Remote Training on Programming

The astronomers have long since learned how to forecast solar eclipses for many years ahead. They use the classic laws of celestial mechanics which need practically no corrections. Some of the important characteristics of a solar eclipse are its start and end times, maximum phase and obscuration.

The start of the eclipse is the moment when Moon's and Sun's visible disks touch.

The end of the eclipse is the moment when the disks do not intersect anymore.

Maximum phase is the moment of the maximum intersection of the disks.

Eclipse obscuration is the fraction of the Sun's area occulted by Moon during the maximum phase. Total solar eclipse corresponds to the obscuration of 100%.

It is known that visible Sun's and Moon's disks are of equal size when observed from the Earth.

Write a program that will determine the characteristics of a solar eclipse using the information about Sun's and Moon's movement.

The information is based on the star map, which is considered to be a rectangle in the Cartesian system (for the sake of simplicity). Sun's and Moon's disks are circles of unit radius. The movement of the celestial bodies is described with equations defining the paths of their centers. Linear equations are used. For both the Sun and the Moon they are in the form:

where x and y are the coordinates of the point at the moment t. Constants A, B, C, D are specific for each celestial body.

For short time frames this model produces satisfactory results.

It is thought that the eclipse will start in the time interval (0 <= t <= 1000). If the eclipse does not start in this interval then the obscuration is equal to zero.

Limitations

–100 <= As, Bs, Cs, Ds, Am, Bm, Cm, Dm  <= 100;

Sun’s and Moon’s pathways are non-parallel.

 

Input (Standard input)

The first line specifies four space-delimited real numbers that describe the movement of the Sun's disk: As  Bs  Cs  Ds are coefficients in the equations (*)

The second line specifies four space-delimited real numbers that describe the movement of the Moon's disk: Am  Bm  Cm  Dm .

Output (Standard output)

The first line must contain the obscurity of the eclipse (in %, 0.01 precision). If this value is non-zero, then the second line must contain three numbers separated by a whitespace: start time, end time and the maximum phase of the eclipse (precisions are 0.0001). Note: use dot «.» as decimal separator.

Sample

Standard input	Standard output
1 0 1 0 1 0 -1 10	100.00 4.0000 6.0000 5.0000
1 0 1 -5.3 2 1 0.5 1.1	0.00