GeographicLib  1.52
Accumulator.hpp
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1 /**
2  * \file Accumulator.hpp
3  * \brief Header for GeographicLib::Accumulator class
4  *
5  * Copyright (c) Charles Karney (2010-2020) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * https://geographiclib.sourceforge.io/
8  **********************************************************************/
9 
10 #if !defined(GEOGRAPHICLIB_ACCUMULATOR_HPP)
11 #define GEOGRAPHICLIB_ACCUMULATOR_HPP 1
12 
14 
15 namespace GeographicLib {
16 
17  /**
18  * \brief An accumulator for sums
19  *
20  * This allows many numbers of floating point type \e T to be added together
21  * with twice the normal precision. Thus if \e T is double, the effective
22  * precision of the sum is 106 bits or about 32 decimal places.
23  *
24  * The implementation follows J. R. Shewchuk,
25  * <a href="https://doi.org/10.1007/PL00009321"> Adaptive Precision
26  * Floating-Point Arithmetic and Fast Robust Geometric Predicates</a>,
27  * Discrete & Computational Geometry 18(3) 305--363 (1997).
28  *
29  * Approximate timings (summing a vector<double>)
30  * - double: 2ns
31  * - Accumulator<double>: 23ns
32  *
33  * In the documentation of the member functions, \e sum stands for the value
34  * currently held in the accumulator.
35  *
36  * Example of use:
37  * \include example-Accumulator.cpp
38  **********************************************************************/
39  template<typename T = Math::real>
41  private:
42  // _s + _t accumulators for the sum.
43  T _s, _t;
44  // Same as Math::sum, but requires abs(u) >= abs(v). This isn't currently
45  // used.
46  static T fastsum(T u, T v, T& t) {
47  GEOGRAPHICLIB_VOLATILE T s = u + v;
48  GEOGRAPHICLIB_VOLATILE T vp = s - u;
49  t = v - vp;
50  return s;
51  }
52  void Add(T y) {
53  // Here's Shewchuk's solution...
54  T u; // hold exact sum as [s, t, u]
55  // Accumulate starting at least significant end
56  y = Math::sum(y, _t, u);
57  _s = Math::sum(y, _s, _t);
58  // Start is _s, _t decreasing and non-adjacent. Sum is now (s + t + u)
59  // exactly with s, t, u non-adjacent and in decreasing order (except for
60  // possible zeros). The following code tries to normalize the result.
61  // Ideally, we want _s = round(s+t+u) and _u = round(s+t+u - _s). The
62  // following does an approximate job (and maintains the decreasing
63  // non-adjacent property). Here are two "failures" using 3-bit floats:
64  //
65  // Case 1: _s is not equal to round(s+t+u) -- off by 1 ulp
66  // [12, -1] - 8 -> [4, 0, -1] -> [4, -1] = 3 should be [3, 0] = 3
67  //
68  // Case 2: _s+_t is not as close to s+t+u as it shold be
69  // [64, 5] + 4 -> [64, 8, 1] -> [64, 8] = 72 (off by 1)
70  // should be [80, -7] = 73 (exact)
71  //
72  // "Fixing" these problems is probably not worth the expense. The
73  // representation inevitably leads to small errors in the accumulated
74  // values. The additional errors illustrated here amount to 1 ulp of the
75  // less significant word during each addition to the Accumulator and an
76  // additional possible error of 1 ulp in the reported sum.
77  //
78  // Incidentally, the "ideal" representation described above is not
79  // canonical, because _s = round(_s + _t) may not be true. For example,
80  // with 3-bit floats:
81  //
82  // [128, 16] + 1 -> [160, -16] -- 160 = round(145).
83  // But [160, 0] - 16 -> [128, 16] -- 128 = round(144).
84  //
85  if (_s == 0) // This implies t == 0,
86  _s = u; // so result is u
87  else
88  _t += u; // otherwise just accumulate u to t.
89  }
90  T Sum(T y) const {
91  Accumulator a(*this);
92  a.Add(y);
93  return a._s;
94  }
95  public:
96  /**
97  * Construct from a \e T. This is not declared explicit, so that you can
98  * write <code>Accumulator<double> a = 5;</code>.
99  *
100  * @param[in] y set \e sum = \e y.
101  **********************************************************************/
102  Accumulator(T y = T(0)) : _s(y), _t(0) {
103  static_assert(!std::numeric_limits<T>::is_integer,
104  "Accumulator type is not floating point");
105  }
106  /**
107  * Set the accumulator to a number.
108  *
109  * @param[in] y set \e sum = \e y.
110  **********************************************************************/
111  Accumulator& operator=(T y) { _s = y; _t = 0; return *this; }
112  /**
113  * Return the value held in the accumulator.
114  *
115  * @return \e sum.
116  **********************************************************************/
117  T operator()() const { return _s; }
118  /**
119  * Return the result of adding a number to \e sum (but don't change \e
120  * sum).
121  *
122  * @param[in] y the number to be added to the sum.
123  * @return \e sum + \e y.
124  **********************************************************************/
125  T operator()(T y) const { return Sum(y); }
126  /**
127  * Add a number to the accumulator.
128  *
129  * @param[in] y set \e sum += \e y.
130  **********************************************************************/
131  Accumulator& operator+=(T y) { Add(y); return *this; }
132  /**
133  * Subtract a number from the accumulator.
134  *
135  * @param[in] y set \e sum -= \e y.
136  **********************************************************************/
137  Accumulator& operator-=(T y) { Add(-y); return *this; }
138  /**
139  * Multiply accumulator by an integer. To avoid loss of accuracy, use only
140  * integers such that \e n &times; \e T is exactly representable as a \e T
141  * (i.e., &plusmn; powers of two). Use \e n = &minus;1 to negate \e sum.
142  *
143  * @param[in] n set \e sum *= \e n.
144  **********************************************************************/
145  Accumulator& operator*=(int n) { _s *= n; _t *= n; return *this; }
146  /**
147  * Multiply accumulator by a number. The fma (fused multiply and add)
148  * instruction is used (if available) in order to maintain accuracy.
149  *
150  * @param[in] y set \e sum *= \e y.
151  **********************************************************************/
153  using std::fma;
154  T d = _s; _s *= y;
155  d = fma(y, d, -_s); // the error in the first multiplication
156  _t = fma(y, _t, d); // add error to the second term
157  return *this;
158  }
159  /**
160  * Reduce accumulator to the range [-y/2, y/2].
161  *
162  * @param[in] y the modulus.
163  **********************************************************************/
165  using std::remainder;
166  _s = remainder(_s, y);
167  Add(0); // This renormalizes the result.
168  return *this;
169  }
170  /**
171  * Test equality of an Accumulator with a number.
172  **********************************************************************/
173  bool operator==(T y) const { return _s == y; }
174  /**
175  * Test inequality of an Accumulator with a number.
176  **********************************************************************/
177  bool operator!=(T y) const { return _s != y; }
178  /**
179  * Less operator on an Accumulator and a number.
180  **********************************************************************/
181  bool operator<(T y) const { return _s < y; }
182  /**
183  * Less or equal operator on an Accumulator and a number.
184  **********************************************************************/
185  bool operator<=(T y) const { return _s <= y; }
186  /**
187  * Greater operator on an Accumulator and a number.
188  **********************************************************************/
189  bool operator>(T y) const { return _s > y; }
190  /**
191  * Greater or equal operator on an Accumulator and a number.
192  **********************************************************************/
193  bool operator>=(T y) const { return _s >= y; }
194  };
195 
196 } // namespace GeographicLib
197 
198 #endif // GEOGRAPHICLIB_ACCUMULATOR_HPP
Header for GeographicLib::Constants class.
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:66
#define GEOGRAPHICLIB_VOLATILE
Definition: Math.hpp:58
An accumulator for sums.
Definition: Accumulator.hpp:40
Accumulator & operator-=(T y)
bool operator>=(T y) const
Accumulator & operator*=(int n)
bool operator!=(T y) const
bool operator<=(T y) const
bool operator==(T y) const
Accumulator & operator*=(T y)
Accumulator & operator=(T y)
Accumulator & operator+=(T y)
Accumulator & remainder(T y)
static T sum(T u, T v, T &t)
Definition: Math.cpp:105
Namespace for GeographicLib.
Definition: Accumulator.cpp:12