James Fisher wrote:
On Tue, Jul 5, 2011 at 8:49 PM, Don <[email protected]
<mailto:[email protected]>> wrote:
James Fisher wrote:
On Tue, Jul 5, 2011 at 12:31 PM, James Fisher
<[email protected] <mailto:[email protected]>
<mailto:[email protected]
<mailto:[email protected]>__>> 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?
The ones defined in decimal are obsolete, they haven't had a conversion
to hex yet.
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?
Yes, it's 80 bit. Currently there's a problem with DMC's floating-point
parser, all those numbers should really be 128 bit (we should be ready
for 128 bit quads).
