from functools import partial
import numpy as np
import jax.numpy as jnp
import jax

@partial(jax.jit, static_argnums=(2,))
def f(m, b, d=128):
    i = jnp.arange(d / 2)
    return jnp.cos(m[:, None] * b ** (-2 * i[None] / d)).sum(axis=1)

@np.vectorize
def fmin(L, b):
    return f(np.arange(L), b).min()

def bmin(L):
    B = 1000 * L
    for k in range(1, 6):
        bs = np.linspace(0, 1, 10**k + 1)[1:] * B  
        ys = fmin(L, bs)
        for b, y in zip(bs, ys):
            if y >= 0:
                B = b
                break
    return B

bmin(1024 * 128)