/*

 * // 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::bessel::i0::bessel_rsqrt_hard;

use crate::bessel::i0_exp;

use crate::double_double::DoubleDouble;

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

use crate::exponents::rational128_exp;

use crate::logs::{fast_log_d_to_dd, log_dd};

use crate::polyeval::f_polyeval3;

/// Modified Bessel of the second kind of order 0

///

/// Max ULP 0.5

pub fn f_k0(x: f64) -> f64 {

    let ix = x.to_bits();

    if ix >= 0x7ffu64 << 52 || ix == 0 {

        // |x| == NaN, x == inf, |x| == 0, x < 0

        if ix.wrapping_shl(1) == 0 {

            // |x| == 0

            return f64::INFINITY;

        }

        if x.is_infinite() {

            return if x.is_sign_positive() { 0. } else { f64::NAN };

        }

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

    }

    let xb = x.to_bits();

    if xb >= 0x40862e42fefa39f0u64 {

        // x >= 709.7827128933841

        return 0.;

    }

    if xb <= 0x3ff0000000000000 {

        // x <= 1

        return k0_small_dd(x).to_f64();

    }

    k0_asympt(x)

}

/**

Computes I0 on interval [0; 1]

as rational approximation I0 = 1 + (x/2)^2 * Pn((x/2)^2)/Qm((x/2)^2))

Generated by Wolfram Mathematica:

```python

<<FunctionApproximations`

ClearAll["Global`*"]

f[x_]:=(BesselI[0,x]-1)/(x/2)^2

g[z_]:=f[2 Sqrt[z]]

{err,approx}=MiniMaxApproximation[g[z],{z,{0.0000000000001,1},5,5},WorkingPrecision->60]

poly=Numerator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

poly=Denominator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

```

**/

#[inline]

fn i0_0_to_1_fast(x: f64) -> DoubleDouble {

    let half_x = x * 0.5; // this is exact

    let eval_x = DoubleDouble::from_exact_mult(half_x, half_x);

    const P: [(u64, u64); 3] = [

        (0xbae20452afd5045b, 0x3ff0000000000000),

        (0xbc5b6ff3f140da20, 0x3fc93c83592c03de),

        (0x3c25b350e9128d49, 0x3f904f33ef2de455),

    ];

    let ps_num = f_polyeval3(

        eval_x.hi,

        f64::from_bits(0x3f433805a2fabaaa),

        f64::from_bits(0x3ee5897e7f554966),

        f64::from_bits(0x3e731401f0bb5de4),

    );

    let mut p_num = DoubleDouble::mul_f64_add(eval_x, ps_num, DoubleDouble::from_bit_pair(P[2]));

    p_num = DoubleDouble::mul_add(eval_x, p_num, DoubleDouble::from_bit_pair(P[1]));

    p_num = DoubleDouble::mul_add(eval_x, p_num, DoubleDouble::from_bit_pair(P[0]));

    const Q: [(u64, u64); 3] = [

        (0x0000000000000000, 0x3ff0000000000000),

        (0x3c323fa63bef2b4e, 0xbfab0df29b4ff089),

        (0x3bfedbdf64ed3110, 0x3f564662064157d2),

    ];

    let ps_den = f_polyeval3(

        eval_x.hi,

        f64::from_bits(0xbef6bdbb484fd0a4),

        f64::from_bits(0x3e8d6ced53309351),

        f64::from_bits(0xbe13cff13854e945),

    );

    let mut p_den = DoubleDouble::mul_f64_add(eval_x, ps_den, DoubleDouble::from_bit_pair(Q[2]));

    p_den = DoubleDouble::mul_add(eval_x, p_den, DoubleDouble::from_bit_pair(Q[1]));

    p_den = DoubleDouble::mul_add(eval_x, p_den, DoubleDouble::from_bit_pair(Q[0]));

    let p = DoubleDouble::div(p_num, p_den);

    let z = DoubleDouble::quick_mult(p, eval_x);

    DoubleDouble::full_add_f64(z, 1.)

}

/**

K0(x) + log(x) * I0(x) = P(x^2)

hence

K0(x) = P(x^2) - log(x)*I0(x)

Generated in Wolfram Mathematica:

```text

<<FunctionApproximations`

ClearAll["Global`*"]

f[x_]:=BesselK[0,x]+Log[x]BesselI[0,x]

g[z_]:=f[Sqrt[z]]

{err,approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},5,5},WorkingPrecision->60]

poly=Numerator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

poly=Denominator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

```

**/

#[inline]

pub(crate) fn k0_small_dd(x: f64) -> DoubleDouble {

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

    const P: [(u64, u64); 6] = [

        (0x3c1be095d044e896, 0x3fbdadb014541eb2),

        (0x3c7321baa1d0a2d9, 0x3fd1b9f19bc9019a),

        (0xbc33ce33a244e5bd, 0x3f94ec39f8744183),

        (0x3bd7008dfc623255, 0x3f3d85175b25727d),

        (0xbb4aa2a1c4905d30, 0x3ed007a860ef3235),

        (0xbae8daa77abd6f7f, 0x3e4839e32c19f31a),

    ];

    let ps_num = f_polyeval3(

        dx.hi,

        f64::from_bits(0x3f3d85175b25727d),

        f64::from_bits(0x3ed007a860ef3235),

        f64::from_bits(0x3e4839e32c19f31a),

    );

    let mut p_num = DoubleDouble::mul_f64_add(dx, ps_num, DoubleDouble::from_bit_pair(P[2]));

    p_num = DoubleDouble::mul_add(dx, p_num, DoubleDouble::from_bit_pair(P[1]));

    p_num = DoubleDouble::mul_add(dx, p_num, DoubleDouble::from_bit_pair(P[0]));

    const Q: [(u64, u64); 3] = [

        (0x0000000000000000, 0x3ff0000000000000),

        (0xbc2a82a292acdc83, 0xbf91be3a25c968d6),

        (0xbb9d2c37183a6496, 0x3f23bac6961619d8),

    ];

    let ps_den = f_polyeval3(

        dx.hi,

        f64::from_bits(0xbeac315b81faa1bf),

        f64::from_bits(0x3e2ab2d2fbae0863),

        f64::from_bits(0xbd9be23550f83df7),

    );

    let mut p_den = DoubleDouble::mul_f64_add(dx, ps_den, DoubleDouble::from_bit_pair(Q[2]));

    p_den = DoubleDouble::mul_add(dx, p_den, DoubleDouble::from_bit_pair(Q[1]));

    p_den = DoubleDouble::mul_add_f64(dx, p_den, f64::from_bits(0x3ff0000000000000));

    let prod = DoubleDouble::div(p_num, p_den);

    let vi_log = fast_log_d_to_dd(x);

    let vi = i0_0_to_1_fast(x);

    let r = DoubleDouble::mul_add(vi_log, -vi, prod);

    let err = r.hi * f64::from_bits(0x3c00000000000000); // 2^-63

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

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

    if ub == lb {

        return r;

    }

    k0_small_hard(x, vi)

}

/**

K0(x) + log(x) * I0(x) = P(x^2)

hence

K0(x) = P(x^2) - log(x)*I0(x)

Generated in Wolfram Mathematica:

```text

<<FunctionApproximations`

ClearAll["Global`*"]

f[x_]:=BesselK[0,x]+Log[x]BesselI[0,x]

g[z_]:=f[Sqrt[z]]

{err,approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},5,5},WorkingPrecision->60]

poly=Numerator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

poly=Denominator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

```

**/

#[cold]

#[inline(never)]

fn k0_small_hard(x: f64, vi: DoubleDouble) -> DoubleDouble {

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

    const P: [(u64, u64); 6] = [

        (0x3c1be095d044e896, 0x3fbdadb014541eb2),

        (0x3c7321baa1d0a2d9, 0x3fd1b9f19bc9019a),

        (0xbc33ce33a244e5bd, 0x3f94ec39f8744183),

        (0x3bd7008dfc623255, 0x3f3d85175b25727d),

        (0xbb4aa2a1c4905d30, 0x3ed007a860ef3235),

        (0xbae8daa77abd6f7f, 0x3e4839e32c19f31a),

    ];

    let mut p_num = DoubleDouble::mul_add(

        dx,

        DoubleDouble::from_bit_pair(P[5]),

        DoubleDouble::from_bit_pair(P[4]),

    );

    p_num = DoubleDouble::mul_add(dx, p_num, DoubleDouble::from_bit_pair(P[3]));

    p_num = DoubleDouble::mul_add(dx, p_num, DoubleDouble::from_bit_pair(P[2]));

    p_num = DoubleDouble::mul_add(dx, p_num, DoubleDouble::from_bit_pair(P[1]));

    p_num = DoubleDouble::mul_add(dx, p_num, DoubleDouble::from_bit_pair(P[0]));

    const Q: [(u64, u64); 6] = [

        (0x0000000000000000, 0x3ff0000000000000),

        (0xbc2a82a292acdc83, 0xbf91be3a25c968d6),

        (0xbb9d2c37183a6496, 0x3f23bac6961619d8),

        (0xbb32032e14c6c2b2, 0xbeac315b81faa1bf),

        (0x3aa1a1dc04bfba96, 0x3e2ab2d2fbae0863),

        (0x3a3e0f678099fcff, 0xbd9be23550f83df7),

    ];

    let mut p_den = DoubleDouble::mul_add(

        dx,

        DoubleDouble::from_bit_pair(Q[5]),

        DoubleDouble::from_bit_pair(Q[4]),

    );

    p_den = DoubleDouble::mul_add(dx, p_den, DoubleDouble::from_bit_pair(Q[3]));

    p_den = DoubleDouble::mul_add(dx, p_den, DoubleDouble::from_bit_pair(Q[2]));

    p_den = DoubleDouble::mul_add(dx, p_den, DoubleDouble::from_bit_pair(Q[1]));

    p_den = DoubleDouble::mul_add_f64(dx, p_den, f64::from_bits(0x3ff0000000000000));

    let prod = DoubleDouble::div(p_num, p_den);

    let v_log = log_dd(x);

    DoubleDouble::mul_add(v_log, -vi, prod)

}

/**

Generated in Wolfram

Computes sqrt(x)*exp(x)*K0(x)=Pn(1/x)/Qm(1/x)

hence

K0(x) = Pn(1/x)/Qm(1/x) / (sqrt(x) * exp(x))

```text

<<FunctionApproximations`

ClearAll["Global`*"]

f[x_]:=Sqrt[x] Exp[x] BesselK[0,x]

g[z_]:=f[1/z]

{err, approx}=MiniMaxApproximation[g[z],{z,{0.0000000000001,1},11,11},WorkingPrecision->60]

poly=Numerator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

poly=Denominator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

```

**/

#[inline]

fn k0_asympt(x: f64) -> f64 {

    let recip = DoubleDouble::from_quick_recip(x);

    let e = i0_exp(x * 0.5);

    let r_sqrt = DoubleDouble::from_sqrt(x);

    const P: [(u64, u64); 12] = [

        (0xbc9a6a11d237114e, 0x3ff40d931ff62706),

        (0x3cdd614ddf4929e5, 0x4040645168c3e483),

        (0xbd1ecf9ea0af6ab2, 0x40757419a703a2ab),

        (0xbd3da3551fb27770, 0x409d4e65365522a2),

        (0xbd564d58855d1a46, 0x40b6dd32f5a199d9),

        (0xbd6cf055ca963a8e, 0x40c4fd2368f19618),

        (0x3d4b6cdfbdc058df, 0x40c68faa11ebcd59),

        (0x3d5b4ce4665bfa46, 0x40bb6fbe08e0a8ea),

        (0xbd4316909063be15, 0x40a1953103a5be31),

        (0x3d12f3f8edf41af0, 0x4074d2cb001e175c),

        (0xbcd7bba36540264f, 0x40316cffcad5f8f9),

        (0xbc6bf28dfdd5d37d, 0x3fc2f487fe78b8d7),

    ];

    let x2 = DoubleDouble::quick_mult(recip, recip);

    let x4 = DoubleDouble::quick_mult(x2, x2);

    let x8 = DoubleDouble::quick_mult(x4, x4);

    let e0 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(P[1]),

        DoubleDouble::from_bit_pair(P[0]),

    );

    let e1 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(P[3]),

        DoubleDouble::from_bit_pair(P[2]),

    );

    let e2 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(P[5]),

        DoubleDouble::from_bit_pair(P[4]),

    );

    let e3 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(P[7]),

        DoubleDouble::from_bit_pair(P[6]),

    );

    let e4 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(P[9]),

        DoubleDouble::from_bit_pair(P[8]),

    );

    let e5 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(P[11]),

        DoubleDouble::from_bit_pair(P[10]),

    );

    let f0 = DoubleDouble::mul_add(x2, e1, e0);

    let f1 = DoubleDouble::mul_add(x2, e3, e2);

    let f2 = DoubleDouble::mul_add(x2, e5, e4);

    let g0 = DoubleDouble::mul_add(x4, f1, f0);

    let p_num = DoubleDouble::mul_add(x8, f2, g0);

    const Q: [(u64, u64); 12] = [

        (0x0000000000000000, 0x3ff0000000000000),

        (0xbcb9e8a5b17e696a, 0x403a485acd64d64a),

        (0x3cd2e2e9c87f71f7, 0x4071518092320ecb),

        (0xbd0d05bdb9431a2f, 0x4097e57e4c22c08e),

        (0x3d5207068ab19ba9, 0x40b2ebadb2db62f9),

        (0xbd64e37674083471, 0x40c1c0e4e9d6493d),

        (0x3d3efb7a9a62b020, 0x40c3b94e8d62cdc7),

        (0x3d47d6ce80a2114b, 0x40b93c2fd39e868e),

        (0xbd1dfda61f525861, 0x40a1877a53a7f8d8),

        (0x3d1236ff523dfcfa, 0x4077c3a10c2827de),

        (0xbcc889997c9b0fe7, 0x4039a1d80b11c4a1),

        (0x3c7ded0e8d73dddc, 0x3fdafe4913722123),

    ];

    let e0 = DoubleDouble::mul_add_f64(

        recip,

        DoubleDouble::from_bit_pair(Q[1]),

        f64::from_bits(0x3ff0000000000000),

    );

    let e1 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(Q[3]),

        DoubleDouble::from_bit_pair(Q[2]),

    );

    let e2 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(Q[5]),

        DoubleDouble::from_bit_pair(Q[4]),

    );

    let e3 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(Q[7]),

        DoubleDouble::from_bit_pair(Q[6]),

    );

    let e4 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(Q[9]),

        DoubleDouble::from_bit_pair(Q[8]),

    );

    let e5 = DoubleDouble::mul_add(

        recip,

        DoubleDouble::from_bit_pair(Q[11]),

        DoubleDouble::from_bit_pair(Q[10]),

    );

    let f0 = DoubleDouble::mul_add(x2, e1, e0);

    let f1 = DoubleDouble::mul_add(x2, e3, e2);

    let f2 = DoubleDouble::mul_add(x2, e5, e4);

    let g0 = DoubleDouble::mul_add(x4, f1, f0);

    let p_den = DoubleDouble::mul_add(x8, f2, g0);

    let z = DoubleDouble::div(p_num, p_den);

    let r = DoubleDouble::div(z, e * r_sqrt * e);

    let err = r.hi * f64::from_bits(0x3c10000000000000); // 2^-62

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

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

    if ub != lb {

        return k0_asympt_hard(x);

    }

    r.to_f64()

}

/**

Generated in Wolfram

Computes sqrt(x)*exp(x)*K0(x)=Pn(1/x)/Qm(1/x)

hence

K0(x) = Pn(1/x)/Qm(1/x) / (sqrt(x) * exp(x))

```text

<<FunctionApproximations`

ClearAll["Global`*"]

f[x_]:=Sqrt[x] Exp[x] BesselK[0,x]

g[z_]:=f[1/z]

{err, approx}=MiniMaxApproximation[g[z],{z,{0.0000000000001,1},14,14},WorkingPrecision->90]

poly=Numerator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

poly=Denominator[approx][[1]];

coeffs=CoefficientList[poly,z];

TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]

```

**/

#[inline(never)]

#[cold]

fn k0_asympt_hard(x: f64) -> f64 {

    static P: [DyadicFloat128; 15] = [

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -127,

            mantissa: 0xa06c98ff_b1382cb2_be520f51_a7b8f970_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -122,

            mantissa: 0xc84d8d0c_7faeef84_e56abccc_3d70f8a2_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -118,

            mantissa: 0xd1a71096_3da22280_35768c9e_0d3ddf42_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -115,

            mantissa: 0xf202e474_3698aabb_05688da0_ea1a088d_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -112,

            mantissa: 0xaaa01830_8138af4d_1137b2dd_11a216f5_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -110,

            mantissa: 0x99e0649f_320bca1a_c7adadb3_f5d8498e_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -109,

            mantissa: 0xb4d81657_de1baf00_918cbc76_c6974e96_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -108,

            mantissa: 0x8a9a28c8_a61c2c7a_12416d56_51c0b3d3_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -108,

            mantissa: 0x88a079f1_d9bd4582_6353316c_3aeb9dc9_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -109,

            mantissa: 0xa82e10eb_9dc6225a_ef6223e7_54aa254d_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -111,

            mantissa: 0xf5fc07fe_6b652e8a_0b9e74ba_d0c56118_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -113,

            mantissa: 0xc5288444_c7354b24_4a4e1663_86488928_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -116,

            mantissa: 0x96d3d226_a220ae6e_d6cca1ae_40f01e27_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -121,

            mantissa: 0xa7ab931b_499c4964_499c1091_4ab9673d_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -129,

            mantissa: 0xf8373d1a_9ff3f9c6_e5cfbe0a_85ccc131_u128,

        },

    ];

    static Q: [DyadicFloat128; 15] = [

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -127,

            mantissa: 0x80000000_00000000_00000000_00000000_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -122,

            mantissa: 0xa05190f4_dcf0d35c_277e0f21_0635c538_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -118,

            mantissa: 0xa8837381_94c38992_86c0995d_5e5fa474_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -115,

            mantissa: 0xc3a4f279_9297e905_f59cc065_75959de8_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -112,

            mantissa: 0x8b05ade4_03432e06_881ce37d_a907216d_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -111,

            mantissa: 0xfd77f85e_35626f21_355ae728_01b78cbe_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -109,

            mantissa: 0x972ed117_254970eb_661121dc_a4462d2f_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -109,

            mantissa: 0xec9d204a_9294ab57_2ef500d5_59d701b7_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -109,

            mantissa: 0xf033522d_cae45860_53a01453_c56da895_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -109,

            mantissa: 0x9a33640c_9896ead5_1ce040d7_b36544f3_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -111,

            mantissa: 0xefe714fa_49da0166_fdf8bc68_57b77fa0_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -113,

            mantissa: 0xd323b84c_214196b0_e25b8931_930fea0d_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -116,

            mantissa: 0xbbb5240b_346642d8_010383cb_1e8a607e_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -120,

            mantissa: 0x88dcfa2a_f9f7d2ab_dd017994_8fae7e87_u128,

        },

        DyadicFloat128 {

            sign: DyadicSign::Pos,

            exponent: -127,

            mantissa: 0xc891477c_526e0f5e_74c4ae9f_9d8732b5_u128,

        },

    ];

    let recip = DyadicFloat128::accurate_reciprocal(x);

    let e = rational128_exp(x);

    let r_sqrt = bessel_rsqrt_hard(x, recip);

    let mut p0 = P[14];

    for i in (0..14).rev() {

        p0 = recip * p0 + P[i];

    }

    let mut q = Q[14];

    for i in (0..14).rev() {

        q = recip * q + Q[i];

    }

    let v = p0 * q.reciprocal();

    let r = v * e.reciprocal() * r_sqrt;

    r.fast_as_f64()

}

#[cfg(test)]

mod tests {

    use super::*;

    #[test]

    fn test_k0() {

        assert_eq!(f_k0(0.11), 2.3332678776741127);

        assert_eq!(f_k0(0.643), 0.7241025575342853);

        assert_eq!(f_k0(0.964), 0.4433737413379138);

        assert_eq!(f_k0(2.964), 0.03621679838808167);

        assert_eq!(f_k0(423.43), 7.784461905543397e-186);

        assert_eq!(f_k0(0.), f64::INFINITY);

        assert_eq!(f_k0(-0.), f64::INFINITY);

        assert!(f_k0(-0.5).is_nan());

        assert!(f_k0(f64::NEG_INFINITY).is_nan());

        assert_eq!(f_k0(f64::INFINITY), 0.);

    }

}