"""
Ex. 2.19 from "A primer on..."
Explore round-off errors by repeated calls to sqrt followed
by the same number of **. Repeated square roots result in 
a number closer and closer to 1.0, and at some point (n~50) it
becomes so close that the machine treats it as 1.0. 
Squaring 1.0 n times then just gives 1.0, and does not bring us back
to 2.0. 
"""

from math import sqrt

for n in range(1, 60):
    r = 2.0
    for i in range(n):
        r = sqrt(r)
    for i in range(n):
        r = r**2
    print(f"{n:d} times sqrt and **2:  {r:.16f}")

"""
Terminal> python repeated_sqrt.py 
1 times sqrt and **2:  2.0000000000000004
2 times sqrt and **2:  1.9999999999999996
3 times sqrt and **2:  1.9999999999999996
4 times sqrt and **2:  1.9999999999999964
5 times sqrt and **2:  1.9999999999999964
6 times sqrt and **2:  1.9999999999999964
7 times sqrt and **2:  1.9999999999999714
...
49 times sqrt and **2:  1.8682459487159784
50 times sqrt and **2:  1.6487212645509468
51 times sqrt and **2:  1.6487212645509468
52 times sqrt and **2:  1.0000000000000000
53 times sqrt and **2:  1.0000000000000000
54 times sqrt and **2:  1.0000000000000000
55 times sqrt and **2:  1.0000000000000000
56 times sqrt and **2:  1.0000000000000000
57 times sqrt and **2:  1.0000000000000000
58 times sqrt and **2:  1.0000000000000000
59 times sqrt and **2:  1.0000000000000000
"""