std.math.TAU

[James Fisher]

On Tue, Jul 5, 2011 at 8:49 PM, Don <nospam at nospam.com> wrote:

> James Fisher wrote:
>
>  On Tue, Jul 5, 2011 at 12:31 PM, James Fisher <jameshfisher at gmail.com<mailto:
>> jameshfisher at gmail.com**>> wrote:
>>
>>    Sorry, I didn't state this very clearly.  Multiplying the
>>    approximation of PI in std.math should yield the exact double of
>>    that approximation, as it should just involve increasing the
>>    exponent by 1.  However, [double the approximation of the constant]
>>    is not necessarily equal to [the approximation of double the
>>    constant].  Does that make sense?
>>
>
> I understand what you're getting at, but actually multiplication by powers
> of 2 is always exact for binary floating point numbers.
> The reason is that the rounding is based on the values after the lowest bit
> of the _significand_. The exponent plays no role.
> Multiplication or division by two doesn't change the significand at all,
> only the exponent, so if the rounding was correct before, it is still
> correct after the multiplication.
>
> Or to put it another way: PI in binary is a infinitely long string of 1s
> and zeros. Multiplying it by two only shifts the string left and right, it
> doesn't change any of the 1s to 0s, etc, so the approximation doesn't change
> either.
>

Great explanation, thanks.

 (I think this is why the constants in math.d <https://github.com/D-**
>> Programming-Language/phobos/**blob/master/std/math.d#L206<https://github.com/D-Programming-Language/phobos/blob/master/std/math.d#L206>>
>> are each defined separately rather than in terms of each other.)
>>
>
> Hmm. I'm not sure why PI_2 and PI_4 are there. They should be defined in
> terms of PI. Probably should fix that.
>

Another thing -- why are some constants defined in decimal, others in hex,
and one (E) with the long 'L' suffix?  And is there a significance to the
number of decimal/hexadecimal places -- e.g., is this the minimum places
required to ensure the closest floating point value for all common hardware
accuracies?
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