4 ms·

> just computing the integrals numerically and then converting to fractions > because the error in result was smaller than the the threshold needed to convert

> just computing the integrals numerically and then converting to fractions > because the error in result was smaller than the the threshold needed to convert it to a fraction. Then why can't they display 9 instead of 9.000000007? What is that threshold if 0.7e-9 not small enough? And, are you telling me that computing integral(sin(x)dx from 0 to pi/3) can be done with better precision than 0.7e-9, whereas arcsin(arccos(arctan(tan(cos(sin(9)))))) cannot? Otherwise, of course I don't know how these devices work internally. But have you designed one, or you're just assuming things?

saagarjha 8y ago
I'm basing this on experience using one; I have no "inside information" on how they work so I'm doing my best with my insight. Here's what I've found just by messing around; if you want details I can post the "experiments" I did to get these: * The largest epsilon that the calculator is willing is confident about showing to the user is about 10^-13. (0.1 + 10^-13 - 0.1 = 10^-13, 0.1 + 10^-14 - 0.1 = 0) * The calculator will show about 10 digits of precision. (0.1 + 10^-10 = .1000000001, 0.1 + 10^-11 = 0.1) * Rounding seems to occur at the last digit of internal precision, perhaps at a 10^-14/10^-15 digit that isn't shown? This might mean the calculator is using double precision. (.5-10^13=0.5, .5-10^-14=1/2) I think the results you got are pretty consistent with this. First of all, the sequence of trigonometric operations you performed are quite lossy, since you're exercising a very small range and you perform them repeatedly, which magnifies the error. Performing the integral is quite accurate; even a simple midpoint Riemann sum with 1000 subdivisions is accurate to one part in 10^12). FYI, you seem to have been shadowbanned (I have chosen to vouch for the one you made here because it was somewhat reasonable and so I could respond to it), which is not surprising considering your other comments.
- ludditus 8y ago
  Thanks for the 10^-13 thing. It makes sense. As for shadowbanning... all I want to say is this. IMNSO opinion, whoever chooses to comment on places like this or like Reddit is an highly impolite, because that's what blogs are for, and commenting is allowed on my blog (moderated, so one wouldn't see a comment right away). I am sick of people ranting here or on reddit like: "oh, look of what crap some guy is saying somewhere," or "he's wrong wrt to this thing" ... but why on Earth aren't such comments made AT THE ORIGINAL BLOG POST? If and only if the OP deletes your comment, only then posting here makes sense: "that narcissistic guy didn't accept comments, so I'll put them here: blah-blah." I can't understand you, people.

saagarjha 8y ago

I'm basing this on experience using one; I have no "inside information" on how they work so I'm doing my best with my insight. Here's what I've found just by messing around; if you want details I can post the "experiments" I did to get these: * The largest epsilon that the calculator is willing is confident about showing to the user is about 10^-13. (0.1 + 10^-13 - 0.1 = 10^-13, 0.1 + 10^-14 - 0.1 = 0) * The calculator will show about 10 digits of precision. (0.1 + 10^-10 = .1000000001, 0.1 + 10^-11 = 0.1) * Rounding seems to occur at the last digit of internal precision, perhaps at a 10^-14/10^-15 digit that isn't shown? This might mean the calculator is using double precision. (.5-10^13=0.5, .5-10^-14=1/2) I think the results you got are pretty consistent with this. First of all, the sequence of trigonometric operations you performed are quite lossy, since you're exercising a very small range and you perform them repeatedly, which magnifies the error. Performing the integral is quite accurate; even a simple midpoint Riemann sum with 1000 subdivisions is accurate to one part in 10^12). FYI, you seem to have been shadowbanned (I have chosen to vouch for the one you made here because it was somewhat reasonable and so I could respond to it), which is not surprising considering your other comments.

ludditus 8y ago
Thanks for the 10^-13 thing. It makes sense. As for shadowbanning... all I want to say is this. IMNSO opinion, whoever chooses to comment on places like this or like Reddit is an highly impolite, because that's what blogs are for, and commenting is allowed on my blog (moderated, so one wouldn't see a comment right away). I am sick of people ranting here or on reddit like: "oh, look of what crap some guy is saying somewhere," or "he's wrong wrt to this thing" ... but why on Earth aren't such comments made AT THE ORIGINAL BLOG POST? If and only if the OP deletes your comment, only then posting here makes sense: "that narcissistic guy didn't accept comments, so I'll put them here: blah-blah." I can't understand you, people.