dsp: add atan2

dsp/src/atan2.rs

@@ -0,0 +1,126 @@
+use super::{abs, shift_round};
+
+/// 2-argument arctangent function.
+///
+/// This implementation uses all integer arithmetic for fast
+/// computation. It is designed to have high accuracy near the axes
+/// and lower away from the axes. It is additionally designed so that
+/// the error changes slowly with respect to the angle.
+///
+/// # Arguments
+///
+/// * `y` - Y-axis component.
+/// * `x` - X-axis component.
+///
+/// # Returns
+///
+/// The angle between the x-axis and the ray to the point (x,y). The
+/// result range is from i32::MIN to i32::MAX, where i32::MIN
+/// corresponds to an angle of -pi and i32::MAX corresponds to an
+/// angle of +pi.
+pub fn atan2(y: i32, x: i32) -> i32 {
+    let y = y >> 16;
+    let x = x >> 16;
+
+    let ux = abs::<i32>(x);
+    let uy = abs::<i32>(y);
+
+    // Uses the general procedure described in the following
+    // Mathematics stack exchange answer:
+    //
+    // https://math.stackexchange.com/a/1105038/583981
+    //
+    // The atan approximation method has been modified to be cheaper
+    // to compute and to be more compatible with integer
+    // arithmetic. The approximation technique used here is
+    //
+    // pi / 4 * x + 0.285 * x * (1 - abs(x))
+    //
+    // which is taken from Rajan 2006: Efficient Approximations for
+    // the Arctangent Function.
+    let (min, max) = if ux < uy { (ux, uy) } else { (uy, ux) };
+
+    if max == 0 {
+        return 0;
+    }
+
+    let ratio = (min << 15) / max;
+
+    let mut angle = {
+        // pi/4, referenced to i16::MAX
+        const PI_4_FACTOR: i32 = 25735;
+        // 0.285, referenced to i16::MAX
+        const FACTOR_0285: i32 = 9339;
+        // 1/pi, referenced to u16::MAX
+        const PI_INVERTED_FACTOR: i32 = 20861;
+
+        let r1 = shift_round(ratio * PI_4_FACTOR, 15);
+        let r2 = shift_round(
+            (shift_round(ratio * FACTOR_0285, 15)) * (i16::MAX as i32 - ratio),
+            15,
+        );
+        (r1 + r2) * PI_INVERTED_FACTOR
+    };
+
+    if uy > ux {
+        angle = (i32::MAX >> 1) - angle;
+    }
+
+    if x < 0 {
+        angle = i32::MAX - angle;
+    }
+
+    if y < 0 {
+        angle *= -1;
+    }
+
+    angle
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use core::f64::consts::PI;
+    use crate::testing::isclose;
+
+    fn angle_to_axis(angle: f64) -> f64 {
+        let angle = angle % (PI / 2.);
+        (PI / 2. - angle).min(angle)
+    }
+
+    #[test]
+    fn absolute_error() {
+        const NUM_VALS: usize = 1_001;
+        let mut test_vals: [f64; NUM_VALS] = [0.; NUM_VALS];
+        let val_bounds: (f64, f64) = (-1., 1.);
+        let val_delta: f64 =
+            (val_bounds.1 - val_bounds.0) / (NUM_VALS - 1) as f64;
+        for i in 0..NUM_VALS {
+            test_vals[i] = val_bounds.0 + i as f64 * val_delta;
+        }
+
+        for &x in test_vals.iter() {
+            for &y in test_vals.iter() {
+                let atol: f64 = 4e-5;
+                let rtol: f64 = 0.127;
+                let actual = (y.atan2(x) as f64 * i16::MAX as f64).round()
+                    / i16::MAX as f64;
+                let tol = atol + rtol * angle_to_axis(actual).abs();
+                let computed = (atan2(
+                    ((y * i16::MAX as f64) as i32) << 16,
+                    ((x * i16::MAX as f64) as i32) << 16,
+                ) >> 16) as f64
+                    / i16::MAX as f64
+                    * PI;
+
+                if !isclose(computed, actual, 0., tol) {
+                    println!("(x, y)   : {}, {}", x, y);
+                    println!("actual   : {}", actual);
+                    println!("computed : {}", computed);
+                    println!("tolerance: {}\n", tol);
+                    assert!(false);
+                }
+            }
+        }
+    }
+}