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from decimal import Decimal
from sympy import series, atan, nsimplify, diff
from sympy.calculus.util import Interval, maximum
from math import floor, factorial
from src.taylor import arctan
import pytest
from sympy.abc import x
def term(N, xval, a):
ivl = Interval(xval - 0.001, a + 0.001) + Interval(a - 0.001, xval + 0.001)
d = diff(atan(x), x, N + 1)
M = maximum(d, x, ivl)
return nsimplify(abs(((M * ((x - a)**(N + 1))/factorial(N + 1)).subs(x, xval))))
def ref_taylor(f, a, N, xval):
return nsimplify(series(f, x=x, x0=a, n=N).removeO().subs(x, xval).doit())
EPSILON = 0.0001
def approx_atan(N, xval):
if floor(xval) == 0:
return ref_taylor(atan(x), 0, N, xval), term(N, xval, 0)
else:
_, center_error = approx_atan(N, floor(xval) - EPSILON)
return ref_taylor(atan(x), floor(xval), N, xval), center_error + term(N, xval, floor(xval) - EPSILON)
@pytest.mark.parametrize('xval', [x/10 for x in range(20)])
def test_arctan(xval):
print('testing', xval)
ref_val1, ref_error = approx_atan(10, xval)
ref_val1 = float(ref_val1)
ref_error = float(ref_error)
ref_val2 = atan(xval)
our_val, our_error = arctan(Decimal(xval))
our_val = float(our_val)
our_error = float(our_error)
errors_diff = our_error - ref_error
error_with_ref1 = abs(ref_val1 - our_val)
error_with_ref2 = abs(ref_val2 - our_val)
epsilon = 0.15
if xval < 0.5:
epsilon = 0.001
elif xval < 1.0:
epsilon = 0.05
assert 0 <= errors_diff < epsilon
assert error_with_ref1 <= our_error
assert error_with_ref2 <= our_error
assert abs(error_with_ref2 - our_error) < epsilon
if __name__ == "__main__":
test_arctan(0.0)
test_arctan(0.5)
test_arctan(0.9)
test_arctan(1.0)
test_arctan(1.5)