Since we're on the subject of floating-point numbers in Lisp/Scheme: https://wukix.com/lisp-decimals
This uses the Lisp ratio type in place of floats, to avoid the usual float issues.
ntoronto, any thoughts?
This uses the Lisp ratio type in place of floats, to avoid the usual float issues.
ntoronto, any thoughts?
Just looking at the Wu-Decimal page, though, I can't tell whether they're internally exact rationals or base-10 floats. If they're base-10 floats, they have all the same issues base-2 floats have. If they're internally exact rationals, I wonder what they do for division, which the set D isn't closed under.
My favorite representation so far pairs a signed float in [2^-512,2^512) with a signed integer to extend the exponent range. The idea is to avoid overflow and underflow, particularly when multiplying thousands of probabilities.
(On average, adding a few thousand log probabilities or densities and then exponentiating the sum yields a number with about 9 digits precision. Worst case for adding log probabilities and exponentiating is about 8 digits precision, and for adding log densities it's 0 digits. Multiplying the same number of probabilities or densities retains about 13 digits in the worst case.)
Nitpick: although base 10 floats have similar issues related to being of limited precision, they are superior to base 2 floats in the aspect of representing decimal numbers without the binary approximation: e.g. 1/10 is nonrepeating as 1.0 x 10^-1 but infinitely repeating as binary, so for IEEE 754 binary32 you get 1.10011001100110011001101 (1.60000002384185791015625) x 2^-4.