ruint/algorithms/div/
reciprocal.rs

//! Reciprocals and division using reciprocals
//! See [MG10].
//!
//! [MG10]: https://gmplib.org/~tege/division-paper.pdf
//! [GM94]: https://gmplib.org/~tege/divcnst-pldi94.pdf
//! [new]: https://gmplib.org/list-archives/gmp-devel/2019-October/005590.html
#![allow(dead_code, clippy::cast_possible_truncation, clippy::cast_lossless)]

use core::num::Wrapping;

pub use self::{reciprocal_2_mg10 as reciprocal_2, reciprocal_mg10 as reciprocal};

/// ⚠️ Computes $\floor{\frac{2^{128} - 1}{\mathtt{d}}} - 2^{64}$.
///
/// Requires $\mathtt{d} ≥ 2^{127}$, i.e. the highest bit of $\mathtt{d}$ must
/// be set.
#[inline(always)]
#[must_use]
pub fn reciprocal_ref(d: u64) -> u64 {
    debug_assert!(d >= (1 << 63));
    let r = u128::MAX / u128::from(d);
    debug_assert!(r >= (1 << 64));
    debug_assert!(r < (1 << 65));
    r as u64
}

/// ⚠️ Computes $\floor{\frac{2^{128} - 1}{\mathsf{d}}} - 2^{64}$.
///
/// Requires $\mathsf{d} ∈ [2^{63}, 2^{64})$, i.e. the highest bit of
/// $\mathsf{d}$ must be set.
///
/// Using [MG10] algorithm 3. See also the [intx] implementation. Here is a
/// direct translation of the algorithm to Python for reference:
///
/// ```python
/// d0 = d % 2
/// d9 = d // 2**55
/// d40 = d // 2**24 + 1
/// d63 = (d + 1) // 2
/// v0 = (2**19 - 3 * 2**8) // d9
/// v1 = 2**11 * v0 - v0**2 * d40 // 2**40 - 1
/// v2 = 2**13 * v1 + v1 * (2**60 - v1 * d40) // 2**47
/// e = 2**96 - v2 * d63 + (v2 // 2) * d0
/// v3 = (2**31 * v2 +v2 * e // 2**65) % 2**64
/// v4 = (v3 - (v3 + 2**64 + 1) * d // 2**64) % 2**64
/// ```
///
/// [MG10]: https://gmplib.org/~tege/division-paper.pdf
/// [intx]: https://github.com/chfast/intx/blob/8b5f4748a7386a9530769893dae26b3273e0ffe2/include/intx/intx.hpp#L683
#[inline]
#[must_use]
pub fn reciprocal_mg10(d: u64) -> u64 {
    const ZERO: Wrapping<u64> = Wrapping(0);
    const ONE: Wrapping<u64> = Wrapping(1);

    // Lookup table for $\floor{\frac{2^{19} -3 ⋅ 2^8}{d_9 - 256}}$
    static TABLE: [u16; 256] = [
        2045, 2037, 2029, 2021, 2013, 2005, 1998, 1990, 1983, 1975, 1968, 1960, 1953, 1946, 1938,
        1931, 1924, 1917, 1910, 1903, 1896, 1889, 1883, 1876, 1869, 1863, 1856, 1849, 1843, 1836,
        1830, 1824, 1817, 1811, 1805, 1799, 1792, 1786, 1780, 1774, 1768, 1762, 1756, 1750, 1745,
        1739, 1733, 1727, 1722, 1716, 1710, 1705, 1699, 1694, 1688, 1683, 1677, 1672, 1667, 1661,
        1656, 1651, 1646, 1641, 1636, 1630, 1625, 1620, 1615, 1610, 1605, 1600, 1596, 1591, 1586,
        1581, 1576, 1572, 1567, 1562, 1558, 1553, 1548, 1544, 1539, 1535, 1530, 1526, 1521, 1517,
        1513, 1508, 1504, 1500, 1495, 1491, 1487, 1483, 1478, 1474, 1470, 1466, 1462, 1458, 1454,
        1450, 1446, 1442, 1438, 1434, 1430, 1426, 1422, 1418, 1414, 1411, 1407, 1403, 1399, 1396,
        1392, 1388, 1384, 1381, 1377, 1374, 1370, 1366, 1363, 1359, 1356, 1352, 1349, 1345, 1342,
        1338, 1335, 1332, 1328, 1325, 1322, 1318, 1315, 1312, 1308, 1305, 1302, 1299, 1295, 1292,
        1289, 1286, 1283, 1280, 1276, 1273, 1270, 1267, 1264, 1261, 1258, 1255, 1252, 1249, 1246,
        1243, 1240, 1237, 1234, 1231, 1228, 1226, 1223, 1220, 1217, 1214, 1211, 1209, 1206, 1203,
        1200, 1197, 1195, 1192, 1189, 1187, 1184, 1181, 1179, 1176, 1173, 1171, 1168, 1165, 1163,
        1160, 1158, 1155, 1153, 1150, 1148, 1145, 1143, 1140, 1138, 1135, 1133, 1130, 1128, 1125,
        1123, 1121, 1118, 1116, 1113, 1111, 1109, 1106, 1104, 1102, 1099, 1097, 1095, 1092, 1090,
        1088, 1086, 1083, 1081, 1079, 1077, 1074, 1072, 1070, 1068, 1066, 1064, 1061, 1059, 1057,
        1055, 1053, 1051, 1049, 1047, 1044, 1042, 1040, 1038, 1036, 1034, 1032, 1030, 1028, 1026,
        1024,
    ];

    debug_assert!(d >= (1 << 63));
    let d = Wrapping(d);

    let d0 = d & ONE;
    let d9 = d >> 55;
    let d40 = ONE + (d >> 24);
    let d63 = (d + ONE) >> 1;
    // let v0 = Wrapping(TABLE[(d9.0 - 256) as usize] as u64);
    let v0 = Wrapping(*unsafe { TABLE.get_unchecked((d9.0 - 256) as usize) } as u64);
    let v1 = (v0 << 11) - ((v0 * v0 * d40) >> 40) - ONE;
    let v2 = (v1 << 13) + ((v1 * ((ONE << 60) - v1 * d40)) >> 47);
    let e = ((v2 >> 1) & (ZERO - d0)) - v2 * d63;
    let v3 = (mul_hi(v2, e) >> 1) + (v2 << 31);
    let v4 = v3 - muladd_hi(v3, d, d) - d;

    v4.0
}

/// ⚠️ Computes $\floor{\frac{2^{192} - 1}{\mathsf{d}}} - 2^{64}$.
///
/// Requires $\mathsf{d} ∈ [2^{127}, 2^{128})$, i.e. the most significant bit
/// of $\mathsf{d}$ must be set.
///
/// Implements [MG10] algorithm 6.
///
/// [MG10]: https://gmplib.org/~tege/division-paper.pdf
#[inline]
#[must_use]
pub fn reciprocal_2_mg10(d: u128) -> u64 {
    debug_assert!(d >= (1 << 127));
    let d1 = (d >> 64) as u64;
    let d0 = d as u64;

    let mut v = reciprocal(d1);
    let mut p = d1.wrapping_mul(v).wrapping_add(d0);
    // OPT: This is checking the carry flag
    if p < d0 {
        v = v.wrapping_sub(1);
        if p >= d1 {
            v = v.wrapping_sub(1);
            p = p.wrapping_sub(d1);
        }
        p = p.wrapping_sub(d1);
    }
    let t = u128::from(v) * u128::from(d0);
    let t1 = (t >> 64) as u64;
    let t0 = t as u64;

    let p = p.wrapping_add(t1);
    // OPT: This is checking the carry flag
    if p < t1 {
        v = v.wrapping_sub(1);
        if (u128::from(p) << 64) | u128::from(t0) >= d {
            v = v.wrapping_sub(1);
        }
    }
    v
}

#[allow(clippy::missing_const_for_fn)] // False positive
#[inline]
#[must_use]
fn mul_hi(a: Wrapping<u64>, b: Wrapping<u64>) -> Wrapping<u64> {
    let a = u128::from(a.0);
    let b = u128::from(b.0);
    let r = a * b;
    Wrapping((r >> 64) as u64)
}

#[allow(clippy::missing_const_for_fn)] // False positive
#[inline]
#[must_use]
fn muladd_hi(a: Wrapping<u64>, b: Wrapping<u64>, c: Wrapping<u64>) -> Wrapping<u64> {
    let a = u128::from(a.0);
    let b = u128::from(b.0);
    let c = u128::from(c.0);
    let r = a * b + c;
    Wrapping((r >> 64) as u64)
}

#[cfg(test)]
mod tests {
    use super::*;
    use proptest::proptest;

    #[test]
    fn test_reciprocal() {
        proptest!(|(n: u64)| {
            let n = n | (1 << 63);
            let expected = reciprocal_ref(n);
            let actual = reciprocal_mg10(n);
            assert_eq!(expected, actual);
        });
    }

    #[test]
    fn test_reciprocal_2() {
        assert_eq!(reciprocal_2_mg10(1 << 127), u64::MAX);
        assert_eq!(reciprocal_2_mg10(u128::MAX), 0);
        assert_eq!(
            reciprocal_2_mg10(0xd555_5555_5555_5555_5555_5555_5555_5555),
            0x3333_3333_3333_3333
        );
        assert_eq!(
            reciprocal_2_mg10(0xd0e7_57b0_2171_5fbe_cba4_ad0e_825a_e500),
            0x39b6_c5af_970f_86b3
        );
        assert_eq!(
            reciprocal_2_mg10(0xae5d_6551_8a51_3208_a850_5491_9637_eb17),
            0x77db_09d1_5c3b_970b
        );
    }
}
ruint/algorithms/div/reciprocal.rs

ruint/algorithms/div/
reciprocal.rs
