---

import math

# Константа округления (Unit Roundoff) для типа данных double (IEEE 754)
# Равна половине машинного эпсилон: 2^(-53) ≈ 1.11e-16
UNIT_ROUNDOFF = 2**(-53)

def ceil_log2(x):
    """Возвращает округленное вверх двоичное логарифмическое значение."""
    return math.ceil(math.log2(x)) if x > 0 else 0

def pairwise_summation(x):
    """Рекурсивное попарное суммирование для уменьшения погрешности."""
    n = len(x)
    if n <= 1:
        return x[0] if n == 1 else 0.0
    elif n <= 128:  # Базовый случай для повышения производительности
        return sum(x)
    else:
        m = n // 2
        return pairwise_summation(x[:m]) + pairwise_summation(x[m:])

def test1():
    harmonic_numbers = [1 / k for k in range(1, 10**6 + 1, 1)]
    n = ceil_log2(len(harmonic_numbers))
    pairwise_sum_result = pairwise_summation(harmonic_numbers)
    pairwise_sum_result_with_abs = pairwise_summation(
        [abs(x) for x in harmonic_numbers]
    )
    
    apriori_abs_error = (
        pairwise_sum_result_with_abs
        * (n * UNIT_ROUNDOFF)
        / (1.0 - 2 * n * UNIT_ROUNDOFF)
    )
    
    print("--- Test 1 (Harmonic Numbers) ---")
    print("Pairwise Summation Result:", pairwise_sum_result)
    print("Pairwise Summation Result with Absolute Values:", pairwise_sum_result_with_abs)
    print(
        "A Priori Absolute Error Estimate:",
        apriori_abs_error,
        pairwise_sum_result_with_abs * (n * UNIT_ROUNDOFF) / (1.0 - n * UNIT_ROUNDOFF),
    )

def test2():
    # Дополненный код для квадратов гармонических чисел
    squared_harmonic_numbers = [1 / (k * k) for k in range(1, 10**3 + 1, 1)]
    n = ceil_log2(len(squared_harmonic_numbers))
    pairwise_sum_result = pairwise_summation(squared_harmonic_numbers)
    pairwise_sum_result_with_abs = pairwise_summation(
        [abs(x) for x in squared_harmonic_numbers]
    )
    
    apriori_abs_error = (
        pairwise_sum_result_with_abs
        * (n * UNIT_ROUNDOFF)
        / (1.0 - 2 * n * UNIT_ROUNDOFF)
    )
    
    print("\n--- Test 2 (Squared Harmonic Numbers) ---")
    print("Pairwise Summation Result:", pairwise_sum_result)
    print("Pairwise Summation Result with Absolute Values:", pairwise_sum_result_with_abs)
    print(
        "A Priori Absolute Error Estimate:",
        apriori_abs_error,
        pairwise_sum_result_with_abs * (n * UNIT_ROUNDOFF) / (1.0 - n * UNIT_ROUNDOFF),
    )

if __name__ == "__main__":
    test1()
    test2()