code: def cos_257() -> Expr:
      r"""Computes $\cos \frac{\pi}{257}$ in square roots
  
      References
      ==========
  
      .. [*] https://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals
      .. [*] https://r-knott.surrey.ac.uk/Fibonacci/simpleTrig.html
      """
  
      def f1(a: Expr, b: Expr) -> tuple[Expr, Expr]:
          return (a + sqrt(a**2 + b)) / 2, (a - sqrt(a**2 + b)) / 2
  
      def f2(a: Expr, b: Expr) -> Expr:
          return (a - sqrt(a**2 + b)) / 2
  
      t1, t2 = f1(S.NegativeOne, Integer(256))
      z1, z3 = f1(t1, Integer(64))
      z2, z4 = f1(t2, Integer(64))
      y1, y5 = f1(z1, 4 * (5 + t1 + 2 * z1))
      y6, y2 = f1(z2, 4 * (5 + t2 + 2 * z2))
      y3, y7 = f1(z3, 4 * (5 + t1 + 2 * z3))
      y8, y4 = f1(z4, 4 * (5 + t2 + 2 * z4))
      x1, x9 = f1(y1, -4 * (t1 + y1 + y3 + 2 * y6))
      x2, x10 = f1(y2, -4 * (t2 + y2 + y4 + 2 * y7))
      x3, x11 = f1(y3, -4 * (t1 + y3 + y5 + 2 * y8))
      x4, x12 = f1(y4, -4 * (t2 + y4 + y6 + 2 * y1))
      x5, x13 = f1(y5, -4 * (t1 + y5 + y7 + 2 * y2))
      x6, x14 = f1(y6, -4 * (t2 + y6 + y8 + 2 * y3))
      x15, x7 = f1(y7, -4 * (t1 + y7 + y1 + 2 * y4))
      x8, x16 = f1(y8, -4 * (t2 + y8 + y2 + 2 * y5))
      v1 = f2(x1, -4 * (x1 + x2 + x3 + x6))
      v2 = f2(x2, -4 * (x2 + x3 + x4 + x7))
      v3 = f2(x8, -4 * (x8 + x9 + x10 + x13))
      v4 = f2(x9, -4 * (x9 + x10 + x11 + x14))
      v5 = f2(x10, -4 * (x10 + x11 + x12 + x15))
      v6 = f2(x16, -4 * (x16 + x1 + x2 + x5))
      u1 = -f2(-v1, -4 * (v2 + v3))
      u2 = -f2(-v4, -4 * (v5 + v6))
      w1 = -2 * f2(-u1, -4 * u2)
      return sqrt(sqrt(2) * sqrt(w1 + 4) / 8 + S.Half)
  
file_path: sympy/sympy/functions/elementary/_trigonometric_special.py
imports: from __future__ import annotations
  from typing import Callable
  from functools import reduce
  from sympy.core.expr import Expr
  from sympy.core.singleton import S
  from sympy.core.intfunc import igcdex
  from sympy.core.numbers import Integer
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.core.cache import cacheit
label_desc: Use image node_id 7 for calling a global function with example usage: cos_257() and returns: sqrt, unknown,  unknown, unknown