Trying to find smallest number

Question

In my program I'm trying to find the smallest number that python can give me. When I kept dividing a number by 2, I got 5 x 10^-324 (5e-324). I thought I could divide this by the biggest number I can use in python. I tried to get the biggest number in python by doing this:

z = 1
    while True:
        try:
            z = z+1
        except OverflowError:
            z = z-1
            break

Here is my full code:

from os import system
x = 76556758478567587
while True:
    x = x/2
    if x/2 == 0.0:
        break

print("Smallest number I can compute:", x)
print()
z = 1
while True:
    try:
        z = z+1
    except OverflowError:
        z = z-1
        break

print(str(x) + " divided by " + str(z) + " is...")
z = x/z
print(z)
system("pause >nul")

Every time I run this it does nothing. I suddenly recognize it's still trying to solve the problem so I open task manager and Python was eating up my CPU like a pack of wolves eating a dead cow.

I know the smallest number in python would be negative but I want to get the the smallest number above zero.

you first algorithm is log(N)-complex, your second one is N-complex. You should multiply by 2 until you reach overflow THEN work your way down. However I'm sensing that the exercise is pointless. — Jean-François Fabre, Sep 28 '16 at 21:17
What? There is no maximum or minimum number in Python. Python will automagically convert your "int" to a "long" which is actually infinite size (realistically w.e. is allocated by the OS). There is a max 'int', but it's fairly meaningless. If that's not satisfying, check out https://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic or search for "big number library" for reference impls. — Eugene K, Sep 28 '16 at 21:25

score 3 · Accepted Answer · edited Jun 20 '20 at 09:12

3

You may use sys.float_info to get the maximum/minimum representable finite float as:

>>> import sys
>>> sys.float_info.min
2.2250738585072014e-308
>>> sys.float_info.max
1.7976931348623157e+308

Python uses double-precision floats, which can hold values from about 10 to the -308 to 10 to the 308 power. Below is the experiment from the python prompt:

# for maximum
>>> 1e308
1e+308
>>> 1e+309
inf   <-- Infinite

You may even get numbers smaller than 1e-308 via denormals, but there is a significant performance hit to this and such numbers are represented with a loss of precision. I found that Python is able to handle 1e-323 but underflows on 1e-324 and returns 0.0 as the value.

# for minimum
>>> 1e-323
1e-323
>>> 1e-324
0.0

You can get denormalized minimum as sys.float_info.min * sys.float_info.epsilon, which comes as 5e-324.

As per the document:

sys.float_info.max is maximum representable finite float

sys.float_info.min is minimum positive normalized float

To get more information check: sys.float_info:

sys.float_info is a struct sequence holding information about the float type. It contains low level information about the precision and internal representation. The values correspond to the various floating-point constants defined in the standard header file float.h for the ‘C’ programming language; see section 5.2.4.2.2 of the 1999 ISO/IEC C standard [C99], ‘Characteristics of floating types’, for details.

edited Jun 20 '20 at 09:12

Community

1
1

answered Sep 28 '16 at 21:17

Moinuddin Quadri

46,825
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The interesting thing is that it seems that you _can_ still divide `sys.float_info.min` ... And I'm not sure _why_. Maybe an IEEE guru can enlighten us? – mgilson Sep 28 '16 at 21:19
How do you suppose that 2.2250738585072014e-308 is the smallest available float when the OP has gotten to 5e-324? – TigerhawkT3 Sep 28 '16 at 21:19
Try: `sys.float_info.min*sys.float_info.epsilon` – brianpck Sep 28 '16 at 21:21
@mgilson isn't this about normalized and unnormalized floats? `2.2e-308` might be divided into `0.000000002e-308`, etc? – Andras Deak -- Слава Україні Sep 28 '16 at 21:24
@AndrasDeak -- Likely. I don't know the difference between a "normalized" and "unnormalized" float :-). You'd think I would have picked that up somewhere ... – mgilson Sep 28 '16 at 21:25
1

@mgilson and I don't even know the terminology, apparently they're called [denormal numbers](https://en.wikipedia.org/wiki/Denormal_number). – Andras Deak -- Слава Україні Sep 28 '16 at 21:27
In order to clarify the doubts here: `sys.float_info.max` is the number with the largest exponent and the full precision in use. As you go above it, you'll loose precision. Similar is for `.min` – Moinuddin Quadri Sep 28 '16 at 21:42
Python uses double-precision floats, which can hold values from about 10 to the -308 to 10 to the 308 power. Updated the answer with the experiment from the python prompt. Hope that will clarify your doubt – Moinuddin Quadri Sep 28 '16 at 21:47
This answer was most helpful because I tried the code and divided what I got with the maximum float. I got 0, this confirmed that I had found the smallest number (above 0) in python. Thanks. – AAAAAAAAA EEEEEEEEEEEEEEEEEEEE Oct 03 '16 at 22:45

score 1 · Answer 2 · answered Sep 28 '16 at 22:08

The smallest floating point number representable in Python is 5e-324, or 2.0**-1075.

The other two answers have each given you half of what you need to find this number. It can be found by multiplying sys.float_info.min (2.2250738585072014e-308, the smallest normalized float) by sys.float_info.epsilon (2.220446049250313e-16, the smallest proportion you can modify a number by while still getting a distinct value).

score 0 · Answer 3 · edited May 23 '17 at 12:19

0

The key word that you're apparently missing is "epsilon." If you search on Python epsilon value then it returns a links to StackOverflow: Value for epsilon in Python near the top of the results in Google.

Obviously it helps if you understand how the term is associated with this concept.

edited May 23 '17 at 12:19

Community

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answered Sep 28 '16 at 21:32

Jim Dennis

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Trying to find smallest number

3 Answers3