How to fix the number of decimals to 500 digits output in Python?
Question:
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?
Answers:
You can’t with the Python float type. It’s dependent on the underlying machine architecture, and in most cases you’re limited to a double-precision float.
However, you can get higher precision with the decimal module:
>>> from decimal import Decimal, getcontext
>>> getcontext().prec = 500
>>> d = Decimal(2)
>>> d.ln()
Decimal('0.69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641868754200148102057068573368552023575813055703267075163507596193072757082837143519030703862389167347112335011536449795523912047517268157493206515552473413952588295045300709532636664265410423915781495204374043038550080194417064167151864471283996817178454695702627163106454615025720740248163777338963855069526066834113727387372292895649354702576265209885969320196505855476470330679365443254763274495125040607')
>>> print(d.ln())
0.69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641868754200148102057068573368552023575813055703267075163507596193072757082837143519030703862389167347112335011536449795523912047517268157493206515552473413952588295045300709532636664265410423915781495204374043038550080194417064167151864471283996817178454695702627163106454615025720740248163777338963855069526066834113727387372292895649354702576265209885969320196505855476470330679365443254763274495125040607
I tried to get 500 digits output I get only 53 decimals output of ln(2) as follows:
The problem is not in the printing. The 500 digit output is the exact value returned from math.log(2).
The return value of math.log(2) is encoded using binary64 which can only represent about 264 different finite values – each of them is a dyadic rational. Mathematically log(2) is an irrational number, thus it is impossible for x to encode the math result exactly.
Instead math.log(2) returns the nearest encodable value.
That value is exactly 0.6931471805599452862267639829951804131269454956054687500…
Printing binary64 with more than 17 significant digits typically does not add important value information.
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
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?
You can’t with the Python float type. It’s dependent on the underlying machine architecture, and in most cases you’re limited to a double-precision float.
However, you can get higher precision with the decimal module:
>>> from decimal import Decimal, getcontext
>>> getcontext().prec = 500
>>> d = Decimal(2)
>>> d.ln()
Decimal('0.69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641868754200148102057068573368552023575813055703267075163507596193072757082837143519030703862389167347112335011536449795523912047517268157493206515552473413952588295045300709532636664265410423915781495204374043038550080194417064167151864471283996817178454695702627163106454615025720740248163777338963855069526066834113727387372292895649354702576265209885969320196505855476470330679365443254763274495125040607')
>>> print(d.ln())
0.69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641868754200148102057068573368552023575813055703267075163507596193072757082837143519030703862389167347112335011536449795523912047517268157493206515552473413952588295045300709532636664265410423915781495204374043038550080194417064167151864471283996817178454695702627163106454615025720740248163777338963855069526066834113727387372292895649354702576265209885969320196505855476470330679365443254763274495125040607
I tried to get 500 digits output I get only 53 decimals output of ln(2) as follows:
The problem is not in the printing. The 500 digit output is the exact value returned from math.log(2).
The return value of math.log(2) is encoded using binary64 which can only represent about 264 different finite values – each of them is a dyadic rational. Mathematically log(2) is an irrational number, thus it is impossible for x to encode the math result exactly.
Instead math.log(2) returns the nearest encodable value.
That value is exactly 0.6931471805599452862267639829951804131269454956054687500…
Printing binary64 with more than 17 significant digits typically does not add important value information.
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