On Mon, 2010-11-29 at 10:24 -0500, Hal Finkel wrote:
On Mon, 2010-11-29 at 11:07 +0100, Mathias Gaunard wrote:
On 28/11/2010 22:30, Hal Finkel wrote:
Hello,
A few months ago, I created an MPL extension type for representing real numbers.
I find that idea very interesting.
The current default is 20 to allow an effective precision near that of a double. Also, continued fractions, being "based on" rational numbers, can exactly represent any rational (for example, 1/3) so the computations can be more accurate than those using a floating-point representation.
It would be nice to know what minimum value is necessary to provide correctly rounded results for IEEE754 floats and doubles.
This depends on the number: For a rational, you need only two integers ;)
Actually, ignore that statement. You only need two integers for a0 + 1/a1, in general you need more. For example, if you look at the test program (basic.cpp in the archive), it provides some examples (where the floating-point output is float, double, long double): 3/2 + 1/4 = [ 1 1 3] 1.75 1.75 1.75 1/7 + 1/15 = [ 0 4 1 3 2 2] 0.209524 0.20952380952381 0.209523809523809524 There are other continued fraction representations where any rational can be represented with a fixed number of integers (a0 + b0/(a1 + ...)), but the algorithms are a little more complicated, and I fear the compile times would increase. -Hal
For a general real number, it depends on the rate of convergence. In my experience, you get about one decimal digit per integer coefficient, so 10 is sufficient for single precision, 20 for double precision (and long double too on an x86, since the extended type there is only 18 digits). The MPL containers generally come in max-size-increments of 10, but it would not be too hard to generate something else.
-Hal
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