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Accurate sums and dot products for Python

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Accurate sums and (dot) products for Python.

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Summing up values in a list can get tricky if the values are floating point numbers; digit cancellation can occur and the result may come out wrong. A classical example is the sum

1.0e16 + 1.0 - 1.0e16

The actual result is 1.0, but in double precision, this will result in 0.0. While in this example the failure is quite obvious, it can get a lot more tricky than that. accupy provides

p, exact, cond = accupy.generate_ill_conditioned_sum(100, 1.0e20)

which, given a length and a target condition number, will produce an array of floating point numbers that is hard to sum up.

accupy has the following methods for summation:

  • accupy.kahan_sum(p): Kahan summation

  • accupy.fsum(p): A vectorization wrapper around math.fsum (which uses Shewchuck's algorithm [1] (see also here)).

  • accupy.ksum(p, K=2): Summation in K-fold precision (from [2])

All summation methods sum the first dimension of a multidimensional NumPy array.

Let's compare them.

Accuracy comparison (sum)

As expected, the naive sum performs very badly with ill-conditioned sums; likewise for numpy.sum which uses pairwise summation. Kahan summation not significantly better; this, too, is expected.

Computing the sum with 2-fold accuracy in accupy.ksum gives the correct result if the condition is at most in the range of machine precision; further increasing K helps with worse conditions.

Shewchuck's algorithm in math.fsum always gives the correct result to full floating point precision.

Runtime comparison (sum)

We compare more and more sums of fixed size (above) and larger and larger sums, but a fixed number of them (below). In both cases, the least accurate method is the fastest (numpy.sum), and the most accurate the slowest (accupy.fsum).

Dot products

accupy has the following methods for dot products:

  • accupy.fdot(p): A transformation of the dot product of length n into a sum of length 2n, computed with math.fsum

  • accupy.kdot(p, K=2): Dot product in K-fold precision (from [2])

Let's compare them.

Accuracy comparison (dot)

accupy can construct ill-conditioned dot products with

x, y, exact, cond = accupy.generate_ill_conditioned_dot_product(100, 1.0e20)

With this, the accuracy of the different methods is compared.

As for sums, is the least accurate, followed by instanced of kdot. fdot is provably accurate up into the last digit

Runtime comparison (dot)

NumPy's is much faster than all alternatives provided by accupy. This is because the bookkeeping of truncation errors takes more steps, but mostly because of NumPy's highly optimized dot implementation.


  1. Richard Shewchuk, Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates, J. Discrete Comput. Geom. (1997), 18(305), 305–363

  2. Takeshi Ogita, Siegfried M. Rump, and Shin'ichi Oishi, Accurate Sum and Dot Product, SIAM J. Sci. Comput. (2006), 26(6), 1955–1988 (34 pages)


accupy needs the C++ Eigen library, provided in Debian/Ubuntu by libeigen3-dev.


accupy is available from the Python Package Index, so with

pip install -U accupy

you can install/upgrade.


To run the tests, just check out this repository and type



To create a new release

  1. bump the __version__ number,

  2. publish to PyPi and GitHub:

    $ make publish


accupy is published under the MIT license.

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