Closed-form for _parabolic_invcdf().

rhettinger · rhettinger · commit e9e5d543fb4e · 2024-04-14T22:06:25.000-05:00
diff --git a/Lib/statistics.py b/Lib/statistics.py
@@ -1729,18 +1729,13 @@ def _simple_s_curve(power):
                       if p <= 1/2 else
                       1 - (2 - 2*p) ** power)
 
-_parabolic_invcdf = _newton_raphson(
-    f_inv_estimate = _simple_s_curve(0.583367470424302),  # (7/12)
-    f = lambda t: -1/4 * t**3 + 3/4 * t + 1/2,
-    f_prime = lambda t: 3/4 * (1.0 - t * t))
-
 _quartic_invcdf = _newton_raphson(
     f_inv_estimate = _simple_s_curve(0.4258865685331),   # (204/479)
     f = lambda t: 3/16 * t**5 - 5/8 * t**3 + 15/16 * t + 1/2,
     f_prime = lambda t: 15/16 * (1.0 - t * t) ** 2)
 
 _triweight_invcdf = _newton_raphson(
-    f_inv_estimate = _simple_s_curve(0.3400218741872791), (17/50)
+    f_inv_estimate = _simple_s_curve(0.3400218741872791),  # (17/50)
     f = lambda t: 35/32 * (-1/7*t**7 + 3/5*t**5 - t**3 + t) + 1/2,
     f_prime = lambda t: 35/32 * (1.0 - t * t) ** 3)
 
@@ -1749,7 +1744,7 @@ def _simple_s_curve(power):
     'logistic': lambda p: log(p / (1 - p)),
     'sigmoid': lambda p: log(tan(p * pi/2)),
     'rectangular': lambda p: 2*p - 1,
-    'parabolic': _parabolic_invcdf,
+    'parabolic': lambda p: 2 * cos((acos(2*p-1) + pi) / 3),
     'quartic': _quartic_invcdf,
     'triweight': _triweight_invcdf,
     'triangular': lambda p: sqrt(2*p) - 1 if p < 1/2 else 1 - sqrt(2 - 2*p),
