import math


def sigmoid(x: float) -> float:
    if x >= 0:
        return 1.0 / (1.0 + math.exp(-x))
    exp_x = math.exp(x)
    return exp_x / (1.0 + exp_x)


def softmax(values: list[float]) -> list[float]:
    if not values:
        return []
    max_v = max(values)
    exps = [math.exp(v - max_v) for v in values]
    total = sum(exps)
    if total <= 0:
        return [1.0 / len(values)] * len(values)
    return [e / total for e in exps]