5 ms·

How much precision can you squeeze out of a table?

62 points by nomemory 6mo ago

PaulHoule 6mo ago
When I was in high school I thought book this was so much fun http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/118/Francis_Scheid_Schaums_Outline_of_Numerical_Analysis%2C_Second_Edition____1989.pdf?sequence=1&isAllowed=y http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/... which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences https://en.wikipedia.org/wiki/Finite_difference https://en.wikipedia.org/wiki/Finite_difference
- yababa_y 6mo ago
  i've been missing this knowledge! thank you for the recommendation.
- Rendello 6mo ago
  The typography and design inside the book is beautiful but the cover looks like an old bargain-store flyer. There's an expression about that, something about judging a book by its cover? I can't remember exactly.
- harperlee 6mo ago
  In order to mitigate link rot, the book is Numerical Analysis Second Edition, from Francis Scheid (Schaum's Outlines)
- animal531 6mo ago
  Ah, very cool. My personal favourite is this one: https://archive.org/details/engineeringformu00giec_0/page/n5/mode/2up https://archive.org/details/engineeringformu00giec_0/page/n5... My father owned a copy of it which I still have to this day. It had a lot of different formulas than the ones we were taught in school, and I liked the range of topics that it covered.
- the_arun 6mo ago
  I just tried a portion of the url & it took me to Bangladesh university - http://182.160.97.198:8080/xmlui/bitstream/handle/ http://182.160.97.198:8080/xmlui/bitstream/handle/ . Intersecting. When I go to root of this url, the error messages are listing the softwares this site is powered by. Generally this is not considered a secure way of protecting a site.
gus_massa 6mo ago
Nitpicking: > You’re never going to get error less than 10E−15 since that’s the error in the tabulated values, If you have like 100 (or 400) values in the table, you can squeeze one more digit. In the simple case, imagine you have the constant pi, with 15 digits, repeated 100 times. If the rounding was done in a random direction like floor(pi * 1E15 + random() - 1/2 ) / 1E15 then you can use statistic to get an average with an error that is like K/sqrt(N) where K is a constant and N is the number of samples. If you were paying attention in you statistic clases, you probably know the value of K, but it's not my case. Anyway, just pick N=100 or N=400 or something like that and you get another digit for "free". Nobody builds a table for a constant and uses a uniform offset for rounding. Anyway, with some smooth function that has no special values in the points where it's sample, the exact value of the cut decimal is quite "random" and you get a similar randomization.
- _0ffh 6mo ago
  Also nobody in his right mind uses lookup tables where the table value is actually the float approximation of the true f(x) - you choose the support values to minimize an error (e.g. mse) of a dense sampling of your interpolated value over x (or, in the limit, the integral of the chosen error function between the true curve and the interpolation of your supports). If you want to e.g. approximate a convex function using linear interpolation, all the tabulated values f'(x) would be <= the true f(x).
  - AlotOfReading 6mo ago
    The true value is far more useful in a lot of cases. If you're building a table indexed by the upper mantissa bits of the float, for example, it's difficult to distribute the error properly across all intervals.
spacebacon 6mo ago
I can't know how to hear anymore about tables

62 points by nomemory 6mo ago

When I was in high school I thought book this was so much fun http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/118/Francis_Scheid_Schaums_Outline_of_Numerical_Analysis%2C_Second_Edition____1989.pdf?sequence=1&isAllowed=y http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/... which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences https://en.wikipedia.org/wiki/Finite_difference https://en.wikipedia.org/wiki/Finite_difference

yababa_y 6mo ago
i've been missing this knowledge! thank you for the recommendation.
Rendello 6mo ago
The typography and design inside the book is beautiful but the cover looks like an old bargain-store flyer. There's an expression about that, something about judging a book by its cover? I can't remember exactly.
harperlee 6mo ago
In order to mitigate link rot, the book is Numerical Analysis Second Edition, from Francis Scheid (Schaum's Outlines)
animal531 6mo ago
Ah, very cool. My personal favourite is this one: https://archive.org/details/engineeringformu00giec_0/page/n5/mode/2up https://archive.org/details/engineeringformu00giec_0/page/n5... My father owned a copy of it which I still have to this day. It had a lot of different formulas than the ones we were taught in school, and I liked the range of topics that it covered.
the_arun 6mo ago
I just tried a portion of the url & it took me to Bangladesh university - http://182.160.97.198:8080/xmlui/bitstream/handle/ http://182.160.97.198:8080/xmlui/bitstream/handle/ . Intersecting. When I go to root of this url, the error messages are listing the softwares this site is powered by. Generally this is not considered a secure way of protecting a site.

gus_massa 6mo ago

Nitpicking: > You’re never going to get error less than 10E−15 since that’s the error in the tabulated values, If you have like 100 (or 400) values in the table, you can squeeze one more digit. In the simple case, imagine you have the constant pi, with 15 digits, repeated 100 times. If the rounding was done in a random direction like floor(pi * 1E15 + random() - 1/2 ) / 1E15 then you can use statistic to get an average with an error that is like K/sqrt(N) where K is a constant and N is the number of samples. If you were paying attention in you statistic clases, you probably know the value of K, but it's not my case. Anyway, just pick N=100 or N=400 or something like that and you get another digit for "free". Nobody builds a table for a constant and uses a uniform offset for rounding. Anyway, with some smooth function that has no special values in the points where it's sample, the exact value of the cut decimal is quite "random" and you get a similar randomization.

_0ffh 6mo ago
Also nobody in his right mind uses lookup tables where the table value is actually the float approximation of the true f(x) - you choose the support values to minimize an error (e.g. mse) of a dense sampling of your interpolated value over x (or, in the limit, the integral of the chosen error function between the true curve and the interpolation of your supports). If you want to e.g. approximate a convex function using linear interpolation, all the tabulated values f'(x) would be <= the true f(x).
- AlotOfReading 6mo ago
  The true value is far more useful in a lot of cases. If you're building a table indexed by the upper mantissa bits of the float, for example, it's difficult to distribute the error properly across all intervals.

spacebacon 6mo ago

I can't know how to hear anymore about tables