Count number of individual operations

Question

Im interested in counting the number of operations required to evaluate a given formula. Let me give you an example.

Lets say I have entered the expression $\dfrac{\operatorname{Sqrt}[\operatorname{Sin}[a*x+b*x-c*y]]}{2*a+b}$

Now lets say you have values for those unknows $a,b,c$. Then you would to calculate $a*x+b*x-c*y$ first which requires 3 multiplications, 1 addition and 1 subtraction. Then you take the $\sin(...)$ and then the square root.

Now you might calculate $2*a+b$ which requires 1 multiplication and 1 addition. In the end you take the quotient. This gives a total number of:

$1\times\sin, 1\times\sqrt{\cdot} \ 1 \times$ quotient, $4 \times $ multiplication, $2 \times$ addition and 1 subtraction.

Well you might call one subtraction the result of 1 multiplication and 1 addition. But that doesnt matter to me.

Since I have most of my formulas stored in Mathematica it would be great to have such a function. That would save a lot of time counting by hand.

Answer 1

Maybe you can use the output of Experimental`OptimizeExpression? For your example:

Experimental`OptimizeExpression[
    Sqrt[Sin[a x + b x - c y]]/(2 a + b), 
    "OptimizationLevel"->2
]

Experimental`OptimizedExpression[
   Block[{Compile`$34, Compile`$35, Compile`$36, Compile`$37, Compile`$38, Compile`$39, Compile`$40, Compile`$41, Compile`$42}, 
      Compile`$34 = 2 a; 
      Compile`$35 = Compile`$34 + b; 
      Compile`$36 = 1/Compile`$35;
          Compile`$37 = a x; 
      Compile`$38 = b x; 
      Compile`$39 = -c y; 
          Compile`$40 = Compile`$37 + Compile`$38 + Compile`$39; 
      Compile`$41 = Sin[Compile`$40]; 
          Compile`$42 = Sqrt[Compile`$41]; 
      Compile`$36 Compile`$42
   ]
]

A version using it might be something like:

opCount[expr_] := Extract[
    Experimental`OptimizeExpression[expr, "OptimizationLevel"->2],
    {1,1},
    Length
] + 1

opCount[Sqrt[Sin[a x + b x - c y]]/(2 a + b)]

10

Answer 2

I'm not sure if this gets you exactly what you want, but I found this in the documentation for LeafCount:

expr = Sqrt[Sin[a x + b x - c y]]/(2 a + b)
leaves = Level[expr, {-1}, Heads -> True]
Tally[leaves]

$\left( \begin{array}{cc} \text{Times} & 5 \\ \text{Power} & 2 \\ \text{Plus} & 2 \\ 2 & 1 \\ a & 2 \\ b & 2 \\ -1 & 2 \\ \text{Sin} & 1 \\ x & 2 \\ c & 1 \\ y & 1 \\ \frac{1}{2} & 1 \\ \end{array} \right)$

Like you mentioned, subtraction is dealt with as a multiplication of a negative number and addition. Also I'm not sure if this is a deal-breaker, but division is dealt with as a multiplication with $(2a+b)^{-1}$ and the square root is dealt with as Power[..., Rational[1, 2]].

You could filter the resulting tally a bit with

Select[Tally[leaves], Or @@ Thread[#[[1]] == {Sin, Times, Power, Plus}] &]

$\left( \begin{array}{cc} \text{Times} & 5 \\ \text{Power} & 2 \\ \text{Plus} & 2 \\ \text{Sin} & 1 \\ \end{array} \right)$