On 3/28/07, Robert Bradshaw <[EMAIL PROTECTED]> wrote:
> > If we decide that we value portability but not necessarily precision,
> > we could use RDF but avoid using C library implementations of the
> > mathematical functions. (Currently, RDF uses the C library for tan(),
> > acos(), asin(), atan(), cosh(), sinh(), tanh(), according to a quick
> > scan of real_double.pyx -- I may have missed some.) We would also
> > want to verify that the GSL implementations are portable across
> > architectures, and do something about 32-bit x86.
>
> I'm leaning towards this too. There area actually fpu_fix_* functions
> in the quad double package to handle 32-bit x86 that might be useful
> to look at.
My impression of the discussion so far is this:
* There is a vague impression that RDF is "vastly faster" than RR (mpfr),
so people are considering changing things so RDF is the default 53-bit
real field in SAGE, purely for efficiency reasons.
* There is no discussions so far about actual benchmarks, and how much
faster RDF really is.
* RDF (=machine floats) mostly uses the C library float
special functions that are potentially very
inaccurate in comparison to mpfr special functions, so the suggestion
is to maybe make RDF use better special functions.
I did a few benchmarks (see below), and 53-bit RR mpfr arithmetic in
the interpreter is maybe 10% slower than RDF, i.e., arithmetic doesn't
make enough of a difference to warrant even considering switching to
RDF from RR. I then did benchmarks with some trig functions and there
RDF is vastly faster, over 20 times faster in one benchmark.
Conclusion: The only justifiable reason to switch to RDF for the
default 53-bit real field is to speed up special function evaluation.
But ironically, the fast special functions fror 53-bit floats have
serious quality and precision issues, so we probably wouldn't switch
unless we made them better, which will necessarily make them much
slower (?).
Benchmarks:
{{{
%time
a = RDF(1.5); b = RDF(27.3)
for i in range(10^6):
c = a*b + a+b + a-b
///
CPU time: 2.09 s, Wall time: 3.33 s
}}}
{{{
%time
a = RR(1.5); b = RR(27.3)
for i in range(10^6):
c = a*b + a+b + a-b
///
CPU time: 2.14 s, Wall time: 3.95 s
}}}
{{{
%time
a = RDF(1.5); b = RDF(27.3)
for i in range(10^6):
c = a*b
///
CPU time: 0.58 s, Wall time: 1.06 s
}}}
{{{
%time
a = RR(1.5); b = RR(27.3)
for i in range(10^6):
c = a*b
///
CPU time: 0.69 s, Wall time: 0.97 s
}}}
{{{
%time
a = RDF(1.5); b = RDF(27.3)
for i in range(10^6):
c = a.sin() * b.cos()
///
CPU time: 1.64 s, Wall time: 3.01 s
}}}
{{{
%time
a = RR(1.5); b = RR(27.3)
for i in range(10^6):
c = a.sin() * b.cos()
///
CPU time: 22.17 s, Wall time: 36.39 s
}}}
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