smallest |cos(x)| ?

[Albert Chan]

(10-09-2021 01:15 PM)ijabbott Wrote:  There should be infinitely many such values for x producing a smaller ε, shouldn't there?

The question is not if there are infinite smaller ε's, but whether ε^2 will underflow.

Going from IP(x) of 9 diigits to 10, it is 10 times harder to get smaller ε.
On the other hand, we have 10 times many more candidates to try.

Both effect tends to cancel out, giving min(ε) not much smaller than machine epsilon.
(this is a conjecture, but experiments tends to support it ...)

(10-07-2021 06:40 PM)Albert Chan Wrote:  Free42 Decimal:

2995.508595197867852874130465957006 COS      → -8.200102230416340029960966876550293e-35
258947733.2551582209163675623249625 COS      → 1.532138639232766081410107492878833e-35

Using convergents / semi-convergents idea, for x < 1E50, these two have smaller ε:

9.510911580848780648392560778912209e40 COS  → -1.328353856527533918552271223903534e-35
9.341730789500356812974471132146251e46 COS  → -1.028415848209791685669808767152452e-35

To show how hard to get similar sized ε, this is true x, before rounded.

30274171828040719778093696476072003369775.5 * pi
= 95109115808487806483925607789122090000000.00000000000000000000000000000000001328​3538...

29735652643654715492245370225672550765119900844.5 * pi
= 93417307895003568129744711321462509999999999999.99999999999999999999999999999999​9989...