pounder_test/dsp/src/atan2.rs

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2021-01-21 23:12:59 +08:00
/// 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();
}
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);
}
}