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Diving into a floating point bug

March 2nd, 2017
tech

Yesterday I refactored some python code that effectively changed:

   floor(n * (100 / 101))
into:
   floor(n * Decimal(100/101))
For most integers this doesn't change anything, but for multiples of 101 it gives you different answers:
  floor(101 * (100 / 101))       -> 100
  floor(101 * Decimal(100/101))  ->  99
In python2, however, both give 100:
  floor(101 * (100.0 / 101))       -> 100
  floor(101 * Decimal(100.0/101))  -> 100
What's going on? First, what's the difference between python2 and python3 here? One of the changes not listed in What's New In Python 3.0 is that floor now returns an int instead of a float: Since these are positive numbers, we can use int() instead of floor() and now python2 and python3 give the same result:
  2.x: int(101 * Decimal(100.0/101)) -> 99
  3.x: int(101 * Decimal(100/101))   -> 99
The root of the problem is that 100/101 is a repeating decimal, 0.99009900..., and so can't be represented exactly as a Decimal or a float. To store it as a Decimal python defaults to 28 decimal digits and rounds it:
> D(100)/D(101)
Decimal('0.9900990099009900990099009901')
Storing it as a float is more complicated. Python uses the platform's hardware support, which is IEEE 754 floating point bascially everywhere, and it gets converted to a binary fraction with 53 bits for the numerator and a power of two for the denominator:
  1114752383012499 / 2**50
Python is aware that there are many numbers for which this is the closest possible floating point approximation, and so to be friendly it selectes the shortest decimal representation when printing it:
> 100/101
0.9900990099009901

Now we have all the pieces to see why 101*Decimal(100/101) is less than 101*(100/101), and, critically, why one is just under 100 while the other is exactly 100.

In the Decimal case python first divides 100 by 101 and gets 1114752383012499 / 2**50. Then it takes the closest representable Decimal to that number, which is Decimal('0.99009900990099009021605525049380958080291748046875'). That's 50 bits of precision because Decimal tries to track and preseve precision in its calculations, and 53 bits of precision is 50 decimal digits of precision. [Update 2017-03-02: this is wrong: 53 bits of precision much less than 50 decimal digits. I'm not sure why Decimal is doing this.] All of the digits starting with 02160... are just noise, but Decimal has no way of knowing that. Then when we multiply by 101 we get Decimal('99.999...58030'), because our Decimal version of the float was slightly less than true 100/101.

On the other hand, when python gets 101 * (100/101) it stays in IEEE 754 land. It multiplies 101 by 1114752383012499 / 2**50, and the closest float to that is exactly 100.

I ran into this bug because I had been trying to break a large change into two behavior-preserving changes, first switching some code to use Decimal and then switching the rest. To fix the bug, I combined the two changes into one, so that we switched from using float and int to using Decimal throughout this module.

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