pxfm/bessel/

k1f.rs

/*
 * // 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
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 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */
use crate::common::f_fmla;
use crate::exponents::core_expf;
use crate::logs::fast_logf;
use crate::polyeval::{f_estrin_polyeval8, f_polyeval3, f_polyeval4};

/// Modified Bessel of the second kind of order 1
///
/// Max ULP 0.5
pub fn f_k1f(x: f32) -> f32 {
    let ux = x.to_bits();
    if ux >= 0xffu32 << 23 || ux == 0 {
        // |x| == 0, |x| == inf, |x| == NaN, x < 0
        if ux.wrapping_shl(1) == 0 {
            return f32::INFINITY;
        }
        if x.is_infinite() {
            return if x.is_sign_positive() { 0. } else { f32::NAN };
        }
        return x + f32::NAN;
    }

    let xb = x.to_bits();

    if xb >= 0x42cbc779u32 {
        // x > 101.889595
        return 0.;
    }

    if xb <= 0x3f800000u32 {
        // x <= 1.0
        if xb <= 0x34000000u32 {
            // |x| <= f32::EPSILON
            let dx = x as f64;
            let leading_term = 1. / dx;
            if xb <= 0x3109705fu32 {
                // |x| <= 2e-9
                // taylor series for tiny K1(x) ~ 1/x + O(x)
                return leading_term as f32;
            }
            // taylor series for small K1(x) ~ 1/x+1/4 (-1+2 EulerGamma-2 Log[2]+2 Log[x]) x + O(x^3)
            const C: f64 = f64::from_bits(0xbff3b5b6028a83d7); // -1+2 EulerGamma-2 Log[2]
            let log_x = fast_logf(x);
            let r = f_fmla(log_x, 2., C);
            let w0 = f_fmla(dx * 0.25, r, leading_term);
            return w0 as f32;
        }
        return k1f_small(x);
    }

    k1f_asympt(x)
}

/**
Computes
I1(x) = x/2 * (1 + 1 * (x/2)^2 + (x/2)^4 * P((x/2)^2))

Generated by Woflram Mathematica:

```text
<<FunctionApproximations`
ClearAll["Global`*"]
f[x_]:=(BesselI[1,x]*2/x-1-1/2(x/2)^2)/(x/2)^4
g[z_]:=f[2 Sqrt[z]]
{err, approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},3,2},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 i1f_small(x: f32) -> f64 {
    let dx = x as f64;
    let x_over_two = dx * 0.5;
    let x_over_two_sqr = x_over_two * x_over_two;
    let x_over_two_p4 = x_over_two_sqr * x_over_two_sqr;

    let p_num = f_polyeval4(
        x_over_two_sqr,
        f64::from_bits(0x3fb5555555555355),
        f64::from_bits(0x3f6ebf07f0dbc49b),
        f64::from_bits(0x3f1fdc02bf28a8d9),
        f64::from_bits(0x3ebb5e7574c700a6),
    );
    let p_den = f_polyeval3(
        x_over_two_sqr,
        f64::from_bits(0x3ff0000000000000),
        f64::from_bits(0xbfa39b64b6135b5a),
        f64::from_bits(0x3f3fa729bbe951f9),
    );
    let p = p_num / p_den;

    let p1 = f_fmla(0.5, x_over_two_sqr, 1.);
    let p2 = f_fmla(x_over_two_p4, p, p1);
    p2 * x_over_two
}

/**
Series for
f(x) := BesselK(1, x) - Log(x)*BesselI(1, x) - 1/x

Generated by Wolfram Mathematica:
```text
<<FunctionApproximations`
ClearAll["Global`*"]
f[x_]:=(BesselK[1, x]-Log[x]BesselI[1,x]-1/x)/x
g[z_]:=f[Sqrt[z]]
{err, approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},3,3},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 k1f_small(x: f32) -> f32 {
    let dx = x as f64;
    let rcp = 1. / dx;
    let x2 = dx * dx;
    let p_num = f_polyeval4(
        x2,
        f64::from_bits(0xbfd3b5b6028a83d6),
        f64::from_bits(0xbfb3fde2c83f7cca),
        f64::from_bits(0xbf662b2e5defbe8c),
        f64::from_bits(0xbefa2a63cc5c4feb),
    );
    let p_den = f_polyeval4(
        x2,
        f64::from_bits(0x3ff0000000000000),
        f64::from_bits(0xbf9833197207a7c6),
        f64::from_bits(0x3f315663bc7330ef),
        f64::from_bits(0xbeb9211958f6b8c3),
    );
    let p = p_num / p_den;

    let lg = fast_logf(x);
    let v_i = i1f_small(x);
    let z = f_fmla(lg, v_i, rcp);
    let z0 = f_fmla(p, dx, z);
    z0 as f32
}

/**
Generated by Wolfram Mathematica:
```text
<<FunctionApproximations`
ClearAll["Global`*"]
f[x_]:=Sqrt[x] Exp[x] BesselK[1,x]
g[z_]:=f[1/z]
{err, approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},7,7},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 k1f_asympt(x: f32) -> f32 {
    let dx = x as f64;
    let recip = 1. / dx;
    let e = core_expf(x);
    let r_sqrt = dx.sqrt();
    let p_num = f_estrin_polyeval8(
        recip,
        f64::from_bits(0x3ff40d931ff6270d),
        f64::from_bits(0x402d250670ed7a6c),
        f64::from_bits(0x404e517b9b494d38),
        f64::from_bits(0x405cb02b7433a838),
        f64::from_bits(0x405a03e606a1b871),
        f64::from_bits(0x4045c98d4308dbcd),
        f64::from_bits(0x401d115c4ce0540c),
        f64::from_bits(0x3fd4213e72b24b3a),
    );
    let p_den = f_estrin_polyeval8(
        recip,
        f64::from_bits(0x3ff0000000000000),
        f64::from_bits(0x402681096aa3a87d),
        f64::from_bits(0x404623ab8d72ceea),
        f64::from_bits(0x40530af06ea802b2),
        f64::from_bits(0x404d526906fb9cec),
        f64::from_bits(0x403281caca389f1b),
        f64::from_bits(0x3ffdb93996948bb4),
        f64::from_bits(0x3f9a009da07eb989),
    );
    let v = p_num / p_den;
    let pp = v / (e * r_sqrt);
    pp as f32
}

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

    #[test]
    fn test_k1f() {
        assert_eq!(f_k1f(0.3), 3.055992);
        assert_eq!(f_k1f(1.89), 0.16180483);
        assert_eq!(f_k1f(5.89), 0.0015156545);
        assert_eq!(f_k1f(101.89), 0.);
        assert_eq!(f_k1f(0.), f32::INFINITY);
        assert_eq!(f_k1f(-0.), f32::INFINITY);
        assert!(f_k1f(-0.5).is_nan());
        assert!(f_k1f(f32::NEG_INFINITY).is_nan());
        assert_eq!(f_k1f(f32::INFINITY), 0.);
    }
}