pxfm 0.1.29

Fast and accurate math
Documentation 
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/*
 * // Copyright (c) Radzivon Bartoshyk 8/2025. All rights reserved.
 * //
 * // Redistribution and use in source and binary forms, with or without modification,
 * // are permitted provided that the following conditions are met:
 * //
 * // 1.  Redistributions of source code must retain the above copyright notice, this
 * // list of conditions and the following disclaimer.
 * //
 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
 * // this list of conditions and the following disclaimer in the documentation
 * // and/or other materials provided with the distribution.
 * //
 * // 3.  Neither the name of the copyright holder nor the names of its
 * // contributors may be used to endorse or promote products derived from
 * // this software without specific prior written permission.
 * //
 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */
use crate::double_double::DoubleDouble;
use crate::dyadic_float::{DyadicFloat128, DyadicSign};
use crate::polyeval::f_polyeval9;

/**
Note expansion generation below: this is negative series expressed in Sage as positive,
so before any real evaluation `x=1/x` should be applied

Generated by SageMath:
```python
def binomial_like(n, m):
    prod = QQ(1)
    z = QQ(4)*(n**2)
    for k in range(1,m + 1):
        prod *= (z - (2*k - 1)**2)
    return prod / (QQ(2)**(2*m) * (ZZ(m).factorial()))

R = LaurentSeriesRing(RealField(300), 'x',default_prec=300)
x = R.gen()

def Pn_asymptotic(n, y, terms=10):
    # now y = 1/x
    return sum( (-1)**m * binomial_like(n, 2*m) / (QQ(2)**(2*m)) * y**(QQ(2)*m) for m in range(terms) )

def Qn_asymptotic(n, y, terms=10):
    return sum( (-1)**m * binomial_like(n, 2*m + 1) / (QQ(2)**(2*m + 1)) * y**(QQ(2)*m + 1) for m in range(terms) )

P = Pn_asymptotic(1, x, 50)
Q = Qn_asymptotic(1, x, 50)

def sqrt_series(s):
    val = S.valuation()
    lc = S[val]  # Leading coefficient
    b = lc.sqrt() * x**(val // 2)

    for _ in range(5):
        b = (b + S / b) / 2
        b = b
    return b

S = (P**2 + Q**2).truncate(50)

b_series = sqrt_series(S).truncate(30)
# see the beta series
print(b_series)
```

See notes/bessel_asympt.ipynb for generation
**/
#[inline]
pub(crate) fn bessel_1_asympt_beta_fast(recip: DoubleDouble) -> DoubleDouble {
    const C: [u64; 10] = [
        0x3ff0000000000000,
        0x3fc8000000000000,
        0xbfc8c00000000000,
        0x3fe9c50000000000,
        0xc01ef5b680000000,
        0x40609860dd400000,
        0xc0abae9b7a06e000,
        0x41008711d41c1428,
        0xc15ab70164c8be6e,
        0x41bc1055e24f297f,
    ];

    // Doing (1/x)*(1/x) instead (1/(x*x)) to avoid spurious overflow/underflow
    let x2 = DoubleDouble::quick_mult(recip, recip);

    let p = f_polyeval9(
        x2.hi,
        f64::from_bits(C[1]),
        f64::from_bits(C[2]),
        f64::from_bits(C[3]),
        f64::from_bits(C[4]),
        f64::from_bits(C[5]),
        f64::from_bits(C[6]),
        f64::from_bits(C[7]),
        f64::from_bits(C[8]),
        f64::from_bits(C[9]),
    );

    DoubleDouble::mul_f64_add_f64(x2, p, f64::from_bits(C[0]))
}

/**
Note expansion generation below: this is negative series expressed in Sage as positive,
so before any real evaluation `x=1/x` should be applied

Generated by SageMath:
```python
def binomial_like(n, m):
    prod = QQ(1)
    z = QQ(4)*(n**2)
    for k in range(1,m + 1):
        prod *= (z - (2*k - 1)**2)
    return prod / (QQ(2)**(2*m) * (ZZ(m).factorial()))

R = LaurentSeriesRing(RealField(300), 'x',default_prec=300)
x = R.gen()

def Pn_asymptotic(n, y, terms=10):
    # now y = 1/x
    return sum( (-1)**m * binomial_like(n, 2*m) / (QQ(2)**(2*m)) * y**(QQ(2)*m) for m in range(terms) )

def Qn_asymptotic(n, y, terms=10):
    return sum( (-1)**m * binomial_like(n, 2*m + 1) / (QQ(2)**(2*m + 1)) * y**(QQ(2)*m + 1) for m in range(terms) )

P = Pn_asymptotic(1, x, 50)
Q = Qn_asymptotic(1, x, 50)

def sqrt_series(s):
    val = S.valuation()
    lc = S[val]  # Leading coefficient
    b = lc.sqrt() * x**(val // 2)

    for _ in range(5):
        b = (b + S / b) / 2
        b = b
    return b

S = (P**2 + Q**2).truncate(50)

b_series = sqrt_series(S).truncate(30)
# see the beta series
print(b_series)
```

See notes/bessel_asympt.ipynb for generation
**/
#[inline]
pub(crate) fn bessel_1_asympt_beta(recip: DoubleDouble) -> DoubleDouble {
    const C: [(u64, u64); 10] = [
        (0x0000000000000000, 0x3ff0000000000000), // 1
        (0x0000000000000000, 0x3fc8000000000000), // 2
        (0x0000000000000000, 0xbfc8c00000000000), // 3
        (0x0000000000000000, 0x3fe9c50000000000), // 4
        (0x0000000000000000, 0xc01ef5b680000000), // 5
        (0x0000000000000000, 0x40609860dd400000), // 6
        (0x0000000000000000, 0xc0abae9b7a06e000), // 7
        (0x0000000000000000, 0x41008711d41c1428), // 8
        (0xbdf7a00000000000, 0xc15ab70164c8be6e),
        (0xbe40e1f000000000, 0x41bc1055e24f297f),
    ];

    // Doing (1/x)*(1/x) instead (1/(x*x)) to avoid spurious overflow/underflow
    let x2 = DoubleDouble::quick_mult(recip, recip);

    let mut p = DoubleDouble::mul_add(
        x2,
        DoubleDouble::from_bit_pair(C[9]),
        DoubleDouble::from_bit_pair(C[8]),
    );

    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[7].1)); // 8
    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[6].1)); // 7
    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[5].1)); // 6
    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[4].1)); // 5
    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[3].1)); // 4
    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[2].1)); // 3
    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[1].1)); // 2
    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[0].1)); // 1
    p
}

/// see [bessel_1_asympt_beta] for more info
pub(crate) fn bessel_1_asympt_beta_hard(recip: DyadicFloat128) -> DyadicFloat128 {
    static C: [DyadicFloat128; 12] = [
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -127,
            mantissa: 0x80000000_00000000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -130,
            mantissa: 0xc0000000_00000000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Neg,
            exponent: -130,
            mantissa: 0xc6000000_00000000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -128,
            mantissa: 0xce280000_00000000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Neg,
            exponent: -125,
            mantissa: 0xf7adb400_00000000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -120,
            mantissa: 0x84c306ea_00000000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Neg,
            exponent: -116,
            mantissa: 0xdd74dbd0_37000000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -110,
            mantissa: 0x84388ea0_e0a14000_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Neg,
            exponent: -105,
            mantissa: 0xd5b80b26_45f372f4_00000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -99,
            mantissa: 0xe082af12_794bf6f1_e1000000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Neg,
            exponent: -92,
            mantissa: 0x94a06149_f30146bc_fe8ed000_00000000_u128,
        },
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -86,
            mantissa: 0xf212edfc_42a62526_4fac2b0c_00000000_u128,
        },
    ];

    let x2 = recip * recip;

    let mut p = C[11];
    for i in (0..11).rev() {
        p = x2 * p + C[i];
    }
    p
}