Groups > comp.lang.java.programmer > #6836

Re: higher precision doubles

From	Jan Burse <janburse@fastmail.fm>
Newsgroups	comp.lang.java.programmer
Subject	Re: higher precision doubles
Date	2011-08-07 00:57 +0200
Organization	albasani.net
Message-ID	<j1kgtg$s7g$1@news.albasani.net> (permalink)
References	<j1hqc1$1ch$1@news.albasani.net> <strictfp-20110806003831@ram.dialup.fu-berlin.de> <j1k4db$1n2$1@news.albasani.net> <OrWdncbuf7nLKqDTnZ2dnUVZ_vKdnZ2d@earthlink.com>

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Patricia Shanahan schrieb:
> In this case the difference is most likely in the reduction step - if it
> had produced 0, the sin result would have been 0.
>

I guess 2*pi is "exact" in the sense that the operation * itself
has no loss, since we only need to adjust the exponent.

I forget to mention, we also get a non-zero value for sin(pi)
in Java, and not only for sin(2*pi).

Easiest reduction is sin(x) = cos(x-pi/2). But this reduction
seems not to be used:

         double x=Math.PI;
         System.out.println("pi="+x+", sin(pi)="+Math.sin(x));
         x=Math.PI-Math.PI/2;
         System.out.println("pi-pi/2="+x+", cos(pi-pi/2)="+Math.cos(x));
         x=Math.PI/2;
         System.out.println("pi/2="+x+", cos(pi/2)="+Math.cos(x));

Gives:
	pi=3.141592653589793, sin(pi)=1.2246467991473532E-16
         pi/2=1.5707963267948966, cos(pi/2)=6.123233995736766E-17
         pi-pi/2=1.5707963267948966, cos(pi-pi/2)=6.123233995736766E-17

So it seems that the error in PI representation seems the
only reason to be not to return zero. So we have sin(x) with
x approximating pi. The error is:

     sin(x) - sin(pi) = sin(pi+(x-pi)) - 0.
                      = - sin(x-pi)
                      ~ - (x - pi)
                      = pi - x

So I guess the PI representation is below the real pi. Since
we get a positive result for sin(x). Lets check:

    Java PI:
         3.141592653589793
    real pi:
         3.141592653589793238462643383...
    Difference:
         0.000000000000000238462643383...

Well the above idea does also not work, we only get:

         x=0.000000000000000238462643383;
         System.out.println("PI-pi="+x+", sin(PI-pi)="+Math.sin(x));

	PI-pi=2.38462643383E-16, sin(PI-pi)=2.38462643383E-16

But not 1.2246467991473532E-16. But we can try a fully exact decimal
development of the approximate machine PI. Using the double to 
BigDecimal conversion we get (a thing I like most with the
BigDecimal package):

         System.out.println("PI="+new BigDecimal(Math.PI));

	PI=3.141592653589793115997963468544185161590576171875

Therefore:

    Java PI:
         3.141592653589793115997963468...
    real pi:
         3.141592653589793238462643383...
    Difference:
         0.000000000000000122464679915...

Everything fits perfectly.

Question is why Go gets zero?

Bye

Small-angle Approximation taken from here:
http://en.wikipedia.org/wiki/Small-angle_approximation

Additional PI Digits taken from here:
http://en.wikipedia.org/wiki/Pi

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higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-06 00:20 +0200
  Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-06 03:35 -0700
    Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-06 13:03 +0200
      Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-06 12:20 -0700
        Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-06 23:30 +0200
          Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-06 16:12 -0700
            Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 01:35 +0200
              Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-06 19:26 -0700
    Re: higher precision doubles supercalifragilisticexpialadiamaticonormalizeringelimatisticantations <supercalifragilisticexpialadiamaticonormalizeringelimatisticantations@averylongandannoyingdomainname.com> - 2011-08-09 00:42 -0400
      Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-09 04:07 -0700
        Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-09 14:00 +0200
          Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-09 09:07 -0700
            Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-10 08:50 +0200
        Re: higher precision doubles supercalifragilisticexpialadiamaticonormalizeringelimatisticantations <supercalifragilisticexpialadiamaticonormalizeringelimatisticantations@averylongandannoyingdomainname.com> - 2011-08-10 20:16 -0400
      Re: higher precision doubles Joshua Cranmer <Pidgeot18@verizon.invalid> - 2011-08-09 10:11 -0500
        Re: higher precision doubles Arne Vajhøj <arne@vajhoej.dk> - 2011-08-09 22:40 -0400
          Re: higher precision doubles Joshua Cranmer <Pidgeot18@verizon.invalid> - 2011-08-09 22:06 -0500
            Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-10 08:53 -0700
  Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-06 21:24 +0200
    Re: higher precision doubles markspace <-@.> - 2011-08-06 13:29 -0700
      Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-06 23:20 +0200
        Re: higher precision doubles markspace <-@.> - 2011-08-06 15:43 -0700
          Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 01:06 +0200
            Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-06 16:21 -0700
              Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 01:34 +0200
                Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-06 21:32 -0700
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 17:24 +0200
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 17:39 +0200
                Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-07 09:26 -0700
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 21:23 +0200
                Re: higher precision doubles Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-08-07 20:48 -0400
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-08 09:05 +0200
                Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-08 05:37 -0700
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-08 19:08 +0200
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-08 19:16 +0200
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-08 19:29 +0200
                Re: higher precision doubles Joshua Cranmer <Pidgeot18@verizon.invalid> - 2011-08-08 21:34 -0500
                Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-08 13:49 -0700
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-09 01:08 +0200
                Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-09 02:44 -0700
                Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-09 12:52 +0200
                Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-09 13:15 -0700
                Re: higher precision doubles BGB <cr88192@hotmail.com> - 2011-08-07 13:51 -0700
              Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 01:59 +0200
        Re: higher precision doubles Joshua Cranmer <Pidgeot18@verizon.invalid> - 2011-08-08 21:03 -0500
    Re: higher precision doubles Eric Sosman <esosman@ieee-dot-org.invalid> - 2011-08-06 17:33 -0400
    Re: higher precision doubles Patricia Shanahan <pats@acm.org> - 2011-08-06 14:51 -0700
      Re: higher precision doubles Jan Burse <janburse@fastmail.fm> - 2011-08-07 00:57 +0200

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