Floats vs rationals in arbitrary precision fractional arithmetic (C/C++)

Question

Since there are two ways of implementing an AP fractional number, one is to emulate the storage and behavior of the double data type, only with more bytes, and the other is to use an existing integer APA implementation for representing a fractional number as a rational i.e. as a pair of integers, numerator and denominator, which of the two ways are more likely to deliver efficient arithmetic in terms of performance? (Memory usage is really of minor concern.)

I'm aware of the existing C/C++ libraries, some of which offer fractional APA with "floats" and other with rationals (none of them features fixed-point APA, however) and of course I could benchmark a library that relies on "float" implementation against one that makes use of rational implementation, but the results would largely depend on implementation details of those particular libraries I would have to choose randomly from the nearly ten available ones. So it's more theoretical pros and cons of the two approaches that I'm interested in (or three if take into consideration fixed-point APA).

Answer 1

The question is what you mean by arbitrary precision that you mention in the title. Does it mean "arbitrary, but pre-determined at compile-time and fixed at run-time"? Or does it mean "infinite, i.e. extendable at run-time to represent any rational number"?

In the former case (precision customizable at compile-time, but fixed afterwards) I'd say that one of the most efficient solutions would actually be fixed-point arithmetic (i.e. none of the two you mentioned).

Firstly, fixed-point arithmetic does not require any dedicated library for basic arithmetic operations. It is just a concept overlaid over integer arithmetic. This means that if you really need a lot of digits after the dot, you can take any big-integer library, multiply all your data, say, by 2^64 and you basically immediately get fixed-point arithmetic with 64 binary digits after the dot (at least as long as arithmetic operations are concerned, with some extra adjustments for multiplication and division). This is typically significantly more efficient than floating-point or rational representations.

Note also that in many practical applications multiplication operations are often accompanied by division operations (as in x = y * a / b) that "compensate" for each other, meaning that often it is unnecessary to perform any adjustments for such multiplications and divisions. This also contributes to efficiency of fixed-point arithmetic.

Secondly, fixed-point arithmetic provides uniform precision across the entire range. This is not true for either floating-point or rational representations, which in some applications could be a significant drawback for the latter two approaches (or a benefit, depending on what you need).

So, again, why are you considering floating-point and rational representations only. Is there something that prevents you from considering fixed-point representation?