smallest |cos(x)| ?

[Albert Chan]

Last month, I had proposed a better complex tan(z) for Free42

We considered HP71B math-rom implementation.

Quote:tan((x,y)) = ( sin(x)*cos(x)/d , sinh(y)*cosh(y)/d )
d is sinh(y)^2+cos(x)^2

Does d ever goes to zero ?
What is smallest cos(x)^2 ? (numerically)

To make cos(x)^2 small, x must be close to (2k+1) * pi/2
This is Python script for searching smallest |cos(x)|, for IEEE double

Code:

from gmpy2 import *
def eps(x, f=next_above):
    'find smallest |cos(x)|, x should be close to (2k+1)*pi/2'
    x2 = f(x); y = cos(x); y2 = cos(x2)
    if abs(y) < abs(y2): x,y, x2,y2, f = x2,y2, x,y, next_below
    while y*y2 > 0:
        x, x2 = x2, f(x2)
        y, y2 = y2, cos(x2)
    if abs(y) < abs(y2): return x, y
    return x2, y2

Looping (2k+1)*pi/2, for k = 1 to 1000000 step 2, k = 29 give tiniest |cos(x)|

>>> eps(const_pi()/2 * 29)
(mpfr('45.553093477052002'), mpfr('-6.1898063658835771e-19'))

Others had search for this too. From Accurate trigonometric functions for large arguments

>>> eps(mpfr('0x1.6ac5b262ca1ffp+849'))
(mpfr('5.3193726483265414e+255'), mpfr('-4.6871659242546277e-19'))

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How to setup a Free42 program to search for this ?