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trig/atan2: refine

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* 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
This commit is contained in:
Robert Jördens 2020-12-20 20:35:26 +01:00
parent 12d5945d81
commit 8d9af70c19
1 changed files with 66 additions and 88 deletions
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154
dsp/src/trig.rs
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@ -1,4 +1,4 @@
use super::{shift_round, Complex}; use super::Complex;
use core::f64::consts::PI; use core::f64::consts::PI;
include!(concat!(env!("OUT_DIR"), "/cossin_table.rs")); include!(concat!(env!("OUT_DIR"), "/cossin_table.rs"));
@ -22,18 +22,25 @@ include!(concat!(env!("OUT_DIR"), "/cossin_table.rs"));
/// represents -pi and, equivalently, +pi. i32::MAX represents one /// represents -pi and, equivalently, +pi. i32::MAX represents one
/// count less than +pi. /// count less than +pi.
pub fn atan2(y: i32, x: i32) -> i32 { pub fn atan2(y: i32, x: i32) -> i32 {
let mut y = y >> 16; let sign = (x < 0, y < 0);
let mut x = x >> 16;
let sign = ((y >> 14) & 2) | ((x >> 15) & 1); let mut y = y.wrapping_abs() as u32;
if sign & 1 == 1 { let mut x = x.wrapping_abs() as u32;
x *= -1;
}
if sign & 2 == 2 {
y *= -1;
}
let y_greater = y > x; 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 // Uses the general procedure described in the following
// Mathematics stack exchange answer: // Mathematics stack exchange answer:
@ -44,47 +51,37 @@ pub fn atan2(y: i32, x: i32) -> i32 {
// to compute and to be more compatible with integer // to compute and to be more compatible with integer
// arithmetic. The approximation technique used here is // arithmetic. The approximation technique used here is
// //
// pi / 4 * x + 0.285 * x * (1 - abs(x)) // pi / 4 * r + C * r * (1 - abs(r))
// //
// which is taken from Rajan 2006: Efficient Approximations for // which is taken from Rajan 2006: Efficient Approximations for
// the Arctangent Function. // the Arctangent Function.
if y_greater { //
core::mem::swap(&mut x, &mut y); // 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);
if x == 0 { // `r` is unsigned Q16.16 and <= 1
return 0; // `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.
// We need to share the 31 available non-sign bits between the let mut angle = ((r << 13)
// atan argument and constant factors used in the atan + ((K * (r >> 1) * ((1 << 15) - (r >> 1))) >> (FP_K + 1)))
// approximation. Sharing the bits roughly equally between them as i32;
// gives good accuracy. Additionally, we cannot increase the
// number of atan argument bits beyond 15 because we must square
// it.
const ATAN_ARGUMENT_BITS: usize = 15;
let ratio = (y << ATAN_ARGUMENT_BITS) / x;
let mut angle = {
const K1: i32 = ((1. / 4. + 0.285 / PI)
* (1 << (31 - ATAN_ARGUMENT_BITS)) as f64)
as i32;
const K2: i32 =
((0.285 / PI) * (1 << (31 - ATAN_ARGUMENT_BITS)) as f64) as i32;
ratio * K1 - K2 * shift_round(ratio * ratio, ATAN_ARGUMENT_BITS)
};
if y_greater { if y_greater {
angle = (i32::MAX >> 1) - angle; angle = (1 << 30) - angle;
} }
if sign & 1 == 1 { if sign.0 {
angle = i32::MAX - angle; angle = i32::MAX - angle;
} }
if sign & 2 == 2 { if sign.1 {
angle *= -1; angle = angle.wrapping_neg();
} }
angle angle
@ -162,7 +159,6 @@ pub fn cossin(phase: i32) -> Complex<i32> {
#[cfg(test)] #[cfg(test)]
mod tests { mod tests {
use super::*; use super::*;
use crate::testing::isclose;
use core::f64::consts::PI; use core::f64::consts::PI;
fn angle_to_axis(angle: f64) -> f64 { fn angle_to_axis(angle: f64) -> f64 {
@ -172,61 +168,43 @@ mod tests {
#[test] #[test]
fn atan2_absolute_error() { fn atan2_absolute_error() {
const NUM_VALS: usize = 1_001; const N: usize = 321;
let mut test_vals: [f64; NUM_VALS] = [0.; NUM_VALS]; let mut test_vals = [0i32; N + 4];
let val_bounds: (f64, f64) = (-1., 1.); let scale = (1i64 << 31) as f64;
let val_delta: f64 = for i in 0..N {
(val_bounds.1 - val_bounds.0) / (NUM_VALS - 1) as f64; test_vals[i] = (scale * (-1. + 2. * i as f64 / N as f64)) as i32;
for i in 0..NUM_VALS {
test_vals[i] = val_bounds.0 + i as f64 * val_delta;
} }
let atol: f64 = 4e-5; assert!(test_vals.contains(&i32::MIN));
let rtol: f64 = 0.127; 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 &x in test_vals.iter() {
for &y in test_vals.iter() { for &y in test_vals.iter() {
let actual = (y.atan2(x) as f64 * i16::MAX as f64).round() let want = (y as f64 / scale).atan2(x as f64 / scale);
/ i16::MAX as f64; let have = atan2(y, x) as f64 * PI / scale;
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) { let err = (have - want).abs();
println!("(x, y) : {}, {}", x, y); abs_err = abs_err.max(err);
println!("actual : {}", actual); rms_err += err * err;
println!("computed : {}", computed); if err > 3e-5 {
println!("tolerance: {}\n", tol); rel_err = rel_err.max(err / angle_to_axis(want));
assert!(false);
} }
} }
} }
rms_err = rms_err.sqrt() / test_vals.len() as f64;
// test min and max explicitly println!("max abs err: {:.2e}", abs_err);
for (x, y) in [ println!("rms abs err: {:.2e}", rms_err);
((i16::MIN as i32 + 1) << 16, -(1 << 16) as i32), println!("max rel err: {:.2e}", rel_err);
((i16::MIN as i32 + 1) << 16, (1 << 16) as i32), assert!(abs_err < 5e-3);
] assert!(rms_err < 3e-3);
.iter() assert!(rel_err < 0.6);
{
let yf = *y as f64 / ((i16::MAX as i32) << 16) as f64;
let xf = *x as f64 / ((i16::MAX as i32) << 16) as f64;
let actual =
(yf.atan2(xf) * i16::MAX as f64).round() / i16::MAX as f64;
let computed = (atan2(*y, *x) >> 16) as f64 / i16::MAX as f64 * PI;
let tol = atol + rtol * angle_to_axis(actual).abs();
if !isclose(computed, actual, 0., tol) {
println!("(x, y) : {}, {}", *x, *y);
println!("actual : {}", actual);
println!("computed : {}", computed);
println!("tolerance: {}\n", tol);
assert!(false);
}
}
} }
#[test] #[test]