In the following example:
import math
x = math.log(2)
print("{:.500f}".format(x))
I tried to get 500 digits output I get only 53 decimals output of ln(2) as follows:
0.69314718055994528622676398299518041312694549560546875000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
How I can fix this problem?
Within the realm of real numbers, which is an infinite set of numbers with arbitrary precision, the floating point numbers are a small subset of numbers with a finite precision. They are the numbers that are represented by a linear combination of powers of two (See Double Precision floating point format).
As Ln(2) is not re-presentable as a floating-point number, a computer finds the nearest number by numerical approximations. In case of Ln(2), this number is:
6243314768165359 * 2^-53 = 0.69314718055994528622676398299518041312694549560546875
If you need to do arbitrary precision arithmetic, you are required to make use of different computational methods. Various software packages exist that allow this. For Python, MPmath is fairly standard:
>>> from mpmath import *
>>> mp.dps = 500
>>> mp.pretty=True
>>> ln(2)
0.69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641868754200148102057068573368552023575813055703267075163507596193072757082837143519030703862389167347112335011536449795523912047517268157493206515552473413952588295045300709532636664265410423915781495204374043038550080194417064167151864471283996817178454695702627163106454615025720740248163777338963855069526066834113727387372292895649354702576265209885969320196505855476470330679365443254763274495125040607