import math
import re

def prepare_expression(expr):
    """Add explicit multiplication for terms like '3x' -> '3*x'."""
    expr = re.sub(r'(\d)(x)', r'\1*\2', expr)
    return expr

def eval_expression(expr, x):
    """Evaluate f(x) for the given expression string."""
    prepared_expr = prepare_expression(expr)
    local_namespace = {
        "x": x,
        "sin": math.sin, "cos": math.cos, "tan": math.tan,
        "exp": math.exp, "sqrt": math.sqrt, "log": math.log, "pi": math.pi
    }
    return eval(prepared_expr, {"__builtins__": {}}, local_namespace)

def arc_length_approx(f_str, a, b, n):
    """
    Approximate the arc length of y = f(x) from x=a to x=b
    using n straight-line (Pythagorean) segments.
    """
    h = (b - a) / n
    x_values = [a + i * h for i in range(n + 1)]
    y_values = [eval_expression(f_str, xv) for xv in x_values]

    total_length = 0
    for i in range(n):
        dx = x_values[i + 1] - x_values[i]
        dy = y_values[i + 1] - y_values[i]
        segment_length = math.sqrt(dx**2 + dy**2)   # Pythagorean theorem
        total_length += segment_length

    return total_length

# User prompts
print("Arc Length Approximation Program")
f_str = input("1. Enter the function f(x): ")
a = float(input("2. Enter the starting x-value: "))
b = float(input("3. Enter the ending x-value: "))
n = int(input("4. Enter the number of steps (n): "))

length = arc_length_approx(f_str, a, b, n)
print(f"\nApproximate arc length using {n} segments: {length:.6f}")