text: """ Helpers for randomized testing """
  
  from __future__ import print_function, division
  
  from random import uniform, Random, randrange, randint
  
  from sympy.core.compatibility import is_sequence, as_int
  from sympy.core.containers import Tuple
  from sympy.core.numbers import comp, I
  from sympy.core.symbol import Symbol
  from sympy.simplify.simplify import nsimplify
  
  
  def random_complex_number(a=2, b=-1, c=3, d=1, rational=False, tolerance=None):
      """
      Return a random complex number.
  
      To reduce chance of hitting branch cuts or anything, we guarantee
      b <= Im z <= d, a <= Re z <= c
  
      When rational is True, a rational approximation to a random number
      is obtained within specified tolerance, if any.
      """
      A, B = uniform(a, c), uniform(b, d)
      if not rational:
          return A + I * B
      return nsimplify(A, rational=True, tolerance=tolerance) + I * nsimplify(
          B, rational=True, tolerance=tolerance
      )
  
  
  def verify_numerically(f, g, z=None, tol=1.0e-6, a=2, b=-1, c=3, d=1):
      """
      Test numerically that f and g agree when evaluated in the argument z.
  
      If z is None, all symbols will be tested. This routine does not test
      whether there are Floats present with precision higher than 15 digits
      so if there are, your results may not be what you expect due to round-
      off errors.
  
      Examples
      ========
  
      >>> from sympy import sin, cos
      >>> from sympy.abc import x
      >>> from sympy.testing.randtest import verify_numerically as tn
      >>> tn(sin(x)**2 + cos(x)**2, 1, x)
      True
      """
      f, g, z = Tuple(f, g, z)
      z = [z] if isinstance(z, Symbol) else (f.free_symbols | g.free_symbols)
      reps = list(zip(z, [random_complex_number(a, b, c, d) for _ in z]))
      z1 = f.subs(reps).n()
      z2 = g.subs(reps).n()
      return comp(z1, z2, tol)
  
  
  def test_derivative_numerically(f, z, tol=1.0e-6, a=2, b=-1, c=3, d=1):
      """
      Test numerically that the symbolically computed derivative of f
      with respect to z is correct.
  
      This routine does not test whether there are Floats present with
      precision higher than 15 digits so if there are, your results may
      not be what you expect due to round-off errors.
  
      Examples
      ========
  
      >>> from sympy import sin
      >>> from sympy.abc import x
      >>> from sympy.testing.randtest import test_derivative_numerically as td
      >>> td(sin(x), x)
      True
      """
      from sympy.core.function import Derivative
  
      z0 = random_complex_number(a, b, c, d)
      f1 = f.diff(z).subs(z, z0)
      f2 = Derivative(f, z).doit_numerically(z0)
      return comp(f1.n(), f2.n(), tol)
  
  
  def _randrange(seed=None):
      """Return a randrange generator. ``seed`` can be
          o None - return randomly seeded generator
          o int - return a generator seeded with the int
          o list - the values to be returned will be taken from the list
            in the order given; the provided list is not modified.
  
      Examples
      ========
  
      >>> from sympy.testing.randtest import _randrange
      >>> rr = _randrange()
      >>> rr(1000) # doctest: +SKIP
      999
      >>> rr = _randrange(3)
      >>> rr(1000) # doctest: +SKIP
      238
      >>> rr = _randrange([0, 5, 1, 3, 4])
      >>> rr(3), rr(3)
      (0, 1)
      """
      if seed is None:
          return randrange
      elif isinstance(seed, int):
          return Random(seed).randrange
      elif is_sequence(seed):
          seed = list(seed)  # make a copy
          seed.reverse()
  
          def give(a, b=None, seq=seed):
              if b is None:
                  a, b = 0, a
              a, b = as_int(a), as_int(b)
              w = b - a
              if w < 1:
                  raise ValueError("_randrange got empty range")
              try:
                  x = seq.pop()
              except IndexError:
                  raise ValueError("_randrange sequence was too short")
              if a <= x < b:
                  return x
              else:
                  return give(a, b, seq)
  
          return give
      else:
          raise ValueError("_randrange got an unexpected seed")
  
  
  def _randint(seed=None):
      """Return a randint generator. ``seed`` can be
          o None - return randomly seeded generator
          o int - return a generator seeded with the int
          o list - the values to be returned will be taken from the list
            in the order given; the provided list is not modified.
  
      Examples
      ========
  
      >>> from sympy.testing.randtest import _randint
      >>> ri = _randint()
      >>> ri(1, 1000) # doctest: +SKIP
      999
      >>> ri = _randint(3)
      >>> ri(1, 1000) # doctest: +SKIP
      238
      >>> ri = _randint([0, 5, 1, 2, 4])
      >>> ri(1, 3), ri(1, 3)
      (1, 2)
      """
      if seed is None:
          return randint
      elif isinstance(seed, int):
          return Random(seed).randint
      elif is_sequence(seed):
          seed = list(seed)  # make a copy
          seed.reverse()
  
          def give(a, b, seq=seed):
              a, b = as_int(a), as_int(b)
              w = b - a
              if w < 0:
                  raise ValueError("_randint got empty range")
              try:
                  x = seq.pop()
              except IndexError:
                  raise ValueError("_randint sequence was too short")
              if a <= x <= b:
                  return x
              else:
                  return give(a, b, seq)
  
          return give
      else:
          raise ValueError("_randint got an unexpected seed")
  