• Posts
  • RSS
  • ◂◂RSS
  • Contact

  • Diving into a floating point bug

    March 2nd, 2017
    tech  [html]
    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:
    • 2.x: Return the floor of x as a float, the largest integer value less than or equal to x.
    • 3.x: Return the floor of x, the largest integer less than or equal to x. If x is not a float, delegates to x.__floor__(), which should return an Integral value.
    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.

    Comment via: google plus, facebook, hacker news

    Recent posts on blogs I like:

    What should we do about network-effect monopolies?

    Many large companies today are software monopolies that give their product away for free to get monopoly status, then do horrible things. Can we do anything about this?

    via benkuhn.net July 5, 2020

    More on the Deutschlandtakt

    The Deutschlandtakt plans are out now. They cover investment through 2040, but even beforehand, there’s a plan for something like a national integrated timetable by 2030, with trains connecting the major cities every 30 minutes rather than hourly. But the…

    via Pedestrian Observations July 1, 2020

    How do cars fare in crash tests they're not specifically optimized for?

    Any time you have a benchmark that gets taken seriously, some people will start gaming the benchmark. Some famous examples in computing are the CPU benchmark specfp and video game benchmarks. With specfp, Sun managed to increase its score on 179.art (a su…

    via Posts on Dan Luu June 30, 2020

    more     (via openring)


  • Posts
  • RSS
  • ◂◂RSS
  • Contact