[Numpy-discussion] Random int64 and float64 numbers
Thu Nov 5 22:32:37 CST 2009
> On Thu, Nov 5, 2009 at 10:42 PM, David Goldsmith
> <email@example.com> wrote:
>> On Thu, Nov 5, 2009 at 3:26 PM, David Warde-Farley <firstname.lastname@example.org>
>>> On 5-Nov-09, at 4:54 PM, David Goldsmith wrote:
>>> > Interesting thread, which leaves me wondering two things: is it
>>> > documented
>>> > somewhere (e.g., at the IEEE site) precisely how many *decimal*
>>> > mantissae
>>> > are representable using the 64-bit IEEE standard for float
>>> > representation
>>> > (if that makes sense);
>>> IEEE-754 says nothing about decimal representations aside from how to
>>> round when converting to and from strings. You have to provide/accept
>>> *at least* 9 decimal digits in the significand for single-precision
>>> and 17 for double-precision (section 5.6). AFAIK implementations will
>>> vary in how they handle cases where a binary significand would yield
>>> more digits than that.
>> I was actually more interested in the opposite situation, where the decimal
>> representation (which is what a user would most likely provide) doesn't have
>> a finite binary expansion: what happens then, something analogous to the
>> decimal "rule of fives"?
> Since according to my calculations there are only about
>>>> 4* 10**17 * 308
More straightforwardly, it's not too far below 2**64.
> double-precision floats, there are huge gaps in the floating point
> representation of the real line.
> Any user input or calculation result just gets converted to the
> closest float.
Yes. But the "huge" gaps are only huge in absolute size; in fractional
error they're always about the same size. They are usually only an
issue if you're representing numbers in a small range with a huge
offset. To take a not-so-random example, you could be representing
times in days since November 17 1858, when what you care about are the
microsecond-scale differences in photon arrival times. Even then
you're probably okay as long as you compute directly (t[i] =
i*dt/86400 + start_t) rather than having some sort of running
accumulator (t[i]=t; t += dt/86400).
Professor Kahan's reasoning for using doubles for most everything is
that they generally have so much more precision than you actually need
that you can get away with being sloppy.
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