smallest |cos(x)| ?

[Albert Chan]

(10-08-2021 07:02 AM)Werner Wrote:  A bit counterintuitive that the smallest COS happens for such a large argument value;
I thought it would lose about 8 digits of precision after the argument reduction.

This give me an idea !

For IP(x) of 9 digits, calculate (pi/2*10^(34-9)), then throwaway rounded integer part

Absolute of fractional part is how close expression to integer.
No fractional part implied x - rounded(x,34) = 0

c:\> spigot -d15 -C abs(frac(pi/2*1e25+.5)-.5)
0/1
1/11
1/12
25/299
26/311
207/2476
233/2787
440/5263
673/8050
9189/109913
28240/337789
65669/785491
1472958/17618591
1538627/18404082
13781974/164851247
539035613/6447602715

x = (2k+1) * pi/2 = [1e8, 1e9)
(2k+1) = (2/pi) * [1e8, 1e9) ≈ 0.6366197724 * [1e8, 1e9)

From the convergents, 2k+1 = 164851247 satisfied.
3*(2k+1) = 494553741 also valid, but produced error of 3ε

82425623.5 PI * COS         → 1.532138639232766081410107492878833e-35
247276870.5 PI * COS       → -4.596415917698298244230322478636498e-35

Best semi-convergent is not as good: 2k+1 = 18404082 + 164851247*3 = 512957823

256478911.5 PI * COS       → -5.589328359211609676578079200248261e-34