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Best simple explanation I've seen so far. I highly recommend Fabien's Game Engine Black Book. I'm halfway through it, and it's really fun. I've only been a sof

Best simple explanation I've seen so far. I highly recommend Fabien's Game Engine Black Book. I'm halfway through it, and it's really fun. I've only been a software dev for 6 years, so looking at how things could be hacked around in the 90s to squeeze every drop of performance out of very constrained devices is fascinating.

s17n 9y ago
I dunno all this talk of "windows" and "buckets" etc doesn't seem particularly simple to me.
- kehrlann 9y ago
  I guess I'm not math-y enough so that I intuitively understand the simple math formula. On the other hand, I like his window image : e.g. I understand better how, the further you get from 0, the more your precision goes down, because "bigger windows, divided in the same number of buckets".
  - conistonwater 9y ago
    So the question is: if you feel like you don't get the math formula, but you get the windows-and-buckets explanation of it, is it still possible that your understanding doesn't match the true underlying concept? Because that is the pitfall with a lot of intuitive explanations, that unless you are sure that the explanations are equivalent to the true thing, you might end up understanding an idea that is close but slightly off. So a puzzle: if two positive numbers in exactly the same window are subtracted, what is the worst-case rounding error you can get in the result?
    - sharpneli 9y ago
      Error relative to the original numbers or to the result?
      - deleted 9y ago
        [deleted]
    - ambicapter 9y ago
      Is it 2(1/2^23)(window size)?
      - conistonwater 9y ago
        It should be zero, exactly zero, i.e., no rounding error. Because if you do long subtraction on the significands, the number of digits you need to store the result is at most the number of digits you had, because the numbers were within a factor of two from each other (so no shifting left or right). Just to check, with sbv in haskell, using the Z3 SMT solver: λ> f (x::SFloat) (y::SFloat) = fpIsNormal x &&& fpIsNormal y &&& y/2 .<= x &&& x .<= 2*y ==> fromSFloat sRNE (x - y) .== (fromSFloat sRNE x :: SDouble) - (fromSFloat sRNE y :: SDouble) f :: SFloat -> SFloat -> SBool λ> prove (forAll_ f) Q.E.D. Or without rounding through double, using the formula for exact rounding error of a sum: λ> g (x::SFloat) (y::SFloat) = fpIsNormal x &&& fpIsNormal y &&& 0 .<= x &&& y .<= x &&& x .<= 2*y ==> (x - ((x-y) + y)) .== 0 g :: SFloat -> SFloat -> SBool λ> prove (forAll_ g) Q.E.D.
    - kllrnohj 9y ago
      > you might end up understanding an idea that is close but slightly off Just like most floating point numbers ba dum tss
    - jessermeyer 9y ago
      To confuse analogies with formal reasoning is a misuse of analogies. They are stepping stones, a kind of 'sniff test' or 'sizing up' of the mind to set the stage for more formal understanding. Is this your point? Because, to re-use your argument, the pitfall with many formal definitions is that they are not easily understandable, whose payoff is often not much more than the analogy for the learner, but requires significantly more resources. So, which would you prefer as an introduction? A "good enough" understanding, or no understanding because the bar of entry was too high?
megaman22 9y ago
It's a fantastic book. Fabien has a gift for taking complex topics and presenting them in a concise and approachable way. I hope he continues on with the project and does books on Doom and Quake.

s17n 9y ago

I dunno all this talk of "windows" and "buckets" etc doesn't seem particularly simple to me.

kehrlann 9y ago
I guess I'm not math-y enough so that I intuitively understand the simple math formula. On the other hand, I like his window image : e.g. I understand better how, the further you get from 0, the more your precision goes down, because "bigger windows, divided in the same number of buckets".
- conistonwater 9y ago
  So the question is: if you feel like you don't get the math formula, but you get the windows-and-buckets explanation of it, is it still possible that your understanding doesn't match the true underlying concept? Because that is the pitfall with a lot of intuitive explanations, that unless you are sure that the explanations are equivalent to the true thing, you might end up understanding an idea that is close but slightly off. So a puzzle: if two positive numbers in exactly the same window are subtracted, what is the worst-case rounding error you can get in the result?
  - sharpneli 9y ago
    Error relative to the original numbers or to the result?
    - deleted 9y ago
      [deleted]
  - ambicapter 9y ago
    Is it 2(1/2^23)(window size)?
    - conistonwater 9y ago
      It should be zero, exactly zero, i.e., no rounding error. Because if you do long subtraction on the significands, the number of digits you need to store the result is at most the number of digits you had, because the numbers were within a factor of two from each other (so no shifting left or right). Just to check, with sbv in haskell, using the Z3 SMT solver: λ> f (x::SFloat) (y::SFloat) = fpIsNormal x &&& fpIsNormal y &&& y/2 .<= x &&& x .<= 2*y ==> fromSFloat sRNE (x - y) .== (fromSFloat sRNE x :: SDouble) - (fromSFloat sRNE y :: SDouble) f :: SFloat -> SFloat -> SBool λ> prove (forAll_ f) Q.E.D. Or without rounding through double, using the formula for exact rounding error of a sum: λ> g (x::SFloat) (y::SFloat) = fpIsNormal x &&& fpIsNormal y &&& 0 .<= x &&& y .<= x &&& x .<= 2*y ==> (x - ((x-y) + y)) .== 0 g :: SFloat -> SFloat -> SBool λ> prove (forAll_ g) Q.E.D.
  - kllrnohj 9y ago
    > you might end up understanding an idea that is close but slightly off Just like most floating point numbers ba dum tss
  - jessermeyer 9y ago
    To confuse analogies with formal reasoning is a misuse of analogies. They are stepping stones, a kind of 'sniff test' or 'sizing up' of the mind to set the stage for more formal understanding. Is this your point? Because, to re-use your argument, the pitfall with many formal definitions is that they are not easily understandable, whose payoff is often not much more than the analogy for the learner, but requires significantly more resources. So, which would you prefer as an introduction? A "good enough" understanding, or no understanding because the bar of entry was too high?

megaman22 9y ago

It's a fantastic book. Fabien has a gift for taking complex topics and presenting them in a concise and approachable way. I hope he continues on with the project and does books on Doom and Quake.