Trac #24425: Fix inherently failing random_expr doctest

The docs for `random_expr` read:
{{{
  This function will often raise an error because it tries to create
  an erroneous expression (such as a division by zero).
}}}
It has the following doctest:
{{{
        sage: from sage.symbolic.random_tests import *
        sage: set_random_seed(53)
        sage: random_expr(50, nvars=3,
coeff_generator=CDF.random_element) # random
        (v1^(0.97134084277 + 0.195868299334*I)/csc(-pi + v1^2 + v3) +
sgn(1/
...
}}}
Despite having a random seed the test run changes with every new builtin
function introduced in `sage/functions` because the global function list
changes. That's why the test was marked random. The problem however is
that the test can even raise an error, as the docs state above.
The"`random`" keyword does not catch this, and it would make the test
useless anyway.

Tests are meant to test the functionality of the associated code so the
test and perhaps `random_expr` should be rewritten such that it allows a
test that does not change with a changed global function list.

URL: https://trac.sagemath.org/24425
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Marc Mezzarobba

Author: Release Manager

src/sage/symbolic/random_tests.py

@@ -46,10 +46,8 @@ def _mk_full_functions():
         integrate, kronecker_delta, log, polylog, real_part, sec,
         sech, sgn, sin, sinh, tan, tanh, unit_step, zeta, zetaderiv]
 
-    Note that this doctest will fail whenever a Pynac function is added or
-    removed.  In that case, it is very likely that the doctests for
-    random_expr will fail as well.  That's OK; just fix the doctest
-    to match the new output.
+    Note that this doctest will produce different output whenever a
+    symbolic function is added or removed.
     """
     items = sorted(symbol_table['functions'].items())
     return [(1.0, f, f.number_of_arguments())
@@ -261,14 +259,16 @@ def random_expr(size, nvars=1, ncoeffs=None, var_frac=0.5,
     EXAMPLES::
 
         sage: from sage.symbolic.random_tests import *
-        sage: set_random_seed(53)
-        sage: random_expr(50, nvars=3, coeff_generator=CDF.random_element) # random
-        (v1^(0.97134084277 + 0.195868299334*I)/csc(-pi + v1^2 + v3) + sgn(1/
-        ((-v3 - 0.760455994772 - 0.554367254855*I)*erf(v3 + 0.982759757946 -
-        0.0352136502348*I)) + binomial(arccoth(v1^pi), 0.760455994772 +
-        0.554367254855*I) + arccosh(2*v2 - (v2 + 0.841911550437 -
-        0.303757179824*I)/sinh_integral(pi) + arccoth(v3 + 0.530133230474 +
-        0.532140303485*I))))/v2
+        sage: some_functions = [arcsinh, arctan, arctan2, arctanh,
+        ....: arg, beta, binomial, ceil, conjugate, cos, cosh, cot, coth,
+        ....: elliptic_pi, erf, exp, factorial, floor, heaviside, imag_part,
+        ....: sech, sgn, sin, sinh, tan, tanh, unit_step, zeta, zetaderiv]
+        sage: my_internal = [(0.6, full_binary, 2), (0.2, full_unary, 1),
+        ....: (0.2, [(1.0,f,f.number_of_arguments()) for f in some_functions])]
+        sage: set_random_seed(1)
+        sage: random_expr(50, nvars=3, internal=my_internal,
+        ....:   coeff_generator=CDF.random_element)
+        (v1^(0.9713408427702117 + 0.195868299334218*I)/cot(-pi + v1^2 + v3) + tan(arctan(v2 + arctan2(-0.35859061674557324 + 0.9407509502498164*I, v3) - 0.8419115504372718 + 0.30375717982404615*I) + arctan2((0.2275357305882964 - 0.8258002386106038*I)/factorial(v2), -v3 - 0.7604559947718565 - 0.5543672548552057*I) + ceil(1/arctan2(v1, v1))))/v2
         sage: random_expr(5, verbose=True) # random
         About to apply <built-in function inv> to [31]
         About to apply sgn to [v1]