| While taking a walk through the woods, a group of Foxen have come upon a | |
| curious sight — a row of **N** wooden poles sticking straight up out of the | |
| ground! Who placed them there, and why? The Foxen have no clue. | |
| Looking at the poles from the side, they can be modeled as vertical line | |
| segments rising upwards from a number line (which represents the ground), with | |
| the _i_th pole at distinct integral position **Pi** and having a real-valued | |
| height of **Hi**. | |
| One of the Foxen, Ozy, is fascinated by the shadows being cast on the ground | |
| by the poles. The sun is shining down on the poles from some point very high | |
| up in the sky, resulting in infinitely many rays of light descending towards | |
| the number line at all possible positions along it, but all travelling in some | |
| uniform direction. Each ray of light stops travelling as soon as it comes into | |
| contact with either a pole or the ground. Any point on the ground which is | |
| incapable of being reached by rays of light (because they would get blocked by | |
| at least one pole before reaching that point) is considered to be covered in | |
| shadows. | |
| The sunlight's direction can be described by a real value _a_, with absolute | |
| value no larger than 80, where _a_ is the signed angle difference (in degrees) | |
| between the rays' direction and a vector pointing directly downwards. As an | |
| example, let's imagine that there's a single pole at position 50 and with a | |
| height of 100. If _a_ = 45, then sunlight is shining diagonally down and to | |
| the right, meaning that the pole obstructs rays of light from being able to | |
| reach any points on the ground in the interval [50, 150], effectively casting | |
| a shadow with length 100 to the right. If _a_ = -45, then sunlight is shining | |
| diagonally down and to the left, causing the pole to cast a shadow with 100 to | |
| the left instead (over the interval [-50, 50]). If _a_ = 0, then sunlight is | |
| shining directly downwards onto the ground, resulting in the pole not casting | |
| any shadow. | |
| Ozy is planning on returning by himself tomorrow in order to observe the poles | |
| again, but he doesn't know at what time of day he'll be able to make the trip. | |
| He does at least have it narrowed down to being within some interval of time, | |
| during which he knows that the sunlight's direction _a_ will range from **A** | |
| and **B**, inclusive. Given that the sunlight's direction _a_ will be a real | |
| number drawn uniformly at random from the interval [**A**, **B**] when Ozy | |
| visits the poles tomorrow, please help him predict the expected total length | |
| of ground which will be covered in shadows at that time. | |
| ### Input | |
| Input begins with an integer **T**, the number of different sets of poles. For | |
| each set of poles, there is first a line containing the space-separated | |
| integers **N**, **A**, and **B**. Then **N** lines follow, the _i_th of which | |
| contains the integer **Pi** and the real number **Hi** separated by a space. | |
| The poles' heights are given with at most 4 digits after the decimal point. | |
| ### Output | |
| For the _i_th set of poles, print a line containing "Case #**i**: " followed | |
| by a single real number, the expected length of ground which will be covered | |
| in shadows. Your output should have at most 10-6 absolute or relative error. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 30 | |
| 1 ≤ **N** ≤ 500,000 | |
| -80 ≤ **A** < **B** ≤ 80 | |
| 0 ≤ **Pi** ≤ 1,000,000,000 | |
| 1 ≤ **Hi** ≤ 1,000,000 | |
| The sum of **N** values across all **T** cases does not exceed 2,000,000. | |
| ### Explanation of Sample | |
| In the first case, the sunlight's direction _a_ is drawn uniformly at random | |
| from the interval [44, 46]. As described above, the length of ground covered | |
| in shadows when _a_ = 45 is exactly 100. When _a_ = 44, the shadow's length is | |
| ~96.57, and when _a_ = 46, its length is ~103.55. However, note that its | |
| expected length for this distribution of possible _a_ values is not equal to | |
| the average of those sample lengths. | |