pounder_test/dsp/src/atan2.rs

/// 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
/// represents -pi and, equivalently, +pi. i32::MAX represents one
/// count less than +pi.
pub fn atan2(y: i32, x: i32) -> i32 {
    let sign = (x < 0, y < 0);

    let mut y = y.wrapping_abs() as u32;
    let mut x = x.wrapping_abs() as u32;

    let y_greater = y > x;
    if y_greater {
        core::mem::swap(&mut y, &mut x);
    }

    let z = (16 - y.leading_zeros() as i32).max(0);

    x >>= z;
    if x == 0 {
        return 0;
    }
    y >>= z;
    let r = (y << 16) / x;
    debug_assert!(r <= 1 << 16);

    // 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 * r + C * r * (1 - abs(r))
    //
    // which is taken from Rajan 2006: Efficient Approximations for
    // the Arctangent Function.
    //
    // The least mean squared error solution is C = 0.279 (no the 0.285 that
    // Rajan uses). K = C*4/pi.
    // Q5 for K provides sufficient correction accuracy while preserving
    // as much smoothness of the quadratic correction as possible.
    const FP_K: usize = 5;
    const K: u32 = (0.35489 * (1 << FP_K) as f64) as u32;
    // debug_assert!(K == 11);

    // `r` is unsigned Q16.16 and <= 1
    // `angle` is signed Q1.31 with 1 << 31 == +- pi
    // Since K < 0.5 and r*(1 - r) <= 0.25 the correction product can use
    // 4 bits for K, and 15 bits for r and 1-r to remain within the u32 range.
    let mut angle = ((r << 13)
        + ((K * (r >> 1) * ((1 << 15) - (r >> 1))) >> (FP_K + 1)))
        as i32;

    if y_greater {
        angle = (1 << 30) - angle;
    }

    if sign.0 {
        angle = i32::MAX - angle;
    }

    if sign.1 {
        angle = angle.wrapping_neg();
        // Negation ends up in slightly faster assembly
        // angle = !angle;
    }

    angle
}

#[cfg(test)]
mod tests {
    use super::*;
    use core::f64::consts::PI;

    fn angle_to_axis(angle: f64) -> f64 {
        let angle = angle % (PI / 2.);
        (PI / 2. - angle).min(angle)
    }

    #[test]
    fn atan2_absolute_error() {
        const N: usize = 321;
        let mut test_vals = [0i32; N + 4];
        let scale = (1i64 << 31) as f64;
        for i in 0..N {
            test_vals[i] = (scale * (-1. + 2. * i as f64 / N as f64)) as i32;
        }

        assert!(test_vals.contains(&i32::MIN));
        test_vals[N] = i32::MAX;
        test_vals[N + 1] = 0;
        test_vals[N + 2] = -1;
        test_vals[N + 3] = 1;

        let mut rms_err = 0f64;
        let mut abs_err = 0f64;
        let mut rel_err = 0f64;

        for &x in test_vals.iter() {
            for &y in test_vals.iter() {
                let want = (y as f64 / scale).atan2(x as f64 / scale);
                let have = atan2(y, x) as f64 * PI / scale;

                let err = (have - want).abs();
                abs_err = abs_err.max(err);
                rms_err += err * err;
                if err > 3e-5 {
                    rel_err = rel_err.max(err / angle_to_axis(want));
                }
            }
        }
        rms_err = rms_err.sqrt() / test_vals.len() as f64;
        println!("max abs err: {:.2e}", abs_err);
        println!("rms abs err: {:.2e}", rms_err);
        println!("max rel err: {:.2e}", rel_err);
        assert!(abs_err < 5e-3);
        assert!(rms_err < 3e-3);
        assert!(rel_err < 0.6);
    }
}