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> I do not see how this can possibly be the case. How do you define rational arithmetic with bounded denominators so that addition is associative? There's only

> I do not see how this can possibly be the case. How do you define rational arithmetic with bounded denominators so that addition is associative? There's only one mathematically natural choiceIn the case of bounded numerator and bounded denominator. You simply define (an/ad) + (bn/bd) = (bd × an + ad × bn) / (bd × ad) where the + on the right hand side is ordinary twos complement addition (and the equality test for an/ad and bn/bd is ad × bn == bd × an, which also naturally accounts for the zero-denominator-due-to-zero-divisors cases; alternatively, you can put everything in lowest terms, but there's no point). > Regardless of efficiency, using rational arithmetic with bounded integers in numerical computing would be extremely unnatural to most analysts Floating-point was invented by numerical analysts for numerical analysts. No wonder they find it most natural. The overwhelming majority software developers are not numerical analysts, and most programming languages do not target numerical analysts. Nobody says that numerical analysts should not use floats (except maybe the unum guy), we're arguing about defaults in languages that explicitly do not have numerical analysts among their core target audience.

enriquto 7y ago
>> I do not see how this can possibly be the case. How do you define rational arithmetic with bounded denominators so that addition is associative? > That's easy. In the case of bounded numerator and bounded denominator, you simply define (an/ad) + (bn/bd) = (bd x an + ad x bn) / (bd x ad) This definition is not complete. What happens when "bd x ad" is larger than the maximum allowed denominator ? > The overwhelming majority software developers are not numerical analysts, the overwhelming majority of young people who learn to program today do machine learning, which is based on mungling huge arrays of floating point numbers. Tell them to use rationals if you dare!
- saithound 7y ago
  > This definition is not complete. What happens when "bd x ad" is larger than the maximum allowed denominator ? All integer operations are twos complement operations, as I'm sure you guessed anyway. > the overwhelming majority of young people who learn to program today do machine learning, which is based on mungling huge arrays of floating point numbers. Tell them to use rationals if you dare! That's very far from factually true, but this kind of argument is not relevant to the Raku defaults in any case. A machine learning library can use whatever optimized number representation its creators wish to use. In fact, the default choice of most language implementors (double-precision floating point) is typically not the representation used in training or inference on deep learning models anyway. The 1080Ti is fast enough only with single-precision floats.
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enriquto 7y ago

>> I do not see how this can possibly be the case. How do you define rational arithmetic with bounded denominators so that addition is associative? > That's easy. In the case of bounded numerator and bounded denominator, you simply define (an/ad) + (bn/bd) = (bd x an + ad x bn) / (bd x ad) This definition is not complete. What happens when "bd x ad" is larger than the maximum allowed denominator ? > The overwhelming majority software developers are not numerical analysts, the overwhelming majority of young people who learn to program today do machine learning, which is based on mungling huge arrays of floating point numbers. Tell them to use rationals if you dare!

saithound 7y ago
> This definition is not complete. What happens when "bd x ad" is larger than the maximum allowed denominator ? All integer operations are twos complement operations, as I'm sure you guessed anyway. > the overwhelming majority of young people who learn to program today do machine learning, which is based on mungling huge arrays of floating point numbers. Tell them to use rationals if you dare! That's very far from factually true, but this kind of argument is not relevant to the Raku defaults in any case. A machine learning library can use whatever optimized number representation its creators wish to use. In fact, the default choice of most language implementors (double-precision floating point) is typically not the representation used in training or inference on deep learning models anyway. The 1080Ti is fast enough only with single-precision floats.

deleted 7y ago

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