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Improve logarithm accuracy near the unit circle #163

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261 changes: 258 additions & 3 deletions src/lib.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -268,9 +268,29 @@ impl<T: Float> Complex<T> {
/// The branch satisfies `-π ≤ arg(ln(z)) ≤ π`.
#[inline]
pub fn ln(self) -> Self {
// formula: ln(z) = ln|z| + i*arg(z)
let (r, theta) = self.to_polar();
Self::new(r.ln(), theta)
let (radius, theta) = self.to_polar();
let low = T::from(0.75).unwrap();
let high = T::from(1.25).unwrap();
if !(radius >= low && radius <= high) {
return Self::new(radius.ln(), theta);
}

let re = self.re.abs();
let im = self.im.abs();
let (a, b) = if re >= im { (re, im) } else { (im, re) };
let one = T::one();
let two = T::from(2.0).unwrap();
let p = a * a;
let q = b * b;
if a == one && q < two * T::min_positive_value() {
return Self::new(b * (b / two), theta);
}
let s = p + q;
let addition_error = (p - s) + q;
let large_square_error = a.mul_add(a, -p);
let small_square_error = b.mul_add(b, -q);
let delta = (((s - one) + addition_error) + large_square_error) + small_square_error;
Self::new(delta.ln_1p() / two, theta)
}

/// Computes the principal value of the square root of `self`.
Expand Down Expand Up @@ -1942,6 +1962,241 @@ pub(crate) mod test {
}
}

#[test]
fn test_ln_near_unit_f64() {
let cases: &[(u64, u64, u64, u64)] = &[
(
0x3feffff5de392baf,
0x3f720281174e0b28,
0x3ed447b4d1091f51,
8,
),
(
0x3fec395ffdd35dd1,
0x3fde28d594f859ff,
0x3c317cb70a8af428,
8,
),
(
0x3fe3333333333333,
0x3fe999999999999a,
0x3c7999999999999a,
8,
),
(
0x3fe6a09e667f3bcd,
0x3fe6a09e667f3bcd,
0x3c93b3efbf5e2229,
8,
),
(
0x3ff0000000000000,
0x3e50000000000000,
0x3c9fffffffffffff,
8,
),
(
0x3ff0000000000000,
0x1e60000000000001,
0x0000000000000001,
0,
),
(
0x3ff0000000000000,
0x0010000000000000,
0x0000000000000000,
0,
),
(
0x3ff0000000000000,
0x0000000000000001,
0x0000000000000000,
0,
),
(
0x3ff0000000000000,
0x0000000000000000,
0x0000000000000000,
0,
),
(
0x3ff0000000000000,
0x2006a09e667f3bcc,
0x000fffffffffffff,
0,
),
(
0x3ff0000000000000,
0x2006a09e667f3bcd,
0x0010000000000001,
8,
),
(
0x3ff0000000000000,
0x2006a09e667f3bce,
0x0010000000000002,
8,
),
(
0x3fefffffffffffff,
0x1e60000000000001,
0xbca0000000000000,
8,
),
(
0x3ff0000000000001,
0x1e60000000000001,
0x3cafffffffffffff,
8,
),
];
for &(real_bits, imag_bits, expected_bits, max_ulp) in cases {
for re_sign in [0, 1 << 63] {
for im_sign in [0, 1 << 63] {
let re = f64::from_bits(real_bits | re_sign);
let im = f64::from_bits(imag_bits | im_sign);
for value in [Complex::new(re, im), Complex::new(im, re)] {
let actual = value.ln();
assert!(
actual.re.to_bits().abs_diff(expected_bits) <= max_ulp,
"input={value:?} actual={:x} expected={expected_bits:x} max_ulp={max_ulp}",
actual.re.to_bits(),
);
assert_eq!(actual.im.to_bits(), value.arg().to_bits());
}
}
}
}
}

#[test]
fn test_ln_fallback_f64() {
let cases = [
(0.0, 0.0),
(3.0, 4.0),
(0.5, 0.5),
(2.0, 3.0),
(f64::from_bits((0.75_f64).to_bits() - 1), 0.0),
(f64::from_bits((1.25_f64).to_bits() + 1), 0.0),
(f64::MAX, f64::MAX),
(f64::MIN_POSITIVE, f64::MIN_POSITIVE),
(f64::from_bits(1), f64::from_bits(1)),
(f64::INFINITY, 0.0),
(f64::INFINITY, 1.0),
(f64::INFINITY, f64::INFINITY),
(f64::INFINITY, f64::NAN),
(f64::NAN, 0.0),
(f64::NAN, 1.0),
(f64::NAN, f64::NAN),
];
for (re, im) in cases {
for re_sign in [1.0, -1.0] {
for im_sign in [1.0, -1.0] {
for value in [
Complex::new(re * re_sign, im * im_sign),
Complex::new(im * im_sign, re * re_sign),
] {
let (radius, theta) = value.to_polar();
let expected_re = Float::ln(radius);
let actual = value.ln();
if expected_re.is_nan() {
assert!(actual.re.is_nan());
} else {
assert_eq!(actual.re.to_bits(), expected_re.to_bits());
}
if theta.is_nan() {
assert!(actual.im.is_nan());
} else {
assert_eq!(actual.im.to_bits(), theta.to_bits());
}
}
}
}
}
}

#[test]
fn test_ln_near_unit_f32() {
let cases: &[(u32, u32, u32, u32)] = &[
(0x3f7fffaf, 0x3b901409, 0x36a25a15, 8),
(0x3f7ffa52, 0x3c57b2e6, 0xac83195c, 8),
(0x3f19999a, 0x3f4ccccd, 0x32cccccd, 8),
(0x3f800000, 0x39800000, 0x33000000, 8),
(0x3f800000, 0x1a3504f4, 0x00000001, 0),
(0x3f800000, 0x00800000, 0x00000000, 0),
(0x3f800000, 0x00000001, 0x00000000, 0),
(0x3f800000, 0x00000000, 0x00000000, 0),
(0x3f800000, 0x203504f2, 0x007ffffe, 0),
(0x3f800000, 0x203504f3, 0x00800000, 0),
(0x3f800000, 0x203504f4, 0x00800001, 8),
(0x3f7fffff, 0x1a3504f4, 0xb3800000, 8),
(0x3f800001, 0x1a3504f4, 0x33ffffff, 8),
];
for &(real_bits, imag_bits, expected_bits, max_ulp) in cases {
for re_sign in [0, 1 << 31] {
for im_sign in [0, 1 << 31] {
let re = f32::from_bits(real_bits | re_sign);
let im = f32::from_bits(imag_bits | im_sign);
for value in [Complex::new(re, im), Complex::new(im, re)] {
let actual = value.ln();
assert!(
actual.re.to_bits().abs_diff(expected_bits) <= max_ulp,
"input={value:?} actual={:x} expected={expected_bits:x} max_ulp={max_ulp}",
actual.re.to_bits(),
);
assert_eq!(actual.im.to_bits(), value.arg().to_bits());
}
}
}
}
}

#[test]
fn test_ln_fallback_f32() {
let cases = [
(0.0, 0.0),
(3.0, 4.0),
(0.5, 0.5),
(2.0, 3.0),
(f32::from_bits((0.75_f32).to_bits() - 1), 0.0),
(f32::from_bits((1.25_f32).to_bits() + 1), 0.0),
(f32::MAX, f32::MAX),
(f32::MIN_POSITIVE, f32::MIN_POSITIVE),
(f32::from_bits(1), f32::from_bits(1)),
(f32::INFINITY, 0.0),
(f32::INFINITY, 1.0),
(f32::INFINITY, f32::INFINITY),
(f32::INFINITY, f32::NAN),
(f32::NAN, 0.0),
(f32::NAN, 1.0),
(f32::NAN, f32::NAN),
];
for (re, im) in cases {
for re_sign in [1.0, -1.0] {
for im_sign in [1.0, -1.0] {
for value in [
Complex::new(re * re_sign, im * im_sign),
Complex::new(im * im_sign, re * re_sign),
] {
let (radius, theta) = value.to_polar();
let expected_re = Float::ln(radius);
let actual = value.ln();
if expected_re.is_nan() {
assert!(actual.re.is_nan());
} else {
assert_eq!(actual.re.to_bits(), expected_re.to_bits());
}
if theta.is_nan() {
assert!(actual.im.is_nan());
} else {
assert_eq!(actual.im.to_bits(), theta.to_bits());
}
}
}
}
}
}

#[test]
fn test_powc() {
let a = Complex::new(2.0, -3.0);
Expand Down