[Numpy-discussion] Re: numpy.sin

[Robert K. <rob...@gm...>]

Alan G Isaac wrote:
> On Thu, 25 May 2006, Robert Kern apparently wrote: 
> 
>>What continuity? This is floating-point arithmetic. 
> 
> Sure, but a continuity argument suggests (in the absence of 
> specific floating point reasons to doubt it) that a better 
> approximation at one point will mean better approximations 
> nearby.  E.g.,
> 
>>>>epsilon = 0.00001
>>>>sin(100*pi+epsilon)
> 
> 9.999999976550551e-006
> 
>>>>sin((100*pi+epsilon)%(2*pi))
> 
> 9.9999999887966145e-006
> 
> Compare to the bc result of
> 9.9999999998333333e-006
> 
> bc 1.05
> Copyright 1991, 1992, 1993, 1994, 1997, 1998 Free Software Foundation, Inc.
> This is free software with ABSOLUTELY NO WARRANTY.
> For details type `warranty'.
> scale = 50
> epsilon = 0.00001
> s(100*pi + epsilon)
> .00000999999999983333333333416666666666468253968254

You aren't using bc correctly.

bc 1.06
Copyright 1991-1994, 1997, 1998, 2000 Free Software Foundation, Inc.
This is free software with ABSOLUTELY NO WARRANTY.
For details type `warranty'.
100*pi
0

If you know that you are epsilon from n*2*π (the real number, not the floating
point one), you should just be calculating sin(epsilon). Usually, you do not
know this, and % (2*pi) will not tell you this. (100*pi + epsilon) is not the
same thing as (100*π + epsilon).

FWIW, for the calculation that you did in bc, numpy.sin() gives the same results
(up to the last digit):

>>> from numpy import *
>>> sin(0.00001)
9.9999999998333335e-06

You wanted to know if something there is something exploitable to improve the
accuracy of numpy.sin(). In general, there is not. However, if you know the
difference between your value and an integer multiple of the real number 2*π,
then you can do your floating-point calculation on that difference. However, you
will not in general get an improvement by using % (2*pi) to calculate that
difference.

-- 
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless enigma
 that is made terrible by our own mad attempt to interpret it as though it had
 an underlying truth."
  -- Umberto Eco