code: def distance_paraboloid(
      point: List[int], rel_str: str, TOL: float = 1e-10
  ):
      """
      Returns the minimum distance from the given point to the relation given by `rel_str` which has the form:
      z = A x^2 + B x y + C y^2 + D x + E y + F
      Uses method of Lagrange multipliers.
      """
      rel_str = rel_str.split(" = ")[-1]
      expr = sympy.parse_expr(
          rel_str.replace("^", "**"),
          transformations=SYMPY_TRANSFORMATIONS,
      )
      x, y = list(expr.free_symbols)
      if x.name == "y":
          x, y = y, x
      z = Symbol("z")
      x0, y0, z0 = point
      f = (x - x0) ** 2 + (y - y0) ** 2 + (z - z0) ** 2
      g = z - expr
      if abs(g.subs([(x, x0), (y, y0), (z, z0)])) < TOL:
          return float(0.0)
      λ = Symbol("λ")
      # The code below is meant to be equivalent to
      # `sols = sympy.solve([eq_x, eq_y, eq_z, g], [x, y, z, λ])`
      # but sympy.solve was failing to find any solution on many inputs
      # as well as not finding some solutions
      # so this breaks it down for the special case of `f - λ g` which is at most quadratic.
  
      # Set up the equations from method of Lagrange multipliers
      eq_x = diff(f, x) - λ * diff(g, x)
      eq_y = diff(f, y) - λ * diff(g, y)
      eq_z = diff(f, z) - λ * diff(g, z)
      # Solve for each variable individually
      has_xy = y in eq_x.free_symbols  # has xy term
      if has_xy:
          sols_x = sympy.solve(eq_x, [x, y, λ])
          sols_y = sympy.solve(eq_y, [x, y, λ])
          sols_z = sympy.solve(eq_z, [z, λ])
      else:
          sols_x = sympy.solve(eq_x, [x, λ])
          sols_y = sympy.solve(eq_y, [y, λ])
          sols_z = sympy.solve(eq_z, [z, λ])
      try:
          # Put the solutions together
  
          # Extract x,y,z resp. from tuples
          sols_lst_xyz = [
              [_[0] for _ in lst] for lst in [sols_x, sols_y, sols_z]
          ]
  
          # Extract solutions for λ from tuples
          sols_lst_λλλ = [
              [_[-1] for _ in lst] for lst in [sols_x, sols_y, sols_z]
          ]
  
          # Get list of possible solution tuples and corresponding solutions for λ
          sols_xyz = list(product(*sols_lst_xyz))
          vals_λ = list(product(*sols_lst_λλλ))
  
          sols = []
          # Try each possible combined solution for x, y, z, λ
          for sol_xyz, val_λs in zip(sols_xyz, vals_λ):
              val_λs = list(
                  set(filter(lambda _: not _.is_symbol, val_λs))
              )  # get distinct values for λ if there are any
              if (
                  len(val_λs) > 1
              ):  # there can be at most one distinct value for λ
                  continue
              val_λ = val_λs[0] if val_λs else λ
              sol_x, sol_y, sol_z = sol_xyz
              if not val_λ.is_symbol:
                  # Substitute in values of λ
                  sol_x = sol_x.subs(λ, val_λ)
                  sol_y = sol_y.subs(λ, val_λ)
                  sol_z = sol_z.subs(λ, val_λ)
                  g_λ = g.subs(λ, val_λ)
              else:
                  g_λ = g
  
              # Substitute in solutions for x, y, z
              if has_xy:
                  g_λ = g_λ.subs([(x, sol_x), (z, sol_z)])
                  sol_ys = sympy.solve(sol_x - sol_y, y)
                  for sol_y in sol_ys:
                      g_λy = g_λ.subs(y, sol_y)
                      sol_xy = sol_x.subs(y, sol_y)
                      syms = list(g_λy.free_symbols)
                      if len(syms) > 1:  # underdetermined system
                          continue
                      sym = syms[0]
                      vals = [
                          sympy.N(_) for _ in sympy.solveset(g_λy, sym)
                      ]
                      sols.extend(
                          [
                              (
                                  sol_xy.subs(sym, _),
                                  sol_y.subs(sym, _),
                                  sol_z.subs(sym, _),
                              )
                              for _ in vals
                          ]
                      )
              else:
                  g_λ = g_λ.subs([(x, sol_x), (y, sol_y), (z, sol_z)])
                  syms = list(g_λ.free_symbols)
                  if len(syms) > 1:  # underdetermined system
                      continue
                  # Solve for remaining variable
                  sym = syms[0]
                  vals = [sympy.N(_) for _ in sympy.solveset(g_λ, sym)]
                  sols.extend(
                      [
                          (
                              sol_x.subs(sym, _),
                              sol_y.subs(sym, _),
                              sol_z.subs(sym, _),
                          )
                          for _ in vals
                      ]
                  )
      except ZeroDivisionError:
          # This shouldn't happen, but has happened for a prior implementation of
          # `distance_paraboloid`, so catch it conservatively:
          print("Failed to compute minimum distance.")
          # pdb.set_trace()
          return float(0.0)
      poly_f = sympy.Poly(f, [x, y, z])
      # Evaluate f on found solutions
      try:
          dist_vals = list(map(lambda _: sympy.N(poly_f.eval(_)), sols))
      except sympy.polys.polyerrors.UnificationFailed:
          # Forgot to substitute all variables in some expression.
          # This shouldn't happen, but has happened for a prior implementation of
          # `distance_paraboloid`, so catch it conservatively:
          print("sympy error: Unification failed.")
          # pdb.set_trace()
          return float(0.0)
      # Get the minimum nonnegative real value
      try:
          dist_val = min(
              [
                  sympy.re(_)
                  for _ in dist_vals
                  if abs(sympy.im(_)) < TOL and sympy.re(_) >= 0
              ]
          )
      except ValueError:
          # A real solution should exist, but if not (eg. numerical error exceeds TOL):
          print("Failed to compute minimum distance.")
          print([eq_x, eq_y, eq_z, g])
          print(sols)
          # pdb.set_trace()
          return float(0.0)
      return np.sqrt(float(dist_val))
  
imports: from collections import defaultdict
  from dataclasses import dataclass, field
  from itertools import combinations_with_replacement, product
  import math
  from math import comb
  import numpy
  import numpy.typing
  import random
  import sympy
  from sympy import Symbol, Poly, diff
  from sympy.parsing.sympy_parser import standard_transformations, implicit_multiplication_application
  from typing import List, Optional, Tuple, Dict
  from helm.benchmark.adaptation.adapters.adapter_factory import ADAPT_GENERATION
  from helm.benchmark.adaptation.adapter_spec import AdapterSpec
  from helm.benchmark.window_services.tokenizer_service import TokenizerService
  from helm.common.authentication import Authentication
  from helm.proxy.services.server_service import ServerService
  from .scenario import Scenario, Instance, Reference, TRAIN_SPLIT, TEST_SPLIT, CORRECT_TAG, Input, Output
label_desc: Use image node_id 16 for calling a global function with example usage: distance_paraboloid(point, rel_str, TOL) and returns: np, float, float, float, float