If I understand correctly, there is no cost in representing pow(x, n) as x*x*x*x... for any positive integer n, as long as it's done correctly. C compilers don't like to change how you write out calculations unless they're asked too. So x*x*x*x will not generate the same machine code as (x*x)*(x*x). The second case is preferable, as it will result in y = x*x, sol = y*y, rather than sol = x*x*x*x. This removes the need for one computation. Also, the second way apparently results in better precision for the end result (not sure why).
I am all for writing all positive integer powers as multiplication, provided we can get the parenthesis convention correct. So x**4 -> (x*x)*(x*x), or x**11 -> x*((x*x)*(x*x)*x)*((x*x)*(x*x)*x), etc... If others are supportive of this, I'll submit a PR. On Fri, Aug 29, 2014 at 2:00 PM, Aaron Meurer <asmeu...@gmail.com> wrote: > I think we should print pow using repeated multiplication. People > might not know about --ffast-math, not realize that we are using pow > and that it is needed, or not want other optimizations that it > provides. > > Is there a reason to put a limit on the power (5 was suggested here, > 10 on the pull request)? > > Aaron Meurer > > On Fri, Aug 29, 2014 at 9:38 AM, Jason Moore <moorepa...@gmail.com> wrote: > > Sorry, it wasn't merged. He found that the --fast-math flag in the > complier > > takes care of this. > > > > > > Jason > > moorepants.info > > +01 530-601-9791 > > > > > > On Fri, Aug 29, 2014 at 10:37 AM, Jason Moore <moorepa...@gmail.com> > wrote: > >> > >> Here is some work on the pow issue: > >> https://github.com/sympy/sympy/pull/7519 > >> > >> Looks like it was merged so the ccode printer should print x*x*x... for > >> less that 10 x's. > >> > >> > >> Jason > >> moorepants.info > >> +01 530-601-9791 > >> > >> > >> On Fri, Aug 29, 2014 at 7:33 AM, Jason Moore <moorepa...@gmail.com> > wrote: > >>> > >>> > >>> > >>> > >>> Jason > >>> moorepants.info > >>> +01 530-601-9791 > >>> > >>> > >>> On Fri, Aug 29, 2014 at 2:38 AM, James Crist <crist...@umn.edu> wrote: > >>>> > >>>> I was planning on going to bed, but ended up working on this instead. > I > >>>> have no self control... > >>>> > >>>> Anyway, I've uncovered some things: > >>>> > >>>> 1. Addition of the restrict keyword to tell the compiler we're not > >>>> aliasing offers marginal gains. Gain a couple microseconds here and > there. > >>>> This requires a c99 compiler, but it's 2014, everyone should have one > by > >>>> now. > >>>> > >>>> 2. Inlining the function call resulted in smaller gains than 1, but > >>>> still *slightly* measurable. I suspect that for larger expression > sizes this > >>>> will be negligible to none. > >>>> > >>>> 3. Here's the big one: For small powers, pow(c, n) is considerably > >>>> slower than c*c*c*c... Changing the ccode Pow handler to print all > pows less > >>>> than 5 (arbitrary number) out as multiplication I was able to > match/beat > >>>> (slightly) all of jason's benchmarks with the C + numpy ufuncs. > >>> > >>> > >>> Oh yes! I knew that. In fact, I feel like I read in the current code > >>> somewhere. I forget, but that seems like a standard way we should be > >>> handling pows in C. Nice find! > >>> > >>>> > >>>> > >>>> > >>>> On Thursday, August 28, 2014 1:38:30 PM UTC-5, Tim Lahey wrote: > >>>>> > >>>>> On why Fortran is faster, Fortran semantics ensure that function > >>>>> arguments never alias, this allows the optimizer to make assumptions > about > >>>>> the function and the arguments. This the main advantage of Fortran > over C. > >>>>> But, because of this, it can lead to more memory usage. I know that > the > >>>>> newer C++ standards have a keyword to mark arguments to indicate > that they > >>>>> won't be aliased, but that requires that the code generator and the > compiler > >>>>> support them. > >>>>> > >>>>> Cheers, > >>>>> > >>>>> Tim. > >>>>> > >>>>> On 2014-08-28, at 2:17 PM, Jason Moore <moore...@gmail.com> wrote: > >>>>> > >>>>> > Jim and others, > >>>>> > > >>>>> > Here are the benchmarks I made yesterday: > >>>>> > > >>>>> > http://www.moorepants.info/blog/fast-matrix-eval.html > >>>>> > > >>>>> > The working code is here: > >>>>> > https://gist.github.com/moorepants/6ef8ab450252789a1411 > >>>>> > > >>>>> > Any feedback is welcome. > >>>>> > > >>>>> > > >>>>> > Jason > >>>>> > moorepants.info > >>>>> > +01 530-601-9791 > >>>>> > > >>>>> > > >>>>> > On Wed, Aug 27, 2014 at 11:44 PM, James Crist <cris...@umn.edu> > >>>>> > wrote: > >>>>> > I was wondering about that. I wasn't sure if the overhead from > >>>>> > looping through the inputs multiple times would outweigh > improvements from > >>>>> > fast C loops. Glad that in your case it does. > >>>>> > > >>>>> > I've thrown a WIP PR up: https://github.com/sympy/sympy/pull/7929 > >>>>> > > >>>>> > For some reason, creating the functions in python with numpy calls > >>>>> > still seems to be faster (for micro-benchmarks). This probably has > something > >>>>> > to do with function complexity (the example function above is > simple), but > >>>>> > I'd still think it'd be faster in pure C. I tried inlining the > call, which > >>>>> > was a small improvement, but it was still slower than the pure > numpy-python > >>>>> > version. Something to look into. > >>>>> > > >>>>> > > >>>>> > On Wed, Aug 27, 2014 at 10:28 PM, Jason Moore <moore...@gmail.com> > >>>>> > wrote: > >>>>> > Yeh, but if you simply create a ufunc for each expression in a > matrix > >>>>> > you still get substantial speedups. I wrote a bunch of test cases > that I'll > >>>>> > post to my blog tomorrow. > >>>>> > > >>>>> > > >>>>> > Jason > >>>>> > moorepants.info > >>>>> > +01 530-601-9791 > >>>>> > > >>>>> > > >>>>> > On Wed, Aug 27, 2014 at 11:26 PM, James Crist <cris...@umn.edu> > >>>>> > wrote: > >>>>> > Not yet. I wrote it this morning during an extremely boring > meeting, > >>>>> > and haven't had a chance to clean it up. This doesn't solve your > problem > >>>>> > about broadcasting a matrix calculation though... > >>>>> > > >>>>> > > >>>>> > On Wed, Aug 27, 2014 at 10:23 PM, Jason Moore <moore...@gmail.com> > >>>>> > wrote: > >>>>> > Awesome. I was working on this today but it looks like you've by > >>>>> > passed what I had working. Do you have a PR with this? > >>>>> > > >>>>> > > >>>>> > Jason > >>>>> > moorepants.info > >>>>> > +01 530-601-9791 > >>>>> > > >>>>> > > >>>>> > On Wed, Aug 27, 2014 at 11:11 PM, Matthew Rocklin < > mroc...@gmail.com> > >>>>> > wrote: > >>>>> > Cool > >>>>> > > >>>>> > > >>>>> > On Wed, Aug 27, 2014 at 8:07 PM, James Crist <cris...@umn.edu> > wrote: > >>>>> > I still need to do some cleanups and add tests, but I finally have > >>>>> > this working and thought I'd share. I'm really happy with this: > >>>>> > > >>>>> > In [1]: from sympy import * > >>>>> > > >>>>> > In [2]: a, b, c = symbols('a, b, c') > >>>>> > > >>>>> > In [3]: expr = (sin(a) + sqrt(b)*c**2)/2 > >>>>> > > >>>>> > In [4]: from sympy.utilities.autowrap import ufuncify > >>>>> > > >>>>> > In [5]: func = ufuncify((a, b, c), expr) > >>>>> > > >>>>> > In [6]: func(1, 2, 3) > >>>>> > Out[6]: 6.7846965230828769 > >>>>> > > >>>>> > In [7]: func([1, 2, 3, 4, 5], [6, 7, 8, 9, 10], 3) > >>>>> > Out[7]: array([ 11.44343933, 12.36052961, 12.79848207, > >>>>> > 13.12159875, 13.75078733]) > >>>>> > > >>>>> > In [8]: from numpy import arange > >>>>> > > >>>>> > In [9]: a = arange(10).reshape((2, 5)) > >>>>> > > >>>>> > In [10]: c = arange(10, 20).reshape((2, 5)) > >>>>> > > >>>>> > In [11]: b = 25 > >>>>> > > >>>>> > In [12]: func(a, b, c) > >>>>> > Out[12]: > >>>>> > array([[ 250. , 302.92073549, 360.45464871, 422.57056 , > >>>>> > 489.62159875], > >>>>> > [ 562.02053786, 639.86029225, 722.8284933 , 810.49467912, > >>>>> > 902.70605924]]) > >>>>> > > >>>>> > In [13]: type(func) > >>>>> > Out[13]: numpy.ufunc > >>>>> > > >>>>> > This now does everything a numpy `ufunc` does normally, as it *is* > a > >>>>> > ufunc. Codegen is hooked up to numpy api. Type conversion and > broadcasting > >>>>> > are done automagically. > >>>>> > > >>>>> > Caveats: only functions with a single output are accepted (this > could > >>>>> > be changed to accept multi-output without much effort though). > Also, as with > >>>>> > all unfuncs, input/outputs must all be scalars (no matrix/Indexed > operations > >>>>> > allowed). > >>>>> > > >>>>> > -- > >>>>> > You received this message because you are subscribed to the Google > >>>>> > Groups "sympy" group. > >>>>> > To unsubscribe from this group and stop receiving emails from it, > >>>>> > send an email to sympy+un...@googlegroups.com. > >>>>> > To post to this group, send email to sy...@googlegroups.com. > >>>>> > Visit this group at http://groups.google.com/group/sympy. > >>>>> > To view this discussion on the web visit > >>>>> > > https://groups.google.com/d/msgid/sympy/76e0fbbe-5ce4-43b7-855b-6ac821f6b8ae%40googlegroups.com > . > >>>>> > For more options, visit https://groups.google.com/d/optout. > >>>>> > > >>>>> > > >>>>> > -- > >>>>> > You received this message because you are subscribed to the Google > >>>>> > Groups "sympy" group. > >>>>> > To unsubscribe from this group and stop receiving emails from it, > >>>>> > send an email to sympy+un...@googlegroups.com. > >>>>> > To post to this group, send email to sy...@googlegroups.com. > >>>>> > Visit this group at http://groups.google.com/group/sympy. > >>>>> > To view this discussion on the web visit > >>>>> > > https://groups.google.com/d/msgid/sympy/CAJ8oX-EHZXbd5aFFNRy7gJ0hcydpAsG2qxv7Py65DQ9cA9VUUA%40mail.gmail.com > . > >>>>> > > >>>>> > For more options, visit https://groups.google.com/d/optout. > >>>>> > > >>>>> > > >>>>> > -- > >>>>> > You received this message because you are subscribed to a topic in > >>>>> > the Google Groups "sympy" group. > >>>>> > To unsubscribe from this topic, visit > >>>>> > https://groups.google.com/d/topic/sympy/azVZHLOv9Vc/unsubscribe. > >>>>> > To unsubscribe from this group and all its topics, send an email to > >>>>> > sympy+un...@googlegroups.com. > >>>>> > > >>>>> > To post to this group, send email to sy...@googlegroups.com. > >>>>> > Visit this group at http://groups.google.com/group/sympy. > >>>>> > To view this discussion on the web visit > >>>>> > > https://groups.google.com/d/msgid/sympy/CAP7f1AieaeoOFtc_S4XPxWOX2jr2zmda9VCRpWpzHMTGLkmHPQ%40mail.gmail.com > . > >>>>> > > >>>>> > For more options, visit https://groups.google.com/d/optout. > >>>>> > > >>>>> > > >>>>> > -- > >>>>> > You received this message because you are subscribed to the Google > >>>>> > Groups "sympy" group. > >>>>> > To unsubscribe from this group and stop receiving emails from it, > >>>>> > send an email to sympy+un...@googlegroups.com. > >>>>> > To post to this group, send email to sy...@googlegroups.com. > >>>>> > Visit this group at http://groups.google.com/group/sympy. > >>>>> > To view this discussion on the web visit > >>>>> > > https://groups.google.com/d/msgid/sympy/CAJ2L7mfL_xO%3DO-ZRMx-zfpZzJKJ-%2BUdTzSCz5jYf%2B%3DdovR%2B_7Q%40mail.gmail.com > . > >>>>> > > >>>>> > For more options, visit https://groups.google.com/d/optout. > >>>>> > > >>>>> > > >>>>> > -- > >>>>> > You received this message because you are subscribed to a topic in > >>>>> > the Google Groups "sympy" group. > >>>>> > To unsubscribe from this topic, visit > >>>>> > https://groups.google.com/d/topic/sympy/azVZHLOv9Vc/unsubscribe. > >>>>> > To unsubscribe from this group and all its topics, send an email to > >>>>> > sympy+un...@googlegroups.com. > >>>>> > To post to this group, send email to sy...@googlegroups.com. > >>>>> > Visit this group at http://groups.google.com/group/sympy. > >>>>> > To view this discussion on the web visit > >>>>> > > https://groups.google.com/d/msgid/sympy/CAP7f1AjcHrsopXjwK5uYdALeSrokxLMwA7xebTikHyhwL-%2BOVg%40mail.gmail.com > . > >>>>> > > >>>>> > For more options, visit https://groups.google.com/d/optout. > >>>>> > > >>>>> > > >>>>> > -- > >>>>> > You received this message because you are subscribed to the Google > >>>>> > Groups "sympy" group. > >>>>> > To unsubscribe from this group and stop receiving emails from it, > >>>>> > send an email to sympy+un...@googlegroups.com. > >>>>> > To post to this group, send email to sy...@googlegroups.com. > >>>>> > Visit this group at http://groups.google.com/group/sympy. > >>>>> > To view this discussion on the web visit > >>>>> > > https://groups.google.com/d/msgid/sympy/CAJ2L7me73iJmkWm%3D_LiyWrsuOCZm%2B4OZbqD%2BkwwScWWx23HVdg%40mail.gmail.com > . > >>>>> > > >>>>> > For more options, visit https://groups.google.com/d/optout. > >>>>> > > >>>>> > > >>>>> > -- > >>>>> > You received this message because you are subscribed to the Google > >>>>> > Groups "sympy" group. > >>>>> > To unsubscribe from this group and stop receiving emails from it, > >>>>> > send an email to sympy+un...@googlegroups.com. > >>>>> > To post to this group, send email to 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