Given two normalized vectors in 3d space, how can I get a value from $-1$ to $1$ based on their angle without using arc cosine?

With use of arc cosine, I think this would give me the correct result. But since arc cosine is a computational expensive function in computer programming, I need to avoid it.

$$\frac{cos^{-1}(a * b)}{180^°}-1$$

It is acceptable that the solution doesn't return the same values as the formula above. It's just important that the output depends (not necessarily linear) on the angle and is in the range from $-1$ to $1$.


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