smallest |cos(x)| ?

[Albert Chan]

Thanks, Werner !

I found another approach.
Instead of search min(|cos(x)|), I search min(|x - round(x,34)|), where x = (2k+1)*pi/2 = (k+0.5)*pi
Searching k all the way to 100 million, we found a winner: x = 82,425,623.5 pi

This required arbitrary precision package, with more digits of pi
We use 100 digits of pi, search 1 million of k at a time.

The code is a 1 liner

eps.bat Wrote:@gawk "BEGIN{n=1e6; while(n--) print}" | rpn =100 pi sp %1 1e6x .5+ x s d =34 - abs [ rp r + s d = - abs $s k1

rpn is a sample program, for MAPM. (+ - * are exact, rounding must be called explicitly)
It is not programmable, thus the need for gawk to produce 1 million blank lines.

=100 pi sp # p = 100-digits-of-pi
%1 1e6x .5+ x s # start = (%1*1e6 + 0.5) * p
d =34 - abs # eps1 = abs(start - round(start,34))

[ # start a pipe, for each line of input, perform below commands
rp r + s # start += p
d = - abs # eps2 = abs(start - round(start,34))
$s k1 # sort (eps1, eps2) in ascending order, then drop the big one

c:\> eps 0
8.200102230416340029960966876550294E-35

c:\> eps 82
1.532138639232766081410107492878833E-35

Then, locate x within the million k, and confirm:

c:\> rpn
=100
pi 953.5x ?68 d =-34 ?
2995.5085951978678528741304659570060000820010223041634002996096687655
2995.508595197867852874130465957006
- abs ?34
8.200102230416340029960966876550294E-35

pi 82425623.5x ?68 d =-34 ?
2.5894773325515822091636756232496249999999998467861360767233918589893E+8
2.589477332551582209163675623249625E+8
- abs ?34
1.532138639232766081410107492878833E-35

|cos(x)| = sin(ε) = ε - ε^3/3! + ... ≈ ε

Free42 Decimal:

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

Intel Decimal128 angle reduction code is very good