smallest |cos(x)| ?

[Werner]

(10-08-2021 07:54 PM)Albert Chan Wrote:  

(10-08-2021 02:16 PM)Werner Wrote:  instead of doing (k+0.5)*pi, which has 2 rounding errors (one for pi, one for *), do:
->RAD(180*k + 90), which has only one ...

"Bad" PI does not matter. (Besides, ->RAD might use the same PI)
Consider 34th digit of stored PI = 3, but should be 2.884...

x = (k+0.5)*PI
rel_error(x) = rel_error(PI) ≈ (0.116 ULP) / (PI*10^33 ULP) ≈ 3.686E-35

If x has |cos(x)| minimized, x scaled to 34 digits, is very close to integer.
Say, combined relative error = 3.7E-35 (as long as error < 5E-35, we are safe)

ulp_error(x) < 10^34 ULP * 3.7E-35 = 0.37 ULP

If we assume multiply has correct rounding, 0.37 ULP will have removed.
Searching for min(|cos(x)|, my OP eps() code can be replaced with plain cos(x)

Somehow, this is not true.. the division by 2 (or mult. by 0.5 ..) may introduce an error of one ULP, due to unbiased rounding, as is the case for k=0.
For the following values of k , (k+0.5)*pi is not the same as k*180+90 ->RAD, and the latter is correct:
k=0, 2, 8, 15, 18, 25, ..
consequently, (k+0.5)*pi does not yield the lowest |cos(x)|, but the ->RAD version does.

Cheers, Werner