Cool! Great work! On Friday, 29 August 2014 22:48:23 UTC+2, James Crist wrote: > > 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. >
Compilation will be slower for large n. > 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. > I prefer either to pass -ffast-math flag (setting compiler flags is already an issue since we need to indicate optimization level, right?) or write a specialized callback to be inlined (tuned to an Intel i7): https://github.com/bjodah/cinterpol/blob/master/cInterpol/power.c and tell sympy to generate code using it: https://github.com/sympy/sympy/pull/7517 > > On Fri, Aug 29, 2014 at 2:00 PM, Aaron Meurer <[email protected] > <javascript:>> 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 <[email protected] >> <javascript:>> 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 <[email protected] >> <javascript:>> 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 <[email protected] >> <javascript:>> wrote: >> >>> >> >>> >> >>> >> >>> >> >>> Jason >> >>> moorepants.info >> >>> +01 530-601-9791 >> >>> >> >>> >> >>> On Fri, Aug 29, 2014 at 2:38 AM, James Crist <[email protected] >> <javascript:>> 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 <[email protected]> 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 <[email protected]> >> >>>>> > 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 <[email protected] >> > >> >>>>> > 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 <[email protected]> >> >>>>> > 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 <[email protected] >> > >> >>>>> > 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 < >> [email protected]> >> >>>>> > wrote: >> >>>>> > Cool >> >>>>> > >> >>>>> > >> >>>>> > On Wed, Aug 27, 2014 at 8:07 PM, James Crist <[email protected]> >> 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 [email protected]. >> >>>>> > To post to this group, send email to [email protected]. >> >>>>> > 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 [email protected]. >> >>>>> > To post to this group, send email to [email protected]. >> >>>>> > 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 >> >>>>> > [email protected]. >> >>>>> > >> >>>>> > To post to this group, send email to [email protected]. >> >>>>> > 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 [email protected]. >> >>>>> > To post to this group, send email to [email protected]. >> >>>>> > 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 >> >>>>> > [email protected]. >> >>>>> > To post to this group, send email to [email protected]. >> >>>>> > 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. 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