# computing pi in ten different ways
VECTOR piapprox SIZE 10
VAR x y

# the double-precision internal representation of pi (M_PI)
piapprox[1] = pi

# four times the arc-tangent of the unity
piapprox[2] = 4*atan(1)

# the abscissae x where tan(x) = 0 in the range [3:3.5]
piapprox[3] = root(tan(x), x, 3, 3.5)

# the square of the numerical integral of the gaussian curve 
piapprox[4] = integral(exp(-x^2), x, -10, 10)^2

# the numerical integral of a circle inscribed in a unit square
piapprox[5] = integral(integral((x^2+y^2) < 1, y, -1, 1), x, -1, 1)

# the numerical integral of a circle parametrized with sqrt(1-x^2)
piapprox[6] = integral(integral(1, y, -sqrt(1-x^2), sqrt(1-x^2)), x, -1, 1)

# the gregory-leibniz sum (one hundred thousand terms)
piapprox[7] = 4*sum((-1)^(i+1)/(2*i-1), i, 1, 1e5)

# the abraham sharp sum (twenty-one terms)
piapprox[8] = sum(2*(-1)^i * 3^(1/2-i)/(2*i+1), i, 0, 20)

# this integral is equal to 22/7-pi
piapprox[9] = 22/7-integral((x^4*(1-x)^4)/(1+x^2), x, 0, 1)

# ramanujan-sato series
piapprox[10] = 1/(2*sqrt(2)/(99^2)*sum(gammaf(4*i)/gammaf(i)^4 * (26390*i + 1103)/(396^(4*i)), i, 0, 2))

PRINT_VECTOR "% .16f" piapprox piapprox(i)-pi