THE TORQUE MINIMIZER

Difficulty: Easy Tags: Calculus, Optimization

Story

An Amazon warehouse robot arm's joint controller minimizes a quadratic torque-penalty cost function before every movement, cheap enough to run at control-loop frequency, but it still needs a correct gradient step, not an approximation.

The Math

C(theta) = a * theta^2 + b * theta + c
dC/dtheta = 2*a*theta + b

Given a, b, c and a starting theta_0, take one gradient-descent step with learning rate eta:

theta_1 = theta_0 - eta * dC/dtheta at theta_0

Input Format

a b c theta_0 eta

Output Format

theta_1 and C(theta_1), space-separated, to 6 decimal places.

Constraints

• -100 <= a, b, c, theta_0 <= 100, a != 0, 0 < eta <= 1
• Time limit: 1.0 second.

Example

Input

1.0 -4.0 5.0 0.0 0.1

Output

0.400000 3.560000

Explanation: dC/dtheta at 0 is -4.0, so theta_1 = 0 - 0.1 * (-4.0) = 0.4, and C(0.4) = 1(0.16) - 4(0.4) + 5 = 3.56.