -> [4, 0, -1] -> [4, -1] = 3 should be [3, 0] = 3
//
// Case 2: _s+_t is not as close to s+t+u as it shold be
// [64, 5] + 4 -> [64, 8, 1] -> [64, 8] = 72 (off by 1)
// should be [80, -7] = 73 (exact)
//
// "Fixing" these problems is probably not worth the expense. The
// representation inevitably leads to small errors in the accumulated
// values. The additional errors illustrated here amount to 1 ulp of the
// less significant word during each addition to the Accumulator and an
// additional possible error of 1 ulp in the reported sum.
//
// Incidentally, the "ideal" representation described above is not
// canonical, because _s = round(_s + _t) may not be true. For example,
// with 3-bit floats:
//
// [128, 16] + 1 -> [160, -16] -- 160 = round(145).
// But [160, 0] - 16 -> [128, 16] -- 128 = round(144).
//
if (_s == 0) // This implies t == 0,
_s = u; // so result is u
else
_t += u; // otherwise just accumulate u to t.
}
T Sum(T y) const {
Accumulator a(*this);
a.Add(y);
return a._s;
}
public:
/**
* Construct from a \e T. This is not declared explicit, so that you can
* write <code>Accumulator<double> a = 5;</code>.
*
* @param[in] y set \e sum = \e y.
**********************************************************************/
Accumulator(T y = T(0)) : _s(y), _t(0) {
static_assert(!std::numeric_limits<T>::is_integer,
"Accumulator type is not floating point");
}
/**
* Set the accumulator to a number.
*
* @param[in] y set \e sum = \e y.
**********************************************************************/
Accumulator& operator=(T y) { _s = y; _t = 0; return *this; }
/**
* Return the value held in the accumulator.
*
* @return \e sum.
**********************************************************************/
T operator()() const { return _s; }
/**
* Return the result of adding a number to \e sum (but don't change \e
* sum).
*
* @param[in] y the number to be added to the sum.
* @return \e sum + \e y.
**********************************************************************/
T operator()(T y) const { return Sum(y); }
/**
* Add a number to the accumulator.
*
* @param[in] y set \e sum += \e y.
**********************************************************************/
Accumulator& operator+=(T y) { Add(y); return *this; }
/**
* Subtract a number from the accumulator.
*
* @param[in] y set \e sum -= \e y.
**********************************************************************/
Accumulator& operator-=(T y) { Add(-y); return *this; }
/**
* Multiply accumulator by an integer. To avoid loss of accuracy, use only
* integers such that \e n × \e T is exactly representable as a \e T
* (i.e., ± powers of two). Use \e n = −1 to negate \e sum.
*
* @param[in] n set \e sum *= \e n.
**********************************************************************/
Accumulator& operator*=(int n) { _s *= n; _t *= n; return *this; }
/**
* Multiply accumulator by a number. The fma (fused multiply and add)
* instruction is used (if available) in order to maintain accuracy.
*
* @param[in] y set \e sum *= \e y.
**********************************************************************/
Accumulator& operator*=(T y) {
using std::fma;
T d = _s; _s *= y;
d = fma(y, d, -_s); // the error in the first multiplication
_t = fma(y, _t, d); // add error to the second term
return *this;
}
/**
* Reduce accumulator to the range [-y/2, y/2].
*
* @param[in] y the modulus.
**********************************************************************/
Accumulator& remainder(T y) {
using std::remainder;
_s = remainder(_s, y);
Add(0); // This renormalizes the result.
return *this;
}
/**
* Test equality of an Accumulator with a number.
**********************************************************************/
bool operator==(T y) const { return _s == y; }
/**
* Test inequality of an Accumulator with a number.
**********************************************************************/
bool operator!=(T y) const { return _s != y; }
/**
* Less operator on an Accumulator and a number.
**********************************************************************/
bool operator<(T y) const { return _s < y; }
/**
* Less or equal operator on an Accumulator and a number.
**********************************************************************/
bool operator<=(T y) const { return _s <= y; }
/**
* Greater operator on an Accumulator and a number.
**********************************************************************/
bool operator>(T y) const { return _s > y; }
/**
* Greater or equal operator on an Accumulator and a number.
**********************************************************************/
bool operator>=(T y) const { return _s >= y; }
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_ACCUMULATOR_HPP