import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10)
y = np.cos(x) / 2. + .5
z = np.sin(x) / 20. + .5


def jr_transform(x, baseline=.50, scaling=100):
    x = x - baseline
    above = np.log(scaling*x*(x>0)+1)
    below = np.log(-scaling*x*(x<=0)+1)
    x = np.nan_to_num(above) - np.nan_to_num(below)
    return x / 2. + baseline

fig, (ax, ax1) = plt.subplots(1, 2)
ax.plot(x, y, color='C0')
ax.plot(x, z, color='C1')
yticks = np.linspace(0, 1, 11)
ax.set_yticks(yticks)
ax.set_yticklabels(['.%.2f' % i for i in yticks])

ax.set_title('original')

ax1.plot(x, jr_transform(y), color='C0')
ax1.plot(x, jr_transform(z), color='C1')
ax1.set_yticks(jr_transform(yticks))
ax1.set_yticklabels(['.%.2f' % i for i in yticks])
ax1.set_title('log')
plt.show()