Assume $x$ and $y$ are two vectors in $\mathbb{R}^3$ and we want to compute the acute angle $\alpha\in(0,\pi/2]$ between these two (noncolinear) vectors. There are (at least) two possibilities:

1. In the naive approach, we compute the absolute value of the dot product of the normalized vectors $x$ and $y$ $$\frac{x^Ty}{\|x\|\|y\|}$$ and take the inverse cosine of the result.

2. The less naive approach is based on the fact that $$\|x\times y\|=\|x\|\|y\|\sin\alpha\quad\text{and}\quad|x^Ty|=\|x\|\|y\|\cos\alpha$$ so $\alpha$ is equal to an angle in a right triangle with legs of the length $\|x\times y\|$ and $|x^Ty|$ (for convenience, one can use a variant of the inverse tangent implemented in the atan2 function which is available in most programming languages; the function takes the side lengths of the legs as two arguments).

Now assume that $x$ is given and $y=x+\Delta x$ where $\|\Delta x\|\leq\tau\|x\|$, $\tau\ll 1$, that is, the vectors are almost colinear (for simplicity also of almost same norms). Assume that $\tilde\alpha_1$ and $\tilde\alpha_2$ are, respectively, the angles computed by the naive and the less naive approaches. Recently, I've run several tests which suggest that $$\tag{1} \frac{|\alpha-\tilde\alpha_1|}{\alpha}\leq \epsilon\mathcal{O}(\tau^{-2}) \quad\text{and}\quad \frac{|\alpha-\tilde\alpha_2|}{\alpha}\leq \epsilon\mathcal{O}(\tau^{-1}), $$ where $\epsilon$ is the machine precision. I understand that both algorithms suffer from a certain inaccuracy when $y\approx x$; in particular, both computing the dot and cross products. I suppose this is not due to the error in computing the inverse trigonometric functions as these are usually implemented to give very precise results.

I'm not asking anybody for performing any kind of analysis. I was just wondering whether there is a known reference where the accuracy of the two approaches is considered, if possible revealing why (1) (probably) holds. Thanks a lot in advance.

Assume $x$ and $y$ are two vectors in $\mathbb{R}^3$ and we want to compute the acute angle $\alpha\in(0,\pi/2]$ between these two (noncolinear) vectors. There are (at least) two possibilities:

1. In the naive approach, we compute the absolute value of the dot product of the normalized vectors $x$ and $y$ $$\frac{x^Ty}{\|x\|\|y\|}$$ and take the inverse cosine of the result.

2. The less naive approach is based on the fact that $$\|x\times y\|=\|x\|\|y\|\sin\alpha\quad\text{and}\quad|x^Ty|=\|x\|\|y\|\cos\alpha$$ so $\alpha$ is equal to an angle in a right triangle with legs of the length $\|x\times y\|$ and $|x^Ty|$ (for convenience, one can use a variant of the inverse tangent implemented in the atan2 function which is available in most programming languages; the function takes the side lengths of the legs as two arguments).

Now assume that $x$ is given and $y=x+\Delta x$ where $\|\Delta x\|\leq\tau\|x\|$, $\tau\ll 1$, that is, the vectors are almost colinear (for simplicity also of almost same norms). Assume that $\tilde\alpha_1$ and $\tilde\alpha_2$ are, respectively, the angles computed by the naive and the less naive approaches. Recently, I've run several tests which suggest that $$\tag{1} \frac{|\alpha-\tilde\alpha_1|}{\alpha}\leq \epsilon\mathcal{O}(\tau^{-2}) \quad\text{and}\quad \frac{|\alpha-\tilde\alpha_2|}{\alpha}\leq \epsilon\mathcal{O}(\tau^{-1}), $$ where $\epsilon$ is the machine precision. I understand that both algorithms suffer from a certain inaccuracy when $y\approx x$; in particular, both computing the dot and cross products. I suppose the inverse trigonometric functions are not an issue as these are usually implemented to give very accurate results.

I'm not asking anybody for performing any kind of analysis. I was just wondering whether there is a known reference where the accuracy of the two approaches is considered, if possible revealing why (1) (probably) holds. Thanks a lot in advance.