Try placing sticks a and b as follows:

  #
  #`.  a
  #  `.
  #    `.
x #      \
  #       \  b
  #        \
  #         \
=====================
       y

Hint: It can be proved that the area is maximal when x = y and angle between a and b is 135 degrees.

try 10000 1 may be

                // S(x, a, b) + x * x / 4 max.
        // sqrt((x + a + b) * (x + a - b) * (x + b - a) * (a + b - x)) + x * x
        // 2*x + 1/2 * (-2x * (x * x - (a-b) * (a-b)) + 2x * (-x * x + (a+b) * (a+b))) / sqrt(...) = 0
        // 2*x + (-x * (x * x - (a-b) * (a-b)) + x * (-x * x + (a+b) * (a+b))) / sqrt(...) = 0
        //     delete x
        // 2 + (2*(a*a + b*b) - 2*x*x) / sqrt(...) = 0
        // 1 + ((a*a + b*b) - x * x) / sqrt(...) = 0
        //   x^2 - (a^2 + b^2) = sqrt(...)
        // (x^2 - (a^2 + b^2)) ^ 2 = (x ^ 2 - (a - b) ^ 2) * ((a + b) ^ 2 - x ^ 2)
        // y = x^2
        // (a*a + b*b))^2 - 2*(a*a + b*b) * y + y*y = -y*y + (2*(a*a + b*b)) * y - (a*a - b*b) ^ 2
        // 2*y*y - 4*(a*a + b*b)*y + 2*(a^4 + b^4) = 0
        // y^2 - 2(a^2 + b^2)y + (a^4 + b^4) = 0
        // D/2 = 2(ab)^2
        // y = (a^2+b^2 (+-) ab*sqrt(2))
        // Smax = (2y - (a^2+b^2)) / 4 = (a^2+b^2)/4 + ab/sqrt(2)
        printf("%.18lf\n",  (a * a + b * b) / 4 + (a * b) / sqrt(2.0));

       it's the part of my solution and explanation of this formula.

Hmm, I spent about 3-5 minutes to find this formula :)

Thank you very much.