import math

def p_from_score_macro(S):
    # S = (1/3) * 2p/(1+p)  ->  p = 3S/(2-3S)
    return (3*S) / (2 - 3*S)

def p_from_score_weighted(S):
    # S = 2p^2/(1+p) -> 2p^2 - S p - S = 0
    return (S + math.sqrt(S*S + 8*S)) / 4
  
def p_from_score_micro(S):
    # micro-F1 == accuracy
    return S

# S_low, S_med, S_high from the LB:
S_low, S_med, S_high = 0.0, 0.0, 0.0  # fill these

for name, f in [
    ("macro", p_from_score_macro),
    ("weighted", p_from_score_weighted),
    ("micro", p_from_score_micro),]:

    ps = [f(S_low), f(S_med), f(S_high)]
    print(name, ps, "sum =", sum(ps))