Hacker News Reader

Comment by kqr | Hacker News Reader

> There's a difference between knowing the answer is exactly 40 and knowing only that you've computed a few hundred decimal places after 40 and so far they're all zero, maybe the next one won't be.

I would guess that if you pulled in a random sample of 2000 users of pocket calculators and surveyed their use cases you would find a grand total of 0 of them in which the cost function evaluated on a hundredth-decimal place error is at all meaningful.

In other words, no, that difference is not meaningful to a user of a pocket calculator.

View on HN

What don't seriously curious people deserve pocket calculators?

Are you implying that "seriously curious people" as a group need more than a hundred digits? If so, I don't think I agree.

Otherwise I can't figure out what you mean.

The sine of x is 0 when x is any integer multiple of pi - it's not approximately zero, it's really zero, actual zero. So clearly some formulae can actually be zero. If I'm curious, I might wonder whether eventually the formula e * -x reaches zero for some x

e ** -10 is about 0.000045399929 and presumably you agree that's not zero

e ** -100 is about 3.72 times 10 ** -44, is that still not zero? An IEEE single precision floating point number has a non-zero representation for this, although it is a denormal meaning there's not very much precision left...

e ** -1000 is about 5.075 times 10 ** -435 and so it won't fit in the IEEE single or double precision types. So they both call this zero. Is it zero?

If you take the naive approach you've described, the answer apparently is yes, non-zero numbers are zero. Huh.

[Edited, fixed asterisks]

I'm not particularly worried that you would be unable to recognize patterns or rounding. Sorry you confused your hypothetical self.

And for the record, since we're talking about hundred digit numbers, as an IEEE float that would mean 23 exponent bits and you'd have to go below 10e-4000000 before it rounds to zero. Or 32 exponents bits if you follow some previous software implementations.

> And for the record, since we're talking about hundred digit numbers, as an IEEE float that would mean 23 exponent bits and you'd have to go below 10e-4000000 before it rounds to zero. Or 32 exponents bits if you follow some previous software implementations.

Um, no. Have you confused your not-at-all hypothetical self? Are you mistaking the significand, aka the mantissa for an exponent? The significand in a 32-bit "single precision" IEEE float is 23 bits (with an implied leading 1 bit for normal values)

When I wrote that example I of course tried this, since it so happens that I have been testing some conversions recently so...

  >> (e -1000)
  f32: 0
  f64: 0
  real ~5.07595889754945676529180947957434e-435

Please refer back to this post: https://news.ycombinator.com/item?id=43070827

That's the first thing I said in this conversation. I did not ever suggest that single precision was enough. A hundred digits is beyond octuple precision. Octuple has 19 exponent bits, and in general every step up adds 4 more.

And going further up the comment chain, the original version was your mention of computing 40 followed by hundreds of digits of precision.

Reply on news.ycombinator.com