141–150 of 205 posts
Earlier quoted context omitted.
That's what I was thinking too; If I do 1 / 3, then of course it will have to truncate, and integration errors would still be inevitable.
Actually it's possible to represent 1/3 perfectly accurately, but what about all the numbers that it's theoretically impossible to compute? (Almost all real numbers have this property)
At a glance this reads similar to Interval Arithmetic in that it places bounds on how much error a value carries. Is there something more novel to his approach? https://en.wikipedia.org/wiki/Interval_arithmetic
Earlier quoted context omitted.
Actually it's possible to represent 1/3 perfectly accurately, but what about all the numbers that it's theoretically impossible to compute? (Almost all real numbers have this property)
> Almost all real numbers have this property > Almost all I love this comment because it brings back memories of school. In a layman's terms, I think almost all in this case means all but a finite amount can we say almost all real numbers are irrational? 1 1/1 2/1 3/1 4/1 5/1 ... 2 1/2 2/2 3/2 4/2 5/2 ... 3 ... so clearly we can count all the rational numbers but how many irrational numbers are there? are there (many…
So all of the irrationals that we use or could ever use in calculations are countable. The uncountability of the irrationals comes entirely from the uncomputable ones, which we will probably never see.
The reals are deeply weird.
The floating point error problem has not been solved. This patent describes a floating-point representation that includes fields for storing error information. The standard IEEE floating-point representation has three fields: a sign field, an exponent field, and a mantissa (or significand). This patent proposes reducing the size of other fields and adding additional fields to store error information. The error inform…
Earlier quoted context omitted.
That's what I was thinking too; If I do 1 / 3, then of course it will have to truncate, and integration errors would still be inevitable.
There is inaccuracy, but the point is that it tracks how much inaccuracy there might be. I picture this as being similar to how computers can't trust time for all sorts of reasons, so Google's Spanner uses time ranges and estimates of potential inaccuracy to make it possible to work with that. It will truncate, so you won't really have 1/3, but you'll know it's approximately 0.333, definitely more than 6/20 but defin…
So in that case, "[it] allows representation of real numbers accurate to the last digit" doesn't really hold true.
The floating point error problem has not been solved. This patent describes a floating-point representation that includes fields for storing error information. The standard IEEE floating-point representation has three fields: a sign field, an exponent field, and a mantissa (or significand). This patent proposes reducing the size of other fields and adding additional fields to store error information. The error inform…
> “In the current art, static error analysis requires significant mathematical analysis and cannot determine actual error in real time,” reads a section of the patent. “This work must be done by highly skilled mathematician programmers. Therefore, error analysis is only used for critical projects because of the greatly increased cost and time required. In contrast, the present invention provides error computation in…
I'm the main author of Arb. Note that it's an arbitrary-precision library. It's ~100x times slower than hardware floating-point because of using arbitrary-precision floating-point numbers implemented entirely in software. But if you have to do arbitrary-precision arithmetic to begin with, Arb's error tracking only adds negligible further overhead. For machine precision, I believe ordinary interval arithmetic is the b…
Thanks for the numbers, how did you get that estimate? Did you consider SIMD?
Without reading the patent it sounds a lot like interval arithmetic [1] which sounds like a really good idea at first but is not without its own problems. For example the inverse 1/x for an interval x like [-1,+1] containing 0 consists of two intervals (-∞,-1] and [+1;+∞). [1] https://en.wikipedia.org/wiki/Interval_arithmetic
You could always just return nan in that case, like for normal arithmetic.
Earlier quoted context omitted.
> Almost all real numbers have this property > Almost all I love this comment because it brings back memories of school. In a layman's terms, I think almost all in this case means all but a finite amount can we say almost all real numbers are irrational? 1 1/1 2/1 3/1 4/1 5/1 ... 2 1/2 2/2 3/2 4/2 5/2 ... 3 ... so clearly we can count all the rational numbers but how many irrational numbers are there? are there (many…
The rational numbers are clearly countable. The irrational numbers are uncountable, which means that for every rational number there are infinitely many irrational numbers.
Infinity is funny like that.
The conclusion that there are somehow more irrationals than rationals depends on subtle philosophical points that have no possible proof or disproof and usually get glossed over. Accepting that philosophy also leads to the conclusion that not only do numbers which can in no way ever be represented exist, but there are more of them than numbers which we can explicitly name. Now I ask you, in what sense do they REALLY exist?
Earlier quoted context omitted.
Actually it's possible to represent 1/3 perfectly accurately, but what about all the numbers that it's theoretically impossible to compute? (Almost all real numbers have this property)
> Almost all real numbers have this property > Almost all I love this comment because it brings back memories of school. In a layman's terms, I think almost all in this case means all but a finite amount can we say almost all real numbers are irrational? 1 1/1 2/1 3/1 4/1 5/1 ... 2 1/2 2/2 3/2 4/2 5/2 ... 3 ... so clearly we can count all the rational numbers but how many irrational numbers are there? are there (many…