reformulation_of_the_arctangent_function.sf

#!/usr/bin/ruby

# Formula due to S. M. Abrarov and B. M. Quine.
# Described in their paper: "A reformulated series expansion of the arctangent function"

func arctan(x, z) {
    sum(1..z, {|m|
        sum(1..(2*m - 1), {|n|
            (-1)**n / (2*m - 1) / (1 + (x**2 / 4))**(2*m - 1) * (x/2)**(4*m - 2*n - 1) * binomial(2*m - 1, 2*n - 1)
        })
    }) * -2
}

say atan(1)             # Pi/4 (built-in function)
say arctan(1, 70)       # Pi/4

say "\nFractions converging to Pi/4:"
for z in (1..10) {
    say arctan(1, z).as_frac
}

__END__
0.785398163397448309615660845819875721049292349844
0.785398163397448309615660845819875721049292349844

Fractions converging to Pi/4:
4/5
296/375
36772/46875
1288688/1640625
96641548/123046875
26576092808/33837890625
8637277012172/10997314453125
1079658805128928/1374664306640625
91770997994914276/116846466064453125
43591225139846360008/55502071380615234375