smallest |cos(x)| ?

[Albert Chan]

(10-09-2021 02:09 PM)Albert Chan Wrote:  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 ...)

Perhaps Loch's theorem explain this.

Quote:this can be interpreted as saying that each additional term in the continued fraction representation of a "typical" real number increases the accuracy of the representation by approximately one decimal place.

When I did the convergents, steps required to get q = 2k+1 is roughly the same.
If valid 2k+1 is big, convergents also start with big denominator.

Relative error of final convergent tends towards the same ballpark.

Example, IP(x) of 47 digits. Convergents of y = (pi/2) / 10^(47 - 34):

0/1
1/6366197723675
1/6366197723676
5/31830988618379
...
933595789363114285779172744945397/5943455389076782359664376669525821535544153034
9341730789500356812974471132146251/59471305287309430984490740451345101530239801689

y = 1.570796326794896619231321691639751 ... * 10^-13
p = 9341730789500356812974471132146251
q = 59471305287309430984490740451345101530239801689

x = p * 10^13                   ≈ 9.341730789500356812974471132146251e46
ε = abs(q*y - p) * 10^13 ≈ 1.028415848209791685669808767152452e-35

Update:
Brute force proof, for Free42-Decimal, |cos(x)| > 10^-37
see https://www.hpmuseum.org/forum/thread-17...#pid153539