Docs for fmprb.h uclear (to me) about arb HOT 5 CLOSED

Could the answer be, that if I do things like sin(x), then the prec parameter specifies the internal working precision, but the result will always be correctly rounded to whatever precision is mathematically correct (so it can be both less or more than prec, depending on the operation and the internal algorithm)?

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Or is the result's accuracy always strictly less than prec or always strictly more than prec?

It's really confusing, I can imagine it working either way.

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The prec parameter just specifies the internal working precision used to compute the midpoint. For example, if adding two balls x +/- r, y +/- s, the result is round(x + y, prec) +/- (r + s + eps) where eps is a bound for the error from round(x + y, prec).

If the precision is much higher than the accuracy of the inputs, all that happens is that eps is much smaller than r + s, so the output radius is approximately r + s.

If the precision is much lower than the accuracy of the inputs, all that happens is that eps is much larger than r + s, so the output radius is approximately eps.

The only restrictions are that the precision has to be at least 2 bits, and not so large that overflow occurs (not usually an issue, at least on 64-bit systems).

The only strict guarantee for all functions is that the output ball contains the mathematically exact result. There's not a simple answer for how one should choose the precision in any given situation.

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Thanks, I think that explains it. I created #5 fixing this issue.

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You answered all the questions. The answer to the question what prec to use if I want to have as accurate result as possible is simply: minimize eps. To minimize eps, I just need to use accuracy something like max(r, s) + extra, where extra depends on the algorithm done internally. Then eps will be very small and that's what I wanted.

In situations like evaluating a symbolic expression, I would just set prec everywhere the same, then check the accuracy of the answer, and if it's not enough for me, then double prec.

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