From: Robert Klemme <shortcutter@...>
Date: 2009-12-27T02:45:07+09:00
Subject: Re: Trig value errors

On 26.12.2009 15:43, Rick DeNatale wrote:
> On Sat, Dec 26, 2009 at 4:56 AM, Brian Candler <b.candler@pobox.com> wrote:
> 
>> Floating-point can represent smaller increments the closer you get to
>> zero. So values of 0 +/- dX can be represented much more accurately than
>> 1 +/- dX.
> 
> Strictly speaking they can be represented much more precisely.

Are you sure you meant "precisely" and not "accurately"?  Given the 
Wikipedia article you quote it seems _that_ is what you rather wanted to 
say.

> Precision and accuracy are different things.
> 
> http://en.wikipedia.org/wiki/Accuracy_and_precision

In light of the article the term "precision" (aka "repeatability") is 
pretty meaningless in this case (you could also say "uninteresting") 
because due to the deterministic nature of the implementation of 
trigonometric functions the result is always identical for the same 
input.  It is the accuracy that the whole thread revolves around.  IEEE 
standards mandate a certain accuracy and people who need better accuracy 
need to use specialized tools - either math libraries with higher 
accuracy (or even selectable accuracy) or tools that do symbolic math.

Also, as elsewhere in this thread has been explained, the "correction" 
approach suggested (if the abs is lower than a particular epsilon, 
return 0 instead of the value) is a pretty bad one - much worse than 
general rounding of values because it introduces distorted accuracy. 
Rounding is better because it generally cuts off digits; which is ironic 
because it *reduces* overall accuracy of the function...

Kind regards

	robert


A program that nicely demonstrates the nonlinearity of the error 
introduced by the suggested "fix":

module Math
   def sin2(x, epsilon = 1e-10)
     y = sin(x)
     y.abs < epsilon ? 0.0 : y
   end
end

include Math

# find interesting start value
div = 1.2
x = 1.0
x /= div while sin(x) == sin2(x)

x *= div ** 10

# do the test
100.times do |i|
   y1 = sin(x)
   y2 = sin2(x)
   printf "%6d %20e %20e %20e error: %20e\n", i, x, y1, y2, ((y2-y1)/x).abs
   x /= div
end

-- 
remember.guy do |as, often| as.you_can - without end
http://blog.rubybestpractices.com/