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Re: a more precise distance algorithm

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From Oscar Benjamin <oscar.j.benjamin@gmail.com>
Date Wed, 27 May 2015 23:03:49 +0100
Subject Re: a more precise distance algorithm
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On 27 May 2015 at 19:00, Brian Blais <bblais@gmail.com> wrote:
> On Mon, May 25, 2015 at 11:11 PM, Steven D'Aprano <steve@pearwood.info> wrote:
>>
>> Let's compare three methods.
>>
>> def naive(a, b):
>>     return math.sqrt(a**2 + b**2)
>>
>> def alternate(a, b):
>>     a, b = min(a, b), max(a, b)
>>     if a == 0:  return b
>>     if b == 0:  return a
>>     return a * math.sqrt(1 + b**2 / a**2)
>
>
>>     d1 = naive(a, b)
>>     d2 = alternate(a, b)
>>     d3 = math.hypot(a, b)
>>
>
>> which shows that:
>>
>> (1) It's not hard to find mismatches;
>> (2) It's not obvious which of the three methods is more accurate.
>>
>
> Bottom line: they all suck.  :)
>
> I ran the program you posted, and, like you, got the following two examples:
>
> for fun in [naive, alternate, math.hypot]:
>     print '%.20f' % fun(222.44802484683657,680.255801504161)
>
> 715.70320611153294976248
> 715.70320611153283607564
> 715.70320611153283607564
>
> and
>
> for fun in [naive, alternate, math.hypot]:
>     print '%.20f' % fun(376.47153302262484,943.1877995550265)
>
> 1015.54617837194291496417
> 1015.54617837194280127733
> 1015.54617837194291496417
>
> but when comparing to Wolfram Alpha, which calculates these out many
> more decimal places, we have for the two cases:
>
> 715.7032061115328768204988784125331443593766145937358347357252...
> 715.70320611153294976248
> 715.70320611153283607564
> 715.70320611153283607564
>
> 1015.546178371942943007625196455666280385821355370154991424749...
> 1015.54617837194291496417
> 1015.54617837194280127733
> 1015.54617837194291496417
>
> where all of the methods deviate at the 13/14 decimal place.

So they have 12/13 correct digits after the decimal point. Including
the digits before the decimal point they all have 15/16 correct
decimal digits. This is exactly what you should expect when using
double precision floating point. Feel free to print as many digits as
you like but the extra ones won't add any more accuracy.

The difference between the answers you showed are just 1 ULP:
>>> a = 715.70320611153283607564
>>> b = 715.70320611153294976248
>>> a
715.7032061115328
>>> a + 5e-14
715.7032061115328
>>> a + 6e-14
715.703206111533
>>> a + 6e-14 == b
True

Since you don't show the source numbers we can't know whether the true
result lies between these two or not.


Oscar

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Thread

a more precise distance algorithm ravas <ravas@outlook.com> - 2015-05-25 12:21 -0700
  Re: a more precise distance algorithm felix <felix@epepm.cupet.cu> - 2015-05-25 16:06 -0400
  Re: a more precise distance algorithm Christian Gollwitzer <auriocus@gmx.de> - 2015-05-25 22:27 +0200
    Re: a more precise distance algorithm ravas <ravas@outlook.com> - 2015-05-25 14:03 -0700
  Re: a more precise distance algorithm Gary Herron <gary.herron@islandtraining.com> - 2015-05-25 13:20 -0700
    Re: a more precise distance algorithm ravas <ravas@outlook.com> - 2015-05-25 14:05 -0700
  Re: a more precise distance algorithm Steven D'Aprano <steve@pearwood.info> - 2015-05-26 13:11 +1000
    Re: a more precise distance algorithm ravas <ravas@outlook.com> - 2015-05-25 21:13 -0700
      Re: a more precise distance algorithm Gary Herron <gherron@digipen.edu> - 2015-05-25 22:09 -0700
        Re: a more precise distance algorithm ravas <ravas@outlook.com> - 2015-05-25 22:49 -0700
    Re: a more precise distance algorithm Christian Gollwitzer <auriocus@gmx.de> - 2015-05-26 07:33 +0200
    Re: a more precise distance algorithm Brian Blais <bblais@gmail.com> - 2015-05-27 14:00 -0400
    Re: a more precise distance algorithm Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2015-05-27 23:03 +0100
    Re: a more precise distance algorithm Dennis Lee Bieber <wlfraed@ix.netcom.com> - 2015-05-27 23:04 -0400
  Re: a more precise distance algorithm Ian Kelly <ian.g.kelly@gmail.com> - 2015-05-25 22:42 -0600
    Re: a more precise distance algorithm ravas <ravas@outlook.com> - 2015-05-25 21:59 -0700
  Re: a more precise distance algorithm random832@fastmail.us - 2015-05-26 09:40 -0400
  Re: a more precise distance algorithm random832@fastmail.us - 2015-05-26 09:51 -0400
  Re: a more precise distance algorithm Robin Becker <robin@reportlab.com> - 2015-05-27 14:02 +0100

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