Initial SymPy benchmark suite

Initial SymPy benchmark suite

I've just went through our commit history and searched for commits with
%timeit or %time in their description and extracted benchmarks from
there.

So now we have at least something. Here is how it looks like:

    kirr@evo:~/src/sympy/sympy-git$ ./bin/py.bench sympy/
    ====================== test process starts =======================
    executable:   /usr/bin/python  (2.5.2-final-0)
    using py lib: /home/kirr/src/tools/py/py/py <rev unknown>

    sympy/benchmarks/bench_symbench.py[8] ........
    sympy/core/benchmarks/bench_arit.py[10] ..........
    sympy/core/benchmarks/bench_assumptions.py[2] ..
    sympy/core/benchmarks/bench_basic.py[4] ....
    sympy/core/benchmarks/bench_expand.py[4] ....
    sympy/core/benchmarks/bench_numbers.py[21] .....................
    sympy/core/benchmarks/bench_sympify.py[2] ..
    sympy/functions/elementary/benchmarks/bench_exp.py[1] .
    sympy/functions/special/benchmarks/bench_special.py[1] .
    sympy/integrals/benchmarks/bench_integrate.py[4] ....
    sympy/integrals/benchmarks/bench_trigintegrate.py[2] ..
    sympy/matrices/benchmarks/bench_matrix.py[4] ....
    sympy/polynomials/benchmarks/bench_factor.py[2] ..
    sympy/series/benchmarks/bench_limit.py[1] .
    sympy/solvers/benchmarks/bench_solvers.py[1] .

    =========== tests finished: 68 passed in 64.66 seconds ===========

    ==============================
     *** BENCHMARKING RESULTS ***
    ==============================

    bench_R1                        |      670              ms  |  real(f(f(f(f(f(f(f(f(f(f(i/2)))))))))))
    bench_R2                        |      664              ms  |  Hermite polynomial hermite(15, y)
    bench_R3                        |            731        us  |  a = [bool(f==f) for _ in range(10)]
    bench_R4                        |              6.2      us  |  # we don't have Tuples
    bench_R5                        |       12.9            ms  |  blowup(L, 8); L=uniq(L)
    bench_R6                        |      465              ms  |  sum(trim((x+sin(i))/x+(x-sin(i))/x) for i in xrange(100))
    bench_R10                       |      150              ms  |  v = [-pi,-pi+1/10..,pi]
    bench_R11                       |  3.17                 s   |  a = [random() + random()*I for w in [0..1000]]
    timeit_neg                      |             20.5      us  |  -x
    timeit_Add_x1                   |             28.7      us  |  x+1
    timeit_Add_1x                   |             42.3      us  |  1+x
    timeit_Add_x05                  |             24.3      us  |  x+0.5
    timeit_Add_xy                   |             24.3      us  |  x+y
    timeit_Add_xyz                  |             23.3      us  |  Add(*[x,y,z])
    timeit_Mul_xy                   |             24.1      us  |  x*y
    timeit_Mul_xyz                  |             22.8      us  |  Mul(*[x,y,z])
    timeit_Div_xy                   |             39.4      us  |  x/y
    timeit_Div_2y                   |             46.6      us  |  2/y
    timeit_x_is_integer             |              4.72     us  |  x.is_integer
    timeit_Integer_is_irrational    |              1.06     us  |  i3.is_irrational
    timeit_Symbol_meth_lookup       |              1.07     us  |  x.diff  # no call, just method lookup
    timeit_S_lookup                 |                  765  ns  |  S.Exp1
    timeit_C_lookup                 |                  627  ns  |  C.Add
    timeit_Symbol_eq_xy             |             11.3      us  |  x == y
    timeit_expand_nothing_todo      |            363        us  |  p.expand()
    bench_expand_32                 |  5.66                 s   |  (x+y+z+1)**32  -> expand
    timeit_expand_complex_number_1  |        3.6            ms  |  ((2+3*I)**1000).expand(complex=True)
    timeit_expand_complex_number_2  |        4.94           ms  |  ((2+3*I/4)**1000).expand(complex=True)
    timeit_Integer_create           |              3.42     us  |  Integer(2)
    timeit_Integer_int              |              4.96     us  |  int(i3)
    timeit_neg_one                  |              2.51     us  |  -S.One
    timeit_Integer_neg              |              6.72     us  |  -i3
    timeit_Integer_abs              |              4.21     us  |  abs(i3)
    timeit_Integer_sub              |             13        us  |  i3 - i3
    timeit_abs_pi                   |              2.86     us  |  abs(pi)
    timeit_neg_oo                   |              2.18     us  |  -oo
    timeit_Integer_add_i1           |             10.3      us  |  i3+1
    timeit_Integer_add_ij           |             13        us  |  i3+i4
    timeit_Integer_add_Rational     |             45.3      us  |  i3+r34
    timeit_Integer_mul_i4           |             10.5      us  |  i3*4
    timeit_Integer_mul_ij           |             13.2      us  |  i3*i4
    timeit_Integer_mul_Rational     |             42.7      us  |  i3*r34
    timeit_Integer_eq_i3            |              4.56     us  |  i3==3
    timeit_Integer_ed_Rational      |             33.8      us  |  i3==r34
    timeit_integer_nthroot          |             18.7      us  |  integer_nthroot(100, 2)
    timeit_number_igcd_23_17        |             26.7      us  |  igcd(23,17)
    timeit_number_igcd_60_3600      |             24.6      us  |  igcd(60,3600)
    timeit_Rational_add_r1          |             40.8      us  |  r34+1
    timeit_Rational_add_rq          |             36.2      us  |  r34+q45
    timeit_sympify_1                |             12.5      us  |  sympify(1)
    timeit_sympify_x                |              3.39     us  |  sympify(x)
    timeit_exp_subs                 |             22        us  |  e.subs(q, y)
    timeit_Ylm_xy                   |            766        us  |  Ylm(1,1, x,y)
    bench_integrate_sin             |       16              ms  |  integrate(sin(x), x)
    bench_integrate_x1sin           |      887              ms  |  integrate(x**1*sin(x), x)
    bench_integrate_x2sin           |  1.61                 s   |  integrate(x**2*sin(x), x)
    bench_integrate_x3sin           |  3.04                 s   |  integrate(x**3*sin(x), x)
    timeit_trigintegrate_sin3x      |        3.36           ms  |  trigintegrate(sin(x)**3, x)
    timeit_trigintegrate_x2         |            610        us  |  trigintegrate(x**2, x)  # -> None
    timeit_Matrix__getitem_ii       |             16.2      us  |  M[3,3]
    timeit_Matrix__getitem_II       |             18        us  |  M[i3,i3]
    timeit_Matrix__getslice         |       41.2            ms  |  M[:,:]
    timeit_Matrix_zeronm            |       39.6            ms  |  zeros((100, 100))
    timeit_factor_x4_x_1            |       62.4            ms  |  factor(x**4+x+1)
    bench_factor_x56_1              |  8.91                 s   |  factor(x**56-1)
    timeit_limit_1x                 |            175        us  |  limit(1/x, x, oo)
    timeit_linsolve_trivial         |        8.22           ms  |  solve_linear_system(M, *S)

Not ideal, but still worth to have ...

----

The next step will be to implement

  a. storing benchmark data to files, and
  b. comparing results of benchmarks runs.

>>> Anyone want to help with this?

I'm afraid I'll be working hard at work for the whole next week and
don't have time for this.

Signed-off-by: Kirill Smelkov <kirr@landau.phys.spbu.ru>
Signed-off-by: Ondrej Certik <ondrej@certik.cz>