4 ms·

That isn't a computerism. It is a feature of floating point arithmetic. Computers are perfectly able to calculate perfectly. >should get the same right answer

That isn't a computerism. It is a feature of floating point arithmetic. Computers are perfectly able to calculate perfectly. >should get the same right answer in a Jupyter notebook. The only person who should be exposed to the base 2 "Why does my notebook take hours to do a simple task" >There are numerous social consequences of this that are harmful such as the perception that computer programmers are "grinds" and "nerds" and the idea that "idea people" are more worthy than the people that execute, etc. The benfits of floating point arithmetic easily outweigh people having to learn it.

wruza 3y ago
"Why does my notebook take hours to do a simple task" “Because the chip it is running on doesn’t have native base-10 math”? See also “why my notebook [is fast but] yields incorrect results”. The benfits of floating point arithmetic easily outweigh people having to learn it. GP talks about saving non-low-level programmers from base 2 FP, not about removing it. CPUs could use an additional block (or mode) of base 10 exponent FP. This and other geeky issues make programmers programmers instead of making everyone a programmer. The consequence of this is much heavier than any benefits of base 2 FP.
- constantcrying 3y ago
  >“Because the chip it is running on doesn’t have native base-10 math”? Base 10 fixes exactly zero of the problems with floating point. Decimal floating point exists as a standard already. It is even part of the upcomming C standard. But again decimal arithmetic is just as weird as floating point arithmetic. The change of base s irrelevant, except for a few niche applications. >The consequence of this is much heavier than any benefits of base 2 FP. Totally false. Decimals don't fix floats. They are just as weird. Changing the base is irrelevant to the inherent properties of floats and using base 10 instead of base 2 just means that with a very high chance you get something even worse. If you do not understand the basics of floating point arithmetic you should not be programming software. Tough world out there for people who refuse to learn, I know.
  - PaulHoule 3y ago
    No way. The problem with binary floating point is that it is input and output as decimal floating point. Every floating point number is really a fraction of the form M ------- b^E When you write x = 0.1 you are really asking for 1/10. If b=2, however, you can only get a denominator of 1/2, 1/4, 1/8 or something like that. 1/10 just doesn't exist in that number system, but there is a number A that you get when you ask for 0.1 that round-trips back to 0.1, and the same is true for 0.2 (B) and 0.3 (C). The trouble is those substitute numbers aren't the real numbers and A != B + C This has nothing to do with numeric precision, it's always going to be off a bit even if you are using 1024-bit floats or 1048576-bit floats. The problem punches above its weight because it is targeted right at two fault lines of the mind: (1) people flinch at inconsistencies, 0.1 + 0.2 = 0.3 is an identity and when basic identities are wrong people feel uncomfortable and don't want to proceed; if you are working with an accountant, for instance, and they see something that is not consistent in a way they've been trained has to be consistent, they will just stop until it is consistent. (2) A certain kind of laziness leads people to not get to the root of a problem like this and instead waste a huge amount of time and energy into non-solutions (rounding!) that are just like pushing a bubble around under a rug. Note that decimal floating point does not require that you use BCD. That is, the mantissa and the exponent of a floating point number are just integers, and unlike the floats, base 7 and base 192 integers are the same numbers. An obvious idea is to use base 2 for the mantissa and exponent, just have the exponent be base 10. One difficulty is the cost of sliding two numbers so they have the same exponent before you add them, for instance to add 43 723 --- 766 you have to express the 723 and 43 in the same base to add them (multiply/add by ten) and multiplication by powers of ten is a lot hard with base 2 math than with base 10 math. You also can represent decimal floating point numbers with a decimal mantissa and face the tradeoff of BCD numbers being wasteful of bits but the factors of 10 being easy to deal with. There is an efficiency gap between hardware binary floating point and hardware decimal floating point but it's not as bad as you might think.
    - constantcrying 3y ago
      The problems with floats are independent of the basis. >1/10 just doesn't exist in that number system And 1/3 doesn't exist for b=10. I really have no idea what you are on about. The number system you want does not exist, it is a mathematical theorem, floats are the best approximation of real numbers and choosing b=10 is dumb outside of specific applications. Decimal floating point fixes nothing it just moves around the issues where they less affect numbers in base 10. It is exactly as broken as base 2. You still violate basic identities in b=10, just different ones. Again, the number system you want doesn't exist and it can't. You can never approximate the real numbers with constant bits, division and arithmetic consistency.

wruza 3y ago

"Why does my notebook take hours to do a simple task" “Because the chip it is running on doesn’t have native base-10 math”? See also “why my notebook [is fast but] yields incorrect results”. The benfits of floating point arithmetic easily outweigh people having to learn it. GP talks about saving non-low-level programmers from base 2 FP, not about removing it. CPUs could use an additional block (or mode) of base 10 exponent FP. This and other geeky issues make programmers programmers instead of making everyone a programmer. The consequence of this is much heavier than any benefits of base 2 FP.

constantcrying 3y ago
>“Because the chip it is running on doesn’t have native base-10 math”? Base 10 fixes exactly zero of the problems with floating point. Decimal floating point exists as a standard already. It is even part of the upcomming C standard. But again decimal arithmetic is just as weird as floating point arithmetic. The change of base s irrelevant, except for a few niche applications. >The consequence of this is much heavier than any benefits of base 2 FP. Totally false. Decimals don't fix floats. They are just as weird. Changing the base is irrelevant to the inherent properties of floats and using base 10 instead of base 2 just means that with a very high chance you get something even worse. If you do not understand the basics of floating point arithmetic you should not be programming software. Tough world out there for people who refuse to learn, I know.
- PaulHoule 3y ago
  No way. The problem with binary floating point is that it is input and output as decimal floating point. Every floating point number is really a fraction of the form M ------- b^E When you write x = 0.1 you are really asking for 1/10. If b=2, however, you can only get a denominator of 1/2, 1/4, 1/8 or something like that. 1/10 just doesn't exist in that number system, but there is a number A that you get when you ask for 0.1 that round-trips back to 0.1, and the same is true for 0.2 (B) and 0.3 (C). The trouble is those substitute numbers aren't the real numbers and A != B + C This has nothing to do with numeric precision, it's always going to be off a bit even if you are using 1024-bit floats or 1048576-bit floats. The problem punches above its weight because it is targeted right at two fault lines of the mind: (1) people flinch at inconsistencies, 0.1 + 0.2 = 0.3 is an identity and when basic identities are wrong people feel uncomfortable and don't want to proceed; if you are working with an accountant, for instance, and they see something that is not consistent in a way they've been trained has to be consistent, they will just stop until it is consistent. (2) A certain kind of laziness leads people to not get to the root of a problem like this and instead waste a huge amount of time and energy into non-solutions (rounding!) that are just like pushing a bubble around under a rug. Note that decimal floating point does not require that you use BCD. That is, the mantissa and the exponent of a floating point number are just integers, and unlike the floats, base 7 and base 192 integers are the same numbers. An obvious idea is to use base 2 for the mantissa and exponent, just have the exponent be base 10. One difficulty is the cost of sliding two numbers so they have the same exponent before you add them, for instance to add 43 723 --- 766 you have to express the 723 and 43 in the same base to add them (multiply/add by ten) and multiplication by powers of ten is a lot hard with base 2 math than with base 10 math. You also can represent decimal floating point numbers with a decimal mantissa and face the tradeoff of BCD numbers being wasteful of bits but the factors of 10 being easy to deal with. There is an efficiency gap between hardware binary floating point and hardware decimal floating point but it's not as bad as you might think.
  - constantcrying 3y ago
    The problems with floats are independent of the basis. >1/10 just doesn't exist in that number system And 1/3 doesn't exist for b=10. I really have no idea what you are on about. The number system you want does not exist, it is a mathematical theorem, floats are the best approximation of real numbers and choosing b=10 is dumb outside of specific applications. Decimal floating point fixes nothing it just moves around the issues where they less affect numbers in base 10. It is exactly as broken as base 2. You still violate basic identities in b=10, just different ones. Again, the number system you want doesn't exist and it can't. You can never approximate the real numbers with constant bits, division and arithmetic consistency.