Practice Recursion and Abstraction with the exercise "Derivative time !!!

Learning Opportunities

This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.

Recursion

Statement

Goal

Story:
Louise is learning calculus in her maths courses and she likes the idea. When doing her homework, she is not sure and she wants to check her answers with yours. Could you create a program so that we can check the answers?

It's recommended to finish Part 1 first:
https://www.codingame.com/training/medium/derivative-time---part1

Calculate the partial derivative of a given formula.

For example, in Test1 "crazy chain"
Line1: formula: ln(ln(ln(ln x)))
Line2: vars: x
Line3: vars' values: x 15.16

a formula ln(ln(ln(ln x))) and x means to calculate
d(f(x))/dx,
which should be 1/x*1/(ln x)*1/ln(ln x)*1/ln(ln(ln x)),
and then use the value x=15.16 to resolve it.
So the answer is 174.23.

Note:
Formula: "e" and "pi" could appear in the formula.

Priority: [ln, sin, cos] > [^] > [*] > [+]
[ln, sin, cos] will be followed by one space if no parenthesis
Power follows an order from right to left
e.g.: 2^3^4 = 2^(3^4)
Negative power has no parenthesis
e.g.: 5*sin x*y^-2 = 5*(sin x)*(y^(-2))

To simplify, x^x form will only has same expression at floor and power position.
e.g.: (sin x)^(sin x) could appear, but (x+2)^(x*3) will not appear

vars may be in other forms other than x, y, and z. Similar to identifiers in many programming languages, the var would be some letter (or ASCII letter) followed by letters, numbers or underscore.

link about calculus rules: https://en.wikipedia.org/wiki/Differentiation_rules
The rules needed here:
a'=0
(a*x)'=a
(x^a)'=a*x^(a-1) (when a is not 0)
(a^x)'=ln(a)*a^x
(x^x)'=(ln(x)+1)*x^x
(ln x)'=x^-1
(sin x)'=cos x
(cos x)'=-sin x
(u+v)'=u'+v'
(u*v)'=u'*v+v'*u
df/dx=df/dt*dt/dx (the chain rule)
e^(pi*i)+1=0 (just kidding, we will not need this one :) )

Input

Line1: formula, built by vars (e.g. x, y, z, Gamma, x1...), constants (integers, floats, "e", "pi"), and operators (ln, sin, cos, ^, *, +).
Line2: list of vars for partial derivative, separated by space, length of the list will be 1, 2 or 3.
Line3: vars' values, paired and separated by space, value will be float with 2 digits.

Output

The result (always float with 2 digits, rounding to nearest)

Constraints

Line2 would give a list from 1 to 3 different/same vars to do partial derivative.

Example

Input

ln(ln(ln(ln x)))
x
x 15.16

Output

174.23

Solve it

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