/*

 * // Copyright (c) Radzivon Bartoshyk 7/2025. 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.

 */

#![allow(clippy::excessive_precision)]

use crate::bessel::alpha1::{

    bessel_1_asympt_alpha, bessel_1_asympt_alpha_fast, bessel_1_asympt_alpha_hard,

};

use crate::bessel::beta1::{

    bessel_1_asympt_beta, bessel_1_asympt_beta_fast, bessel_1_asympt_beta_hard,

};

use crate::bessel::i0::bessel_rsqrt_hard;

use crate::bessel::j1_coeffs::{J1_COEFFS, J1_ZEROS, J1_ZEROS_VALUE};

use crate::bessel::j1_coeffs_taylor::J1_COEFFS_TAYLOR;

use crate::common::f_fmla;

use crate::double_double::DoubleDouble;

use crate::dyadic_float::{DyadicFloat128, DyadicSign};

use crate::polyeval::{f_polyeval8, f_polyeval9, f_polyeval12, f_polyeval19};

use crate::rounding::CpuCeil;

use crate::sin_helper::{sin_dd_small, sin_dd_small_fast, sin_f128_small};

use crate::sincos_reduce::{AngleReduced, rem2pi_any, rem2pi_f128};

/// Bessel of the first kind of order 1

///

/// Note about accuracy:

/// - Close to zero Bessel have tiny values such that testing against MPFR must be done exactly

///   in the same precision, since any nearest representable number have ULP > 0.5,

///   for example `J1(0.000000000000000000000000000000000000023509886)` in single precision

///   have 0.7 ULP for any number with extended precision that would be represented in f32

///   Same applies to J1(4.4501477170144018E-309) in double precision and some others subnormal numbers

pub fn f_j1(x: f64) -> f64 {

    let ux = x.to_bits().wrapping_shl(1);

    if ux >= 0x7ffu64 << 53 || ux <= 0x7960000000000000u64 {

        // |x| <= f64::EPSILON, |x| == inf, x == NaN

        if ux <= 0x72338c9356bb0314u64 {

            // |x| <= 0.000000000000000000000000000000001241

            // J1(x) ~ x/2+O[x]^3

            return x * 0.5;

        }

        if ux <= 0x7960000000000000u64 {

            // |x| <= f64::EPSILON

            // J1(x) ~ x/2-x^3/16+O[x]^5

            let quad_part_x = x * 0.125; // exact. x / 8

            return f_fmla(quad_part_x, -quad_part_x, 0.5) * x;

        }

        if x.is_infinite() {

            return 0.;

        }

        return x + f64::NAN; // x == NaN

    }

    let ax: u64 = x.to_bits() & 0x7fff_ffff_ffff_ffff;

    if ax < 0x4052a6784230fcf8u64 {

        // |x| < 74.60109

        if ax < 0x3feccccccccccccd {

            // |x| < 0.9

            return j1_maclaurin_series_fast(x);

        }

        return j1_small_argument_fast(x);

    }

    j1_asympt_fast(x)

}

/*

   Evaluates:

   J1 = sqrt(2/(PI*x)) * beta(x) * cos(x - 3*PI/4 - alpha(x))

   discarding 1*PI/2 using identities gives:

   J1 = sqrt(2/(PI*x)) * beta(x) * sin(x - PI/4 - alpha(x))

   to avoid squashing small (-PI/4 - alpha(x)) into a large x actual expansion is:

   J1 = sqrt(2/(PI*x)) * beta(x) * sin((x mod 2*PI) - PI/4 - alpha(x))

*/

#[inline]

fn j1_asympt_fast(x: f64) -> f64 {

    let origin_x = x;

    static SGN: [f64; 2] = [1., -1.];

    let sign_scale = SGN[x.is_sign_negative() as usize];

    let x = x.abs();

    const SQRT_2_OVER_PI: DoubleDouble = DoubleDouble::new(

        f64::from_bits(0xbc8cbc0d30ebfd15),

        f64::from_bits(0x3fe9884533d43651),

    );

    const MPI_OVER_4: DoubleDouble = DoubleDouble::new(

        f64::from_bits(0xbc81a62633145c07),

        f64::from_bits(0xbfe921fb54442d18),

    );

    let recip = if x.to_bits() > 0x7fd000000000000u64 {

        DoubleDouble::quick_mult_f64(DoubleDouble::from_exact_safe_div(4.0, x), 0.25)

    } else {

        DoubleDouble::from_recip(x)

    };

    let alpha = bessel_1_asympt_alpha_fast(recip);

    let beta = bessel_1_asympt_beta_fast(recip);

    let AngleReduced { angle } = rem2pi_any(x);

    // Without full subtraction cancellation happens sometimes

    let x0pi34 = DoubleDouble::full_dd_sub(MPI_OVER_4, alpha);

    let r0 = DoubleDouble::full_dd_add(angle, x0pi34);

    let m_sin = sin_dd_small_fast(r0);

    let z0 = DoubleDouble::quick_mult(beta, m_sin);

    let r_sqrt = DoubleDouble::from_rsqrt_fast(x);

    let scale = DoubleDouble::quick_mult(SQRT_2_OVER_PI, r_sqrt);

    let p = DoubleDouble::quick_mult(scale, z0);

    let err = f_fmla(

        p.hi,

        f64::from_bits(0x3be0000000000000), // 2^-65

        f64::from_bits(0x3a60000000000000), // 2^-89

    );

    let ub = p.hi + (p.lo + err);

    let lb = p.hi + (p.lo - err);

    if ub == lb {

        return p.to_f64() * sign_scale;

    }

    j1_asympt(origin_x, recip, r_sqrt, angle)

}

/*

   Evaluates:

   J1 = sqrt(2/(PI*x)) * beta(x) * cos(x - 3*PI/4 - alpha(x))

   discarding 1*PI/2 using identities gives:

   J1 = sqrt(2/(PI*x)) * beta(x) * sin(x - PI/4 - alpha(x))

   to avoid squashing small (-PI/4 - alpha(x)) into a large x actual expansion is:

   J1 = sqrt(2/(PI*x)) * beta(x) * sin((x mod 2*PI) - PI/4 - alpha(x))

*/

fn j1_asympt(x: f64, recip: DoubleDouble, r_sqrt: DoubleDouble, angle: DoubleDouble) -> f64 {

    let origin_x = x;

    static SGN: [f64; 2] = [1., -1.];

    let sign_scale = SGN[x.is_sign_negative() as usize];

    const SQRT_2_OVER_PI: DoubleDouble = DoubleDouble::new(

        f64::from_bits(0xbc8cbc0d30ebfd15),

        f64::from_bits(0x3fe9884533d43651),

    );

    const MPI_OVER_4: DoubleDouble = DoubleDouble::new(

        f64::from_bits(0xbc81a62633145c07),

        f64::from_bits(0xbfe921fb54442d18),

    );

    let alpha = bessel_1_asympt_alpha(recip);

    let beta = bessel_1_asympt_beta(recip);

    // Without full subtraction cancellation happens sometimes

    let x0pi34 = DoubleDouble::full_dd_sub(MPI_OVER_4, alpha);

    let r0 = DoubleDouble::full_dd_add(angle, x0pi34);

    let m_sin = sin_dd_small(r0);

    let z0 = DoubleDouble::quick_mult(beta, m_sin);

    let scale = DoubleDouble::quick_mult(SQRT_2_OVER_PI, r_sqrt);

    let r = DoubleDouble::quick_mult(scale, z0);

    let p = DoubleDouble::from_exact_add(r.hi, r.lo);

    let err = f_fmla(

        p.hi,

        f64::from_bits(0x3bc0000000000000), // 2^-67

        f64::from_bits(0x39c0000000000000), // 2^-99

    );

    let ub = p.hi + (p.lo + err);

    let lb = p.hi + (p.lo - err);

    if ub == lb {

        return p.to_f64() * sign_scale;

    }

    j1_asympt_hard(origin_x)

}

/**

Generated in Sollya:

```text

pretty = proc(u) {

  return ~(floor(u*1000)/1000);

};

bessel_j1 = library("./cmake-build-release/libbessel_sollya.dylib");

f = bessel_j1(x)/x;

d = [0, 0.921];

w = 1;

pf = fpminimax(f, [|0,2,4,6,8,10,12,14,16,18,20,22,24|], [|107, 107, 107, 107, 107, D...|], d, absolute, floating);

w = 1;

or_f = bessel_j1(x);

pf1 = pf * x;

err_p = -log2(dirtyinfnorm(pf1*w-or_f, d));

print ("relative error:", pretty(err_p));

for i from 0 to degree(pf) by 2 do {

    print("'", coeff(pf, i), "',");

};

```

See ./notes/bessel_sollya/bessel_j1_at_zero_fast.sollya

**/

#[inline]

pub(crate) fn j1_maclaurin_series_fast(x: f64) -> f64 {

    const C0: DoubleDouble = DoubleDouble::from_bit_pair((0x3b30e9e087200000, 0x3fe0000000000000));

    let x2 = DoubleDouble::from_exact_mult(x, x);

    let p = f_polyeval12(

        x2.hi,

        f64::from_bits(0xbfb0000000000000),

        f64::from_bits(0x3f65555555555555),

        f64::from_bits(0xbf0c71c71c71c45e),

        f64::from_bits(0x3ea6c16c16b82b02),

        f64::from_bits(0xbe3845c87ec0cbef),

        f64::from_bits(0x3dc27e0313e8534c),

        f64::from_bits(0xbd4443dd2d0305d0),

        f64::from_bits(0xbd0985a435fe9aa1),

        f64::from_bits(0x3d10c82d92c46d30),

        f64::from_bits(0xbd0aa3684321f219),

        f64::from_bits(0x3cf8351f29ac345a),

        f64::from_bits(0xbcd333fe6cd52c9f),

    );

    let mut z = DoubleDouble::mul_f64_add(x2, p, C0);

    z = DoubleDouble::quick_mult_f64(z, x);

    // squaring error (2^-56) + poly error 2^-75

    let err = f_fmla(

        x2.hi,

        f64::from_bits(0x3c70000000000000), // 2^-56

        f64::from_bits(0x3b40000000000000), // 2^-75

    );

    let ub = z.hi + (z.lo + err);

    let lb = z.hi + (z.lo - err);

    if ub == lb {

        return z.to_f64();

    }

    j1_maclaurin_series(x)

}

/**

Generated in Sollya:

```text

pretty = proc(u) {

  return ~(floor(u*1000)/1000);

};

bessel_j1 = library("./cmake-build-release/libbessel_sollya.dylib");

f = bessel_j1(x)/x;

d = [0, 0.921];

w = 1;

pf = fpminimax(f, [|0,2,4,6,8,10,12,14,16,18,20,22,24|], [|107, 107, 107, 107, 107, D...|], d, absolute, floating);

w = 1;

or_f = bessel_j1(x);

pf1 = pf * x;

err_p = -log2(dirtyinfnorm(pf1*w-or_f, d));

print ("relative error:", pretty(err_p));

for i from 0 to degree(pf) by 2 do {

    print("'", coeff(pf, i), "',");

};

```

See ./notes/bessel_sollya/bessel_j1_at_zero.sollya

**/

pub(crate) fn j1_maclaurin_series(x: f64) -> f64 {

    let origin_x = x;

    static SGN: [f64; 2] = [1., -1.];

    let sign_scale = SGN[x.is_sign_negative() as usize];

    let x = x.abs();

    const CL: [(u64, u64); 5] = [

        (0xb930000000000000, 0x3fe0000000000000),

        (0x39c8e80000000000, 0xbfb0000000000000),

        (0x3c05555554f3add7, 0x3f65555555555555),

        (0xbbac71c4eb0f8c94, 0xbf0c71c71c71c71c),

        (0xbb3f56b7a43206d4, 0x3ea6c16c16c16c17),

    ];

    let dx2 = DoubleDouble::from_exact_mult(x, x);

    let p = f_polyeval8(

        dx2.hi,

        f64::from_bits(0xbe3845c8a0ce5129),

        f64::from_bits(0x3dc27e4fb7789ea2),

        f64::from_bits(0xbd4522a43f633af1),

        f64::from_bits(0x3cc2c97589d53f97),

        f64::from_bits(0xbc3ab8151dca7912),

        f64::from_bits(0x3baf08732286d1d4),

        f64::from_bits(0xbb10ac65637413f4),

        f64::from_bits(0xbae4d8336e4f779c),

    );

    let mut p_e = DoubleDouble::mul_f64_add(dx2, p, DoubleDouble::from_bit_pair(CL[4]));

    p_e = DoubleDouble::mul_add(dx2, p_e, DoubleDouble::from_bit_pair(CL[3]));

    p_e = DoubleDouble::mul_add(dx2, p_e, DoubleDouble::from_bit_pair(CL[2]));

    p_e = DoubleDouble::mul_add(dx2, p_e, DoubleDouble::from_bit_pair(CL[1]));

    p_e = DoubleDouble::mul_add(dx2, p_e, DoubleDouble::from_bit_pair(CL[0]));

    let p = DoubleDouble::quick_mult_f64(p_e, x);

    let err = f_fmla(

        p.hi,

        f64::from_bits(0x3bd0000000000000), // 2^-66

        f64::from_bits(0x3a00000000000000), // 2^-95

    );

    let ub = p.hi + (p.lo + err);

    let lb = p.hi + (p.lo - err);

    if ub != lb {

        return j1_maclaurin_series_hard(origin_x);

    }

    p.to_f64() * sign_scale

}

/**

Taylor expansion at 0

Generated by SageMath:

```python

def print_expansion_at_0():

    print(f"static C: [DyadicFloat128; 13] = ")

    from mpmath import mp, j1, taylor, expm1

    poly = taylor(lambda val: j1(val), 0, 26)

    real_i = 0

    print("[")

    for i in range(1, len(poly), 2):

        print_dyadic(poly[i])

        real_i = real_i + 1

    print("],")

    print("];")

mp.prec = 180

print_expansion_at_0()

```

**/

#[cold]

#[inline(never)]

fn j1_maclaurin_series_hard(x: f64) -> f64 {

    static SGN: [f64; 2] = [1., -1.];

    let sign_scale = SGN[x.is_sign_negative() as usize];

    let x = x.abs();

    static C: [DyadicFloat128; 13] = [

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -128,

            mantissa: 0x80000000_00000000_00000000_00000000_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Neg,

            exponent: -131,

            mantissa: 0x80000000_00000000_00000000_00000000_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -136,

            mantissa: 0xaaaaaaaa_aaaaaaaa_aaaaaaaa_aaaaaaab_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Neg,

            exponent: -142,

            mantissa: 0xe38e38e3_8e38e38e_38e38e38_e38e38e4_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -148,

            mantissa: 0xb60b60b6_0b60b60b_60b60b60_b60b60b6_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Neg,

            exponent: -155,

            mantissa: 0xc22e4506_72894ab6_cd8efb11_d33f5618_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -162,

            mantissa: 0x93f27dbb_c4fae397_780b69f5_333c725b_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Neg,

            exponent: -170,

            mantissa: 0xa91521fb_2a434d3f_649f5485_f169a743_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -178,

            mantissa: 0x964bac6d_7ae67d8d_aec68405_485dea03_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Neg,

            exponent: -187,

            mantissa: 0xd5c0f53a_fe6fa17f_8c7b0b68_39691f4e_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -196,

            mantissa: 0xf8bb4be7_8e7896b0_58fee362_01a4370c_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Neg,

            exponent: -205,

            mantissa: 0xf131bdf7_cff8d02e_e1ef6820_f9d58ab6_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -214,

            mantissa: 0xc5e72c48_0d1aec75_3caa2e0d_edd008ca_u128,

        },

    ];

    let rx = DyadicFloat128::new_from_f64(x);

    let dx = rx * rx;

    let mut p = C[12];

    for i in (0..12).rev() {

        p = dx * p + C[i];

    }

    (p * rx).fast_as_f64() * sign_scale

}

/// This method on small range searches for nearest zero or extremum.

/// Then picks stored series expansion at the point end evaluates the poly at the point.

#[inline]

pub(crate) fn j1_small_argument_fast(x: f64) -> f64 {

    static SIGN: [f64; 2] = [1., -1.];

    let sign_scale = SIGN[x.is_sign_negative() as usize];

    let x_abs = f64::from_bits(x.to_bits() & 0x7fff_ffff_ffff_ffff);

    // let avg_step = 74.60109 / 47.0;

    // let inv_step = 1.0 / avg_step;

    const INV_STEP: f64 = 0.6300176043004198;

    let fx = x_abs * INV_STEP;

    const J1_ZEROS_COUNT: f64 = (J1_ZEROS.len() - 1) as f64;

    let idx0 = unsafe { fx.min(J1_ZEROS_COUNT).to_int_unchecked::<usize>() };

    let idx1 = unsafe {

        fx.cpu_ceil()

            .min(J1_ZEROS_COUNT)

            .to_int_unchecked::<usize>()

    };

    let found_zero0 = DoubleDouble::from_bit_pair(J1_ZEROS[idx0]);

    let found_zero1 = DoubleDouble::from_bit_pair(J1_ZEROS[idx1]);

    let dist0 = (found_zero0.hi - x_abs).abs();

    let dist1 = (found_zero1.hi - x_abs).abs();

    let (found_zero, idx, dist) = if dist0 < dist1 {

        (found_zero0, idx0, dist0)

    } else {

        (found_zero1, idx1, dist1)

    };

    if idx == 0 {

        return j1_maclaurin_series_fast(x);

    }

    let r = DoubleDouble::full_add_f64(-found_zero, x_abs);

    // We hit exact zero, value, better to return it directly

    if dist == 0. {

        return f64::from_bits(J1_ZEROS_VALUE[idx]) * sign_scale;

    }

    let is_zero_too_close = dist.abs() < 1e-3;

    let c = if is_zero_too_close {

        &J1_COEFFS_TAYLOR[idx - 1]

    } else {

        &J1_COEFFS[idx - 1]

    };

    let p = f_polyeval19(

        r.hi,

        f64::from_bits(c[5].1),

        f64::from_bits(c[6].1),

        f64::from_bits(c[7].1),

        f64::from_bits(c[8].1),

        f64::from_bits(c[9].1),

        f64::from_bits(c[10].1),

        f64::from_bits(c[11].1),

        f64::from_bits(c[12].1),

        f64::from_bits(c[13].1),

        f64::from_bits(c[14].1),

        f64::from_bits(c[15].1),

        f64::from_bits(c[16].1),

        f64::from_bits(c[17].1),

        f64::from_bits(c[18].1),

        f64::from_bits(c[19].1),

        f64::from_bits(c[20].1),

        f64::from_bits(c[21].1),

        f64::from_bits(c[22].1),

        f64::from_bits(c[23].1),

    );

    let mut z = DoubleDouble::mul_f64_add(r, p, DoubleDouble::from_bit_pair(c[4]));

    z = DoubleDouble::mul_add(z, r, DoubleDouble::from_bit_pair(c[3]));

    z = DoubleDouble::mul_add(z, r, DoubleDouble::from_bit_pair(c[2]));

    z = DoubleDouble::mul_add(z, r, DoubleDouble::from_bit_pair(c[1]));

    z = DoubleDouble::mul_add(z, r, DoubleDouble::from_bit_pair(c[0]));

    let err = f_fmla(

        z.hi,

        f64::from_bits(0x3c70000000000000), // 2^-56

        f64::from_bits(0x3bf0000000000000), // 2^-64

    );

    let ub = z.hi + (z.lo + err);

    let lb = z.hi + (z.lo - err);

    if ub == lb {

        return z.to_f64() * sign_scale;

    }

    j1_small_argument_dd(sign_scale, r, c)

}

fn j1_small_argument_dd(sign_scale: f64, r: DoubleDouble, c0: &[(u64, u64); 24]) -> f64 {

    let c = &c0[15..];

    let p0 = f_polyeval9(

        r.to_f64(),

        f64::from_bits(c[0].1),

        f64::from_bits(c[1].1),

        f64::from_bits(c[2].1),

        f64::from_bits(c[3].1),

        f64::from_bits(c[4].1),

        f64::from_bits(c[5].1),

        f64::from_bits(c[6].1),

        f64::from_bits(c[7].1),

        f64::from_bits(c[8].1),

    );

    let c = c0;

    let mut p_e = DoubleDouble::mul_f64_add(r, p0, DoubleDouble::from_bit_pair(c[14]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[13]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[12]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[11]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[10]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[9]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[8]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[7]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[6]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[5]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[4]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[3]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[2]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[1]));

    p_e = DoubleDouble::mul_add(p_e, r, DoubleDouble::from_bit_pair(c[0]));

    let p = DoubleDouble::from_exact_add(p_e.hi, p_e.lo);

    let err = f_fmla(

        p.hi,

        f64::from_bits(0x3c10000000000000), // 2^-62

        f64::from_bits(0x3a00000000000000), // 2^-95

    );

    let ub = p.hi + (p.lo + err);

    let lb = p.hi + (p.lo - err);

    if ub != lb {

        return j1_small_argument_path_hard(sign_scale, r, c);

    }

    p.to_f64() * sign_scale

}

#[cold]

#[inline(never)]

fn j1_small_argument_path_hard(sign_scale: f64, r: DoubleDouble, c: &[(u64, u64); 24]) -> f64 {

    let mut p = DoubleDouble::from_bit_pair(c[23]);

    for i in (0..23).rev() {

        p = DoubleDouble::mul_add(r, p, DoubleDouble::from_bit_pair(c[i]));

        p = DoubleDouble::from_exact_add(p.hi, p.lo);

    }

    p.to_f64() * sign_scale

}

/*

   Evaluates:

   J1 = sqrt(2/(PI*x)) * beta(x) * cos(x - 3*PI/4 - alpha(x))

   discarding 1*PI/2 using identities gives:

   J1 = sqrt(2/(PI*x)) * beta(x) * sin(x - PI/4 - alpha(x))

   to avoid squashing small (-PI/4 - alpha(x)) into a large x actual expansion is:

   J1 = sqrt(2/(PI*x)) * beta(x) * sin((x mod 2*PI) - PI/4 - alpha(x))

   This method is required for situations where x*x or 1/(x*x) will overflow

*/

#[cold]

#[inline(never)]

fn j1_asympt_hard(x: f64) -> f64 {

    static SGN: [f64; 2] = [1., -1.];

    let sign_scale = SGN[x.is_sign_negative() as usize];

    let x = x.abs();

    const SQRT_2_OVER_PI: DyadicFloat128 = DyadicFloat128 {

        sign: DyadicSign::Pos,

        exponent: -128,

        mantissa: 0xcc42299e_a1b28468_7e59e280_5d5c7180_u128,

    };

    const MPI_OVER_4: DyadicFloat128 = DyadicFloat128 {

        sign: DyadicSign::Neg,

        exponent: -128,

        mantissa: 0xc90fdaa2_2168c234_c4c6628b_80dc1cd1_u128,

    };

    let x_dyadic = DyadicFloat128::new_from_f64(x);

    let recip = DyadicFloat128::accurate_reciprocal(x);

    let alpha = bessel_1_asympt_alpha_hard(recip);

    let beta = bessel_1_asympt_beta_hard(recip);

    let angle = rem2pi_f128(x_dyadic);

    let x0pi34 = MPI_OVER_4 - alpha;

    let r0 = angle + x0pi34;

    let m_sin = sin_f128_small(r0);

    let z0 = beta * m_sin;

    let r_sqrt = bessel_rsqrt_hard(x, recip);

    let scale = SQRT_2_OVER_PI * r_sqrt;

    let p = scale * z0;

    p.fast_as_f64() * sign_scale

}

#[cfg(test)]

mod tests {

    use super::*;

    #[test]

    fn test_j1() {

        assert_eq!(f_j1(0.000000000000000000000000000000001241), 6.205e-34);

        assert_eq!(f_j1(0.0000000000000000000000000000004321), 2.1605e-31);

        assert_eq!(f_j1(0.00000000000000000004321), 2.1605e-20);

        assert_eq!(f_j1(73.81695991658546), -0.06531447184607607);

        assert_eq!(f_j1(0.01), 0.004999937500260416);

        assert_eq!(f_j1(0.9), 0.4059495460788057);

        assert_eq!(

            f_j1(162605674999778540000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.),

            0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008686943178258183

        );

        assert_eq!(f_j1(3.831705970207517), -1.8501090915423025e-15);

        assert_eq!(f_j1(-3.831705970207517), 1.8501090915423025e-15);

        assert_eq!(f_j1(-6.1795701510782757E+307), 8.130935041593236e-155);

        assert_eq!(

            f_j1(0.000000000000000000000000000000000000008827127),

            0.0000000000000000000000000000000000000044135635

        );

        assert_eq!(

            f_j1(-0.000000000000000000000000000000000000008827127),

            -0.0000000000000000000000000000000000000044135635

        );

        assert_eq!(f_j1(5.4), -0.3453447907795863);

        assert_eq!(

            f_j1(77.743162408196766932633181568235159),

            0.09049267898021947

        );

        assert_eq!(

            f_j1(84.027189586293545175976760219782591),

            0.0870430264022591

        );

        assert_eq!(f_j1(f64::NEG_INFINITY), 0.0);

        assert_eq!(f_j1(f64::INFINITY), 0.0);

        assert!(f_j1(f64::NAN).is_nan());

    }

}