from typing import Callable
import math


def line_length(
    fnc: Callable[[int | float], int | float],
    x_start: int | float,
    x_end: int | float,
    steps: int = 100,
) -> float:

    """
    Approximates the arc length of a line segment by treating the curve as a
    sequence of linear lines and summing their lengths
    :param fnc: a function which defines a curve
    :param x_start: left end point to indicate the start of line segment
    :param x_end: right end point to indicate end of line segment
    :param steps: an accuracy gauge; more steps increases accuracy
    :return: a float representing the length of the curve

    >>> def f(x):
    ...    return x
    >>> f"{line_length(f, 0, 1, 10):.6f}"
    '1.414214'

    >>> def f(x):
    ...    return 1
    >>> f"{line_length(f, -5.5, 4.5):.6f}"
    '10.000000'

    >>> def f(x):
    ...    return math.sin(5 * x) + math.cos(10 * x) + x * x/10
    >>> f"{line_length(f, 0.0, 10.0, 10000):.6f}"
    '69.534930'
    """

    x1 = x_start
    fx1 = fnc(x_start)
    length = 0.0

    for i in range(steps):

        # Approximates curve as a sequence of linear lines and sums their length
        x2 = (x_end - x_start) / steps + x1
        fx2 = fnc(x2)
        length += math.hypot(x2 - x1, fx2 - fx1)

        # Increment step
        x1 = x2
        fx1 = fx2

    return length