-- Module: Mathlib.Analysis.Complex.ExponentialBounds
Goal:
h :
  |1 -
        (((((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *
                                                                  7 -
                                                                1) *
                                                              8 -
                                                            1) *
                                                          9 -
                                                        1) *
                                                      10 -
                                                    1) *
                                                  11 -
                                                1) *
                                              12 -
                                            1) *
                                          13 -
                                        1) *
                                      14 -
                                    1) *
                                  15 -
                                1) *
                              16 -
                            1) *
                          17 -
                        1) *
                      18 -
                    1) *
                  19 -
                1) *
              20 -
            1) *
          21| ≤
    1 / HPow.hPow (α := ℝ) (β := ℕ) 10 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 *
              18 *
            19 *
          20 *
        21 -
      HDiv.hDiv (α := ℝ) |1| 22 * (HAdd.hAdd (α := ℝ) 22 1 / 22)
⊢ |rexp 1 -
        expNear 21 1
          ((((((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *
                                                                    7 -
                                                                  1) *
                                                                8 -
                                                              1) *
                                                            9 -
                                                          1) *
                                                        10 -
                                                      1) *
                                                    11 -
                                                  1) *
                                                12 -
                                              1) *
                                            13 -
                                          1) *
                                        14 -
                                      1) *
                                    15 -
                                  1) *
                                16 -
                              1) *
                            17 -
                          1) *
                        18 -
                      1) *
                    19 -
                  1) *
                20 -
              1) *
            21)| ≤
    HPow.hPow (α := ℝ) (β := ℕ) |1| 21 / (↑(Nat.factorial 21) : ℝ) *
      (1 / HPow.hPow (α := ℝ) (β := ℕ) 10 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 *
                17 *
              18 *
            19 *
          20 *
        21)

Tactic:
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_

Premises used: Real.exp_approx_end'