taylor_series.py

#!/usr/bin/env python3
"""
Task: Solve this formula

sinus(x)=x-((x**3)/3!)+((x**5)/5!)-((x**7)/7!)+...

https://en.wikipedia.org/wiki/Taylor_series
"""


from math import factorial, pow, sin, tau
from itertools import count, cycle
from operator import add, sub, mul
from functools import reduce

import matplotlib.pyplot as plt


def pow(x, y):
    """
    simple own implementation of pow
    """
    return x ** y


def factorial(x):
    """
    simple own implementation of factorial
    functional style
    """
    return reduce(mul, range(2, x + 1), 1)


def sinus(x, degree=21):
    """
    First overcomplicated idea. Just wanted to use itertools
    as much I can.
    """
    if n == 1:
        return x
    y = 0
    ops = cycle((add, sub))
    for factor, op, _ in zip(count(3, step=2), ops, range(degree)):
        y = op(y, pow(x, factor) / factorial(factor))
    return x - y


def sinus2(x, degree=21):
    """
    Straight forward implementation, which looks easier
    and should run faster. The name tau is 2 * pi
    The modulo operation makes it possible to use x < -2pi and x > 2pi
    """
    x %= tau
    sign = -1
    y = 0
    for factor in range(3, degree + 1, 2):
        y += sign * pow(x, factor) / factorial(factor)
        sign *= -1
    return x + y


def cosine(x, degree=21):
    """
    Almost the same like sinus2
    """
    x %= tau
    sign = -1
    y = 0
    for factor in range(2, degree + 1, 2):
        y += sign * (pow(x, factor) / factorial(factor))
        sign *= -1
    return  1 + y


def visualize(degree):
    legend = []
    base = [x / 1000 for x in range(-10_000, 10_000, 100)]
    for n in degree:
        legend.append(f'degree={n}')
        plt.plot(base, [sinus(x, n) for x in base])
    legend.append('math.sin(x)')
    plt.plot(base, [sin(x) for x in base])
    plt.legend(legend, loc='upper right', framealpha=0.2, fontsize=8)
    plt.title('Taylor series')
    plt.xlabel('x')
    plt.ylabel('y')
    plt.xlim(-10, 10)
    plt.ylim(-10, 10)
    plt.grid()
    plt.savefig('Taylor series.png', dpi=300)
    plt.show()


if __name__ == '__main__':
    visualize([1, 3, 5, 7, 9, 11, 13])