# cython: language_level=3

# L-15 MCS 507 Mon 25 Sep 2023 : integral4pi_typed.pyx

"""
This script applies the composite trapezoidal rule
to the integral of sqrt(1-x^2) for x from 0 to 1,
to obtain an approximation for Pi.
Type declarations have been added so the for loop
will be compiled into pure C code.
"""

from math import sqrt

def circle(double xvl):
    """
    Returns the y corresponding to xvl
    on the upper half of the unit circle.
    """
    return sqrt(1-xvl**2)

def integral4pi(int nbvals):
    """
    Approximates Pi with the trapezoidal
    rule with nbvals subintervals of [0,1].
    """
    cdef int i
    cdef double step, result
    step = 1.0/nbvals
    result = (circle(0)+circle(1))/2
    for i in range(nbvals):
        result += circle(i*step)
    return 4*result*step