What Every Computer Scientist Should Know About Floating-Point Arithmetic (1991)

Re:What Every Computer Scientist Should Know About Floating-... by bjourne on Tuesday June 30, 2020 @05:33AM (#23688298)

What do you mean by algebraic types? If you mean algebraic number then yes, it is true that rationals cannot express all algebraic numbers such as the square root of prime numbers. But neither can floats! In fact, floats are limited to either 2^32 or 2^64 numbers (give or take a few) but rationals can express as many numbers as you have bits of memory in your system. I.e the number of numbers rationals can express is in practice infinite.

Neither does floats support transcendental functions. Those functions are implemented using Taylor series approximation and the same technique could be used for implementing rational transcendental functions. To boot, you'd get much higher precision.

Yes, rationals with arbitrary precision are more expensive to compute with than precision-limited floats. But who cares? It is not the 80s anymore.

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Re:What Every Computer Scientist Should Know About Floating-... by jcranmer on Tuesday June 30, 2020 @08:44AM (#23689431)

> Yes, rationals with arbitrary precision are more expensive to compute with than precision-limited floats. But who cares? It is not the 80s anymore.

The people who care are the people who now either have to buy 10000× the computing power or solve only 1/10000× the largest problem size to solve their needs, usually without meaningfully improving their results. Put another way, using rationals would turn your 2020-era computer into a 1980-era computer in terms of performance on your linear algebra benchmarks.

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