trig/atan2: refine
* use dynamic scaling of the inputs to get accurate ratios (effectively floating point) to maintain accuracy for small arguments * this also allows shifting later and keep more bits * use u32 ratio to keep one more bit * merge the corner case unittests into the big test value list * print rms, absolute and axis-relative angle * simplify the correction expression to get rid of one multiplication * use 5 bit for the correction constant and 15 bits for r * least squares optimal correction constant, this lowers the max error below 5e-5
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152
dsp/src/trig.rs
152
dsp/src/trig.rs
@ -1,4 +1,4 @@
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use super::{shift_round, Complex};
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use super::Complex;
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use core::f64::consts::PI;
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include!(concat!(env!("OUT_DIR"), "/cossin_table.rs"));
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@ -22,18 +22,25 @@ include!(concat!(env!("OUT_DIR"), "/cossin_table.rs"));
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/// represents -pi and, equivalently, +pi. i32::MAX represents one
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/// count less than +pi.
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pub fn atan2(y: i32, x: i32) -> i32 {
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let mut y = y >> 16;
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let mut x = x >> 16;
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let sign = (x < 0, y < 0);
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let sign = ((y >> 14) & 2) | ((x >> 15) & 1);
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if sign & 1 == 1 {
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x *= -1;
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}
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if sign & 2 == 2 {
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y *= -1;
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}
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let mut y = y.wrapping_abs() as u32;
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let mut x = x.wrapping_abs() as u32;
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let y_greater = y > x;
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if y_greater {
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core::mem::swap(&mut y, &mut x);
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}
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let z = (16 - y.leading_zeros() as i32).max(0);
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x >>= z;
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if x == 0 {
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return 0;
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}
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y >>= z;
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let r = (y << 16) / x;
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debug_assert!(r <= 1 << 16);
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// Uses the general procedure described in the following
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// Mathematics stack exchange answer:
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@ -44,47 +51,37 @@ pub fn atan2(y: i32, x: i32) -> i32 {
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// to compute and to be more compatible with integer
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// arithmetic. The approximation technique used here is
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//
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// pi / 4 * x + 0.285 * x * (1 - abs(x))
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// pi / 4 * r + C * r * (1 - abs(r))
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//
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// which is taken from Rajan 2006: Efficient Approximations for
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// the Arctangent Function.
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if y_greater {
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core::mem::swap(&mut x, &mut y);
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}
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//
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// The least mean squared error solution is C = 0.279 (no the 0.285 that
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// Rajan uses). K = C*4/pi.
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// Q5 for K provides sufficient correction accuracy while preserving
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// as much smoothness of the quadratic correction as possible.
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const FP_K: usize = 5;
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const K: u32 = (0.35489 * (1 << FP_K) as f64) as u32;
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// debug_assert!(K == 11);
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if x == 0 {
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return 0;
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}
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// We need to share the 31 available non-sign bits between the
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// atan argument and constant factors used in the atan
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// approximation. Sharing the bits roughly equally between them
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// gives good accuracy. Additionally, we cannot increase the
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// number of atan argument bits beyond 15 because we must square
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// it.
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const ATAN_ARGUMENT_BITS: usize = 15;
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let ratio = (y << ATAN_ARGUMENT_BITS) / x;
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let mut angle = {
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const K1: i32 = ((1. / 4. + 0.285 / PI)
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* (1 << (31 - ATAN_ARGUMENT_BITS)) as f64)
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// `r` is unsigned Q16.16 and <= 1
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// `angle` is signed Q1.31 with 1 << 31 == +- pi
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// Since K < 0.5 and r*(1 - r) <= 0.25 the correction product can use
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// 4 bits for K, and 15 bits for r and 1-r to remain within the u32 range.
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let mut angle = ((r << 13)
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+ ((K * (r >> 1) * ((1 << 15) - (r >> 1))) >> (FP_K + 1)))
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as i32;
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const K2: i32 =
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((0.285 / PI) * (1 << (31 - ATAN_ARGUMENT_BITS)) as f64) as i32;
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ratio * K1 - K2 * shift_round(ratio * ratio, ATAN_ARGUMENT_BITS)
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};
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if y_greater {
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angle = (i32::MAX >> 1) - angle;
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angle = (1 << 30) - angle;
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}
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if sign & 1 == 1 {
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if sign.0 {
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angle = i32::MAX - angle;
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}
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if sign & 2 == 2 {
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angle *= -1;
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if sign.1 {
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angle = angle.wrapping_neg();
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}
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angle
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@ -162,7 +159,6 @@ pub fn cossin(phase: i32) -> Complex<i32> {
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#[cfg(test)]
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mod tests {
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use super::*;
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use crate::testing::isclose;
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use core::f64::consts::PI;
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fn angle_to_axis(angle: f64) -> f64 {
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@ -172,61 +168,43 @@ mod tests {
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#[test]
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fn atan2_absolute_error() {
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const NUM_VALS: usize = 1_001;
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let mut test_vals: [f64; NUM_VALS] = [0.; NUM_VALS];
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let val_bounds: (f64, f64) = (-1., 1.);
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let val_delta: f64 =
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(val_bounds.1 - val_bounds.0) / (NUM_VALS - 1) as f64;
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for i in 0..NUM_VALS {
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test_vals[i] = val_bounds.0 + i as f64 * val_delta;
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const N: usize = 321;
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let mut test_vals = [0i32; N + 4];
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let scale = (1i64 << 31) as f64;
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for i in 0..N {
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test_vals[i] = (scale * (-1. + 2. * i as f64 / N as f64)) as i32;
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}
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let atol: f64 = 4e-5;
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let rtol: f64 = 0.127;
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assert!(test_vals.contains(&i32::MIN));
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test_vals[N] = i32::MAX;
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test_vals[N + 1] = 0;
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test_vals[N + 2] = -1;
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test_vals[N + 3] = 1;
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let mut rms_err = 0f64;
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let mut abs_err = 0f64;
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let mut rel_err = 0f64;
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for &x in test_vals.iter() {
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for &y in test_vals.iter() {
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let actual = (y.atan2(x) as f64 * i16::MAX as f64).round()
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/ i16::MAX as f64;
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let tol = atol + rtol * angle_to_axis(actual).abs();
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let computed = (atan2(
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((y * i16::MAX as f64) as i32) << 16,
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((x * i16::MAX as f64) as i32) << 16,
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) >> 16) as f64
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/ i16::MAX as f64
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* PI;
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let want = (y as f64 / scale).atan2(x as f64 / scale);
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let have = atan2(y, x) as f64 * PI / scale;
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if !isclose(computed, actual, 0., tol) {
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println!("(x, y) : {}, {}", x, y);
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println!("actual : {}", actual);
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println!("computed : {}", computed);
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println!("tolerance: {}\n", tol);
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assert!(false);
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let err = (have - want).abs();
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abs_err = abs_err.max(err);
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rms_err += err * err;
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if err > 3e-5 {
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rel_err = rel_err.max(err / angle_to_axis(want));
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}
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}
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}
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// test min and max explicitly
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for (x, y) in [
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((i16::MIN as i32 + 1) << 16, -(1 << 16) as i32),
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((i16::MIN as i32 + 1) << 16, (1 << 16) as i32),
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]
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.iter()
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{
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let yf = *y as f64 / ((i16::MAX as i32) << 16) as f64;
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let xf = *x as f64 / ((i16::MAX as i32) << 16) as f64;
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let actual =
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(yf.atan2(xf) * i16::MAX as f64).round() / i16::MAX as f64;
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let computed = (atan2(*y, *x) >> 16) as f64 / i16::MAX as f64 * PI;
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let tol = atol + rtol * angle_to_axis(actual).abs();
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if !isclose(computed, actual, 0., tol) {
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println!("(x, y) : {}, {}", *x, *y);
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println!("actual : {}", actual);
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println!("computed : {}", computed);
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println!("tolerance: {}\n", tol);
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assert!(false);
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}
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}
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rms_err = rms_err.sqrt() / test_vals.len() as f64;
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println!("max abs err: {:.2e}", abs_err);
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println!("rms abs err: {:.2e}", rms_err);
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println!("max rel err: {:.2e}", rel_err);
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assert!(abs_err < 5e-3);
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assert!(rms_err < 3e-3);
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assert!(rel_err < 0.6);
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}
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#[test]
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