Replace FP division with fixed-point shift in atan2

[jserv]

The double-precision division was replaced with bit shift operation in twin_atan2_first_quadrant(), using right shift by 15 bits equivalent to division by 32768.

Summary by Bito

This pull request optimizes the `twin_atan2_first_quadrant` function by replacing a double-precision division with a fixed-point bit shift operation, enhancing performance for angle calculations in the twin angle system.

[weihsinyeh]

Does this commit also aim to fix the incorrect casting of the variable angle from twin_angle_t to double?

[jserv]

Does this commit also aim to fix the incorrect casting of the variable angle from twin_angle_t to double?

Yes, intentionally.

[weihsinyeh]

Then, why didn't you write this purpose on this commit log ?

[jouae]

Since TWIN_ANGLE_360=4096 = 2^12by simple simplification, the original expression can be represented as

$$\frac{\text{angle} * 4096}{32768} = \text{angle} * 2^{12} *2^{-15} = \text{angle} * 2^{-3}$$

which is equivalent to

+/* Fixed-point conversion: angle * TWIN_ANGLE_360 / 32768 */
+    return (twin_angle_t) (angle >> 3);

[jouae]

When the value range of angle is $[-9092,9092]$, using the original expression:

(angle * TWIN_ANGLE_360) >> 15

with TWIN_ANGLE_360 = 4096 can cause overflow. The maximum possible result of the multiplication is:

$$9092 * 4096 = 37,249,024$$

This value exceeds the limits of 16-bit, it may result in integer overflow and unpredictable behavior.

To avoid this issue, the multiplication and division can be replaced with a simple bit shift:

angle >> 3