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

Anne Archibald archibald at astron.nl
Fri Dec 11 11:22:55 EST 2015

Actually, GCC implements 128-bit floats in software and provides them as
__float128; there are also quad-precision versions of the usual functions.
The Intel compiler provides this as well, I think, but I don't think
Microsoft compilers do. A portable quad-precision library might be less

The cleanest way to add extended precision to numpy is by adding a
C-implemented dtype. This can be done in an extension module; see the
quaternion and half-precision modules online.


On Fri, Dec 11, 2015, 16:46 Charles R Harris <charlesr.harris at gmail.com>

> 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 :(.
> Chuck
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