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Date: Sat, 06 Aug 2011 14:51:21 -0700
From: Patricia Shanahan <pats@acm.org>
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Subject: Re: higher precision doubles
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On 8/6/2011 12:24 PM, Jan Burse wrote:
> Stefan Ram schrieb:
>> Jan Burse<janburse@fastmail.fm> writes:
>>> IEEE allows to internally calculate with additional bits.
>>> Is it possible to have awailable these ops? Like an add
>>> with this higher precision? How could we store such a
>>> result? Would there be a wrapper like Double?
>>
>> See also:
>>
>> http://en.wikipedia.org/wiki/Strictfp
>>
>> .
>>
>
> Motivating example, in Go we have:
>
> func main() {
> x := math.Sin(2*math.Pi)
> fmt.Printf("x = %.30f, is zero = %v\n", x, x == 0)
> }
>
> x = 0.000000000000000000000000000000, is zero = true
>
> In Java we have:
>
>
> public static double zero1() {
> return Math.sin(2*Math.PI);
> }
>
> public static double zero2() {
> return StrictMath.sin(2*StrictMath.PI);
> }
>
> public static void main(String[] args) {
> System.out.println("zero1="+zero1());
> System.out.println("zero2="+zero2());
> }
>
> zero1=-2.4492935982947064E-16
> zero2=-2.4492935982947064E-16

I'm not sure I understand how this motivates extra precision. Would you
have been happier if the result had been something order 10^-24? Or if
this particular calculation had got 0, but some other even multiple of
PI had got an order 10^-24 result?

There are two issues here, algorithm and arithmetic precision. I do not
know exactly how Java calculates sin, but the usual approach is to have
a polynomial approximation for a range of angles close to zero, and a
reduction algorithm that maps calculating the sin of a general angle to
calculating the sin of an angle in that range (possibly with a need to
negate the result).

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

Without a change in the algorithm, or its objective, I do not think
extra precision would eliminate all cases of this sort of difference.

Patricia