Problem:
Score: 300 points

Problem Statement

Consider an analog clock whose hour and minute hands are A and B centimeters long, respectively.

An endpoint of the hour hand and an endpoint of the minute hand are fixed at the same point, around which each hand rotates clockwise at constant angular velocity. It takes the hour and minute hands 12 hours and 1 hour to make one full rotation, respectively.

At 0 o'clock, the two hands overlap each other. H hours and M minutes later, what is the distance in centimeters between the unfixed endpoints of the hands?

Constraints

All values in input are integers.

1 \leq A, B \leq 1000

0 \leq H \leq 11

0 \leq M \leq 59

Input

Input is given from Standard Input in the following format:

A B H M

Output

Print the answer without units. Your output will be accepted when its absolute or relative error from the correct value is at most 10^{-9}.

Sample Input 1

3 4 9 0

Sample Output 1

5.00000000000000000000

The two hands will be in the positions shown in the figure below, so the answer is 5 centimeters.

Sample Input 2

3 4 10 40

Sample Output 2

4.56425719433005567605

The two hands will be in the positions shown in the figure below. Note that each hand always rotates at constant angular velocity.

Solution in Rust (1.42.0) (judge verdict: Accepted)

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::Integer;
#[allow(unused_imports)]
use proconio::derive_readable;
use proconio::fastout;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::cmp;

use std::f64::consts::PI;

#[allow(dead_code)]
const EPS: f64 = 1e-8;

#[derive(Copy, Clone, Debug)]
struct Point {
  x: f64,
  y: f64,
}
impl Point {
  #[allow(dead_code)]
  fn new(x: f64, y: f64) -> Self {
    Point { x: x, y: y }
  }
  #[allow(dead_code)]
  // divide a segment into; ratio : 1 - ratio
  fn division(self, other: Self, ratio: f64) -> Self {
    self + (other - self) * ratio
  }
  #[allow(dead_code)]
  fn dist(self, other: Self) -> f64 {
    (other - self).abs()
  }
  #[allow(dead_code)]
  fn line_with_normal(self, vec: Vect) -> Line {
    Line::new(vec.x, vec.y, -self.x * vec.x - self.y * vec.y)
  }
  #[allow(dead_code)]
  fn line_toward(self, vec: Vect) -> Line {
    self.line_with_normal(vec.normal_vect())
  }
  #[allow(dead_code)]
  fn line_between(self, other: Point) -> Line {
    self.line_toward(other - self)
  }
  #[allow(dead_code)]
  fn bisector(self, other: Point) -> Line {
    let midpoint = self.division(other, 0.5);
    midpoint.line_with_normal(other - self)
  }
  #[allow(dead_code)]
  fn dist_to_line(self, line: Line) -> f64 {
    (line.a * self.x + line.b * self.y + line.c).abs()
  }
  #[allow(dead_code)]
  fn apollonius(self, other: Point, r1: f64, r2: f64) -> Option<Curve> {
    use Curve::*;
    if self.dist(other) < EPS || r1 < EPS || r2 < EPS {
      return None;
    }
    if (r1 - r2).abs() < EPS {
      return Some(CLine(self.bisector(other)));
    }
    let p1 = self.division(other, r1 / (r1 + r2));
    let p2 = self.division(other, r1 / (r1 - r2));
    Some(CCircle(Circle::new(
      p1.division(p2, 0.5),
      p1.dist(p2) / 2.0,
    )))
  }
  #[allow(dead_code)]
  fn perpendicular(self, line: Line) -> Line {
    self.line_toward(line.normal_vect())
  }
}
impl<T: Into<Vect>> std::ops::Add<T> for Point {
  type Output = Point;
  fn add(self, other: T) -> Point {
    let other = other.into();
    Point {
      x: self.x + other.x,
      y: self.y + other.y,
    }
  }
}
impl std::ops::Sub<Point> for Point {
  type Output = Vect;
  fn sub(self, other: Point) -> Vect {
    Vect {
      x: self.x - other.x,
      y: self.y - other.y,
    }
  }
}
impl<T: Into<Vect>> std::ops::Sub<T> for Point {
  type Output = Point;
  fn sub(self, other: T) -> Point {
    let other = other.into();
    Point {
      x: self.x - other.x,
      y: self.y - other.y,
    }
  }
}
impl From<(f64, f64)> for Point {
  fn from(p: (f64, f64)) -> Self {
    Point::new(p.0, p.1)
  }
}

#[derive(Copy, Clone, Debug)]
struct Vect {
  x: f64,
  y: f64,
}
impl Vect {
  #[allow(dead_code)]
  fn new(x: f64, y: f64) -> Vect {
    Vect { x: x, y: y }
  }
  #[allow(dead_code)]
  fn from_angle(theta: f64) -> Vect {
    Vect {
      x: theta.cos(),
      y: theta.sin(),
    }
  }
  #[allow(dead_code)]
  fn norm(self) -> f64 {
    self.x.powi(2) + self.y.powi(2)
  }
  #[allow(dead_code)]
  fn abs(self) -> f64 {
    self.norm().sqrt()
  }
  #[allow(dead_code)]
  fn normal_vect(self) -> Vect {
    Vect {
      x: -self.y,
      y: self.x,
    }
  }
  #[allow(dead_code)]
  fn normalized(self) -> Vect {
    self / self.abs()
  }
}
impl<T: Into<Vect>> std::ops::Add<T> for Vect {
  type Output = Vect;
  fn add(self, other: T) -> Vect {
    let other = other.into();
    Vect {
      x: self.x + other.x,
      y: self.y + other.y,
    }
  }
}
impl<T: Into<Vect>> std::ops::Sub<T> for Vect {
  type Output = Vect;
  fn sub(self, other: T) -> Vect {
    let other = other.into();
    Vect {
      x: self.x - other.x,
      y: self.y - other.y,
    }
  }
}
impl std::ops::Mul<f64> for Vect {
  type Output = Vect;
  fn mul(self, other: f64) -> Vect {
    Vect {
      x: self.x * other,
      y: self.y * other,
    }
  }
}
impl std::ops::Div<f64> for Vect {
  type Output = Vect;
  fn div(self, other: f64) -> Vect {
    Vect {
      x: self.x / other,
      y: self.y / other,
    }
  }
}
impl<T: Into<Vect>> std::ops::AddAssign<T> for Vect {
  fn add_assign(&mut self, other: T) {
    *self = *self + other;
  }
}
impl<T: Into<Vect>> std::ops::SubAssign<T> for Vect {
  fn sub_assign(&mut self, other: T) {
    *self = *self - other;
  }
}
impl std::ops::MulAssign<f64> for Vect {
  fn mul_assign(&mut self, other: f64) {
    *self = *self * other;
  }
}
impl From<(f64, f64)> for Vect {
  fn from(p: (f64, f64)) -> Self {
    Vect::new(p.0, p.1)
  }
}

#[derive(Copy, Clone, Debug)]
struct Line {
  // ax + by + c = 0
  a: f64,
  b: f64,
  c: f64,
}
impl Line {
  #[allow(dead_code)]
  fn new(a: f64, b: f64, c: f64) -> Self {
    let scale = (a.powi(2) + b.powi(2)).sqrt();
    Line {
      a: a / scale,
      b: b / scale,
      c: c / scale,
    }
  }
  #[allow(dead_code)]
  fn intersection(self, other: Line) -> Option<Point> {
    let det = self.a * other.b - self.b * other.a;
    if det.abs() < EPS {
      return None;
    }
    Some(Point {
      x: (-other.b * self.c + self.b * other.c) / det,
      y: (other.a * self.c - self.a * other.c) / det,
    })
  }
  #[allow(dead_code)]
  fn normal_vect(self) -> Vect {
    Vect::new(self.a, self.b)
  }
}

#[derive(Copy, Clone, Debug)]
struct Circle {
  center: Point,
  radius: f64,
}
impl Circle {
  #[allow(dead_code)]
  fn new(center: Point, radius: f64) -> Self {
    Circle {
      center: center,
      radius: radius,
    }
  }
  #[allow(dead_code)]
  fn intersection_with_line(self, line: Line) -> Vec<Point> {
    let perpendicular = self.center.perpendicular(line);
    let foot = line.intersection(perpendicular).unwrap();
    let dist = self.center.dist_to_line(line);
    if (dist - self.radius).abs() < EPS {
      return vec![foot];
    }
    if dist > self.radius {
      return vec![];
    }
    let base = (self.radius.powi(2) - dist.powi(2)).sqrt();
    let v = perpendicular.normal_vect() * base;
    vec![foot + v, foot - v]
  }
  #[allow(dead_code)]
  fn intersection_with_circle(self, other: Circle) -> Vec<Point> {
    let dist = self.center.dist(other.center);
    if dist < EPS {
      return vec![];
    }
    fn diff(c: Circle) -> f64 {
      c.radius.powi(2) - c.center.x.powi(2) - c.center.y.powi(2)
    }
    let c = (diff(self) - diff(other)) / 2.0;
    let v = self.center - other.center;
    let line = Line::new(v.x, v.y, c);
    self.intersection_with_line(line)
  }
}

#[derive(Copy, Clone, Debug)]
enum Curve {
  CLine(Line),
  CCircle(Circle),
}
impl Curve {
  #[allow(dead_code)]
  fn intersection(self, other: Curve) -> Vec<Point> {
    use Curve::*;
    match (self, other) {
      (CLine(l1), CLine(l2)) => l1.intersection(l2).into_iter().collect(),
      (CLine(l), CCircle(c)) | (CCircle(c), CLine(l)) => c.intersection_with_line(l),
      (CCircle(c1), CCircle(c2)) => c1.intersection_with_circle(c2),
    }
  }
}

#[fastout]
fn main() {
  input! {
    a: f64,
    b: f64,
    h: f64,
    m: f64,
  }
  let h_rad = 2.0 * PI * (h / 12.0 + m / 60.0 / 12.0);
  let m_rad = 2.0 * PI * m / 60.0;
  let h_vec = Vect::from_angle(h_rad) * a;
  let m_vec = Vect::from_angle(m_rad) * b;

  println!("{:.15}", (h_vec - m_vec).abs());
}