[Numpy-discussion] Fast vectorized arithmetic with ~32 significant digits under Numpy

Charles R Harris charlesr.harris at gmail.com
Fri Dec 11 10:46:26 EST 2015

On Fri, Dec 11, 2015 at 6:25 AM, Thomas Baruchel <baruchel at gmx.com> wrote:

> From time to time it is asked on forums how to extend precision of
> computation on Numpy array. The most common answer
> given to this question is: use the dtype=object with some arbitrary
> precision module like mpmath or gmpy.
> See
> http://stackoverflow.com/questions/6876377/numpy-arbitrary-precision-linear-algebra
> or http://stackoverflow.com/questions/21165745/precision-loss-numpy-mpmath
> or
> http://stackoverflow.com/questions/15307589/numpy-array-with-mpz-mpfr-values
> While this is obviously the most relevant answer for many users because it
> will allow them to use Numpy arrays exactly
> as they would have used them with native types, the wrong thing is that
> from some point of view "true" vectorization
> will be lost.
> With years I got very familiar with the extended double-double type which
> has (for usual architectures) about 32 accurate
> digits with faster arithmetic than "arbitrary precision types". I even
> used it for research purpose in number theory and
> I got convinced that it is a very wonderful type as long as such precision
> is suitable.
> I often implemented it partially under Numpy, most of the time by trying
> to vectorize at a low-level the libqd library.
> But I recently thought that a very nice and portable way of implementing
> it under Numpy would be to use the existing layer
> of vectorization on floats for computing the arithmetic operations by
> "columns containing half of the numbers" rather than
> by "full numbers". As a proof of concept I wrote the following file:
> https://gist.github.com/baruchel/c86ed748939534d8910d
> I converted and vectorized the Algol 60 codes from
> http://szmoore.net/ipdf/documents/references/dekker1971afloating.pdf
> (Dekker, 1971).
> A test is provided at the end; for inverting 100,000 numbers, my type is
> about 3 or 4 times faster than GMPY and almost
> 50 times faster than MPmath. It should be even faster for some other
> operations since I had to create another np.ones
> array for testing this type because inversion isn't implemented here
> (which could of course be done). You can run this file by yourself
> (maybe you will have to discard mpmath or gmpy if you don't have it).
> I would like to discuss about the way to make available something related
> to that.
> a) Would it be relevant to include that in Numpy ? (I would think to some
> "contribution"-tool rather than including it in
> the core of Numpy because it would be painful to code all ufuncs; on the
> other hand I am pretty sure that many would be happy
> to perform several arithmetic operations by knowing that they can't use
> cos/sin/etc. on this type; in other words, I am not
> sure it would be a good idea to embed it as an every-day type but I think
> it would be nice to have it quickly available
> in some way). If you agree with that, in which way should I code it (the
> current link only is a "proof of concept"; I would
> be very happy to code it in some cleaner way)?
> b) Do you think such attempt should remain something external to Numpy
> itself and be released on my Github account without being
> integrated to Numpy?

I think astropy does something similar for time and dates. There has also
been some talk of adding a user type for ieee 128 bit doubles. I've looked
once for relevant code for the latter and, IIRC, the available packages
were GPL :(.

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