Why slower than gsl on the same precision? And Which results are right?

[yixin-09]

I simplely test the clausen function of mpmath and gsl. The code is below:

> mp.prec = 53
time_begin = time.time()
print "mpmath results:"
print clsin(2,3.1415926535897327)
print "time1:"
print (time.time() - time_begin)
time_begin = time.time()
print "gsl results:"
print gsl_sf_clausen(3.1415926535897327)
print "time2"
print (time.time() - time_begin)

The results are below:

> mpmath results:
4.19482951966115e-14
time1:
0.00391387939453
gsl results:
4.18544219761e-14
time2
2.69412994385e-05

From the results I found on the same precision, the results are very different(mpmath:4.19... but gsl:4.18...), I want to know which one is more accurarcy?
Any the mpmath more than 100 times slower than gsl_sf_clausen, why?

Thank you

[KCErb]

Not an expert on Clausen functions, but for the record Mathematica gives an answer closer to mpmath: http://www.wolframalpha.com/input/?i=Clausen+Function(2,+3.1415926535897327)

[sven-mayer]

Any idea why the function is so slow? Any fix for this problem?

[skirpichev]

@sven-mayer, not sure how you run above benchmarks (perhaps, with pygsl). With ctypes I got 1000x difference:

$ python -m timeit -u usec -s 'from ctypes import CDLL, c_double;l=CDLL("libgsl.so");l.gsl_sf_clausen.restype = c_double' 'l.gsl_sf_clausen(c_double(3.1415926535897327))'
100000 loops, best of 5: 2.91 usec per loop
$ python -m timeit -u usec -s 'import mpmath;c=mpmath.fp.clsin' 'c(2,3.1415926535897327)'
200 loops, best of 5: 1.03e+03 usec per loop
$ python -m timeit -u usec -s 'import mpmath;c=mpmath.clsin' 'c(2,3.1415926535897327)'
50 loops, best of 5: 8.39e+03 usec per loop

Any idea why the function is so slow?

After all, this is a pure-python function... And the algorithm used in the gsl seems to be inaccurate for this value.