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0.1 and 0.2 Returns 0.30000000000000004 (2018)

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Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#111

Post by dataflow »

crazygringo wrote: post #76
enriquto wrote: post #66

Earlier quoted context omitted.

> It seems like such a useful and intuitive idea I have to wonder why it isn't a primitive in any of the common programming languages. It is basically useless for numerical computation when you perform iterations. Even good convergent algorithms can diverge with interval arithmetic. As you accumulate operations on the same numbers, their intervals become larger and larger, growing eventually to infinity. It has some…

But if the intervals are growing to infinity, then should you be trusting your result at all? Are there really cases where current FP arithmetic gives an accurate result, but where the error bounds of interval arithmetic would grow astronomically? It seems like you'd have to trust FP rounding to always cancel itself out in the long run instead of potentially accumulating more and more bias with each iteration. Is tha…

There was a time when I thought (like you) that everybody should be using interval arithmetic, but then I came across a counterexample that convinced me I was wrong. I don't remember the precise example, but maybe the following will do the same for you.

Say x = 4.0 ± 1.0. What is x / x?

It should be x / x = 1.0 ± 0.0, but interval arithmetic will give you [3/5, 5/3].

Notice the interval is objectively wrong, as the result cannot be anything other than 1.0. Now imagine what happens if you do this a few more iterations. Your interval will diverge to (0, +∞), becoming useless.

The moral of the story (which may be more obvious in hindsight): interval arithmetic is a local operation; error analysis is a global operation. Naturally the former cannot substitute for the latter.

Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#112

Post by seoaeu »

crazygringo wrote: post #76
enriquto wrote: post #66

Earlier quoted context omitted.

> It seems like such a useful and intuitive idea I have to wonder why it isn't a primitive in any of the common programming languages. It is basically useless for numerical computation when you perform iterations. Even good convergent algorithms can diverge with interval arithmetic. As you accumulate operations on the same numbers, their intervals become larger and larger, growing eventually to infinity. It has some…

But if the intervals are growing to infinity, then should you be trusting your result at all? Are there really cases where current FP arithmetic gives an accurate result, but where the error bounds of interval arithmetic would grow astronomically? It seems like you'd have to trust FP rounding to always cancel itself out in the long run instead of potentially accumulating more and more bias with each iteration. Is tha…

This feels like the central limit theorem. Accumulating a bunch of random variables is going to produce a normal distribution, and a normal distribution has an infinite interval.

Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#113

Post by generalizations »

dataflow wrote: post #111
crazygringo wrote: post #76

Earlier quoted context omitted.

But if the intervals are growing to infinity, then should you be trusting your result at all? Are there really cases where current FP arithmetic gives an accurate result, but where the error bounds of interval arithmetic would grow astronomically? It seems like you'd have to trust FP rounding to always cancel itself out in the long run instead of potentially accumulating more and more bias with each iteration. Is tha…

There was a time when I thought (like you) that everybody should be using interval arithmetic, but then I came across a counterexample that convinced me I was wrong. I don't remember the precise example, but maybe the following will do the same for you. Say x = 4.0 ± 1.0. What is x / x? It should be x / x = 1.0 ± 0.0, but interval arithmetic will give you [3/5, 5/3]. Notice the interval is objectively wrong , as the…

Your example only proves your point if every instance of x is the same x, with the same objective value. i.e., what if x/x is actually (x=5.0)/(x=3.0) ?

Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#114

Post by dataflow »

generalizations wrote: post #113
dataflow wrote: post #111

Earlier quoted context omitted.

There was a time when I thought (like you) that everybody should be using interval arithmetic, but then I came across a counterexample that convinced me I was wrong. I don't remember the precise example, but maybe the following will do the same for you. Say x = 4.0 ± 1.0. What is x / x? It should be x / x = 1.0 ± 0.0, but interval arithmetic will give you [3/5, 5/3]. Notice the interval is objectively wrong , as the…

Your example only proves your point if every instance of x is the same x, with the same objective value. i.e., what if x/x is actually (x=5.0)/(x=3.0) ?

Then you're computing x/y and not x/x...

Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#115

Post by dragonwriter »

dataflow wrote: post #111
crazygringo wrote: post #76

Earlier quoted context omitted.

But if the intervals are growing to infinity, then should you be trusting your result at all? Are there really cases where current FP arithmetic gives an accurate result, but where the error bounds of interval arithmetic would grow astronomically? It seems like you'd have to trust FP rounding to always cancel itself out in the long run instead of potentially accumulating more and more bias with each iteration. Is tha…

There was a time when I thought (like you) that everybody should be using interval arithmetic, but then I came across a counterexample that convinced me I was wrong. I don't remember the precise example, but maybe the following will do the same for you. Say x = 4.0 ± 1.0. What is x / x? It should be x / x = 1.0 ± 0.0, but interval arithmetic will give you [3/5, 5/3]. Notice the interval is objectively wrong , as the…

> Say x = 4.0 ± 1.0. What is x / x?

> It should be x / x = 1.0 ± 0.0, but interval arithmetic will give you [3/5, 5/3].

This is the “dependency problem” which is eliminated in many cases, and mitigated in others, by rewriting so identical (vs. merely in components) values appear only once; when you might need a numerical answer at one point (if possible) but to use the value in further computations, this can be done by storing and manipulating values symbolically and extracting numerical results as needed.

Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#116

Post by thaumasiotes »

yarg wrote: post #67
OskarS wrote: post #42

Earlier quoted context omitted.

Yeah, you call tan() on that number, and suddenly your interval is like most of the number line. Actually, you don't even have to be that fancy: if the number is close to epsilon, the error bars on 1/x would be huge.

Sure, but what's the use case for mathematics where you don't know what side of an asymptote you're on?

>> if the number is close to epsilon, the error bars on 1/x would be huge.

> Sure, but what's the use case for mathematics where you don't know what side of an asymptote you're on?

Knowing which side of the asymptote you're on does not solve this problem or even ameliorate it.

Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#118

Post by dataflow »

enriquto wrote: post #86
crazygringo wrote: post #76

Earlier quoted context omitted.

But if the intervals are growing to infinity, then should you be trusting your result at all? Are there really cases where current FP arithmetic gives an accurate result, but where the error bounds of interval arithmetic would grow astronomically? It seems like you'd have to trust FP rounding to always cancel itself out in the long run instead of potentially accumulating more and more bias with each iteration. Is tha…

> But if the intervals are growing to infinity, then should you be trusting your result at all? Most often, yes; the probability distribution of your number inside that interval is not uniform, it is most likely very concentrated around a specific number inside the interval, not necessarily its center. After a few million iterations, the probability of the correct number being close to the boundary of the interval is…

I suggest adding an example to demonstrate this effect because I don't think it's necessarily obvious for someone who hasn't already seen it.

Re: 0.1 and 0.2 Returns 0.30000000000000004 (2018)

#119

Post by blake1 »

seoaeu wrote: post #112
crazygringo wrote: post #76

Earlier quoted context omitted.

But if the intervals are growing to infinity, then should you be trusting your result at all? Are there really cases where current FP arithmetic gives an accurate result, but where the error bounds of interval arithmetic would grow astronomically? It seems like you'd have to trust FP rounding to always cancel itself out in the long run instead of potentially accumulating more and more bias with each iteration. Is tha…

This feels like the central limit theorem. Accumulating a bunch of random variables is going to produce a normal distribution, and a normal distribution has an infinite interval.

Yeah, but the standard deviation goes by sqrt(n) for many operations, and there is significant autocorrelation. Interval arithmetic will give you worst-case bounds, which will quickly get fantastically pessimistic.

Numerical analysis was a field invented to give realistic error bounds.

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