From: jzakiya <jzakiya@...>
Date: 2009-12-28T13:30:06+09:00
Subject: Re: Trig value errors

On Dec 27, 10:56 pm, John W Higgins <wish...@gmail.com> wrote:
> [Note:  parts of this message were removed to make it a legal post.]
>
> On Sun, Dec 27, 2009 at 6:00 PM, jzakiya <jzak...@mail.com> wrote:
> > i.e.  (1) 1 = cos^2 + sin^2
>
> This also means that 0 = 1 - cos^2 - sin^2
>
>
>
>
>
> > To 'fix totally' these errors in sin|cos near the x|y axis define sin1|
> > cos1 here using Ruby's native cos|sin.
>
> > def sin1(x)
> >  y=(1-cos(x)**2)**0.5        # calculate (2)
> >  return 0.0 if y==0
> >  return sin(x) < 0 ? -y : y  # for correct quadrant sign
> > end
>
> > def cos1(x)
> >  y=(1-sin(x)**2)**0.5        # calculate (2)
> >  return 0.0 if y==0
> >  return cos(x) < 0 ? -y : y  # for correct quadrant sign
> > end
>
> You are simply unable to apparently grasp how mathematics works with
> floating point numbers. So lets make a very simple example which shows that
> your method fails as badly as does the standard methods.
>
> First, to show that the standard methods show no greater precision lets test
> 87 degrees shall we.
>
> 1.0 - (sin(PI/180*87) ** 2 + cos(PI/180*87) ** 2)
> => 1.11022302462516e-16
>
> So we fail with the standard and yours works
>
> 1.0 - (sin1(PI/2/90*87) ** 2 + cos1(PI/2/90*87) ** 2)
> => 0.0
>
> But wait shall we - lets look at 67 degrees ok?
>
> 1.0 - (sin(PI/180*67) ** 2 + cos(PI/180*67) ** 2)
> => 0.0
>
> 1.0 - (sin1(PI/180*67) ** 2 + cos1(PI/180*67) ** 2)
> => 2.22044604925031e-16
>
> Oh no - we have a problem - your calculation fails when the standard works.
>
> There is no way to accurate get these values to ever work with floating
> point mathematics. There will always be something that fails because all you
> are doing is trying to cover up one inaccurate calculation with yet another
> inaccurate calculation.
>
> If you hide yourself in a box that is only 1 or 0 shockingly you can make
> x^2 + y^2 = 1 because you are gaming the values. But you fail to appreciate
> that the numbers will always have an element of imprecision that you have no
> control over and you cannot make work to exact numbers when dealing with
> floating points.
>
> Yet again, the standard is the standard because a lot of VERY smart people
> took the time to analyze the entire spectrum of results instead of limiting
> themselves myopically to one particular area.
>
> John

All I can say is....

Against stupidity the Gods themselves contend in vain. :(