Update test_estimation.py

tests/3.A/test_estimation.py

 from decimal import Decimal
 from sympy import series, atan, nsimplify, diff
 from sympy.calculus.util import Interval, maximum
-import math
-
+from math import floor, factorial
 from src.taylor import arctan
 
 import pytest
@@ -12,20 +11,47 @@ 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))/math.factorial(N + 1)).subs(x, xval))))
-
+    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())
 
-def approx_atan(N, xval, a):
-    return ref_taylor(atan(x), a, N, xval), term(N, xval, a)
-
-@pytest.mark.parametrize('q', range(20))
-def test_arctan(q):
-    xval = q / 10
-    _, error = approx_atan(10, xval, 0)
-    v, e = arctan(Decimal(xval))
-    if xval < 0.9:
-        assert 0 <= float(e) - float(error) < 0.003
-    assert abs(math.atan(xval) - float(v)) <= float(e)
+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)