deep_time/

utils.rs

use crate::Real;

/// Clamps an `i128` to the representable range of `i64`.
#[inline]
pub(crate) const fn clamp_i128_to_i64(x: i128) -> i64 {
    if x > i64::MAX as i128 {
        i64::MAX
    } else if x < i64::MIN as i128 {
        i64::MIN
    } else {
        x as i64
    }
}

/// Ultra-high-accuracy sine approximation that can be evaluated at compile time.
///
/// This is a private helper function used internally for high-precision
/// astronomical time transformations (primarily TDB-TT and related
/// relativistic corrections). It provides a balance of accuracy, speed,
/// and `const fn` compatibility without requiring any runtime tables or
/// external dependencies.
///
/// # Algorithm
///
/// 1. **Range reduction**  
///    The input angle is reduced to the interval `[-π, π]` using floating-point
///    modulo, followed by a second reduction to `[0, π/2]` by exploiting the
///    identities `sin(-x) = -sin(x)` and `sin(π - x) = sin(x)`.
///
/// 2. **Taylor series (Horner form)**  
///    The sine is computed using the Taylor series for `sin(x)` around zero,
///    truncated after the `x¹⁵` term and evaluated with Horner's method for
///    optimal numerical stability and minimal operations:
///
///    ```text
///    sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + x⁹/9! − x¹¹/11! + x¹³/13! − x¹⁵/15!
///    ```
///
///    All powers of `x²` are accumulated via repeated multiplication by `y = x²`.
///
/// # Accuracy
///
/// - **Maximum absolute error**: ≈ **6.02 × 10⁻¹²** radians over the entire
///   real line (the worst case occurs near odd multiples of `π/2`).
/// - This is more than **100×** better than the previous 7-term version and
///   over **9,000×** better than the original 5-term implementation.
/// - For all practical TDB-TT and astronomical time-scale work the error is
///   completely negligible — it is smaller than the inherent uncertainty of
///   most input ephemerides.
///
/// # Performance & Const-fn Properties
///
/// - Entirely `const fn` compatible (no heap allocation, no panics on valid input).
/// - Uses only multiplication, addition/subtraction, and a single modulo.
/// - Horner evaluation requires only **8 multiplications and 8 additions** after
///   range reduction — still extremely cheap.
/// - No conditional branches inside the polynomial evaluation.
///
/// # Limitations
///
/// - For extremely large arguments (`|x| ≳ 10¹⁴`), the floating-point modulo
///   operation loses precision because the mantissa of `f64` can no longer
///   represent the fractional part of `x / (2π)` accurately. In practice this
///   is irrelevant for all astronomical time arguments encountered in TDB-TT
///   calculations.
/// - The function does not handle `NaN` or `Infinity` specially; they propagate
///   according to IEEE 754 rules.
///
/// # Design Rationale
///
/// The implementation uses a plain Taylor series (instead of a minimax or
/// Chebyshev polynomial) because:
/// - The coefficients are simple exact fractions (`1/n!`).
/// - The series is trivial to extend or truncate.
/// - Horner form already gives near-optimal accuracy for the chosen degree
///   on `[0, π/2]`.
///
/// One more term (`+x¹⁷/17!`) can be added if future requirements ever demand
/// sub-10⁻¹³ accuracy.
///
/// # See Also
///
/// - Previous versions in git history (5-term, 7-term) for regression testing.
/// - `core::f64::consts::PI` and the `f!` macro for literal typing.
pub const fn sin_approx(x: Real) -> Real {
    const PI: Real = f!(core::f64::consts::PI);
    const TWO_PI: Real = f!(2.0) * PI;

    // === Range reduction to [-π, π] ===
    // Uses the mathematical identity sin(x) = sin(x mod 2π).
    // The two-step adjustment guarantees the result lies in [-π, π].
    let mut x = x % TWO_PI;
    if x < f!(0.0) {
        x += TWO_PI;
    }
    if x > PI {
        x -= TWO_PI;
    }

    // === Sign handling and reduction to [0, π/2] ===
    // sin(-x) = -sin(x)  and  sin(π - x) = sin(x)
    let sign = if x < f!(0.0) { f!(-1.0) } else { f!(1.0) };
    let x = x.abs();

    let x = if x > PI / f!(2.0) { PI - x } else { x };

    // === Taylor series via Horner's method (up to x¹⁵ term) ===
    // y = x²
    // p(y) = 1 − y/3! + y²/5! − y³/7! + y⁴/9! − y⁵/11! + y⁶/13! − y⁷/15!
    // sin(x) = x · p(y)
    //
    // Horner form is used for:
    //   • minimal number of operations (8 muls + 8 adds)
    //   • excellent floating-point rounding properties
    //   • trivial extensibility
    let y = x * x;

    // Start with the highest-degree coefficient and work downwards.
    // All denominators are exact factorials and are exactly representable
    // as f64 constants.
    let p = f!(-1.0) / f!(1307674368000.0); // −1/15!
    let p = p * y + f!(1.0) / f!(6227020800.0); // +1/13!
    let p = p * y + f!(-1.0) / f!(39916800.0); // −1/11!
    let p = p * y + f!(1.0) / f!(362880.0); // +1/9!
    let p = p * y + f!(-1.0) / f!(5040.0); // −1/7!
    let p = p * y + f!(1.0) / f!(120.0); // +1/5!
    let p = p * y + f!(-1.0) / f!(6.0); // −1/3!
    let p = p * y + f!(1.0); // +1

    sign * (x * p)
}

/// `const fn` implementation of floor for `Real`.
///
/// This is identical to `std::f64::floor` (including signed-zero
/// preservation) while remaining fully `const fn` on stable Rust with `#![no_std]`.
pub(crate) const fn floor_f(x: Real) -> Real {
    if x.is_nan() || x.is_infinite() {
        x
    } else if x == f!(0.0) {
        x // preserves +0.0 or -0.0
    } else {
        let i = x as i64;
        let truncated = f!(i);
        if x >= f!(0.0) || truncated == x {
            truncated
        } else {
            truncated - f!(1.0)
        }
    }
}

const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
const LG7: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */

/// The natural logarithm of `x` (f64).
pub const fn log(mut x: f64) -> f64 {
    let x1p54 = f64::from_bits(0x4350000000000000); // 0x1p54 === 2 ^ 54

    let mut ui = x.to_bits();
    let mut hx: u32 = (ui >> 32) as u32;
    let mut k: i32 = 0;

    if (hx < 0x00100000) || ((hx >> 31) != 0) {
        /* x < 2**-126  */
        if ui << 1 == 0 {
            return -1. / (x * x); /* log(+-0)=-inf */
        }
        if hx >> 31 != 0 {
            return (x - x) / 0.0; /* log(-#) = NaN */
        }
        /* subnormal number, scale x up */
        k -= 54;
        x *= x1p54;
        ui = x.to_bits();
        hx = (ui >> 32) as u32;
    } else if hx >= 0x7ff00000 {
        return x;
    } else if hx == 0x3ff00000 && ui << 32 == 0 {
        return 0.;
    }

    /* reduce x into [sqrt(2)/2, sqrt(2)] */
    hx += 0x3ff00000 - 0x3fe6a09e;
    k += ((hx >> 20) as i32) - 0x3ff;
    hx = (hx & 0x000fffff) + 0x3fe6a09e;
    ui = ((hx as u64) << 32) | (ui & 0xffffffff);
    x = f64::from_bits(ui);

    let f: f64 = x - 1.0;
    let hfsq: f64 = 0.5 * f * f;
    let s: f64 = f / (2.0 + f);
    let z: f64 = s * s;
    let w: f64 = z * z;
    let t1: f64 = w * (LG2 + w * (LG4 + w * LG6));
    let t2: f64 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
    let r: f64 = t2 + t1;
    let dk: f64 = k as f64;
    s * (hfsq + r) + dk * LN2_LO - hfsq + f + dk * LN2_HI
}

// musl-style Table-driven Goldschmidt sqrt for f64
// Translated from musl/src/math/sqrt.c and sqrt_data.c
// Provides correctly rounded sqrt(x) matching IEEE 754 / libm quality

const RSQRT_TAB: [u16; 128] = [
    0xb451, 0xb2f0, 0xb196, 0xb044, 0xaef9, 0xadb6, 0xac79, 0xab43, 0xaa14, 0xa8eb, 0xa7c8, 0xa6aa,
    0xa592, 0xa480, 0xa373, 0xa26b, 0xa168, 0xa06a, 0x9f70, 0x9e7b, 0x9d8a, 0x9c9d, 0x9bb5, 0x9ad1,
    0x99f0, 0x9913, 0x983a, 0x9765, 0x9693, 0x95c4, 0x94f8, 0x9430, 0x936b, 0x92a9, 0x91ea, 0x912e,
    0x9075, 0x8fbe, 0x8f0a, 0x8e59, 0x8daa, 0x8cfe, 0x8c54, 0x8bac, 0x8b07, 0x8a64, 0x89c4, 0x8925,
    0x8889, 0x87ee, 0x8756, 0x86c0, 0x862b, 0x8599, 0x8508, 0x8479, 0x83ec, 0x8361, 0x82d8, 0x8250,
    0x81c9, 0x8145, 0x80c2, 0x8040, 0xff02, 0xfd0e, 0xfb25, 0xf947, 0xf773, 0xf5aa, 0xf3ea, 0xf234,
    0xf087, 0xeee3, 0xed47, 0xebb3, 0xea27, 0xe8a3, 0xe727, 0xe5b2, 0xe443, 0xe2dc, 0xe17a, 0xe020,
    0xdecb, 0xdd7d, 0xdc34, 0xdaf1, 0xd9b3, 0xd87b, 0xd748, 0xd61a, 0xd4f1, 0xd3cd, 0xd2ad, 0xd192,
    0xd07b, 0xcf69, 0xce5b, 0xcd51, 0xcc4a, 0xcb48, 0xca4a, 0xc94f, 0xc858, 0xc764, 0xc674, 0xc587,
    0xc49d, 0xc3b7, 0xc2d4, 0xc1f4, 0xc116, 0xc03c, 0xbf65, 0xbe90, 0xbdbe, 0xbcef, 0xbc23, 0xbb59,
    0xba91, 0xb9cc, 0xb90a, 0xb84a, 0xb78c, 0xb6d0, 0xb617, 0xb560,
];

#[inline]
const fn mul32(a: u32, b: u32) -> u32 {
    ((a as u64).wrapping_mul(b as u64) >> 32) as u32
}

#[inline]
const fn mul64(a: u64, b: u64) -> u64 {
    let ahi = a >> 32;
    let alo = a & 0xffffffff;
    let bhi = b >> 32;
    let blo = b & 0xffffffff;
    ahi.wrapping_mul(bhi)
        .wrapping_add(ahi.wrapping_mul(blo) >> 32)
        .wrapping_add(alo.wrapping_mul(bhi) >> 32)
}

/// Computes sqrt(x) using the table-driven Goldschmidt iteration
/// from musl libc. Correctly rounded to nearest-even for all f64 inputs.
/// const, no std, no alloc friendly.
pub const fn sqrt(x: f64) -> f64 {
    let mut ix = x.to_bits();
    let mut top = ix >> 52;

    // Special cases: subnormal, inf, nan, negative, zero
    if top.wrapping_sub(0x001) >= 0x7fe {
        if ix << 1 == 0 {
            return x; // ±0.0
        }
        if ix == 0x7ff0_0000_0000_0000 {
            return x; // +inf
        }
        if ix > 0x7ff0_0000_0000_0000 {
            // negative or NaN → quiet NaN, preserve sign bit for -inf/-num
            let nan_bits = 0x7ff8_0000_0000_0000 | (ix & 0x8000_0000_0000_0000);
            return f64::from_bits(nan_bits);
        }
        // Subnormal: normalize by multiplying by 2^52
        let scale = f64::from_bits(0x4330_0000_0000_0000); // 2^52
        ix = (x * scale).to_bits();
        top = (ix >> 52).wrapping_sub(52);
    }

    let even = top & 1;
    let mut m = (ix << 11) | 0x8000_0000_0000_0000u64;
    if even != 0 {
        m >>= 1;
    }
    let top = (top.wrapping_add(0x3ff)) >> 1; // result exponent (biased)

    // Table-driven initial reciprocal sqrt estimate + Goldschmidt iterations
    // All vars u64 to match C closely; mul32/mul64 return u64 for simplicity
    let three: u64 = 0xc000_0000;
    let i = ((ix >> 46) % 128) as usize;
    let mut r: u64 = (RSQRT_TAB[i] as u64) << 16;

    let mut s: u64 = mul32((m >> 32) as u32, r as u32) as u64;
    let mut d: u64 = mul32(s as u32, r as u32) as u64;
    let mut u: u64 = three - d;
    r = (mul32(r as u32, u as u32) << 1) as u64;
    s = (mul32(s as u32, u as u32) << 1) as u64;

    d = mul32(s as u32, r as u32) as u64;
    u = three - d;
    r = (mul32(r as u32, u as u32) << 1) as u64;

    r <<= 32;
    s = mul64(m, r);
    d = mul64(s, r);
    u = (three << 32) - d;
    s = mul64(s, u);

    // Final adjustment and rounding decision
    s = (s - 2) >> 9;

    let d0 = (m << 42).wrapping_sub(s.wrapping_mul(s));
    let d1 = s.wrapping_sub(d0);
    let _d2 = d1.wrapping_add(s).wrapping_add(1);

    if (d1 >> 63) != 0 {
        s = s.wrapping_add(1);
    }
    s &= 0x000f_ffff_ffff_ffff;
    s |= (top as u64) << 52;

    f64::from_bits(s)
}

const SPLIT: f64 = 134217728. + 1.; // 0x1p27 + 1 === (2 ^ 27) + 1

const fn sq(x: f64) -> (f64, f64) {
    let xh: f64;
    let xl: f64;
    let xc: f64;

    xc = x * SPLIT;
    xh = x - xc + xc;
    xl = x - xh;
    let hi = x * x;
    let lo = xh * xh - hi + 2. * xh * xl + xl * xl;
    (hi, lo)
}

pub const fn hypot(mut x: f64, mut y: f64) -> f64 {
    let x1p700 = f64::from_bits(0x6bb0000000000000); // 0x1p700 === 2 ^ 700
    let x1p_700 = f64::from_bits(0x1430000000000000); // 0x1p-700 === 2 ^ -700

    let mut uxi = x.to_bits();
    let mut uyi = y.to_bits();
    let uti;
    let ex: i64;
    let ey: i64;
    let mut z: f64;

    /* arrange |x| >= |y| */
    uxi &= -1i64 as u64 >> 1;
    uyi &= -1i64 as u64 >> 1;
    if uxi < uyi {
        uti = uxi;
        uxi = uyi;
        uyi = uti;
    }

    /* special cases */
    ex = (uxi >> 52) as i64;
    ey = (uyi >> 52) as i64;
    x = f64::from_bits(uxi);
    y = f64::from_bits(uyi);
    /* note: hypot(inf,nan) == inf */
    if ey == 0x7ff {
        return y;
    }
    if ex == 0x7ff || uyi == 0 {
        return x;
    }
    /* note: hypot(x,y) ~= x + y*y/x/2 with inexact for small y/x */
    /* 64 difference is enough for ld80 double_t */
    if ex - ey > 64 {
        return x + y;
    }

    /* precise sqrt argument in nearest rounding mode without overflow */
    /* xh*xh must not overflow and xl*xl must not underflow in sq */
    z = 1.;
    if ex > 0x3ff + 510 {
        z = x1p700;
        x *= x1p_700;
        y *= x1p_700;
    } else if ey < 0x3ff - 450 {
        z = x1p_700;
        x *= x1p700;
        y *= x1p700;
    }
    let (hx, lx) = sq(x);
    let (hy, ly) = sq(y);
    z * sqrt(ly + lx + hy + hx)
}

#[cfg(feature = "std")]
#[cfg(test)]
mod tests {
    use super::sqrt;
    use std::{eprintln, f64, vec, vec::Vec};

    #[test]
    fn test_special_cases() {
        assert_eq!(sqrt(0.0), 0.0);
        assert_eq!(sqrt(-0.0), -0.0);
        assert!(sqrt(f64::INFINITY).is_infinite() && sqrt(f64::INFINITY) > 0.0);
        assert!(sqrt(f64::NEG_INFINITY).is_nan());
        assert!(sqrt(-1.0).is_nan());
        assert!(sqrt(f64::NAN).is_nan());
        // signaling nan? but in practice quiet
    }

    #[test]
    fn test_perfect_squares() {
        for i in 0..100u32 {
            let x = (i * i) as f64;
            let r = sqrt(x);
            assert!((r - i as f64).abs() < 1e-10 || r.is_nan());
        }
    }

    #[test]
    fn test_random_vs_std() {
        // 5M deterministic LCG random normals in [1,2) — exercises table + Goldschmidt fully
        let mut failures = 0u32;
        let mut state: u64 = 0x123456789abcdef0;
        for _ in 0..5_000_000 {
            state = state.wrapping_mul(6364136223846793005).wrapping_add(1);
            let bits = (state & 0x000f_ffff_ffff_ffff) | 0x3ff0_0000_0000_0000; // positive normal [1,2)
            let val = f64::from_bits(bits);
            let r1 = sqrt(val);
            let r2 = val.sqrt();
            if r1.to_bits() != r2.to_bits() {
                failures += 1;
                if failures < 3 {
                    eprintln!(
                        "Mismatch at {:016x}: ours={:016x} std={:016x}",
                        bits,
                        r1.to_bits(),
                        r2.to_bits()
                    );
                }
            }
        }
        assert_eq!(
            failures, 0,
            "Found {} mismatches in 5M random normals [1,2)",
            failures
        );
    }

    #[test]
    fn test_subnormals_random() {
        // 100k random subnormals (exp=0) — critical for normalize path
        let mut failures = 0u32;
        let mut state: u64 = 0xdeadbeefcafebabe;
        for _ in 0..100_000 {
            state = state.wrapping_mul(6364136223846793005).wrapping_add(1);
            // subnormal: exp=0, random mantissa (low 52 bits)
            let bits = state & 0x000f_ffff_ffff_ffff; // clears sign + exp
            let val = f64::from_bits(bits);
            if val == 0.0 {
                continue;
            } // skip zero
            let r1 = sqrt(val);
            let r2 = val.sqrt();
            if r1.to_bits() != r2.to_bits() {
                failures += 1;
                if failures < 3 {
                    eprintln!(
                        "Subnormal mismatch at {:016x}: ours={:016x} std={:016x}",
                        bits,
                        r1.to_bits(),
                        r2.to_bits()
                    );
                }
            }
        }
        assert_eq!(
            failures, 0,
            "Found {} mismatches in 100k random subnormals",
            failures
        );
    }

    #[test]
    fn test_boundaries() {
        // Critical boundaries: min/max normal, subnormal boundary, overflow edge, powers of 2
        let boundaries: [f64; 8] = [
            f64::MIN_POSITIVE,                         // 2^-1022 (smallest normal)
            f64::from_bits(0x0010_0000_0000_0000),     // 2^-1021
            f64::from_bits(0x000f_ffff_ffff_ffff),     // largest subnormal
            2.0f64.powi(-1074),                        // smallest positive subnormal (2^-1074)
            f64::MAX,                                  // ~1.8e308
            f64::from_bits(0x7fe0_0000_0000_0000),     // largest finite < inf
            2.0f64.powi(1023),                         // 2^1023 (largest power of 2)
            2.0f64.powi(-1022) * (1.0 + f64::EPSILON), // just above min normal
        ];
        for &x in &boundaries {
            let r1 = sqrt(x);
            let r2 = x.sqrt();
            assert_eq!(r1.to_bits(), r2.to_bits(), "Boundary mismatch for {:e}", x);
            // Also check sqrt(x*x) ~ |x| for positive x (within rounding), but skip underflow cases
            if x > 0.0 && x.is_finite() && x > 1e-200 {
                let xx = x * x;
                if xx.is_finite() && xx.is_normal() {
                    let r = sqrt(xx);
                    let rel = ((r - x).abs() / x).max(0.0);
                    assert!(
                        rel < 1e-14 || r.is_nan(),
                        "sqrt(x*x) not close to x for {}",
                        x
                    );
                }
            }
        }
    }

    #[test]
    fn test_known_hard_cases() {
        // Known hard-to-round / exact / boundary cases — all must match std bit-exactly
        let cases: &[f64] = &[
            2.0,
            0.5,
            4.0,
            9.0,
            0.0,
            f64::INFINITY,
            1.0e-300,                              // very small normal
            f64::from_bits(0x0010_0000_0000_0001), // just above min normal
            1.0 + f64::EPSILON,                    // next after 1.0
            f64::from_bits(0x7fefffffffffffff),    // largest finite
        ];
        for &x in cases {
            let r = sqrt(x);
            // bit-exact check vs Rust std (the gold standard for this platform)
            assert_eq!(r.to_bits(), x.sqrt().to_bits(), "Bit mismatch for {:e}", x);
        }
    }

    // Manual nextUp / nextDown (unstable in this Rust version)
    fn next_up(x: f64) -> f64 {
        if x.is_nan() || x == f64::INFINITY {
            return x;
        }
        if x == 0.0 {
            return f64::from_bits(1);
        }
        let bits = x.to_bits();
        if x > 0.0 {
            f64::from_bits(bits + 1)
        } else {
            f64::from_bits(bits - 1)
        }
    }
    fn next_down(x: f64) -> f64 {
        if x.is_nan() || x == f64::NEG_INFINITY {
            return x;
        }
        if x == -0.0 || x == 0.0 {
            return f64::from_bits(0x8000_0000_0000_0001);
        }
        let bits = x.to_bits();
        if x > 0.0 {
            f64::from_bits(bits - 1)
        } else {
            f64::from_bits(bits + 1)
        }
    }

    #[test]
    fn test_powers_of_two() {
        // All representable powers of 2 (even exponents must be exact, odd use std)
        for exp in -1074i32..=1023 {
            let x = if exp >= -1022 {
                2.0f64.powi(exp)
            } else {
                // subnormal 2^exp = 2^(exp + 1074) * 2^-1074
                f64::from_bits(1u64 << (exp + 1074))
            };
            if !x.is_finite() || x == 0.0 {
                continue;
            }
            let r1 = sqrt(x);
            let r2 = x.sqrt();
            assert_eq!(
                r1.to_bits(),
                r2.to_bits(),
                "Power-of-2 mismatch for 2^{}",
                exp
            );
            // For even exponents, result should be exactly 2^(exp/2) when representable
            if exp % 2 == 0 {
                let expected_exp = exp / 2;
                if expected_exp >= -1022 {
                    let expected = 2.0f64.powi(expected_exp);
                    assert_eq!(
                        r1.to_bits(),
                        expected.to_bits(),
                        "Even power-of-2 not exact for 2^{}",
                        exp
                    );
                }
            }
        }
    }

    #[test]
    fn test_nextafter_edges() {
        // nextUp / nextDown around critical points (0, 1, min_normal, max)
        let mut edges: Vec<f64> = vec![
            f64::from_bits(1),                     // smallest positive subnormal
            f64::from_bits(0x0000_0000_0000_0002), // next subnormal
            next_down(f64::MIN_POSITIVE),          // largest subnormal
            f64::MIN_POSITIVE,                     // smallest normal
            next_up(f64::MIN_POSITIVE),
            next_down(1.0),
            1.0,
            next_up(1.0),
            next_down(f64::MAX),
            f64::MAX,
        ];
        // Also a few negative edges (should all produce NaN)
        edges.push(next_up(-f64::MIN_POSITIVE)); // negative smallest normal-ish
        for &x in &edges {
            let r1 = sqrt(x);
            let r2 = x.sqrt();
            assert_eq!(
                r1.to_bits(),
                r2.to_bits(),
                "nextafter edge mismatch for {:e} (bits {:016x})",
                x,
                x.to_bits()
            );
        }
    }

    #[test]
    fn test_negative_subnormals() {
        // All negative subnormals must produce NaN (sign bit set in result)
        let mut state: u64 = 0xfeedface_deadbeef;
        for _ in 0..10_000 {
            state = state.wrapping_mul(6364136223846793005).wrapping_add(1);
            let bits = (state & 0x000f_ffff_ffff_ffff) | 0x8000_0000_0000_0000; // negative subnormal
            let val = f64::from_bits(bits);
            if val == 0.0 {
                continue;
            }
            let r = sqrt(val);
            assert!(
                r.is_nan(),
                "Negative subnormal did not produce NaN: {:e}",
                val
            );
            // sign bit should be set (negative NaN)
            assert!(
                r.to_bits() & 0x8000_0000_0000_0000 != 0,
                "NaN sign bit not set for negative subnormal"
            );
        }
    }
}