You’re doing some construction work, and, to save money,
you’re using some discount, “irregular” construction materials.
In particular, you have some blocks that are mostly
rectangular, but with one edge that’s curvy. As illustrated
below, you’re going to use these irregular blocks between
stacks of ordinary blocks, so they won’t shift sideways or
rotate. You’ll put one irregular block on the bottom, with its
curvy edge pointing up, and another above it, with it’s curvy
edge pointing down. You just need to know how well these blocks
fit together. You define the fit quality as the maximum
vertical gap between the upper edge of the bottom block and the
lower edge of the top block when the upper block is just
touching the lower one.
All blocks are one unit wide. You’ve modeled the curvy edges
as cubic polynomials, with the left edge of the block at
$x = 0$ and the right edge
at $x = 1$.
Input
Each test case is given on two lines, with each line
containing four real numbers. The numbers on the first line,
$b_0~ b_1~ b_2~ b_3$,
describe the shape of the upper edge of the bottom block. This
edge is shaped just like the polynomial $b_0 + b_1 x + b_2 x^2 + b_3 x^3$ for
$0 \leq x \leq 1$. The
numbers on the next input line, $t_0~ t_1~ t_2~ t_3$, describe the
bottom edge of the block that’s going on top. This edge is
shaped just like the polynomial $t_0 + t_1 x + t_2 x^2 + t_3 x^3$ for
$0 \leq x \leq 1$. No
input value will have magnitude greater than $50 000$. There are at most 500 test
cases. Input ends at end of file.
Output
For each test case, print out a single line giving the fit
quality. An answer is considered correct if its absolute or
relative error is at most $10^{7}$.
Sample Input 1 
Sample Output 1 
1.000000 12.904762 40.476190 28.571429
3.000000 11.607143 34.424603 22.817460
2.000000 10.845238 16.964286 10.119048
3.000000 4.190476 3.571429 2.380952

4.396074
6.999999
