"""Delay-aware admissibility rule for attachable residual modules."""
import math

def critical_delay(a, gain):
    if gain <= a:
        return math.inf
    return math.acos(-a / gain) / math.sqrt(gain * gain - a * a)

def max_admissible_gain(a, delay):
    if delay <= 0:
        return math.inf
    lo, hi = a * (1 + 1e-12), max(2 * a, a + 1 / delay)
    while critical_delay(a, hi) > delay:
        hi *= 2
    for _ in range(100):
        mid = (lo + hi) / 2
        if critical_delay(a, mid) > delay:
            lo = mid
        else:
            hi = mid
    return (lo + hi) / 2

def select_modules(gains, benefits, a, delay, minimum_gain=0.0):
    limit = max_admissible_gain(a, delay)
    order = sorted(range(len(gains)), key=lambda i: benefits[i] / max(gains[i], 1e-12), reverse=True)
    chosen, total = [], 0.0
    for i in order:
        if total + gains[i] < limit:
            chosen.append(i); total += gains[i]
    if total < minimum_gain:
        return [], 0.0, limit
    return chosen, total, limit