code: def __sympy_debug():
      # helper function so we don't import os globally
      import os
  
      debug_str = os.getenv("SYMPY_DEBUG", "False")
      if debug_str in ("True", "False"):
          return eval(debug_str)
      else:
          raise RuntimeError(
              "unrecognized value for SYMPY_DEBUG: %s" % debug_str
          )
  
imports: import sys
  from sympy.release import __version__
  from sympy.core.cache import lazy_function
  from .core import sympify, SympifyError, cacheit, Basic, Atom, preorder_traversal, S, Expr, AtomicExpr, UnevaluatedExpr, Symbol, Wild, Dummy, symbols, var, Number, Float, Rational, Integer, NumberSymbol, RealNumber, igcd, ilcm, seterr, E, I, nan, oo, pi, zoo, AlgebraicNumber, comp, mod_inverse, Pow, integer_nthroot, integer_log, trailing, Mul, prod, Add, Mod, Rel, Eq, Ne, Lt, Le, Gt, Ge, Equality, GreaterThan, LessThan, Unequality, StrictGreaterThan, StrictLessThan, vectorize, Lambda, WildFunction, Derivative, diff, FunctionClass, Function, Subs, expand, PoleError, count_ops, expand_mul, expand_log, expand_func, expand_trig, expand_complex, expand_multinomial, nfloat, expand_power_base, expand_power_exp, arity, PrecisionExhausted, N, evalf, Tuple, Dict, gcd_terms, factor_terms, factor_nc, evaluate, Catalan, EulerGamma, GoldenRatio, TribonacciConstant, bottom_up, use, postorder_traversal, default_sort_key, ordered, num_digits
  from .logic import to_cnf, to_dnf, to_nnf, And, Or, Not, Xor, Nand, Nor, Implies, Equivalent, ITE, POSform, SOPform, simplify_logic, bool_map, true, false, satisfiable
  from .assumptions import AppliedPredicate, Predicate, AssumptionsContext, assuming, Q, ask, register_handler, remove_handler, refine
  from .polys import Poly, PurePoly, poly_from_expr, parallel_poly_from_expr, degree, total_degree, degree_list, LC, LM, LT, pdiv, prem, pquo, pexquo, div, rem, quo, exquo, half_gcdex, gcdex, invert, subresultants, resultant, discriminant, cofactors, gcd_list, gcd, lcm_list, lcm, terms_gcd, trunc, monic, content, primitive, compose, decompose, sturm, gff_list, gff, sqf_norm, sqf_part, sqf_list, sqf, factor_list, factor, intervals, refine_root, count_roots, all_roots, real_roots, nroots, ground_roots, nth_power_roots_poly, cancel, reduced, groebner, is_zero_dimensional, GroebnerBasis, poly, symmetrize, horner, interpolate, rational_interpolate, viete, together, BasePolynomialError, ExactQuotientFailed, PolynomialDivisionFailed, OperationNotSupported, HeuristicGCDFailed, HomomorphismFailed, IsomorphismFailed, ExtraneousFactors, EvaluationFailed, RefinementFailed, CoercionFailed, NotInvertible, NotReversible, NotAlgebraic, DomainError, PolynomialError, UnificationFailed, GeneratorsError, GeneratorsNeeded, ComputationFailed, UnivariatePolynomialError, MultivariatePolynomialError, PolificationFailed, OptionError, FlagError, minpoly, minimal_polynomial, primitive_element, field_isomorphism, to_number_field, isolate, round_two, prime_decomp, prime_valuation, galois_group, itermonomials, Monomial, lex, grlex, grevlex, ilex, igrlex, igrevlex, CRootOf, rootof, RootOf, ComplexRootOf, RootSum, roots, Domain, FiniteField, IntegerRing, RationalField, RealField, ComplexField, PythonFiniteField, GMPYFiniteField, PythonIntegerRing, GMPYIntegerRing, PythonRational, GMPYRationalField, AlgebraicField, PolynomialRing, FractionField, ExpressionDomain, FF_python, FF_gmpy, ZZ_python, ZZ_gmpy, QQ_python, QQ_gmpy, GF, FF, ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX, EXRAW, construct_domain, swinnerton_dyer_poly, cyclotomic_poly, symmetric_poly, random_poly, interpolating_poly, jacobi_poly, chebyshevt_poly, chebyshevu_poly, hermite_poly, hermite_prob_poly, legendre_poly, laguerre_poly, apart, apart_list, assemble_partfrac_list, Options, ring, xring, vring, sring, field, xfield, vfield, sfield
  from .series import Order, O, limit, Limit, gruntz, series, approximants, residue, EmptySequence, SeqPer, SeqFormula, sequence, SeqAdd, SeqMul, fourier_series, fps, difference_delta, limit_seq
  from .functions import factorial, factorial2, rf, ff, binomial, RisingFactorial, FallingFactorial, subfactorial, carmichael, fibonacci, lucas, motzkin, tribonacci, harmonic, bernoulli, bell, euler, catalan, genocchi, andre, partition, mobius, sqrt, root, Min, Max, Id, real_root, Rem, cbrt, re, im, sign, Abs, conjugate, arg, polar_lift, periodic_argument, unbranched_argument, principal_branch, transpose, adjoint, polarify, unpolarify, sin, cos, tan, sec, csc, cot, sinc, asin, acos, atan, asec, acsc, acot, atan2, exp_polar, exp, ln, log, LambertW, sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh, acoth, asech, acsch, floor, ceiling, frac, Piecewise, piecewise_fold, piecewise_exclusive, erf, erfc, erfi, erf2, erfinv, erfcinv, erf2inv, Ei, expint, E1, li, Li, Si, Ci, Shi, Chi, fresnels, fresnelc, gamma, lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma, multigamma, dirichlet_eta, zeta, lerchphi, polylog, stieltjes, Eijk, LeviCivita, KroneckerDelta, SingularityFunction, DiracDelta, Heaviside, bspline_basis, bspline_basis_set, interpolating_spline, besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn, jn_zeros, hn1, hn2, airyai, airybi, airyaiprime, airybiprime, marcumq, hyper, meijerg, appellf1, legendre, assoc_legendre, hermite, hermite_prob, chebyshevt, chebyshevu, chebyshevu_root, chebyshevt_root, laguerre, assoc_laguerre, gegenbauer, jacobi, jacobi_normalized, Ynm, Ynm_c, Znm, elliptic_k, elliptic_f, elliptic_e, elliptic_pi, beta, mathieus, mathieuc, mathieusprime, mathieucprime, riemann_xi, betainc, betainc_regularized
  from .ntheory import nextprime, prevprime, prime, primepi, primerange, randprime, Sieve, sieve, primorial, cycle_length, composite, compositepi, isprime, divisors, proper_divisors, factorint, multiplicity, perfect_power, pollard_pm1, pollard_rho, primefactors, totient, divisor_count, proper_divisor_count, divisor_sigma, factorrat, reduced_totient, primenu, primeomega, mersenne_prime_exponent, is_perfect, is_mersenne_prime, is_abundant, is_deficient, is_amicable, abundance, npartitions, is_primitive_root, is_quad_residue, legendre_symbol, jacobi_symbol, n_order, sqrt_mod, quadratic_residues, primitive_root, nthroot_mod, is_nthpow_residue, sqrt_mod_iter, discrete_log, quadratic_congruence, binomial_coefficients, binomial_coefficients_list, multinomial_coefficients, continued_fraction_periodic, continued_fraction_iterator, continued_fraction_reduce, continued_fraction_convergents, continued_fraction, egyptian_fraction
  from .concrete import product, Product, summation, Sum
  from .discrete import fft, ifft, ntt, intt, fwht, ifwht, mobius_transform, inverse_mobius_transform, convolution, covering_product, intersecting_product
  from .simplify import simplify, hypersimp, hypersimilar, logcombine, separatevars, posify, besselsimp, kroneckersimp, signsimp, nsimplify, FU, fu, sqrtdenest, cse, epath, EPath, hyperexpand, collect, rcollect, radsimp, collect_const, fraction, numer, denom, trigsimp, exptrigsimp, powsimp, powdenest, combsimp, gammasimp, ratsimp, ratsimpmodprime
  from .sets import Set, Interval, Union, EmptySet, FiniteSet, ProductSet, Intersection, DisjointUnion, imageset, Complement, SymmetricDifference, ImageSet, Range, ComplexRegion, Complexes, Reals, Contains, ConditionSet, Ordinal, OmegaPower, ord0, PowerSet, Naturals, Naturals0, UniversalSet, Integers, Rationals
  from .solvers import solve, solve_linear_system, solve_linear_system_LU, solve_undetermined_coeffs, nsolve, solve_linear, checksol, det_quick, inv_quick, check_assumptions, failing_assumptions, diophantine, rsolve, rsolve_poly, rsolve_ratio, rsolve_hyper, checkodesol, classify_ode, dsolve, homogeneous_order, solve_poly_system, solve_triangulated, pde_separate, pde_separate_add, pde_separate_mul, pdsolve, classify_pde, checkpdesol, ode_order, reduce_inequalities, reduce_abs_inequality, reduce_abs_inequalities, solve_poly_inequality, solve_rational_inequalities, solve_univariate_inequality, decompogen, solveset, linsolve, linear_eq_to_matrix, nonlinsolve, substitution
  from .matrices import ShapeError, NonSquareMatrixError, GramSchmidt, casoratian, diag, eye, hessian, jordan_cell, list2numpy, matrix2numpy, matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2, rot_axis3, symarray, wronskian, zeros, MutableDenseMatrix, DeferredVector, MatrixBase, Matrix, MutableMatrix, MutableSparseMatrix, banded, ImmutableDenseMatrix, ImmutableSparseMatrix, ImmutableMatrix, SparseMatrix, MatrixSlice, BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity, Inverse, MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace, Transpose, ZeroMatrix, OneMatrix, blockcut, block_collapse, matrix_symbols, Adjoint, hadamard_product, HadamardProduct, HadamardPower, Determinant, det, diagonalize_vector, DiagMatrix, DiagonalMatrix, DiagonalOf, trace, DotProduct, kronecker_product, KroneckerProduct, PermutationMatrix, MatrixPermute, Permanent, per, rot_ccw_axis1, rot_ccw_axis2, rot_ccw_axis3, rot_givens
  from .geometry import Point, Point2D, Point3D, Line, Ray, Segment, Line2D, Segment2D, Ray2D, Line3D, Segment3D, Ray3D, Plane, Ellipse, Circle, Polygon, RegularPolygon, Triangle, rad, deg, are_similar, centroid, convex_hull, idiff, intersection, closest_points, farthest_points, GeometryError, Curve, Parabola
  from .utilities import flatten, group, take, subsets, variations, numbered_symbols, cartes, capture, dict_merge, prefixes, postfixes, sift, topological_sort, unflatten, has_dups, has_variety, reshape, rotations, filldedent, lambdify, threaded, xthreaded, public, memoize_property, timed
  from .integrals import integrate, Integral, line_integrate, mellin_transform, inverse_mellin_transform, MellinTransform, InverseMellinTransform, laplace_transform, laplace_correspondence, laplace_initial_conds, inverse_laplace_transform, LaplaceTransform, InverseLaplaceTransform, fourier_transform, inverse_fourier_transform, FourierTransform, InverseFourierTransform, sine_transform, inverse_sine_transform, SineTransform, InverseSineTransform, cosine_transform, inverse_cosine_transform, CosineTransform, InverseCosineTransform, hankel_transform, inverse_hankel_transform, HankelTransform, InverseHankelTransform, singularityintegrate
  from .tensor import IndexedBase, Idx, Indexed, get_contraction_structure, get_indices, shape, MutableDenseNDimArray, ImmutableDenseNDimArray, MutableSparseNDimArray, ImmutableSparseNDimArray, NDimArray, tensorproduct, tensorcontraction, tensordiagonal, derive_by_array, permutedims, Array, DenseNDimArray, SparseNDimArray
  from .parsing import parse_expr
  from .calculus import euler_equations, singularities, is_increasing, is_strictly_increasing, is_decreasing, is_strictly_decreasing, is_monotonic, finite_diff_weights, apply_finite_diff, differentiate_finite, periodicity, not_empty_in, AccumBounds, is_convex, stationary_points, minimum, maximum
  from .algebras import Quaternion
  from .printing import pager_print, pretty, pretty_print, pprint, pprint_use_unicode, pprint_try_use_unicode, latex, print_latex, multiline_latex, mathml, print_mathml, python, print_python, pycode, ccode, print_ccode, smtlib_code, glsl_code, print_glsl, cxxcode, fcode, print_fcode, rcode, print_rcode, jscode, print_jscode, julia_code, mathematica_code, octave_code, rust_code, print_gtk, preview, srepr, print_tree, StrPrinter, sstr, sstrrepr, TableForm, dotprint, maple_code, print_maple_code
  from .plotting import plot, textplot, plot_backends, plot_implicit, plot_parametric
  from .interactive import init_session, init_printing, interactive_traversal
label_desc: Use image node_id 1 for calling a global function with example usage: __sympy_debug() and returns: eval