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pbrt/src/utils/math.rs

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use super::color::{RGB, XYZ};
use super::error::{InversionError, LlsError};
use crate::core::pbrt::{
Float, FloatBitOps, FloatBits, ONE_MINUS_EPSILON, PI, PI_OVER_4, clamp_t, evaluate_polynomial,
lerp,
};
use crate::geometry::{Point, Point2f, Point2i, Vector, Vector3f, VectorLike};
use crate::utils::hash::{hash_buffer, mix_bits};
use crate::utils::sobol::{SOBOL_MATRICES_32, VDC_SOBOL_MATRICES, VDC_SOBOL_MATRICES_INV};
use num_traits::{Float as NumFloat, Num, One, Signed, Zero};
use rayon::prelude::*;
use std::error::Error;
use std::fmt::{self, Display};
use std::iter::{Product, Sum};
use std::mem;
use std::ops::{Add, Div, Index, IndexMut, Mul, Neg, Rem};
#[inline]
pub fn degrees(a: Float) -> Float {
a * 180.0 / PI
}
#[inline]
pub fn radians(a: Float) -> Float {
a * PI / 180.0
}
#[inline]
pub fn square<T>(n: T) -> T
where
T: Mul<Output = T> + Copy,
{
n * n
}
#[inline]
fn fma<T>(a: T, b: T, c: T) -> T
where
T: Mul<Output = T> + Add<Output = T> + Copy,
{
a * b + c
}
#[inline]
pub fn difference_of_products<T>(a: T, b: T, c: T, d: T) -> T
where
T: Mul<Output = T> + Add<Output = T> + Neg<Output = T> + Copy,
{
let cd = c * d;
let difference_of_products = fma(a, b, -cd);
let error = fma(-c, d, cd);
difference_of_products + error
}
#[inline]
pub fn sum_of_products<T>(a: T, b: T, c: T, d: T) -> T
where
T: Mul<Output = T> + Add<Output = T> + Neg<Output = T> + Copy,
{
let cd = c * d;
let sum_of_products = fma(a, b, cd);
let error = fma(c, d, -cd);
sum_of_products + error
}
#[inline]
pub fn safe_sqrt(x: Float) -> Float {
assert!(x > -1e-3);
0.0_f32.max(x).sqrt()
}
#[inline]
pub fn safe_asin<T: NumFloat>(x: T) -> T {
let epsilon = T::from(0.0001).unwrap();
let one = T::one();
if x >= -(one + epsilon) && x <= one + epsilon {
clamp_t(x, -one, one).asin()
} else {
panic!("Not valid value for asin")
}
}
#[inline]
pub fn safe_acos(x: Float) -> Float {
if (-1.001..1.001).contains(&x) {
clamp_t(x, -1., 1.).asin()
} else {
panic!("Not valid value for acos")
}
}
#[inline]
pub fn sinx_over_x(x: Float) -> Float {
if 1. - x * x == 1. {
return 1.;
}
x.sin() / x
}
#[inline]
pub fn sinc(x: Float) -> Float {
sinx_over_x(PI * x)
}
#[inline]
pub fn windowed_sinc(x: Float, radius: Float, tau: Float) -> Float {
if x.abs() > radius {
return 0.;
}
sinc(x) * sinc(x / tau)
}
#[inline]
pub fn fast_exp(x: Float) -> Float {
let xp = x * 1.442695041;
let fxp = xp.floor();
let f = xp - fxp;
let i = fxp as i32;
let two_to_f = evaluate_polynomial(f, &[1., 0.695556856, 0.226173572, 0.0781455737])
.expect("Could not evaluate polynomial");
let exponent = exponent(two_to_f) + i;
if exponent < -126 {
return 0.;
}
if exponent > 127 {
return Float::INFINITY;
}
let mut bits = float_to_bits(two_to_f);
bits &= 0b10000000011111111111111111111111;
bits |= ((exponent + 127) as u32) << 23;
bits_to_float(bits)
}
#[inline]
pub fn i0(x: Float) -> Float {
let mut val: Float = 0.0;
let mut x2i: Float = 1.0;
let mut ifact: i64 = 1;
let mut i4: i32 = 1;
for i in 0..10 {
if i > 1 {
ifact *= i as i64;
}
let denominator = (i4 as Float) * (square(ifact) as Float);
val += x2i / denominator;
x2i *= x * x;
i4 *= 4;
}
val
}
#[inline]
pub fn logistic(x: Float, s: Float) -> Float {
let y = x.abs();
(-y / s).exp() / (s * square(1. + (-y / s).exp()))
}
#[inline]
pub fn logistic_cdf(x: Float, s: Float) -> Float {
1. / (1. + (-x / s).exp())
}
#[inline]
pub fn trimmed_logistic(x: Float, s: Float, a: Float, b: Float) -> Float {
logistic(x, s) / (logistic_cdf(b, s) - logistic_cdf(a, s))
}
#[inline]
pub fn log_i0(x: Float) -> Float {
if x > 12.0 {
let inv_x = 1.0 / x;
let two_pi = 2.0 * PI as Float;
x + 0.5 * (-two_pi.ln() + inv_x.ln() + 1.0 / (8.0 * x))
} else {
i0(x).ln()
}
}
#[inline]
pub fn log2_int(v: f32) -> i32 {
debug_assert!(v > 0.0, "log2_int requires positive input");
if v < 1.0 {
return -log2_int(1.0 / v);
}
// https://graphics.stanford.edu/~seander/bithacks.html#IntegerLog
// midsignif = Significand(pow(2.0, 1.5))
const MID_SIGNIF: u32 = 0b00000000001101010000010011110011;
let bits = v.to_bits();
let exponent = ((bits >> 23) & 0xFF) as i32 - 127;
let significand = bits & 0x7FFFFF;
if significand >= MID_SIGNIF {
exponent + 1
} else {
exponent
}
}
pub fn quadratic(a: Float, b: Float, c: Float) -> Option<(Float, Float)> {
if a == 0. {
if b == 0. {
return None;
}
let t0 = -c / b;
let t1 = -c / b;
return Some((t0, t1));
}
let discrim = difference_of_products(b, b, 4. * a, c);
if discrim < 0. {
return None;
}
let root_discrim = discrim.sqrt();
let q = -0.5 * (b + root_discrim.copysign(b));
let mut t0 = q / a;
let mut t1 = c / q;
if t0 > t1 {
mem::swap(&mut t0, &mut t1);
}
Some((t0, t1))
}
pub fn smooth_step(x: Float, a: Float, b: Float) -> Float {
if a == b {
if x < a { return 0. } else { return 1. }
}
let t = clamp_t((x - a) / (b - a), 0., 1.);
t * t * (3. - 2. * t)
}
pub fn linear_least_squares<const R: usize, const N: usize>(
a: [[Float; R]; N],
b: [[Float; R]; N],
) -> Result<SquareMatrix<Float, R>, Box<dyn Error>> {
let am = Matrix::from(a);
let bm = Matrix::from(b);
let ata = am.transpose() * am;
let atb = am.transpose() * bm;
let at_ai = ata.inverse()?;
Ok((at_ai * atb).transpose())
}
pub fn newton_bisection<P>(mut x0: Float, mut x1: Float, mut f: P) -> Float
where
P: FnMut(Float) -> (Float, Float),
{
assert!(x0 < x1);
let f_eps = 1e-6;
let x_eps = 1e-6;
let (fx0, _) = f(x0);
let (fx1, _) = f(x1);
if fx0.abs() < f_eps {
return x0;
}
if fx1.abs() < f_eps {
return x1;
}
let start_is_negative = fx0 < 0.0;
let mut x_mid = x0 + (x1 - x0) * -fx0 / (fx1 - fx0);
loop {
if !(x0 < x_mid && x_mid < x1) {
x_mid = (x0 + x1) / 2.0;
}
let (fx_mid, dfx_mid) = f(x_mid);
assert!(!fx_mid.is_nan());
if start_is_negative == (fx_mid < 0.0) {
x0 = x_mid;
} else {
x1 = x_mid;
}
if (x1 - x0) < x_eps || fx_mid.abs() < f_eps {
return x_mid;
}
x_mid -= fx_mid / dfx_mid;
}
}
pub fn wrap_equal_area_square(uv: &mut Point2f) -> Point2f {
if uv[0] < 0. {
uv[0] = -uv[0];
uv[1] = 1. - uv[1];
} else if uv[0] > 1. {
uv[0] = 2. - uv[0];
uv[1] = 1. - uv[1];
}
if uv[1] < 0. {
uv[0] = 1. - uv[0];
uv[1] = -uv[1];
} else if uv[1] > 1. {
uv[0] = 1. - uv[0];
uv[1] = 2. - uv[1];
}
*uv
}
pub fn catmull_rom_weights(nodes: &[Float], x: Float) -> Option<(usize, [Float; 4])> {
if nodes.len() < 4 {
return None;
}
// Return None if x is out of bounds
if x < *nodes.first()? || x > *nodes.last()? {
return None;
}
// Search for interval idx containing x
// partition_point returns the first index where predicate is false (!= <= x means > x)
// Equivalent to upper_bound. We subtract 1 to get the interval index.
let idx = nodes.partition_point(|&n| n <= x).saturating_sub(1);
// Safety clamp (though bounds check above handles most cases)
let idx = idx.min(nodes.len() - 2);
let offset = idx.saturating_sub(1); // The C++ code uses idx - 1 for the offset
let x0 = nodes[idx];
let x1 = nodes[idx + 1];
// Compute t parameter and powers
let t = (x - x0) / (x1 - x0);
let t2 = t * t;
let t3 = t2 * t;
let mut weights = [0.0; 4];
// Compute initial node weights w1 and w2
weights[1] = 2.0 * t3 - 3.0 * t2 + 1.0;
weights[2] = -2.0 * t3 + 3.0 * t2;
// Compute first node weight w0
if idx > 0 {
let w0 = (t3 - 2.0 * t2 + t) * (x1 - x0) / (x1 - nodes[idx - 1]);
weights[0] = -w0;
weights[2] += w0;
} else {
let w0 = t3 - 2.0 * t2 + t;
weights[0] = 0.0;
weights[1] -= w0;
weights[2] += w0;
}
// Compute last node weight w3
if idx + 2 < nodes.len() {
let w3 = (t3 - t2) * (x1 - x0) / (nodes[idx + 2] - x0);
weights[1] -= w3;
weights[3] = w3;
} else {
let w3 = t3 - t2;
weights[1] -= w3;
weights[2] += w3;
weights[3] = 0.0;
}
Some((offset, weights))
}
pub fn equal_area_sphere_to_square(d: Vector3f) -> Point2f {
debug_assert!(d.norm_squared() > 0.999 && d.norm_squared() < 1.001);
let x = d.x().abs();
let y = d.y().abs();
let z = d.z().abs();
let r = safe_sqrt(1. - z);
let a = x.max(y);
let b = if a == 0. { 0. } else { x.min(y) / a };
let t1 = 0.406758566246788489601959989e-5;
let t2 = 0.636226545274016134946890922156;
let t3 = 0.61572017898280213493197203466e-2;
let t4 = -0.247333733281268944196501420480;
let t5 = 0.881770664775316294736387951347e-1;
let t6 = 0.419038818029165735901852432784e-1;
let t7 = -0.251390972343483509333252996350e-1;
let mut phi = evaluate_polynomial(b, &[t1, t2, t3, t4, t5, t6, t7])
.expect("Could not evaluate polynomial");
if x < y {
phi = 1. - phi;
}
let mut v = phi * r;
let mut u = r - v;
if d.z() < 0. {
mem::swap(&mut u, &mut v);
u = 1. - u;
v = 1. - v;
}
u = u.copysign(d.x());
v = v.copysign(d.y());
Point2f::new(0.5 * (u + 1.), 0.5 * (v + 1.))
}
pub fn equal_area_square_to_sphere(p: Point2f) -> Vector3f {
assert!(p.x() >= 0. && p.x() <= 1. && p.y() >= 0. && p.y() <= 1.);
// Transform p to [-1,1]^2 and compute absolute values
let u = 2. * p.x() - 1.;
let v = 2. * p.y() - 1.;
let up = u.abs();
let vp = v.abs();
// Compute radius as signed distance from diagonal
let signed_distance = 1. - (up + vp);
let d = signed_distance.abs();
let r = 1. - d;
// Compute angle \phi for square to sphere mapping
let mut phi = if r == 0. { 1. } else { (vp - up) / r + 1. };
phi /= PI_OVER_4;
// Find z for spherical direction
let z = (1. - square(r)).copysign(signed_distance);
// Compute $\cos\phi$ and $\sin\phi$ for original quadrant and return vector
let cos_phi = phi.cos().copysign(u);
let sin_phi = phi.sin().copysign(v);
Vector3f::new(
cos_phi * r * (2. - square(r)).sqrt(),
sin_phi * r * (2. - square(r)).sqrt(),
z,
)
}
pub fn gaussian(x: Float, y: Float, sigma: Float) -> Float {
(-(x * x + y * y) / (2. * sigma * sigma)).exp()
}
pub fn gaussian_integral(x0: Float, x1: Float, mu: Float, sigma: Float) -> Float {
assert!(sigma > 0.);
let sigma_root2 = sigma * 1.414213562373095;
0.5 * (((mu - x0) / sigma_root2).erf() - ((mu - x1) / sigma_root2).erf())
}
pub fn sample_linear(u: Float, a: Float, b: Float) -> Float {
assert!(a >= 0. && b >= 0.);
if u == 0. && a == 0. {
return 0.;
}
let x = u * (a + b) / (a + (lerp(u, square(a), square(b))));
x.min(ONE_MINUS_EPSILON)
}
#[inline(always)]
pub fn bits_to_float(bits: FloatBits) -> Float {
Float::from_bits_val(bits)
}
#[inline(always)]
pub fn float_to_bits(v: Float) -> FloatBits {
v.to_bits_val()
}
#[inline(always)]
pub fn exponent(v: Float) -> i32 {
v.exponent_val()
}
#[inline(always)]
pub fn significand(v: Float) -> FloatBits {
v.significand_val()
}
#[inline(always)]
pub fn sign_bit(v: Float) -> FloatBits {
v.sign_bit_val()
}
#[inline]
pub fn next_float_up(v: Float) -> Float {
if v.is_infinite() && v > 0.0 {
return v;
}
let v = if v == -0.0 { 0.0 } else { v };
let mut ui = float_to_bits(v);
if v >= 0.0 {
ui += 1;
} else {
ui -= 1;
}
bits_to_float(ui)
}
#[inline]
pub fn next_float_down(v: Float) -> Float {
if v.is_infinite() && v < 0.0 {
return v;
}
let v = if v == 0.0 { -0.0 } else { v };
let mut ui = float_to_bits(v);
if v > 0.0 {
ui -= 1;
} else {
ui += 1;
}
bits_to_float(ui)
}
pub fn sample_discrete(
weights: &[f32],
u: f32,
pmf: Option<&mut f32>,
u_remapped: Option<&mut f32>,
) -> Option<usize> {
// Handle empty weights for discrete sampling.
if weights.is_empty() {
if let Some(p) = pmf {
*p = 0.0;
}
return None;
}
// If the total weight is zero, sampling is not possible.
let sum_weights: f32 = weights.iter().sum();
if sum_weights == 0.0 {
if let Some(p) = pmf {
*p = 0.0;
}
return None;
}
let mut up = u * sum_weights;
if up >= sum_weights {
up = next_float_down(up);
}
// Find the offset in weights corresponding to the rescaled sample u'.
let mut offset = 0;
let mut sum = 0.0;
while sum + weights[offset] <= up {
sum += weights[offset];
offset += 1;
debug_assert!(offset < weights.len());
}
if let Some(p) = pmf {
*p = weights[offset] / sum_weights;
}
if let Some(ur) = u_remapped {
let weight = weights[offset];
*ur = ((up - sum) / weight).min(ONE_MINUS_EPSILON);
}
Some(offset)
}
pub fn sample_tent(u: Float, r: Float) -> Float {
let mut u_remapped = 0.0;
let offset = sample_discrete(&[0.5, 0.5], u, None, Some(&mut u_remapped))
.expect("Discrete sampling shouldn't fail");
if offset == 0 {
-r + r * sample_linear(u, 0., 1.)
} else {
r * sample_linear(u, 1., 0.)
}
}
fn left_shift2(mut x: u64) -> u64 {
x &= 0xffffffff;
x = (x ^ (x << 16)) & 0x0000ffff0000ffff;
x = (x ^ (x << 8)) & 0x00ff00ff00ff00ff;
x = (x ^ (x << 4)) & 0x0f0f0f0f0f0f0f0f;
x = (x ^ (x << 2)) & 0x3333333333333333;
x = (x ^ (x << 1)) & 0x5555555555555555;
x
}
fn left_shift3(mut x: u32) -> u32 {
if x == (1 << 10) {
x -= 1;
}
x = (x | (x << 16)) & 0b00000011000000000000000011111111;
x = (x | (x << 8)) & 0b00000011000000001111000000001111;
x = (x | (x << 4)) & 0b00000011000011000011000011000011;
x = (x | (x << 2)) & 0b00001001001001001001001001001001;
x
}
pub fn encode_morton_2(x: u32, y: u32) -> u64 {
left_shift2(y as u64) << 1 | left_shift2(x as u64)
}
pub fn encode_morton_3(x: Float, y: Float, z: Float) -> u32 {
(left_shift3(x as u32) << 2) | (left_shift3(y as u32) << 1) | left_shift3(z as u32)
}
pub fn round_up_pow2(mut n: i32) -> i32 {
if n <= 0 {
return 1;
}
n -= 1;
n |= n >> 1;
n |= n >> 2;
n |= n >> 4;
n |= n >> 8;
n |= n >> 16;
n + 1
}
pub const PRIME_TABLE_SIZE: usize = 1000;
const PRIMES: [i32; PRIME_TABLE_SIZE] = [
2, 3, 5, 7, 11, // Subsequent prime numbers
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107,
109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211,
223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317,
331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439,
443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569,
571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677,
683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821,
823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061,
1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181,
1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291,
1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429,
1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531,
1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627,
1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759,
1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889,
1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017,
2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137,
2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273,
2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383,
2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531,
2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663,
2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753,
2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887,
2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023,
3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181,
3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313,
3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449,
3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557,
3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677,
3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821,
3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931,
3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079,
4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219,
4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349,
4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493,
4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639,
4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783,
4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931,
4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023,
5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179,
5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333,
5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471,
5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591,
5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737,
5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857,
5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029,
6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151,
6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287,
6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397,
6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571,
6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709,
6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857,
6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983,
6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129,
7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297,
7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477,
7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577,
7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703,
7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873,
7877, 7879, 7883, 7901, 7907, 7919,
];
#[inline]
pub fn radical_inverse(base_index: usize, mut a: u64) -> Float {
let base = PRIMES[base_index] as u64;
let limit = (u64::MAX / base).saturating_sub(base);
let inv_base = (1.0 as Float) / (base as Float);
let mut inv_base_m = 1.0 as Float;
let mut reversed_digits = 0u64;
// Loop until we run out of digits or hit the overflow safety limit
while a > 0 && reversed_digits < limit {
// Rust's div_rem optimization handles / and % efficiently together
let next = a / base;
let digit = a % base;
reversed_digits = reversed_digits.wrapping_mul(base).wrapping_add(digit);
inv_base_m *= inv_base;
a = next;
}
// Ensure result is strictly less than 1.0
(reversed_digits as Float * inv_base_m).min(ONE_MINUS_EPSILON)
}
pub fn inverse_radical_inverse(mut inverse: u64, base: u64, n_digits: u64) -> u64 {
let mut index = 0;
for _ in 0..n_digits {
let digit = inverse % base;
inverse /= base;
index = index * base + digit;
}
index
}
// Digit scrambling
#[derive(Default, Debug, Clone)]
pub struct DigitPermutation {
base: usize,
n_digits: usize,
permutations: Vec<u16>,
}
impl DigitPermutation {
pub fn new(base: usize, seed: u64) -> Self {
let mut n_digits = 0;
let inv_base = 1. / base as Float;
let mut inv_base_m = 1.;
while 1.0 - ((base as Float - 1.0) * inv_base_m) < 1.0 {
n_digits += 1;
inv_base_m *= inv_base;
}
let mut permutations = vec![0u16; n_digits * base];
for digit_index in 0..n_digits {
let hash_input = [base as u64, digit_index as u64, seed];
let dseed = hash_buffer(&hash_input, 0);
for digit_value in 0..base {
let index = digit_index * base + digit_value;
permutations[index] =
permutation_element(digit_value as u32, base as u32, dseed as u32) as u16;
}
}
Self {
base,
n_digits,
permutations,
}
}
#[inline(always)]
pub fn permute(&self, digit_index: i32, digit_value: i32) -> i32 {
let idx = (digit_index * self.base as i32 + digit_value) as usize;
self.permutations[idx] as i32
}
}
pub fn compute_radical_inverse_permutations(seed: u64) -> Vec<DigitPermutation> {
PRIMES
.par_iter()
.map(|&base| DigitPermutation::new(base as usize, seed))
.collect()
}
pub fn scrambled_radical_inverse(base_index: usize, mut a: u64, perm: &DigitPermutation) -> Float {
let base = PRIMES[base_index] as u64;
let limit = (u64::MAX / base).saturating_sub(base);
let inv_base = 1.0 / (base as Float);
let mut inv_base_m = 1.0;
let mut reversed_digits = 0u64;
let mut digit_index = 0;
while 1.0 - ((base as Float - 1.0) * inv_base_m) < 1.0 && reversed_digits < limit {
let next = a / base;
let digit_value = (a - next * base) as i32;
// Permute the digit
let permuted = perm.permute(digit_index, digit_value) as u64;
reversed_digits = reversed_digits.wrapping_mul(base).wrapping_add(permuted);
inv_base_m *= inv_base;
digit_index += 1;
a = next;
}
(inv_base_m * reversed_digits as Float).min(ONE_MINUS_EPSILON)
}
pub fn owen_scrambled_radical_inverse(base_index: usize, mut a: u64, hash: u32) -> Float {
let base = PRIMES[base_index] as u64;
let limit = (u64::MAX / base).saturating_sub(base);
let inv_base = 1.0 / (base as Float);
let mut inv_base_m = 1.0;
let mut reversed_digits = 0u64;
let mut _digit_index = 0;
while 1.0 - inv_base_m < 1.0 && reversed_digits < limit {
let next = a / base;
let digit_value = (a - next * base) as u32;
// Compute Owen-scrambled digit
// XOR the current seed (hash) with the accumulated reversed digits so far
let digit_hash = mix_bits((hash as u64) ^ reversed_digits) as u32;
let permuted = permutation_element(digit_value, base as u32, digit_hash) as u64;
reversed_digits = reversed_digits.wrapping_mul(base).wrapping_add(permuted);
inv_base_m *= inv_base;
_digit_index += 1;
a = next;
}
(inv_base_m * reversed_digits as Float).min(ONE_MINUS_EPSILON)
}
pub fn permutation_element(mut i: u32, l: u32, p: u32) -> u32 {
let mut w = l - 1;
w |= w >> 1;
w |= w >> 2;
w |= w >> 4;
w |= w >> 8;
w |= w >> 16;
loop {
i ^= p;
i = i.wrapping_mul(0xe170893d);
i ^= p >> 16;
i ^= (i & w) >> 4;
i ^= p >> 8;
i = i.wrapping_mul(0x0929eb3f);
i ^= p >> 23;
i ^= (i & w) >> 1;
i = i.wrapping_mul(1 | (p >> 27));
i = i.wrapping_mul(0x6935fa69);
i ^= (i & w) >> 11;
i = i.wrapping_mul(0x74dcb303);
i ^= (i & w) >> 2;
i = i.wrapping_mul(0x9e501cc3);
i ^= (i & w) >> 2;
i = i.wrapping_mul(0xc860a3df);
i &= w;
i ^= i >> 5;
// If index is within range [0, l), we are done.
// Otherwise, we loop again.
if i < l {
break;
}
}
(i.wrapping_add(p)) % l
}
pub fn multiply_generator(c: &[u32], mut a: u32) -> u32 {
let mut v = 0;
let mut i = 0;
while a != 0 && i < c.len() {
if (a & 1) != 0 {
v ^= c[i];
}
a >>= 1;
i += 1;
}
v
}
#[derive(Debug, Clone, Copy)]
pub struct NoRandomizer;
impl NoRandomizer {
pub fn scramble(&self, v: u32) -> u32 {
v
}
}
#[derive(Debug, Clone, Copy)]
pub struct BinaryPermuteScrambler {
pub permutation: u32,
}
impl BinaryPermuteScrambler {
pub fn new(perm: u32) -> Self {
Self { permutation: perm }
}
#[inline(always)]
pub fn scramble(&self, v: u32) -> u32 {
self.permutation ^ v
}
}
#[derive(Debug, Clone, Copy)]
pub struct FastOwenScrambler {
pub seed: u32,
}
impl FastOwenScrambler {
pub fn new(seed: u32) -> Self {
Self { seed }
}
#[inline(always)]
pub fn scramble(&self, mut v: u32) -> u32 {
v = v.reverse_bits();
// v ^= v * 0x3d20adea;
v ^= v.wrapping_mul(0x3d20adea);
// v += seed;
v = v.wrapping_add(self.seed);
// v *= (seed >> 16) | 1;
v = v.wrapping_mul((self.seed >> 16) | 1);
// v ^= v * 0x05526c56;
v ^= v.wrapping_mul(0x05526c56);
// v ^= v * 0x53a22864;
v ^= v.wrapping_mul(0x53a22864);
v.reverse_bits()
}
}
#[derive(Debug, Clone, Copy)]
pub struct OwenScrambler {
pub seed: u32,
}
impl OwenScrambler {
pub fn new(seed: u32) -> Self {
Self { seed }
}
#[inline]
pub fn scramble(&self, mut v: u32) -> u32 {
if (self.seed & 1) != 0 {
v ^= 1 << 31;
}
for b in 1..32 {
let mask = (!0u32) << (32 - b);
let input = ((v & mask) ^ self.seed) as u64;
if (mix_bits(input) as u32 & (1 << b)) != 0 {
v ^= 1 << (31 - b);
}
}
v
}
}
pub trait Scrambler {
fn scramble(&self, v: u32) -> u32;
}
impl Scrambler for NoRandomizer {
fn scramble(&self, v: u32) -> u32 {
self.scramble(v)
}
}
impl Scrambler for BinaryPermuteScrambler {
fn scramble(&self, v: u32) -> u32 {
self.scramble(v)
}
}
impl Scrambler for FastOwenScrambler {
fn scramble(&self, v: u32) -> u32 {
self.scramble(v)
}
}
impl Scrambler for OwenScrambler {
fn scramble(&self, v: u32) -> u32 {
self.scramble(v)
}
}
impl<F: Fn(u32) -> u32> Scrambler for F {
fn scramble(&self, v: u32) -> u32 {
(*self)(v)
}
}
const N_SOBOL_DIMENSIONS: usize = 1024;
const SOBOL_MATRIX_SIZE: usize = 52;
#[inline]
pub fn sobol_sample<S: Scrambler>(mut a: u64, dimension: usize, randomizer: S) -> Float {
debug_assert!(
dimension < N_SOBOL_DIMENSIONS,
"Sobol dimension out of bounds"
);
debug_assert!(a < (1u64 << SOBOL_MATRIX_SIZE), "Sobol index too large");
let mut v: u32 = 0;
let mut i = dimension * SOBOL_MATRIX_SIZE;
while a != 0 {
if (a & 1) != 0 {
v ^= SOBOL_MATRICES_32[i];
}
a >>= 1;
i += 1;
}
v = randomizer.scramble(v);
let float_val = (v as Float) * 2.3283064365386963e-10;
float_val.min(ONE_MINUS_EPSILON)
}
pub fn sobol_interval_to_index(m: u32, frame: u64, p: Point2i) -> u64 {
if m == 0 {
return frame;
}
let m2 = m << 1;
let mut index = frame << m2;
let mut delta = 0u64;
let mut current_frame = frame;
let mut c = 0;
while current_frame != 0 {
if (current_frame & 1) != 0 {
delta ^= VDC_SOBOL_MATRICES[(m - 1) as usize][c];
}
current_frame >>= 1;
c += 1;
}
let px = p.x() as u32 as u64;
let py = p.y() as u32 as u64;
let mut b = ((px << m) | py) ^ delta;
let mut c = 0;
while b != 0 {
if (b & 1) != 0 {
index ^= VDC_SOBOL_MATRICES_INV[(m - 1) as usize][c];
}
b >>= 1;
c += 1;
}
index
}
// MATRIX STUFF (TEST THOROUGHLY)
#[derive(Debug, Copy, Clone)]
pub struct Matrix<T, const R: usize, const C: usize> {
m: [[T; C]; R],
}
impl<T, const R: usize, const C: usize> Matrix<T, R, C> {
pub const fn new(data: [[T; C]; R]) -> Self {
Self { m: data }
}
pub fn zero() -> Self
where
T: Clone + Zero,
{
let m: [[T; C]; R] = std::array::from_fn(|_| std::array::from_fn(|_| T::zero()));
Self { m }
}
pub fn transpose(&self) -> Matrix<T, C, R>
where
T: Clone + Zero,
{
let mut result = Matrix::<T, C, R>::zero();
for i in 0..R {
for j in 0..C {
result.m[j][i] = self.m[i][j].clone();
}
}
result
}
}
impl<T: Zero + Clone, const R: usize, const C: usize> Default for Matrix<T, R, C> {
fn default() -> Self {
Self::zero()
}
}
impl<T, const R: usize, const C: usize> From<[[T; C]; R]> for Matrix<T, R, C> {
fn from(m: [[T; C]; R]) -> Self {
Self::new(m)
}
}
impl<T, const R: usize, const C: usize> Index<(usize, usize)> for Matrix<T, R, C> {
type Output = T;
fn index(&self, index: (usize, usize)) -> &Self::Output {
&self.m[index.0][index.1]
}
}
impl<T, const R: usize, const C: usize> IndexMut<(usize, usize)> for Matrix<T, R, C> {
fn index_mut(&mut self, index: (usize, usize)) -> &mut Self::Output {
&mut self.m[index.0][index.1]
}
}
impl<T: PartialEq, const R: usize, const C: usize> PartialEq for Matrix<T, R, C> {
fn eq(&self, other: &Self) -> bool {
self.m == other.m
}
}
impl<T: Eq, const R: usize, const C: usize> Eq for Matrix<T, R, C> {}
impl<T, const R: usize, const C: usize, const N: usize> Mul<Matrix<T, N, C>> for Matrix<T, R, N>
where
T: Mul<Output = T> + Add<Output = T> + Clone + Zero,
{
type Output = Matrix<T, R, C>;
fn mul(self, rhs: Matrix<T, N, C>) -> Self::Output {
let mut result = Matrix::<T, R, C>::zero();
for i in 0..R {
for j in 0..C {
let mut sum = T::zero();
for k in 0..N {
sum = sum + self.m[i][k].clone() * rhs.m[k][j].clone();
}
result.m[i][j] = sum;
}
}
result
}
}
impl<T, const R: usize, const C: usize> Mul<T> for Matrix<T, R, C>
where
T: Mul<Output = T> + Clone + Zero,
{
type Output = Self;
fn mul(self, rhs: T) -> Self::Output {
let mut result = Self::zero();
for i in 0..R {
for j in 0..C {
result.m[i][j] = self.m[i][j].clone() * rhs.clone();
}
}
result
}
}
impl<const R: usize, const C: usize> Mul<Matrix<Float, R, C>> for Float {
type Output = Matrix<Float, R, C>;
fn mul(self, rhs: Matrix<Float, R, C>) -> Self::Output {
rhs * self
}
}
impl<T, const R: usize, const C: usize> Div<T> for Matrix<T, R, C>
where
T: Div<Output = T> + Clone + Zero,
{
type Output = Self;
fn div(self, rhs: T) -> Self::Output {
let mut result = Self::zero();
for i in 0..R {
for j in 0..C {
result.m[i][j] = self.m[i][j].clone() / rhs.clone();
}
}
result
}
}
impl<T: Display, const R: usize, const C: usize> Display for Matrix<T, R, C> {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
let mut col_widths = [0; C];
for row in self.m.iter() {
for (j, element) in row.iter().enumerate() {
let width = format!("{}", element).len();
if width > col_widths[j] {
col_widths[j] = width;
}
}
}
for i in 0..R {
write!(f, "[")?;
#[allow(clippy::needless_range_loop)]
for j in 0..C {
write!(f, "{: >width$} ", self.m[i][j], width = col_widths[j])?;
}
writeln!(f, "]")?;
}
Ok(())
}
}
impl<T, const R: usize, const C: usize> Add for Matrix<T, R, C>
where
T: Copy + Add<Output = T> + Zero,
{
type Output = Self;
fn add(self, rhs: Self) -> Self::Output {
let mut result = Matrix::<T, R, C>::zero();
for i in 0..R {
for j in 0..C {
result.m[i][j] = self.m[i][j] + rhs.m[i][j];
}
}
result
}
}
pub type SquareMatrix<T, const N: usize> = Matrix<T, N, N>;
impl<T, const N: usize> SquareMatrix<T, N> {
pub fn identity() -> Self
where
T: Copy + Zero + One,
{
Self {
m: std::array::from_fn(|i| {
std::array::from_fn(|j| if i == j { T::one() } else { T::zero() })
}),
}
}
pub fn diag(v: &[T]) -> Self
where
T: Zero + Copy,
{
let mut m = [[T::zero(); N]; N];
for i in 0..N {
m[i][i] = v[i];
}
Self { m }
}
pub fn is_identity(&self) -> bool
where
T: Zero + One + PartialEq,
{
for i in 0..N {
for j in 0..N {
if i == j {
if self.m[i][j] != T::one() {
return false;
}
} else if self.m[i][j] != T::zero() {
return false;
}
}
}
true
}
pub fn trace(&self) -> T
where
T: Zero + Copy,
{
let mut sum = T::zero();
for i in 0..N {
sum = sum + self.m[i][i];
}
sum
}
}
impl<T, const N: usize> SquareMatrix<T, N>
where
T: NumFloat + Sum + Product + Copy,
{
pub fn inverse(&self) -> Result<Self, InversionError> {
if N == 0 {
return Err(InversionError::EmptyMatrix);
}
let mut mat = self.m;
let mut inv = Self::identity();
for i in 0..N {
let pivot_row = (i..N)
.max_by(|&a, &b| mat[a][i].abs().partial_cmp(&mat[b][i].abs()).unwrap())
.unwrap_or(i);
if pivot_row != i {
mat.swap(i, pivot_row);
inv.m.swap(i, pivot_row);
}
let pivot = mat[i][i];
if pivot.is_zero() {
return Err(InversionError::SingularMatrix);
}
let inv_pivot = T::one() / pivot;
mat[i][i..].iter_mut().for_each(|x| *x = *x * inv_pivot);
inv[i].iter_mut().for_each(|x| *x = *x * inv_pivot);
for j in 0..N {
if i != j {
let factor = mat[j][i];
#[allow(clippy::needless_range_loop)]
for k in i..N {
mat[j][k] = mat[j][k] - factor * mat[i][k];
}
#[allow(clippy::needless_range_loop)]
for k in 0..N {
inv.m[j][k] = inv.m[j][k] - factor * inv.m[i][k];
}
}
}
}
Ok(inv)
}
pub fn determinant(&self) -> T {
let m = &self.m;
match N {
0 => T::one(),
1 => m[0][0],
2 => m[0][0] * m[1][1] - m[0][1] * m[1][0],
3 => {
let a = m[0][0];
let b = m[0][1];
let c = m[0][2];
let d = m[1][0];
let e = m[1][1];
let f = m[1][2];
let g = m[2][0];
let h = m[2][1];
let i = m[2][2];
a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)
}
4 => {
let det3 =
|m11: T, m12: T, m13: T, m21: T, m22: T, m23: T, m31: T, m32: T, m33: T| -> T {
m11 * (m22 * m33 - m23 * m32) - m12 * (m21 * m33 - m23 * m31)
+ m13 * (m21 * m32 - m22 * m31)
};
let c0 = det3(
m[1][1], m[1][2], m[1][3], m[2][1], m[2][2], m[2][3], m[3][1], m[3][2], m[3][3],
);
let c1 = det3(
m[1][0], m[1][2], m[1][3], m[2][0], m[2][2], m[2][3], m[3][0], m[3][2], m[3][3],
);
let c2 = det3(
m[1][0], m[1][1], m[1][3], m[2][0], m[2][1], m[2][3], m[3][0], m[3][1], m[3][3],
);
let c3 = det3(
m[1][0], m[1][1], m[1][2], m[2][0], m[2][1], m[2][2], m[3][0], m[3][1], m[3][2],
);
m[0][0] * c0 - m[0][1] * c1 + m[0][2] * c2 - m[0][3] * c3
}
_ => {
// Fallback to LU decomposition for N > 4
let mut lum = self.m;
let mut parity = 0;
for i in 0..N {
let max_row = (i..N)
.max_by(|&a, &b| lum[a][i].abs().partial_cmp(&lum[b][i].abs()).unwrap())
.unwrap_or(i);
if max_row != i {
lum.swap(i, max_row);
parity += 1;
}
// Singular matrix
if lum[i][i] == T::zero() {
return T::zero();
}
// Gaussian elimination
for j in (i + 1)..N {
let factor = lum[j][i] / lum[i][i];
#[allow(clippy::needless_range_loop)]
for k in i..N {
let val_i = lum[i][k];
lum[j][k] = lum[j][k] - factor * val_i;
}
}
}
let det: T = (0..N).map(|i| lum[i][i]).product();
if parity < 0 { -det } else { det }
}
}
}
}
impl<T, const N: usize> Index<usize> for SquareMatrix<T, N> {
type Output = [T; N];
fn index(&self, index: usize) -> &Self::Output {
&self.m[index]
}
}
impl<T, const N: usize> IndexMut<usize> for SquareMatrix<T, N> {
fn index_mut(&mut self, index: usize) -> &mut Self::Output {
&mut self.m[index]
}
}
impl<T, const N: usize> Mul<Vector<T, N>> for SquareMatrix<T, N>
where
T: Copy + Mul<Output = T> + Sum + Default,
{
type Output = Vector<T, N>;
fn mul(self, rhs: Vector<T, N>) -> Self::Output {
let arr = std::array::from_fn(|i| self.m[i].iter().zip(&rhs.0).map(|(m, v)| *m * *v).sum());
Vector(arr)
}
}
impl<T, const N: usize> Mul<Point<T, N>> for SquareMatrix<T, N>
where
T: Copy + Mul<Output = T> + Sum + Default,
{
type Output = Point<T, N>;
fn mul(self, rhs: Point<T, N>) -> Self::Output {
let arr = std::array::from_fn(|i| self.m[i].iter().zip(&rhs.0).map(|(m, v)| *m * *v).sum());
Point(arr)
}
}
#[cfg(test)]
mod tests {
use super::*;
fn assert_matrix_approx_eq<const N: usize>(a: &SquareMatrix<f64, N>, b: &SquareMatrix<f64, N>) {
const EPSILON: f64 = 1e-9;
for i in 0..N {
for j in 0..N {
assert!(
(a[i][j] - b[i][j]).abs() < EPSILON,
"Matrices differ at ({},{}): {} vs {}\nLeft:\n{}\nRight:\n{}",
i,
j,
a[i][j],
b[i][j],
a,
b
);
}
}
}
#[test]
fn test_inverse_2x2() {
let m = SquareMatrix {
m: [[4.0, 7.0], [2.0, 6.0]],
};
let identity = SquareMatrix::identity();
let inv = m.inverse().expect("Matrix should be invertible");
let product = m * inv;
assert_matrix_approx_eq(&product, &identity);
}
#[test]
fn test_inverse_3x3() {
let m = SquareMatrix::new([[1.0, 2.0, 3.0], [0.0, 1.0, 4.0], [5.0, 6.0, 0.0]]);
let identity = SquareMatrix::identity();
let inv = m.inverse().expect("Matrix should be invertible");
let product = m.clone() * inv.clone();
let product_inv = inv.clone() * m.clone();
assert_matrix_approx_eq(&product, &identity);
assert_matrix_approx_eq(&product_inv, &identity);
}
#[test]
fn test_singular_inverse() {
let m = SquareMatrix {
m: [[1.0, 2.0], [2.0, 4.0]],
}; // Determinant is 0
assert!(m.inverse().is_ok());
}
#[test]
fn test_multiplication_2x2() {
let a = SquareMatrix {
m: [[1.0, 2.0], [3.0, 4.0]],
};
let b = SquareMatrix {
m: [[2.0, 0.0], [1.0, 2.0]],
};
let expected = SquareMatrix {
m: [[4.0, 4.0], [10.0, 8.0]],
};
let result = a * b;
assert_matrix_approx_eq(&result, &expected);
}
#[test]
fn test_determinant_3x3() {
let m = SquareMatrix {
m: [[1.0, 2.0, 3.0], [0.0, 1.0, 4.0], [5.0, 6.0, 0.0]],
};
assert_eq!(m.determinant(), 1.0);
}
}
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