mini challenge: find the smallest cosine of an integer

[Albert Chan]

(10-24-2021 06:56 AM)PeterP Wrote:  Alas, things get really slow when you add the test for 0.5. And just doing Pi-3 actually only gives you about 1-2 extra digits and the error still accumulates quickly.

A novel way is use SIN(X)

>PI, SIN(PI) @ REM sum = 3.141592653589793238462643 (25 digits of pi)
3.14159265359 -2.06761537357E-13

This program get numerator of convergents of pi
With internal 25 digits of stored pi, all convergents correct

Building of convergents is fast, using formula, P = P0 + k*P1
(P's are all integers, no error is introduced building P's)

10 P0=0 @ P1=1 @ P=P0 @ S=2
20 P=P+P1 @ S2=ABS(SIN(P))
30 IF S>S2 THEN S=S2 @ DISP P @ P0=P1 @ P1=P @ P=P0
40 IF P<1.E+12 THEN 20

>RUN
1
3
22
333
355
103993
104348
208341
312689
833719
1146408
4272943
5419351
80143857
165707065
245850922
411557987
1068966896
2549491779
6167950454
14885392687
21053343141

sin(2x) = 2*sin(x)*cos(x). If |cos(x)| is small, so does |sin(2x)|
We pick convergents/semi-convergents with even numerator.

2x = 1+3, 1+3*3, 1+3*5, 22, 333+355, 333+355*3, 333+355*5 ... 333+355*291, 104348, ...

x = 2, 5, 8, 11, 344, 699, 1054 ... 51819, 52174, ...