What if float32 had 31 exponent bits and no mantissa?

continuous-float32.py

import matplotlib.pyplot as plt
import numpy as np


def bits_to_float32(n, E=8, M=23):
    sign = (n >> (E + M)) & 1
    exponent = ((n >> M) & (2 ** E - 1)) - (2 ** (E - 1) - 1)
    exponent *= 2 ** (8 - E)
    mantissa = n & (2 ** M - 1)
    return ((-1) ** sign) * (1 + mantissa / (2 ** M)) * (2 ** exponent)

stride = 2**12
bins = np.linspace(-4, 4, 2 ** 10)

dist32f = [bits_to_float32(n) for n in range(0, 2**32, stride)]
dist32c = [bits_to_float32(n, 31, 0) for n in range(0, 2**32, stride)]

plt.style.use('dark_background')
plt.hist(
    [dist32c, dist32f],
    bins,
    label=['Exp-8', '32-bit Float'],
    color=['red', 'blue',],
    alpha=0.75,
    histtype="step",
)
plt.xlim(bins[0], bins[-1])

plt.yscale('log')
plt.xlabel('Amplitude')
plt.ylabel('Resolution')
plt.legend()
plt.title('Audio Bit Depth')

plt.show()