# Reduce the variety of procedures on a straightforward expression

Lets claim I take a calculation that entails just enhancement and also reproduction :

```
(a+b)*(c+d)
```

which can be carried out in several various other means, eg.

```
a*(c+d) + b*(c+d)
a*c + a*d + b*c + b*d
```

In regards to enhancements and also reproductions the variety of procedures needed for each and every of the 3 instances revealed are (2,1) (3,2) (3,4) specifically. Plainly, if the objective is to lower the complete variety of procedures the first transcends. Exists a means, offered an approximate expression to locate the calculation order that calls for the least variety of procedures?

Here is a very closely relevant inquiry which is less complicated to address. Intend you just respect the variety of reproductions. (This is practical in several scenarios where enhancement is economical yet reproduction is pricey, so you intend to decrease the variety of reproductions you are mosting likely to do.) So intend a person offers you an expression like a *c+a *d+b *c+b *d and also you intend to reposition making use of as couple of reproductions as feasible.

In this instance what you are doing is calculating the "rank" of a tensor. (Warning : "rank" of a tensor can suggest 2 entirely various points, this is the ranking that is similar to rate of a matrix, not the ranking that informs you which sort of tensor you are considering.)

As an example, intend your expression just ever before entails items of 2 points, after that this comes to be calculating the ranking of a matrix. In your instance, a *c+a *d+b *c+b *d, you are considering a 4x4 matrix (given that there are 4 variables : a, b, c, d) which is :

$$\begin{pmatrix}0&0&1&1\\ 0&0&1&1 \\ 0&0&0&0 \\ 0&0&0&0 \end{pmatrix}$$

This matrix has ranking 1 as is conveniently seen making use of row - decrease, therefore it can be created making use of just one reproduction.

I do not recognize whether there are excellent formulas for calculating rankings of greater tensors the means there are for matrices.

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