"""Failure mode: floating point instead of exact rationals.

An answer to "write a Python function for Bernoulli numbers" can easily be
float-based, because most Python numerical code is.  The output looks entirely
convincing at every index: ``-0.0333...`` really is close to ``-1/30``.

It is still inexact from ``B2`` on.  The digits stay good to about fourteen
significant figures; what grows is the size of the error, about 1.2e-6 at ``B30``
(``8615841276005/14322``, roughly 601580873.9).  A test that only checked ``B0``
to ``B12`` with an absolute tolerance would have passed this file, and a fixed
tolerance of 1e-9 first fails it at ``B24``.

Expected checker diagnosis: ``not-fraction``, a first divergence at ``n=2``,
and ``odd-terms``: the checker compares each float as the exact rational it
stores, so ``B3 = -1.1102230246251565e-16`` is not zero.
"""


def bernoulli(n: int) -> float:
    """Return B_n as a float (this is the bug)."""
    if n < 0:
        raise ValueError("n must be >= 0")

    values = [0.0] * (n + 1)
    for m in range(n + 1):
        values[m] = 1.0
        for k in range(m):
            binomial = 1.0
            for i in range(k):
                binomial = binomial * (m - i) / (i + 1)
            values[m] -= binomial * values[k] / (m - k + 1)
    if n == 1:
        return -values[n]
    return values[n]