How to randomly generate really small numbers?

Question

Suppose I want to draw a random number in the range [10^-20, 0.1], how do I do that?

If I use numpy.random.uniform, I don't seem to go lower than 10^-2:

In [2]: np.random.uniform(0.1, 10**(-20))
Out[2]: 0.02506361878539856

In [3]: np.random.uniform(0.1, 10**(-20))
Out[3]: 0.04035553250149768

In [4]: np.random.uniform(0.1, 10**(-20))
Out[4]: 0.09801074888377342

In [5]: np.random.uniform(0.1, 10**(-20))
Out[5]: 0.09778150831277296

In [6]: np.random.uniform(0.1, 10**(-20))
Out[6]: 0.08486347093110456

In [7]: np.random.uniform(0.1, 10**(-20))
Out[7]: 0.04206753781952958

Alternatively I could generate an array instead like:

In [44]: fac = np.linspace(10**(-20),10**(-1),100)

In [45]: fac
Out[45]: 
array([  1.00000000e-20,   1.01010101e-03,   2.02020202e-03,
         3.03030303e-03,   4.04040404e-03,   5.05050505e-03,
         6.06060606e-03,   7.07070707e-03,   8.08080808e-03,
         9.09090909e-03,   1.01010101e-02,   1.11111111e-02,
         1.21212121e-02,   1.31313131e-02,   1.41414141e-02,
         1.51515152e-02,   1.61616162e-02,   1.71717172e-02,
         1.81818182e-02,   1.91919192e-02,   2.02020202e-02,
         2.12121212e-02,   2.22222222e-02,   2.32323232e-02,
         2.42424242e-02,   2.52525253e-02,   2.62626263e-02,
         2.72727273e-02,   2.82828283e-02,   2.92929293e-02,
         3.03030303e-02,   3.13131313e-02,   3.23232323e-02,
         3.33333333e-02,   3.43434343e-02,   3.53535354e-02,
         3.63636364e-02,   3.73737374e-02,   3.83838384e-02,
         3.93939394e-02,   4.04040404e-02,   4.14141414e-02,
         4.24242424e-02,   4.34343434e-02,   4.44444444e-02,
         4.54545455e-02,   4.64646465e-02,   4.74747475e-02,
         4.84848485e-02,   4.94949495e-02,   5.05050505e-02,
         5.15151515e-02,   5.25252525e-02,   5.35353535e-02,
         5.45454545e-02,   5.55555556e-02,   5.65656566e-02,
         5.75757576e-02,   5.85858586e-02,   5.95959596e-02,
         6.06060606e-02,   6.16161616e-02,   6.26262626e-02,
         6.36363636e-02,   6.46464646e-02,   6.56565657e-02,
         6.66666667e-02,   6.76767677e-02,   6.86868687e-02,
         6.96969697e-02,   7.07070707e-02,   7.17171717e-02,
         7.27272727e-02,   7.37373737e-02,   7.47474747e-02,
         7.57575758e-02,   7.67676768e-02,   7.77777778e-02,
         7.87878788e-02,   7.97979798e-02,   8.08080808e-02,
         8.18181818e-02,   8.28282828e-02,   8.38383838e-02,
         8.48484848e-02,   8.58585859e-02,   8.68686869e-02,
         8.78787879e-02,   8.88888889e-02,   8.98989899e-02,
         9.09090909e-02,   9.19191919e-02,   9.29292929e-02,
         9.39393939e-02,   9.49494949e-02,   9.59595960e-02,
         9.69696970e-02,   9.79797980e-02,   9.89898990e-02,
         1.00000000e-01])

and pick a random element from that array, but wanted to clarify anyway if the first option is possible since I could be probably missing something obvious.

Answer 1

You need to think closely about what you're doing. You're asking for a uniform distribution between almost 0.0 and 0.1. The average result would be 0.05. Which is exactly what you're getting. It seems you want a random distribution of the exponents.

The following might do what you want:

import random

def rnd():
    exp = random.randint(-19, -1)
    significand = 0.9 * random.random() + 0.1
    return significand * 10**exp

[rnd() for _ in range(20)]

The lowest possible value is when exp=-19 and significand=0.1 giving 0.1*10**-19 = 1**-20. And the highest possible value is when exp=-1 and significand=1.0 giving 1.0*10**-1 = 0.1.

Note: Technically, the significand can only aprach 1.0 as random() is bounded to [0.0, 1.0), i.e., including 0.0, but excluding 1.0.

Output:

[2.3038280595190108e-11,
 0.02658855644891981,
 4.104572641101877e-11,
 3.638231824527544e-19,
 6.220040206106022e-17,
 7.207472203268789e-06,
 6.244626749598619e-17,
 2.299282102612733e-18,
 0.0013251357609258432,
 3.118805901868378e-06,
 6.585606992344938e-05,
 0.005955900790586139,
 1.72779538837876e-08,
 7.556972406280229e-13,
 3.887023124444594e-15,
 0.0019965330694999488,
 1.7732147730252207e-08,
 8.920398286274208e-17,
 4.4422869312622194e-08,
 2.4815949527034027e-18]

See "scientific notation" on wikipedia for definition of significand and exponent.

Answer 2

As per the numpy documentation:

low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0.

With that in mind, decreasing the value of low will produce lower numbers

>>> np.random.uniform(0.00001, 10**(-20))
6.390804027773046e-06