Lower bound for the expected squared error when sending a real [0,1] number using a single bit

LB code

import kmeans1d
import random
from math import sqrt

pts = 1000000

w = 0.270279 #this is an approximation, the real formula is implicit
a = (-2*w**2+w-2*sqrt(-(w-1)*w)+1)/(4*w**2-6*w+2)

nrZeros = int(pts*w)
nrMid = pts-2*nrZeros

inc = (1-2*a)/nrMid
start = a + inc/2

x = [0] * nrZeros + [1] * nrZeros + [start+i*inc for i in range(nrMid)]

k = 2

clusters, centroids = kmeans1d.cluster(x, k)

err = 0
for i,xi in enumerate(x):
    err += (xi-centroids[clusters[i]])**2
    
print('Err:', err/len(x))