f=(x,W=[],Q=['('],z=1,h=p=>'+-*/^^))('.indexOf(p)>>1,C=n=>{for(;h(q=Q.pop())<h(n);W.push(q=='^'?Math.pow(a,b):eval(`a${q}b`)))a=W.pop(b=W.pop());z&&Q.push(q,n)})=>((x+')').replace(/\d+|\S/g,t=>t>'('?t>'('?~h(t)?z&&t=='-'?z=-z:C(t,z=1):(W.push(z*t),z=0):C(t,z=0):(Q.push(t),z=1)),W.pop())

// More readable
U=(x,W=[],Q=['('],z=1,
  h=p=>'+-*/^^))('.indexOf(p)>>1,
  C=n=>{
    for(;h(q=Q.pop())<h(n);
        W.push(q=='^'?Math.pow(a,b):eval(`a${q}b`)))
      a=W.pop(b=W.pop());
    z&&Q.push(q,n)
  }
)=>(
  (x+')')
  .replace(/\d+|\S/g,t=>
       t>'('
       ?t>'('
       ?~h(t)
       ?z&&t=='-'?z=-z:C(t,z=1)
       :(W.push(z*t),z=0)
       :C(t,z=0)
       :(Q.push(t),z=1)
  ),
  W.pop()
)

// TEST
console.log=(...x)=>O.innerHTML+=x.join` `+'\n'

console.log(f('1 - 3 + 4')) // -6
console.log(f('2-2-2')) // 2
console.log(f('(2 + 2 * 3 + 3) / 3 + 3')) // 4
console.log(f('3 + 2 + 1 ^ 3')) // 216
console.log(f('-5^2')) // 25
console.log(f('32 / 8 * 3 - 1')) // 2

// Explained
X=(x,W=[],Q=['('],z=1,
  h=p=> // operator priority '+-:1, */:3, ^:5, ):7, (:9. If an operand then -1
     '+-*/^^))('.indexOf(p)>>1,
  C=n=>{ // Evaluate operators present on stack if they have upper priority, then push new operator on stack
    //console.log('Operand '+n)
    while( h(q = Q.pop()) < h(n) ) // pop operator from op stack and compare priority with current
    {
      // Pop operands from stack and push result
      b = W.pop();
      a = W.pop();
      r = q=='^' ? Math.pow(a,b) : eval('a'+q+'b')
      // console.log('Evaluate '+a+q+b+'='+r)
      W.push(r);
    }
    // if z == 0 do nothing, because the current operands are '(' and ')' that must be discarded
    // else Push again the last operator popped and the current one
    z && Q.push(q, n) // 
  }
)=>(
  (x+')')
  .replace(/[\d.]+|\S/g,t=> {
    //console.log('Q:'+Q,'W:'+W,'T:'+t,'U:'+h(t),'Z:'+z), // display status
    if (t == '(')
    { // open parenthesis
      z = 1
      Q.push(t) // push a operator, its the highest priority
    }
    else if (t == ')')
    { //close parenthesis
      z = 0
      C(t)
    }
    else if (h(t) < 0)
    { // operand
      W.push(z*t) // push operand with right sign
      z = 0 // set z to 0 to mark that we just pushed an operand, so next '-' (if present) is a binary operator 
    }
    else
    { // operator
      if (z && t=='-') // the minus is an unary operator (the only unary operator allowed is '-', '+' will not work)
        z =-z // change the sign
      else
        z = 1, // no unary minus
        C(t)
    }
  }),
  W.pop() // return the value at top of operand stack
)

<pre id=O></pre>